Beyond Computable Tractability
The Limits of Intractable Problems and Computational Complexity in the Physical Universe, Quantum CryptanalytiMyth
Last updated
The Limits of Intractable Problems and Computational Complexity in the Physical Universe, Quantum CryptanalytiMyth
Last updated
Dr. Lin Wang (Duke Quantum Computing), Dr. Dawn M. Lipscomb (Physics), Dylan Rosario (Cybernetics & Cryptography)
This paper explores the concept of computational intractability and its implications within theoretical and practical domains. Intractable problems, characterized by exponential or super-exponential time complexity, pose a unique challenge to current and future computational capabilities, often exceeding feasible technological and energy limits. We analyze complexity classes such as NP, EXP, and NEXP, illustrating these bounds through real-world applications like cryptography, combinatorial optimization, and biological simulations. Physical limits, including Landauer's Principle and the Bekenstein Bound, are used to illustrate the energy and storage constraints of computation. Even advances in quantum computing encounter fundamental hurdles, particularly for tasks demanding or more operations. This study emphasizes that some problems, due to their extreme resource requirements, remain fundamentally unsolvable, marking an intersection between theoretical mathematics, physical laws, and the limits of feasible computation.
Why "knowing what is tractable" is important
Intractable Definition
Universal Constants
Importance of differentiating between tractable and intractable problems.
Understanding Complexity Classes
Implications for real-world problems, e.g., cryptography, logistics, simulations.
Physical and Theoretical Constraints
Bekenstein Bound and information storage limitations.
Time requirements in terms of the age of the universe.
Quantum Computing: Potential and Limitations
Limitations of coherence, error correction, and practical scaling.
Case Studies in Intractability
Combinatorial optimization (e.g., traveling salesman problem).
Large-scale simulations in biology and physics.
Feasibility Analysis of Extreme Computation
Energy, material, and time requirements for planet-scale computation.
Tractable vs. Intractable Problems: A Comparative Framework
Practical examples of what is feasible versus theoretically impossible.
The Future of Computing and Computational Limits
Hypothetical scenarios in advanced technology (quantum, theoretical supercomputers).
Conclusion: The Significance of Intractability in Science and Engineering
Computational intractability represents the boundary between what is practically solvable with current and foreseeable computing power and what remains beyond our reach due to resource limitations and the nature of certain problems. This introduction aims to define intractability, explain its significance, and touch upon the universal constants and principles that impose fundamental limits on computation.
Intractability’s implications stretch beyond theoretical interest, impacting practical fields like cryptography, where intractable problems such as RSA factoring underpin encryption. Combinatorial optimization tasks in logistics and biological simulations also confront these computational boundaries. The intractable nature of these problems necessitates heuristic or approximate solutions, with true exact solutions lying beyond reach.
Ultimately, intractability highlights the need for continual innovation in algorithm design and hardware development. But as complexity continues to escalate, the realization of absolute computational limits grows more salient, illustrating a profound intersection between theoretical mathematics, physical laws, and the boundaries of human technology.
Understanding what is tractable—that is, what can be solved within reasonable time and resources—guides decision-making in technology, science, and everyday applications. By distinguishing between problems that are solvable with polynomial-time algorithms and those that require exponential or super-exponential resources, we can allocate computational efforts more effectively. Tractable problems lend themselves to practical applications in areas like real-time data processing, logistics optimization, and cryptographic verification. When an algorithm’s complexity is known to be tractable, developers and engineers can confidently design systems to solve these problems within expected parameters.
In contrast, identifying intractable problems is equally crucial, as it allows us to focus on heuristic or approximate methods that can provide good-enough solutions within the constraints of time and resources. Fields like cryptography rely on the intractability of certain problems—such as integer factorization and discrete logarithms—to create secure encryption schemes that protect sensitive information. Misjudging intractability can lead to wasted resources, unsatisfactory outcomes, or compromised security, highlighting the need for a thorough understanding of what lies within the boundary of tractability.
The intractability of a problem can be seen as a measure of its resistance to feasible computation. Problems in the NP-hard or EXP classes, for example, are intractable for classical computers because their solution paths involve exploring a vast, combinatorial number of possibilities. This growth rate is fundamentally distinct from polynomial-time algorithms, which scale more linearly with input size and therefore remain practical for larger instances. By defining and identifying intractability, we recognize the constraints that limit computation and delineate the boundaries where exact solutions must give way to approximations.
The limits of computation are not just a matter of technology but are bound by universal constants and physical laws that dictate the maximum amount of information that can be processed, stored, or transferred within finite energy and time constraints. These constants set theoretical ceilings on the computational power we can achieve, even with hypothetical or futuristic technology.
The excitement surrounding quantum computing’s potential has spilled over from academia and industry into mainstream public consciousness, a process hastened by media outlets and marketing campaigns that often oversimplify or exaggerate. This mainstream portrayal has led to a widespread misunderstanding of quantum computing’s capabilities, particularly with regard to breaking cryptographic systems or solving currently intractable problems. For the public, the phrase “quantum computer” conjures images of a machine that can compute anything at near-instant speeds, but the limitations, complexities, and actual science are routinely omitted from these narratives. This disconnect not only risks creating unrealistic expectations but also undermines a crucial understanding of cybersecurity, mathematics, and the limitations of computational power.
One of the core misconceptions popularized by the media is the idea that quantum computers will inevitably crack any encryption algorithm, rendering all cryptographic systems obsolete. The general public, unfamiliar with the intricacies of complexity classes, doesn’t distinguish between the specific kinds of problems quantum computers are suited to address and those they fundamentally cannot solve. As a result, people may be left with the impression that all their online data and communications will one day be laid bare by a sufficiently powerful quantum computer. This fear, while rooted in some factual basis, misses a vital detail: only certain types of cryptographic algorithms, particularly those that rely on the difficulty of factoring large numbers (like RSA), are vulnerable to quantum algorithms like Shor’s. Other cryptographic protocols, including those based on lattice problems or hash functions, are believed to be resilient to quantum attacks. These “quantum-resistant” algorithms are specifically designed to withstand the types of calculations quantum computers excel at.
The danger of these misconceptions goes beyond a misunderstanding of technology. By overhyping the capabilities of quantum computers, the public may be lulled into a false sense of both fear and security. Some may assume that encryption will become obsolete, leading to hasty policy changes or weakened encryption standards. Others might believe that quantum computers will unlock limitless potential for artificial intelligence or instantly optimize complex logistics problems, disregarding the actual constraints imposed by physics and mathematics. This could result in misguided investments, policy missteps, and a public that is poorly prepared for the real challenges posed by quantum computing’s advancements.
In cybersecurity, this misunderstanding could lead to prematurely abandoning secure systems in favor of “quantum-proof” solutions that may not be necessary or cost-effective in the short term. As the public absorbs the notion that cryptographic security is a temporary barrier at best, they might overlook the real threats posed by more traditional cyberattacks that don’t require quantum capabilities. Currently, cybersecurity vulnerabilities largely stem from user behavior, weak passwords, phishing attacks, and social engineering. Shifting the public’s focus away from these practical concerns to hypothetical quantum threats could dilute essential cybersecurity awareness and preparedness.
From a broader societal perspective, the public’s blind faith in quantum computing’s promises risks perpetuating a misunderstanding of computational limits and the role of mathematics in defining these boundaries. The tendency to regard technology as a solution to all problems erodes an appreciation for mathematical rigor and physical principles, leading to a society that prioritizes hype over scientific literacy. When the public loses sight of the fact that intractable problems are defined by their inherent complexity, not merely by a temporary lack of technology, they are set up for disappointment and, ultimately, disillusionment when quantum computers fail to meet these inflated expectations.
Quantum computers offer a marginal increase in computability over classical binary computing as a result of the qubit vs standard classical bit size.
Ultimately, the hype around quantum computing, when taken at face value by the public, feeds into a culture of “technological salvation,” a belief that advanced machines will inevitably conquer all of nature’s mysteries. This belief disregards the reality that some problems are insurmountable, regardless of how advanced technology becomes. By fostering a realistic understanding of quantum computing’s true capabilities—and its fundamental limits—the public can be better prepared to appreciate the advancements that quantum computing will bring without falling prey to the false narrative that it can accomplish the impossible.
The hype around quantum computing is driven in part by companies and researchers who capitalize on the term "quantum" as a sort of "magical mystery word" designed to captivate audiences—especially those with limited technical knowledge. Companies aiming to attract investment and researchers vying for visibility frequently invoke the promise of quantum computing’s power to solve "impossible" problems as a compelling narrative. However, this message often drifts far from reality, creating an illusion that quantum computing will be a silver bullet capable of tackling all computational challenges, even those considered intractable. This framing skews expectations, especially among investors who may lack a strong technical foundation, and fuels a false narrative that distracts from quantum computing’s genuine but focused potential.
Quantum computing, indeed, represents an exciting frontier, with quantum bits (qubits) allowing certain types of parallelism and entanglement that classical bits cannot achieve. However, the real computational gains are specific to certain types of problems, notably those within specific mathematical domains, like integer factorization and discrete logarithmic problems. Quantum algorithms such as Shor’s and Grover’s are revolutionary but limited in scope, offering exponential or quadratic speed-ups in narrow problem spaces rather than across the board. The complexity classes that define truly intractable problems, like those in the double-exponential class (e.g., O(22n)), remain out of reach even for theoretical quantum systems. So while a quantum computer might indeed accelerate some operations, the idea that it can "solve the impossible" is based more on wishful thinking than technical feasibility.
Moreover, researchers are incentivized to embellish quantum computing’s potential for personal gain. Quantum computing is a competitive research field, where funding and recognition are tightly linked to promising breakthroughs. Researchers who can demonstrate or even hint at a connection between their work and the solution of complex, high-profile problems—like cracking cryptographic protocols or optimizing massive logistics networks—gain significant attention. This creates a landscape where researchers may emphasize the "potential" of quantum methods over their realistic applications, portraying themselves as pioneers of a field that can supposedly outsmart fundamental mathematical constraints.
In reality, portraying quantum computing as a solution for all intractable problems is more than just a harmless exaggeration—it’s a distortion of what the technology can actually achieve. Real quantum computing advancements are already being made, and its applications to specialized fields, like material science and specific optimization problems, are legitimate. However, the broader claim that quantum will make all difficult problems easy is a misleading message that overlooks the basic principles of computational complexity and the physical limits of computation itself.
The takeaway? The overhyped claims around quantum computing aren’t just inaccurate; they threaten to overshadow its true potential and risk creating disappointment when the technology ultimately doesn’t meet these inflated expectations. Instead, a realistic appreciation of quantum computing as a transformative but limited tool could enable us to leverage its actual strengths without succumbing to false promises.
Quantum algorithms demonstrate significant improvements in solving certain classes of problems, particularly those that are computationally intensive for classical computers. The implications on brute-force iteration are especially notable, as quantum computing can transform exponential-time problems into polynomial or sub-exponential-time problems, thus rendering previously intractable problems more accessible.
Here we presents a detailed comparison between classical bits and quantum qubits, focusing on their representations, computational capacities, and implications on computational complexity, particularly in the context of brute-force iteration. By exploring the foundational principles and mathematical formulations, we illustrate how quantum computing offers a paradigm shift in tackling computationally intensive problems.
The advent of quantum computing introduces a new computational paradigm that challenges traditional notions of computation based on classical bits. Understanding the distinctions between classical bits and quantum qubits is crucial for appreciating the potential advantages of quantum algorithms, especially for problems deemed intractable by classical means. We describe the theoretical underpinnings of bits and qubits, explore their computational capabilities, and examine the implications on complexity classes and brute-force computation.
A classical bit is the fundamental unit of information in classical computing systems. It is binary in nature, meaning it can exist in one of two possible states, commonly denoted as $0$ or $1$.
An $n$-bit classical system can represent any one of $2^n$ possible configurations. Each configuration corresponds to a unique combination of bits:
However, at any given moment, the system can only be in one of these $2^n$ states.
Classical computation manipulates bits through logical operations defined by Boolean algebra. The computational capacity is directly tied to the number of bits and the speed of processing. For exhaustive search algorithms, the time complexity scales exponentially with the number of bits:
where $T(n)$ represents the time required to process all possible states for input size $n$.
A quantum bit, or qubit, is the basic unit of quantum information. Unlike classical bits, qubits exploit quantum mechanical phenomena such as superposition and entanglement, allowing them to exist in multiple states simultaneously.
A qubit can be represented as a linear combination (superposition) of the basis states $|0\rangle$ and $|1\rangle$:
where $\alpha, \beta \in \mathbb{C}$ are complex probability amplitudes satisfying the normalization condition:
For a system of $n$ qubits, the state vector resides in a $2^n$-dimensional Hilbert space:
where $|i\rangle$ represents the computational basis states (e.g., $|00\ldots0\rangle, |00\ldots1\rangle, \ldots, |11\ldots1\rangle$) and $c_i$ are complex coefficients with the normalization condition:
Entanglement is another key feature where qubits become correlated in such a way that the state of one qubit cannot be described independently of the others.
The ability of qubits to exist in superposition allows a quantum computer to process a vast amount of information simultaneously. An $n$-qubit register can represent all $2^n$ possible states at once, providing a form of inherent parallelism. However, due to the probabilistic nature of quantum measurement, extracting information requires careful algorithm design to gain computational advantages.
P (Polynomial Time): The class of decision problems solvable by a deterministic Turing machine within polynomial time $O(n^k)$, where $k$ is a constant.
NP (Nondeterministic Polynomial Time): The class of decision problems where a given solution can be verified in polynomial time by a deterministic Turing machine.
EXP (Exponential Time): The class of decision problems solvable in exponential time $O(2^{n^k})$.
These classes illustrate the scaling of computational resources with input size $n$. For example, in brute-force search, the time complexity is:
BQP (Bounded-Error Quantum Polynomial Time): The class of decision problems solvable by a quantum computer in polynomial time with an error probability of at most $\frac{1}{3}$ for all instances.
Formally, a language $L$ is in BQP if there exists a uniform family of quantum circuits ${Q_n}$ of size polynomial in $n$ such that:
In classical computation, the number of possible states for an $n$-bit system is:
For quantum systems, while the number of basis states is the same $2^n$, the superposition principle allows simultaneous consideration of these states. The complexity of certain algorithms can be reduced. For instance, Grover's algorithm searches an unsorted database in:
This represents a quadratic speedup over the classical brute-force search.
In classical algorithms, brute-force iteration requires examining each possible configuration individually. For a problem with $N = 2^n$ possible configurations, the time complexity is:
As $n$ increases, the computational resources required become prohibitive.
Quantum algorithms can leverage superposition and interference to explore multiple configurations simultaneously. Grover's algorithm, for example, can find a marked item in an unsorted database with high probability in:
This quadratic reduction in time complexity significantly impacts problems where exhaustive search is the only known solution.
Quantum computing offers varying degrees of speedup depending on the problem:
Quadratic Speedup: As seen in Grover's algorithm for unstructured search problems.
Exponential Speedup: As in Shor's algorithm for integer factorization, which operates in polynomial time:
compared to the best-known classical algorithms with sub-exponential time complexity:
These speedups have profound implications for fields like cryptography, where the security of many systems relies on the difficulty of problems like integer factorization.
Quantum computing introduces new considerations in cryptography, particularly in its potential to disrupt some current encryption standards while remaining unable to break others due to inherent complexity limits. Quantum algorithms like Shor’s have specific strengths that pose a threat to certain cryptographic protocols, especially those that rely on factoring large numbers or calculating discrete logarithms. However, quantum computing does not render all cryptographic systems vulnerable, particularly those designed with quantum resistance in mind.
While quantum computing does pose a valid challenge to certain encryption standards, it is not a panacea that can easily break all cryptographic systems. Quantum computing’s limitations in terms of error correction, coherence, and scalability make it an intriguing tool, but not a universal cryptographic threat. The careful development of post-quantum cryptographic standards and an awareness of complexity limitations will be essential as the field of quantum technology progresses.
Shor’s algorithm, developed by mathematician Peter Shor in 1994, enables quantum computers to factorize large numbers in polynomial time. This makes RSA (Rivest–Shamir–Adleman) encryption, which depends on the difficulty of factoring the product of large primes, vulnerable to quantum attacks. Similarly, elliptic-curve cryptography (ECC), which relies on the discrete logarithm problem, would also be compromised by Shor’s algorithm. This vulnerability has motivated a push toward "post-quantum cryptography"—cryptographic algorithms that are not susceptible to Shor's algorithm.
For example, with RSA and ECC encryption, key sizes could be dramatically increased to counteract quantum capabilities, but this quickly becomes impractical as even moderately-sized quantum computers would significantly reduce the effectiveness of such encryption in the future.
For symmetric key cryptography (e.g., AES), quantum computing provides only a quadratic speedup using Grover’s algorithm. Grover’s algorithm can effectively reduce the time required for brute-force key searching, meaning that a quantum computer could search a space of 2n possible keys in 2n/2 time. This translates to a 128-bit key effectively becoming as secure as a 64-bit key against quantum attacks.
Fortunately, this means symmetric cryptography is more resistant to quantum threats. Increasing key sizes for AES, for example, from 128 to 256 bits is sufficient to maintain security, as a 256-bit key remains computationally secure even with Grover’s speedup.
Post-quantum cryptography focuses on developing encryption methods that would remain secure even against a fully developed quantum computer. Lattice-based cryptography, hash-based cryptography, and multivariate polynomial cryptography are examples of areas being explored for quantum-resistant algorithms. Unlike RSA or ECC, these methods are not easily compromised by known quantum algorithms like Shor’s or Grover’s.
The National Institute of Standards and Technology (NIST) has been working on standardizing post-quantum algorithms, expected to be finalized in the coming years. The development and adoption of these algorithms aim to future-proof cryptographic systems from the potential impact of quantum computing.
Despite theoretical breakthroughs, the practical application of quantum attacks is limited by current quantum computing technology. Quantum computers today are still relatively small, noisy, and susceptible to errors. For example, Shor's algorithm requires a vast number of qubits, fault tolerance, and error correction, making practical attacks on cryptographic systems unfeasible with current quantum hardware.
Even with advancements, quantum computers face fundamental scaling challenges due to qubit coherence, error correction overhead, and the high resource costs of maintaining stable quantum states. These challenges mean that building a quantum computer with enough qubits to break commonly used cryptographic standards (like RSA with 2048-bit keys) may still be decades away, if feasible at all.
This inherent intractability suggests that while quantum computers may necessitate shifts in cryptographic strategies, they are not an all-encompassing threat to cryptography. Protocols designed with exponential intractability in mind (for example, cryptographic hashing functions and high-dimensional lattice problems) are structured to withstand attacks from both classical and quantum computers.
Much of the hype around quantum computing suggests it will render all cryptographic systems obsolete, but this ignores the reality of quantum complexity limits. Intractable problems do not become easy just because quantum computation offers a parallel-processing advantage. The fact remains that only certain cryptographic methods are vulnerable to quantum algorithms, and many quantum-resistant methods already exist.
Marketing claims that quantum computing will "solve all" encryption problems often misrepresent the nuanced realities of quantum mechanics and computational theory. They overstate quantum computing’s ability to scale, disregarding the exponential resource demands required for truly challenging cryptographic attacks. Understanding these limits is crucial, as it ensures that security planning and policy remain grounded in the practical capabilities of quantum systems rather than overhyped projections.
The "Quantum Hype Machine" has reached a fever pitch, fueled by companies and researchers eager to attract funding for their pet projects and quantum "toys" that promise the impossible. Their narrative paints quantum computers as god-like, omnipotent machines capable of solving any problem—especially the juicy target of cryptographic systems—by exploiting quantum phenomena. But, this storyline is an absolute fantasy, a scientific fiction with little grounding in the real-world physics and complexity theory that constrain computation. The goal behind these claims isn’t truth or progress but rather the accumulation of funds from investors dazzled by quantum buzzwords and the illusion of technological omnipotence.
The reality is far more sobering. The laws of physics, particularly those governing energy, information, and time, provide hard limits that no quantum machine, no matter how advanced, can overcome. Yet, companies pushing the quantum narrative—deliberately misinforming the public—don’t want you to know this. The spin-doctors in these organizations, with little to no grasp of fundamental mathematics, have created a fictitious world where quantum computers are a “silver bullet” for all problems, including intractable cryptographic challenges. This oversimplified narrative conveniently sidesteps the hard truths about quantum computing’s actual capabilities, leeching resources from valid technological research and diverting it towards smoke-and-mirror projects.
Let’s unpack the fundamental misconceptions behind these bold quantum claims, starting with the simplest principles taught in physics and mathematics—principles that even elementary school students can grasp.
possible states far exceed the storage capacities of any quantum system that could ever exist in the physical universe. This reality alone dismantles the notion that quantum computers could brute-force encryption schemes that rely on such vast problem spaces.
Quantum computers are not the stable, omnipotent machines they are marketed to be. Quantum states are notoriously fragile, and maintaining coherence over long calculations requires error correction schemes that significantly reduce the efficiency of the system. For every qubit added to perform complex operations, a substantial portion must be dedicated to error correction, and this overhead increases exponentially. In essence, the more complex the problem, the more resources are needed just to keep the system functional. This fragility makes large-scale quantum computations far less feasible than the fantasy narrative suggests.
The quantum hype has become a fundraising tool, not a reflection of scientific truth. Companies and researchers mislead investors by claiming that quantum computers can solve problems that are, in fact, impossible to crack. This misrepresentation not only creates unrealistic expectations but also siphons off valuable resources that could be directed toward research with real-world impact. Instead of focusing on practical technologies that could improve cybersecurity, energy efficiency, or medical diagnostics, the funding is funneled into quantum projects that promise breakthroughs far beyond the reach of current physics and computation theory. The narrative of quantum omnipotence is nothing more than a charade.
At the heart of this issue lies the public’s misunderstanding of quantum technology. Quantum computing is often portrayed as a magic bullet that will revolutionize everything from cryptography to climate change modeling. This fantasy is perpetuated by media outlets eager to ride the quantum wave, and by companies whose primary aim is to secure funding. The problem is, while quantum computing is indeed a promising field, it is far from a universal solution. The real-world limits—energy consumption, error correction, storage, and time constraints—are conveniently left out of the quantum success stories that fill headlines. The public, dazzled by buzzwords, is left believing in a future where every cryptographic wall can be breached, every problem solved, with the wave of a quantum wand. Nothing could be further from the truth.
In reality, quantum computers, while a fascinating and promising development, are far from the omnipotent machines portrayed by opportunistic companies and marketers. The physical limits of computation—rooted in basic laws of energy, information, and time—ensure that many problems, particularly in cryptography, will remain intractable, no matter how powerful quantum devices may become. The quantum "silver bullet" narrative does a disservice to legitimate research and innovation, draining resources away from areas where real progress can be made. As long as the hype machine continues to mislead the public and investors, the potential of quantum technology will remain shrouded in false promises, obscuring the important and real advances that are yet to be achieved.
These universal constants underscore the inherent limitations on computation beyond what technology or clever engineering can overcome. They define the limits of both classical and quantum computing systems and imply that certain problems, especially those requiring vast computational resources, may remain unsolvable due to these absolute physical constraints. As we explore new computing paradigms like quantum computing, these constants serve as a reminder of the natural boundaries we face when approaching intractable problems.
Landauer’s Principle:
Bekenstein Bound:
Speed of Light and Data Transfer:
The speed of light, approximately 3108 meters per second, imposes a fundamental limit on how quickly information can travel across distances. In large-scale, distributed computing systems, this limit affects data synchronization and transfer rates, which are crucial for maintaining coherence in parallel computations.For tasks that involve processing information across vast networks, the speed of light restricts real-time data sharing and can prevent certain computations from being completed within practical timeframes.
The Planck Scale:
This limitation means that computational components, like transistors or qubits, cannot be made indefinitely smaller to increase speed or density, setting a finite boundary on hardware miniaturization and the potential for processing density in computational devices.
Computational complexity is the study of how the resources required to solve a problem—typically time and memory—scale with the size of the input. It provides a framework to classify problems based on the feasibility of finding solutions within realistic constraints. By defining and analyzing complexity, we gain insight into which problems can be solved efficiently and which remain out of reach, requiring excessive resources even with the most advanced computing technologies.
In contrast, intractable problems demand resources that grow at exponential or super-exponential rates as the input size increases. For these problems, the number of possible configurations or solutions expands too quickly to be processed in a practical time frame. This rapid growth means that even relatively small instances of intractable problems may require more time or memory than is realistically available, often surpassing the lifespan of the universe or the energy limits of current technology.
Distinguishing tractable from intractable problems enables researchers and engineers to direct resources effectively. When faced with an intractable problem, efforts can be redirected toward heuristic or approximate solutions that offer “good enough” answers without solving the problem exactly. This is particularly important in fields such as cryptography, logistics, and machine learning, where exact solutions may be unattainable due to the computational demands involved. Additionally, understanding intractability helps secure digital systems by leveraging difficult-to-solve problems (like integer factorization in RSA encryption) as the foundation for secure encryption methods. Recognizing which problems fall within the realm of intractability also helps in setting realistic expectations and goals for computing advancements.
Complexity classes categorize problems based on the computational resources required to solve them, typically with respect to time (number of steps) and space (memory required). These classes allow us to identify the inherent difficulty of various problems, aiding in the development of efficient algorithms and the identification of limits in computational power.
Problems in class P can be solved in polynomial time, meaning their required resources scale as a polynomial function of the input size n. Examples include searching, sorting, and arithmetic operations, which are fundamental to computing and are solvable within reasonable constraints.
Polynomial time is generally associated with tractable problems since the resource demands grow manageably with input size. Most practical algorithms aim to achieve polynomial-time solutions, as this class encompasses many of the most critical problems in computing.
Class NP includes decision problems for which a given solution can be verified in polynomial time, even if finding the solution itself may not be feasible in polynomial time. A classic example is the traveling salesman problem, where verifying a route’s optimality is efficient, but finding that route from scratch is computationally intensive.
The question of whether all problems in NP can also be solved in polynomial time is known as the P vs. NP problem, one of the most important open questions in computer science. If P=NP, it would mean that problems for which solutions can be quickly verified can also be solved quickly, which would have significant implications for fields like cryptography and optimization.
NP-complete problems are a subset of NP problems that are as hard as any problem in NP. If a polynomial-time solution were found for an NP-complete problem, it would imply that all problems in NP could also be solved in polynomial time, solving the P vs. NP question affirmatively.
Common examples include the Boolean satisfiability problem (SAT) and the traveling salesman problem in its decision form. These problems are known for their computational difficulty, and no efficient solutions have been discovered, leading to their classification as intractable for large inputs.
For example, solving the knapsack problem by examining every possible combination of items is an exponential process. Problems in EXP are generally considered intractable due to their rapid growth in complexity, and they often appear in optimization and combinatorial fields.
(PSPACE) represents problems solvable within a polynomial amount of memory, regardless of the time required. Many PSPACE-complete problems can require exponential time to solve, though their memory usage remains within polynomial bounds.
This class is relevant for problems like the quantifier-free theory of real closed fields and games with alternating turns, where the solution space is vast, but the memory required for computation is limited. Problems in PSPACE can sometimes be even more complex than those in NP due to their potentially vast solution space.
(NExpTime) includes problems solvable by a nondeterministic Turing machine in exponential time. While nondeterminism theoretically allows faster solution searching, the time complexity remains prohibitive. Problems in NEXPTIME are beyond even the NP-complete class in terms of complexity, as their time requirements increase doubly exponentially.
Succinct circuit satisfiability is a canonical example of an NEXPTIME problem, where the input’s succinct representation adds significant complexity. Problems in NEXPTIME are extremely intractable and are generally not feasible for real-world computation, even with hypothetical or futuristic computational methods.
Non-elementary complexity includes classes where computational requirements exceed any fixed tower of exponentials. For example, Ackermann’s function grows so rapidly that even small inputs produce astronomical outputs.
Problems with non-elementary complexity are essentially theoretical in nature and illustrate the extreme end of intractability. They are rarely encountered outside of abstract theoretical contexts, as even hypothetical systems could not solve problems in these classes within realistic constraints.
Understanding complexity classes enables researchers to categorize problems and approach them with appropriate methods. Problems within the P class can typically be solved exactly and efficiently, while those in NP-complete and EXP classes often require approximate or heuristic solutions. By studying these classes, computer scientists can design algorithms that offer optimal trade-offs between solution accuracy and computational feasibility. Complexity classes also drive advancements in hardware and algorithm design. For example, identifying a problem as NP-complete suggests that instead of seeking an exact solution, researchers might explore parallelization, quantum algorithms, or approximation methods. Complexity classes also inform developments in fields such as cryptography, where intractable problems are harnessed to protect data security. Without these classifications, the allocation of computational resources would lack direction, and computing would be less efficient and less capable of tackling modern problems effectively.
Understanding complexity classes helps delineate the boundaries of computational feasibility. Physical and theoretical constraints underscore why many problems remain intractable, shaping fields like cryptography and scientific modeling while guiding the development of algorithms that strive to balance efficiency with computational practicality.
In computational complexity theory, understanding the classes of problems by their difficulty—how resource requirements scale with input size—is crucial for identifying which problems are solvable with today’s technology and which remain intractable. This classification framework not only aids in determining the resources needed but also sets the boundaries for current computing, influencing fields as diverse as cryptography, logistics, and scientific simulations. Below is an overview of key complexity classes and how their limitations impact real-world applications and are bounded by physical constraints.
Definition: Class P includes problems that can be solved by an algorithm in polynomial time, meaning that as input size n grows, the computational steps required scale by some polynomial function O(nk), where k is a constant.
Examples: Sorting algorithms, searching algorithms, and pathfinding algorithms like Dijkstra’s shortest path. Many fundamental operations in computing are within P, such as matrix multiplication and linear programming.
Implications for Real-World Problems: Problems in P are considered efficiently solvable, as polynomial time is generally manageable with available technology. For example, sorting large datasets or finding optimal paths in networks are critical in logistics and data management, and these tasks are feasible for both large and small-scale implementations.
Physical Constraints: Problems in P can be solved on modern computers without excessive resources, making them ideal for applications that need reliable, repeatable operations. In practical terms, polynomial growth ensures that resource demands grow in a way that remains manageable as hardware capabilities increase.
Definition: Class NP encompasses decision problems where a solution, if provided, can be verified in polynomial time. However, finding the solution may not necessarily be polynomial in time.
Examples: Classic NP-complete problems include the Boolean satisfiability problem (SAT), traveling salesman problem (TSP) in its decision form, and the knapsack problem. Verifying solutions for these problems is computationally feasible, but solving them requires exponentially growing resources as input size increases.
Implications for Real-World Problems: Many optimization and decision-making problems, especially in logistics and manufacturing, fall into NP. Cryptographic systems, like RSA encryption, are based on problems believed to be intractable, where solving (e.g., factoring large numbers) would require non-polynomial time. However, the ability to verify solutions in polynomial time means cryptographic keys can be checked efficiently, maintaining security and usability in practice.
Physical Constraints: Because no known polynomial-time algorithm exists for NP-complete problems, solving them exactly for large inputs is impractical on any classical computer. The exponential growth in time required means that, for large instances, solving these problems could take longer than the universe's lifespan. Approximate and heuristic methods are often employed to provide workable solutions for real-world applications, though they do not guarantee the exact solution.
Examples: Algorithms for exact solutions to the traveling salesman problem using brute force, certain combinatorial optimizations, and some exact quantum simulations are examples of problems in EXP. While small instances might be solvable, scaling up becomes infeasible very quickly.
Implications for Real-World Problems: Exponential-time problems often appear in scientific simulations, logistics, and combinatorial problems, where each additional variable or constraint increases the solution space exponentially. In cryptography, understanding exponential growth is critical for setting key lengths that remain infeasible to break by brute force within a reasonable timeframe.
Physical Constraints: Exponential growth in required resources means that, even for modestly sized inputs, solving these problems might require computational resources beyond the capabilities of current technology. For example, simulating quantum systems of more than a few hundred particles with full accuracy is practically impossible due to the exponential scaling of required resources.
Definition: NEXP refers to problems solvable by a nondeterministic Turing machine in exponential time, meaning a theoretical machine could “guess” a solution path that would take exponential time to verify.
Examples: Succinct circuit satisfiability, where the inputs are described compactly, is a well-known problem in NEXP. Additionally, certain high-dimensional logic problems and verification tasks for complex computations fall into this class.
Implications for Real-World Problems: Problems in NEXP are beyond the reach of classical computing. For instance, determining the satisfiability of extremely large circuits, such as those used in complex digital logic designs, becomes computationally prohibitive. In cryptography, problems in NEXP could be used to construct robust security frameworks but are largely theoretical due to their extreme computational demands.
Physical Constraints: The vast number of computational paths involved in NEXP problems implies that, without nondeterministic capabilities (hypothetical in classical computing), solving these problems exactly would exceed even theoretical computing limits. Machines would require astronomical amounts of memory and processing power, constrained by factors such as the total information that can be stored in the observable universe (as per the Bekenstein bound).
Definition: Class 2-EXP includes problems with double-exponential growth in resources, scaling as O(22n). This class requires resources that grow doubly with each increment in input size, making them orders of magnitude more complex than problems in EXP.
Examples: Certain formal language problems and higher-dimensional tiling problems fall into this class. Problems in 2-EXP are rarely encountered in practical applications because of their extreme computational requirements.
Implications for Real-World Problems: Double-exponential problems, while generally theoretical, illustrate the limits of computation. In fields like logic and theoretical computer science, problems in 2-EXP serve as upper bounds on problem complexity. For cryptography, 2-EXP problems theoretically offer robust security but remain impractical due to their infeasibility even with powerful computers.
Physical Constraints: Problems in 2-EXP cannot realistically be solved or even approached on any conceivable computational architecture. The sheer number of steps required would exceed the physical limits of energy and space in the universe, as described by theoretical principles such as Landauer’s principle for energy efficiency and the Planck scale for information density.
In cryptography, the reliance on intractable problems (often in classes like NP or higher) forms the basis for secure encryption algorithms. The RSA encryption system, for instance, relies on the infeasibility of factoring large numbers in polynomial time, ensuring that, without a breakthrough, encryption cannot be easily broken. Similarly, problems in NP and EXP underpin digital security protocols, safeguarding data against unauthorized access.
In logistics, NP-complete problems like the traveling salesman problem highlight the challenges of optimization in large networks. Solving these problems exactly is impractical for large instances, but approximate solutions offer reasonable alternatives. Supply chains and transportation networks rely on heuristic algorithms to navigate intractable problems, achieving solutions that balance efficiency with computational feasibility.
Scientific simulations, particularly in physics and molecular biology, often encounter EXP and higher classes due to the exponential complexity of interactions at atomic and subatomic levels. Quantum simulations, weather forecasting, and genome sequencing are examples of fields where intractable complexity arises, requiring approximate or reduced models to gain useful insights.
The physical limitations on computation, including energy consumption, storage capacity, and processing speed, establish hard boundaries on solvable problems. Key principles include:
This principle establishes a minimum energy requirement to erase one bit of information, emphasizing that large-scale computations come with a thermodynamic cost. As computational demands grow exponentially, energy requirements follow suit, eventually surpassing practical limits.
This theoretical limit on information density within a finite region of space restricts how much data can be stored and processed, particularly relevant for highly complex problems. For instance, simulating entire molecular structures or quantum states would exceed the Bekenstein bound in even a small region of space.
The Planck scale represents a physical limit on how finely we can measure or manipulate information. At scales smaller than the Planck length (about 1.610−35 meters), quantum gravity effects dominate, making traditional computation impossible. Complex simulations, like those requiring atomic precision, must contend with these limits, illustrating the boundary between solvable and unsolvable problems.
Although quantum computing offers speedups for certain intractable problems, it faces practical limitations like decoherence and error rates. Shor’s algorithm, which could theoretically break RSA encryption in polynomial time, is limited by the number of qubits that can maintain coherence. For extremely large inputs, even a quantum computer faces constraints that make exponential and 2-EXP problems insurmountable.
Solving problems in higher complexity classes would often take longer than the lifespan of the universe. For example, solving a double-exponential (2-EXP) problem with current technology would take an astronomical timeframe, effectively making the solution unreachable.
The physical limitations on computation extend beyond the capabilities of current technology, touching upon fundamental principles that govern energy, storage, and time. Key principles like Landauer’s Principle, the Bekenstein Bound, and quantum limitations reveal the boundaries of what can be computed. This chapter will examine these concepts and their implications on computational feasibility, particularly for intractable problems.
Practical Impact: At room temperature (around 300 K), the energy required per bit operation is roughly 3^10−21 joules. While seemingly minuscule, this energy requirement scales rapidly with the amount of data processed. For example, large-scale data centers and supercomputers process trillions of bits every second, leading to significant energy demands.
Limits to Efficiency: As computing systems become more efficient, Landauer’s Principle highlights a fundamental energy threshold below which no physical system can operate. Intractable problems, which often require an exponential number of bit manipulations, would thus demand energy resources that surpass physical feasibility.
The Bekenstein Bound, introduced by physicist Jacob Bekenstein in 1973, sets a theoretical limit on the amount of information that can be stored within a finite region of space containing a finite amount of energy. The bound is given by: I2REcln(2) where:
R is the radius of the spherical region,
E is the energy contained within that region,
is the reduced Planck’s constant, and
c is the speed of light.
For a sphere the size of the observable universe (radius around 4.41026 meters) and assuming the entire mass-energy of the universe (~1069 joules), the upper bound for information is around 10120 bits.
Implications for High-Storage Problems: This bound presents a concrete limit to how much information any computing system, even a hypothetical one the size of the universe, could handle. High-complexity problems, particularly those in classes like 2-EXP, require information storage that could easily exceed 10120 bits.
Relevance to Cryptography and Big Data: As data processing needs continue to grow in fields like cryptography and data analytics, the Bekenstein Bound indicates that beyond a certain threshold, storage needs outpace what is physically possible. For example, cryptographic protocols with astronomically large key sizes would theoretically demand storage capacities beyond this bound, making them unimplementable.
Storage Limits for Quantum Simulations: For quantum systems, the Bekenstein Bound becomes a critical constraint. Simulating a complex quantum state may require storage of an exponentially large number of bits, reaching or exceeding the physical limit of the observable universe.
As problems move into higher complexity classes, especially EXP, NEXP, and 2-EXP, the time required to compute solutions grows exponentially or even double-exponentially. When scaled up, the time needed to solve these problems quickly exceeds any reasonable time frame, even surpassing the age of the universe.
Double-Exponential Constraints: Double-exponential growth, such as in 2-EXP complexity problems, requires 22n operations, which grows unfathomably fast. For even small inputs, like n=20, the computation would require 10315 operations, a figure so large it dwarfs the number of atoms in the observable universe.
Quantum computing promises to address some forms of computational intractability by leveraging the principles of quantum mechanics. However, it faces substantial limitations, both in terms of practical implementation and fundamental physics.
Quantum Parallelism and Speedups: Quantum computers use qubits, which can exist in superpositions of states, allowing them to process multiple possibilities simultaneously. For certain problems, like factoring large numbers (e.g., RSA), quantum algorithms such as Shor’s algorithm offer polynomial-time solutions where classical algorithms require exponential time.
Limitations in Quantum Coherence: Quantum systems are highly sensitive to environmental interference, leading to decoherence, where the system loses its quantum properties and behaves classically. Maintaining coherence over long computations is a major technical challenge, as qubits must be isolated from external noise.
Error Correction Overhead: Quantum computations require complex error correction methods due to the fragile nature of qubits. Error correction imposes a significant overhead, as it demands additional qubits and resources, reducing the effective power of current quantum systems.
Scaling Challenges: Scaling quantum computers to handle large inputs, especially for problems in classes like NEXP or 2-EXP, remains challenging. Each qubit added exponentially increases the system's computational potential, but maintaining coherence and managing error rates becomes increasingly difficult as qubit counts grow.
Unsolvable Problems for Quantum Computing: While quantum computing can address certain types of intractable problems, others remain fundamentally unsolvable. Problems with double-exponential complexity, for example, require computational resources that even a scalable quantum computer could not provide. Thus, quantum computing does not eliminate computational intractability; it merely redefines the boundaries of what can be efficiently solved.
These fundamental principles demonstrate that intractability is often not merely a function of current technological limitations but is rooted in physical laws. For some classes of problems, no conceivable technological advance can overcome these boundaries, as they are constrained by the laws of thermodynamics, quantum mechanics, and the structure of spacetime itself. In practical terms:
Computing Beyond Feasible Energy Limits: As computations approach the thresholds defined by Landauer’s Principle, the energy demands become unmanageable. This underscores the importance of energy-efficient algorithms and novel architectures in extending the limits of feasible computation.
Data Limits in Cryptography and AI: The Bekenstein Bound indicates that, regardless of technological advances, storing and processing exponentially growing datasets will hit a physical limit. For fields that require massive data storage—like AI, cryptography, and molecular simulations—this limitation calls for innovative approaches to data compression, approximate computation, and energy optimization.
Redefining Intractability with Quantum Computing: Quantum computers, while powerful, have clear limits. They may redefine tractable problems for specific applications but are unlikely to change the fundamental boundaries of intractability for high-complexity problems. Quantum decoherence and error correction remain key challenges that limit the practical scalability of quantum solutions.
Understanding these constraints emphasizes why certain problems are deemed intractable in the strictest sense. As our computational goals and datasets continue to grow, these physical and theoretical limits will remain guiding factors in the development of computational technology and strategy.
Concept
Description
Implications for Computation
Real-World Relevance
Landauer’s Principle
Minimum energy required to erase a single bit of information due to entropy increase
Limits the efficiency of data processing; high-complexity problems requiring enormous data processing could surpass Earth’s energy capacity
Data centers and supercomputers face high energy demands; efficient algorithms and architectures are critical
Bekenstein Bound
Maximum information that can be stored in a finite region of space with finite energy
Sets a physical limit on storage capacity, even with advanced technology
Relevant for fields needing massive data storage like cryptography and AI; complex problems could demand storage beyond universal limits
Time Requirements (Exponential Growth)
Time needed for exponential and double-exponential problems quickly exceeds practical timeframes
For moderately large inputs, computations could take longer than the age of the universe
High-complexity simulations (e.g., physics, climate) are constrained by realistic time limits, impacting scientific studies
Quantum Computing Potential
Quantum computers can leverage superpositions to solve certain problems faster
Addresses some intractable problems like factoring, but faces scalability issues due to coherence and error correction
Promising for fields like cryptography but remains limited for problems requiring double-exponential resources
Quantum Limitations
Quantum coherence and error correction demand resources, reducing effective quantum computing power
Quantum computers struggle with exponential scaling, limiting their effectiveness for complex tasks
Limits feasible applications of quantum computing for high-complexity problems in logistics and molecular simulation
Practical Constraints
Physical laws like thermodynamics and quantum mechanics set boundaries on feasible computation
For some problems, technological advances cannot overcome these constraints, making them theoretically unsolvable
Constrains growth in cryptography, AI, and physics simulations; underscores need for energy-efficient and approximate solutions
Table : This table provides a high-level view of the concepts, their computational impacts, and relevance, making it easier to understand the boundaries that define computational intractability.
Cooling and Infrastructure: A computer of this magnitude would generate enormous amounts of heat, requiring planetary or even stellar-scale cooling infrastructure.
Operations
Aspect
Computational Power
Time Required
Energy Requirement
Physical Size
Greater than the number of atoms in the observable universe
Feasibility
Beyond physical and technological capabilities
Intractable computation refers to problems that are so complex or computationally demanding that they cannot be solved within a reasonable time frame or using feasible resources, even with the most advanced technology. Intractability is often tied to the exponential growth in resources (such as time, memory, and energy) required to solve these problems as they scale, making them practically unsolvable for realistic problem sizes.
Intractable computation represents the limits of what can be feasibly solved given exponential or super-exponential growth in resources. While breakthroughs in computing, such as quantum algorithms, might make specific intractable problems solvable, there will always remain a subset of problems that defy practical computation, thus defining the outer limits of computational feasibility.
Intractable problems often have computational requirements that grow exponentially or super-exponentially with the size of the input. For example, if solving a problem for an input of size n takes 2n steps, then even a small increase in n drastically increases the number of required steps.
Super-exponential growth includes functions like O(22n), where computational requirements double doubly with each increase in input size. This type of growth quickly exceeds any feasible resources as n grows.
In computer science, complexity classes help classify problems by the resources required to solve them. Problems in the NP-complete, NP-hard, and higher complexity classes (such as EXP, NEXP, and PSPACE) often fall under the umbrella of intractable computation for classical computers.
NP-complete problems are those for which a solution can be verified quickly (in polynomial time) but finding a solution is not known to be feasible in polynomial time. Examples include the traveling salesman problem, the knapsack problem, and boolean satisfiability (SAT).
Problems that require exploring a large number of combinations or configurations tend to be intractable. For instance, the traveling salesman problem, which asks for the shortest route through a set of cities, requires examining every possible route as the number of cities grows.
Many cryptographic protocols, such as RSA encryption, are based on problems that are computationally intractable. Breaking RSA, for example, involves factoring large integers, which becomes infeasible for very large numbers due to the exponential growth in required steps.
Simulating quantum systems with classical computers becomes intractable as the number of particles grows, as each particle adds an exponential increase in state possibilities. This complexity is one of the motivations behind quantum computing, which could theoretically handle such simulations more efficiently.
Some problems simply do not have efficient algorithms, meaning that no shortcuts or clever strategies exist to reduce the required resources. Without a polynomial-time solution, these problems require brute-force methods that are infeasible for large inputs.
Solving intractable problems would require an impractical amount of computational resources, including time, memory, and energy. For example, solving a problem with exponential time complexity might require centuries or even millennia to finish, making it irrelevant for practical use.
Energy and physical limits also restrict computation. Even if a problem could be theoretically computed, the energy required might be beyond what is accessible on Earth.
Intractable problems have real-world implications. In fields like cryptography, intractable problems are a cornerstone of security since they prevent adversaries from easily breaking encryption. In operations research and logistics, intractable problems highlight the need for heuristic and approximation algorithms to find “good enough” solutions rather than exact answers.
Quantum Computing:
Quantum computing holds promise for certain types of intractable problems, particularly those that can leverage quantum parallelism. For example, Shor’s algorithm on a quantum computer could theoretically factor large integers (an intractable task for classical computers) in polynomial time. However, quantum computers are not yet advanced enough to handle all intractable problems.
Approximation Algorithms:
For some intractable problems, approximation algorithms provide near-optimal solutions with significantly less computational effort. These algorithms are often employed in machine learning, operations research, and data science to address NP-hard problems.
Despite advances in computing power, some problems remain intractable due to physical and theoretical limits. Even with the best conceivable technology, certain problems would require so many resources (in terms of time, energy, and matter) that they exceed the capabilities of any machine we could realistically build. These problems serve as boundaries of computability, representing tasks that cannot be resolved in practice within our universe's physical constraints.
The timeline of computational capacity from the 1940s to today demonstrates a remarkable evolution from rudimentary machines to the powerful, highly integrated computing systems that shape modern life. In the 1940s, the computational landscape was sparse, with limited machines dedicated primarily to military and scientific purposes. These machines, operating in the range of thousands of FLOPS (floating-point operations per second), required massive amounts of space and resources and were far beyond the reach of the general public. This early period was one of exploration, where computing’s potential was not fully understood outside select fields, and the technology was perceived as high-stakes and almost mystical by those who had limited exposure to it.
By the 1970s and 1980s, as computers began appearing in universities, corporations, and eventually homes, the idea of personal computing started to take shape. Computing capacity reached billions of FLOPS, and personal computers (PCs) emerged with tens of thousands of FLOPS, allowing small businesses and individuals to run increasingly complex tasks. Public perception shifted as well, with computing power starting to be seen as a practical tool rather than a futuristic concept. PCs became symbols of innovation and empowerment, enabling a new era of personal productivity and creativity. Nevertheless, even as computational capacity and accessibility grew, the capabilities of PCs and mainframes still seemed remarkable, with public awe centered around the rapid advancements in hardware and software.
The 2000s and 2010s saw an explosive growth in computational power, with global compute resources reaching exaFLOPS and home PCs reaching trillions of FLOPS. This period marked the dawn of the digital and Internet era, making computing power a staple in homes and offices worldwide. As smartphones, cloud computing, and the Internet of Things (IoT) took hold, computation became both invisible and omnipresent. People now carried devices more powerful than 1980s supercomputers in their pockets, and the general public became accustomed to seamless, high-speed processing. This ubiquity of compute resources led to the perception that computation had no bounds, fueling both excitement and concern about its impacts on privacy, security, and societal norms.
Today, as we look toward the 2030s, projections suggest compute power will reach yottaFLOPS on a global scale, with exaFLOPS available to the average user. The increasing integration of artificial intelligence and anticipated advancements in quantum computing are redefining what is possible, and with them, a mixed public perception has emerged. While there is optimism about breakthroughs in AI, medicine, and other fields, there is also a critical awareness of the limitations of computation, particularly regarding quantum computing’s feasibility for widespread applications. As we stand at the edge of potential new realms of computation, the evolution of computing power continues to be an intriguing journey that has fundamentally reshaped human experience and technological expectations.
Decade
Typical Available on PC
Public Perception of Compute
1940s
None
Computing seen as a military and academic tool; general public has little awareness.
1950s
105 (hundreds of thousands of FLOPS)
None
Mainframes are seen as massive, room-sized machines primarily for government and large corporations.
1960s
107 (millions of FLOPS)
None
Awareness grows with the Space Race; computers are viewed as high-tech, costly tools for scientific research.
1970s
109 (billions of FLOPS)
104 (tens of thousands of FLOPS)
Microcomputers emerge but remain inaccessible to the average person; computing power is seen as revolutionary for businesses and research.
1980s
1011 (hundreds of billions of FLOPS)
106 (millions of FLOPS)
Personal computers begin to enter homes; computing power is still viewed with awe, seen as futuristic and largely professional.
1990s
1013 (tens of trillions of FLOPS)
108 (hundreds of millions of FLOPS)
The PC revolution takes off; computers are seen as essential tools for work, with emerging potential for personal use.
2000s
1015 (quadrillions of FLOPS)
1010 (tens of billions of FLOPS)
Computing becomes commonplace; the Internet and software boom increase public awareness of compute power, now seen as integral to everyday life.
2010s
1018 (exaFLOPS)
1012 (trillions of FLOPS)
Smartphones and cloud computing redefine compute perception; people see compute power as ubiquitous and accessible everywhere.
2020s
1021 (zettaFLOPS)
1015 (quadrillions of FLOPS)
AI and supercomputing enter the public sphere; computing is increasingly perceived as near-limitless, with both excitement and concern about its applications and ethical impact.
2030s
[Projected] 1024 (yottaFLOPS)
[Projected] 1018 (exaFLOPS)
Speculated to involve AI and quantum breakthroughs; the public may view compute power as nearing its limits or entering entirely new realms of possibility.
To identify the minimum size of computable problems that exceeds 1025 operations and is currently beyond the reach of both classical and quantum computers, we must consider the kinds of problems where quantum computers might offer significant advantages and those where they don't yet make a measurable difference.
Complexity Beyond 10^25 Operations
Exceeding Classical Capabilities:
Quantum Computing Perspective:
Quantum Supremacy: Quantum computers have demonstrated supremacy in specific problems, but these are generally crafted to suit quantum abilities (like random number sampling problems), not necessarily useful computationally intense tasks.
General-Purpose Quantum Computing: Quantum computers are not yet capable of general-purpose computing on a scale that competes with classical supercomputers for many practical applications, such as those listed above.
Highly Connected and Complex Systems Simulations:
Biological Systems: Simulating entire biological systems at the molecular level, including interactions between millions of proteins and other biomolecules over time.
Neurological Simulations: Detailed simulations of human brain activity, which involve an enormous number of interactions and are far beyond the current reach of quantum computing.
Large-Scale Combinatorial Optimizations:
Logistics and Planning for Mega Cities: Optimizing logistics for mega cities in real-time, considering millions of variables (vehicles, people, resources).
Global Climate Models: Predicting climate changes with higher accuracy involves simulating the Earth's atmosphere, oceans, and biosphere at a granular level.
Cryptanalysis of High-Density Encryption Algorithms:
Problems in the double-exponential time complexity class and beyond are fundamentally intractable with any foreseeable advancement in technology, either classical or quantum. The difference in complexity between these problems and those solvable today is comparable to the vast difference in scale between the Planck length and the observable universe. As with physical exploration, while theoretical understanding of these complexity classes can advance, practical computation within these classes remains beyond reach.
The gap between the vastness of the universe and the minuteness of the Planck length provides a profound analogy for understanding the range of computational problems, especially in terms of complexity classes that exceed current and near-future computational capabilities. When considering Big O notation, we categorize the complexity of algorithms based on their growth rates relative to the size of the input. For problems that are intractable even for quantum computers or a hypothetical planet-sized classical computer, we can define these complexities in several key areas:
Exponential Time (EXP) and Beyond:
EXP (Exponential Time): Problems in this class require time that grows exponentially with the input size, such as O(2n). These problems are intractable for large inputs on any classical computational architecture.
NEXP (Nondeterministic Exponential Time): These are problems that nondeterministic machines can solve in exponential time. This complexity class encompasses even more computationally intensive problems than EXP, often involving calculations that grow as O(2poly(n)), where poly(n) is a polynomial in n.
Double-Exponential Time and Beyond:
2-EXP (Double-Exponential Time): Problems that require O(2_2n) time to solve. These are incredibly complex problems, typically only theoretical in nature, because even small increases in input size render them utterly infeasible to solve with any existing or conceivable classical or quantum technology.
Tower of Exponentials: For certain types of mathematical and logic problems, such as those involving higher-order logic or certain types of combinatorial games, the complexity can increase to tower functions like O(2^2_2n) and higher. These levels of complexity quickly surpass what could be simulated or computed on any physical system, including those utilizing all resources available on Earth.
Non-Elementary Complexities:
Non-elementary Recursive Functions: These are problems whose complexity exceeds any finite tower of exponentials, growing faster than any fixed iterated exponential function. Problems of this nature generally arise in very deep theoretical computer science, including some areas of decision problems in logic and specific high-complexity versions of tiling problems.
Physical Analogies and Computational Limits
To draw parallels with physical scales:
Quantum Supremacy: Quantum computers might eventually solve some problems believed to be intractable for classical computers, particularly those in BQP (Bounded-Error Quantum Polynomial time), like integer factorization or certain types of discrete logarithms. However, even quantum computers have limits, primarily bound by decoherence and error rates, which make them unsuitable for EXP-class problems and beyond.
Planetary Scale Computing: Even if we harness all computational resources on Earth, the energy requirements and physical limitations (such as cooling and maintenance of data integrity across vast distances) would likely make it impossible to deal with problems in the 2-EXP class or higher.
Here we pose a fundamental high school level educational tool explaining complexity and problem spaces in a table format, it categorizes various computational problem sizes by their complexity and outlines the computational, energy, and physical resources necessary for different types of computers (classical, quantum, and fictional) capable of solving these problems. Energy consumption is equated to the power consumption of typical microwave ovens (1 KW each), and physical compute size estimates grow with complexity.
[0] Compute Power (FLOPS & Watts)
[1] Problem Size
[2] Big-O Notation
[3] Complexity Class
[4] Computer Type
[5] Exponential Capacity (10^n permutations)
[6] Energy Equivalent (# of Microwave Ovens)
[7] Physical Matter Needed
1 TFLOP, 10 TW
Small Problems
O(n)
P (Polynomial)
Classical
10,000
Desktop or Small Cluster
10 PFLOP, 100 TW
Medium Complexity
O(nlogn)
P
Classical
100,000
Large Data Center
100 PFLOP, 1 PW
Complex Simulations
O(n2)
NP
Classical/Quantum
1,000,000
Data Center with Supercomputer
1 EFLOP, 10 PW
Very Large Problems
O(2n)
EXP
Classical/Quantum
10,000,000
Supercomputer Array
10 EFLOP, 100 PW
NP-Complete
O(2n2)
EXP
Advanced Quantum
100,000,000
Large National Supercomputer
100 EFLOP, 1 EW
Nondeterministic EXP
O(22n)
NEXP
Quantum/Fictional
1 Billion
City-Sized Quantum Network
1 ZFLOP, 10 EW
Tower of Exponentials
O(22n)
2-EXP
Fictional
10 Billion
Planet-Sized Quantum Network
10 ZFLOP, 100 EW
Non-Elementary
O(222n)
Non-elementary Recursive
Fictional
100 Billion
Multi-Planetary Compute Network
100 ZFLOP, 1 ZW
Non-Computable Range
O(n!)
N/A
Fictional
1 Trillion
Dyson Sphere around a Star
Table Key:
TFLOP (TeraFLOPS), PFLOP (PetaFLOPS), EFLOP (ExaFLOPS), ZFLOP (ZettaFLOPS): Measure of computational power in floating-point operations per second.
TW (TeraWatts), PW (PetaWatts), EW (ExaWatts), ZW (ZettaWatts): Measure of power consumption.
Problem Size: Describes relative computational task size and typical application areas.
Big-O Notation: Reflects the time complexity of solving problems as the input size grows.
Complexity Class: Classification of the problem’s computational difficulty.
Computer Type: Type of computer theoretically capable of solving such problems, given current or hypothetical technology.
Exponential Capacity (10^n permutations): Illustrates the problem size in terms of permutations or state space.
Energy Equivalent (Microwave Ovens): Number of microwave ovens (1 KW each) equivalent to the power requirements.
Physical Matter Needed: Rough estimate of the physical size and scale required for the computational infrastructure.
This table illustrates that while classical and quantum computers can tackle polynomial and some exponential time problems, non-elementary and factorial-level complexities exceed the capacity of any realistic computing system due to the prohibitive physical, energy, and scalability constraints.
Complexity Class: NP (if solved in P)
Big-O Notation: O(2n/2)
Permutation Space (10^n): 1040
Tractable?: No
Physical Computability: Not feasible with current technology
Energy Requirement: Equivalent to the power grid of a small country
Hardware Requirements: Classical supercomputer or large quantum computer (Shor's algorithm)
Time Needed to Compute: 1,000+ years on classical, days on quantum (hypothetically)
Complexity Class: NP
Big-O Notation: O(2n/2)
Permutation Space (10^n): 1040
Tractable?: No
Physical Computability: Not feasible with current technology
Energy Requirement: Equivalent to the power grid of a small country
Hardware Requirements: Quantum computer (Shor's algorithm)
Time Needed to Compute: 1,000+ years on classical, days on quantum (hypothetically)
Complexity Class: EXP
Big-O Notation: O(2256)
Permutation Space (10^n): 1077
Tractable?: No
Physical Computability: Not feasible with current technology
Energy Requirement: Exceeds Earth’s total energy production
Hardware Requirements: Planetary-scale quantum network
Time Needed to Compute: >Universe’s age on classical, centuries+ on quantum
Complexity Class: BQP
Big-O Notation: O((logn)3)
Permutation Space (10^n): 1010
Tractable?: Yes (for quantum)
Physical Computability: Feasible for small inputs only
Energy Requirement: 1 MW per computation for moderate primes
Hardware Requirements: Quantum computer
Time Needed to Compute: Hours for large integers (e.g., 1024 bits)
Complexity Class: BQP
Big-O Notation: O(2n/2)
Permutation Space (10^n): 1038
Tractable?: Yes (on quantum systems)
Physical Computability: Feasible with advanced quantum systems
Energy Requirement: Power of a small data center
Hardware Requirements: Large-scale quantum computer
Time Needed to Compute: ~4 million years on classical, weeks on quantum
Complexity Class: Unknown
Big-O Notation: Unknown
Permutation Space (10^n): Undefined
Tractable?: No
Physical Computability: Theoretical only
Energy Requirement: Unknown due to unproven nature
Hardware Requirements: Unknown
Time Needed to Compute: Unknown, unsolved
Complexity Class: Unknown
Big-O Notation: Unknown
Permutation Space (10^n): Undefined
Tractable?: No
Physical Computability: Theoretical only
Energy Requirement: Unknown due to unproven nature
Hardware Requirements: Unknown
Time Needed to Compute: Unknown, unsolved
Complexity Class: Unknown
Big-O Notation: Unknown
Permutation Space (10^n): Undefined
Tractable?: No
Physical Computability: Theoretical only
Energy Requirement: Unknown, exponential growth in cases
Hardware Requirements: Unknown
Time Needed to Compute: Unknown, unsolved
Complexity Class: Unknown
Big-O Notation: Unknown
Permutation Space (10^n): Undefined
Tractable?: No
Physical Computability: Theoretical only
Energy Requirement: Unknown due to open nature
Hardware Requirements: Unknown
Time Needed to Compute: Unknown, unsolved
Complexity Class: Unknown
Big-O Notation: Unknown
Permutation Space (10^n): Undefined
Tractable?: No
Physical Computability: Theoretical only
Energy Requirement: Unknown, exponential growth in cases
Hardware Requirements: Unknown
Time Needed to Compute: Unknown, unsolved
Complexity Class: NP (in some cases)
Big-O Notation: O(nlogn)
Permutation Space (10^n): 1018
Tractable?: Yes (polynomial algorithm for some cases)
Physical Computability: Feasible for smaller cases
Energy Requirement: Moderate, equivalent to a small data center
Hardware Requirements: Classical or quantum computer
Time Needed to Compute: Weeks for large graphs
Complexity Class: Unknown
Big-O Notation: Unknown
Permutation Space (10^n): Undefined
Tractable?: No
Physical Computability: Theoretical only
Energy Requirement: Unknown
Hardware Requirements: Unknown
Time Needed to Compute: Unknown, unsolved
Complexity Class: NP-complete
Big-O Notation: O(2n)
Permutation Space (10^n): 1025
Tractable?: No
Energy Requirement: Requires power grid of a city
Hardware Requirements: Supercomputer or advanced quantum computer
Time Needed to Compute: ~106years for n>50 on classical
The limitations of solving problems requiring 1040 operations or more are governed by a combination of technological constraints, physical resources, and time considerations. Even with the most advanced computing systems conceivable, these factors create fundamental barriers that prevent such computations from being feasible.
Quantum computers, though theoretically powerful, face limitations in qubit coherence and error correction. Qubits are extremely sensitive to environmental interference, and maintaining their coherence over extended computations is challenging. Error correction algorithms add computational overhead, limiting the effective number of calculations that can be done before errors make the results unreliable. Achieving coherence over the necessary number of qubits for a problem of 1040 operations are beyond the reality of currently available or fictitious Quantum technology.
For large-scale computations, even the data transfer rates between processing units become a bottleneck. High-performance interconnects (e.g., InfiniBand, fiber optics) can achieve remarkable speeds but are still limited by distance and the need to maintain synchronization across a vast number of nodes. For a problem requiring 1040 operations, moving data between components alone could consume enormous time and energy, creating logistical and technical limits to solving the problem effectively.
A computation requiring 1040 operations would necessitate an extraordinary amount of power. Current supercomputers, like Frontier, operate at around 20-30 megawatts for petaFLOP-scale performance. Scaling this to the exa-FLOP or zetta-FLOP range increases power needs to levels unsustainable on Earth. The heat generated by such a system would require cooling on an unprecedented scale. For instance, Earth’s energy consumption in 2022 was about 173 petawatts; a computer tackling 1040 operations might need a significant portion of this. Dissipating this heat would require extensive cooling infrastructure, possibly exceeding the scale of feasible engineering on Earth.
Building a computer capable of 1040 operations would demand vast quantities of raw materials, such as silicon, rare earth elements, and specialized superconducting materials for quantum devices. The planet may not have enough accessible resources to construct such a machine. Physically, a computer of this magnitude could require facilities spanning many kilometers, if not larger, to house processors, memory, cooling systems, and support infrastructure. This scale goes beyond the logistics of current fabrication and operational capabilities.
Moore's Law, which predicts exponential growth in computing power, has slowed as transistor sizes approach atomic limits. Even if growth continued, it would take far longer than the foreseeable future to reach the compute capabilities necessary for 1040operations.For example, even a hypothetical computer running at 1030 operations per second (beyond any existing capability) would still require 1010 seconds (over 300 years) to complete 1040 operations.
Computational systems have limited lifespans due to hardware degradation, particularly when operating at high intensities. Replacing or maintaining components over the course of decades, or even centuries, introduces logistical challenges and costs that render such prolonged computations infeasible.
Quantum mechanics and thermodynamics impose limits on computation, such as the Bekenstein bound, which limits the amount of information that can be stored in a finite region of space with finite energy. These principles imply that for a problem size like 10^40 operations, even if we could build a computational system of planetary size, it would be incapable of storing and processing the necessary information due to these physical limits.
These problems represent a sample of problems known to be intractable, many of which are of notable cryptographic and mathematical problems, including complexity class, Big-O notation, permutation space, and the resources required for computation.
Problem
Complexity Class
Big-O Notation
Permutation Space (10^n)
Intractable?
Physical Computability
Energy Requirement
Hardware Requirements
Time Needed to Compute
Integer Factorization (RSA)
NP (if solved in P)
O(2n/2)
1040
Yes
Not feasible with current tech
Equivalent to power grid of small country
Classical supercomputer or large quantum computer (Shor's)
1,000+ years on classical, days on quantum (hypothetically)
Discrete Logarithm Problem
NP
O(2n/2)
1040
Yes
Not feasible with current tech
Equivalent to power grid of small country
Quantum computer (Shor's)
1,000+ years on classical, days on quantum (hypothetically)
AES3-256 Symmetric Key Search
EXP
O(2256)
1077
Yes
Not feasible with current tech
Exceeds Earth’s total energy production
Planetary-scale quantum network
>Universe’s age on classical, centuries+ on quantum
Shor’s Algorithm for Factoring
BQP
O((logn)3)
1010
No (for quantum)
Feasible for small inputs only
1 MW per computation for moderate primes
Quantum computer
Hours for large integers (e.g., 1024 bits)
Grover’s Algorithm (AES Key Search)
BQP
O(2n/2)
1038
Yes (classically)
Feasible with advanced quantum systems
Power of small data center
Large-scale quantum computer
~4 million years on classical, weeks on quantum
Riemann Hypothesis
Unknown
Unknown
Undefined
Yes
Theoretical only
Unknown due to unproven nature
Unknown
Unknown, unsolved
Twin Prime Conjecture
Unknown
Unknown
Undefined
Yes
Theoretical only
Unknown due to unproven nature
Unknown
Unknown, unsolved
P vs. NP
Unknown
Unknown
Undefined
Yes
Theoretical only
Unknown, exponential growth in cases
Unknown
Unknown, unsolved
Navier-Stokes Existence (Millennium)
Unknown
Unknown
Undefined
Yes
Theoretical only
Unknown due to open nature
Unknown
Unknown, unsolved
Birch and Swinnerton-Dyer Conjecture
Unknown
Unknown
Undefined
Yes
Theoretical only
Unknown, exponential growth in cases
Unknown
Unknown, unsolved
Graph Isomorphism Problem
NP (in some cases)
O(nlogn)
1018
No (polynomial algorithm for some cases)
Feasible for smaller cases
Moderate, equivalent to small data center
Classical/quantum computer
Weeks for large graphs
Goldbach Conjecture
Unknown
Unknown
Undefined
Yes
Theoretical only
Unknown
Unknown
Unknown, unsolved
3-SAT (Boolean Satisfiability)
NP-complete
O(2n)
1025
Yes
Intractable for large n
Requires power grid of a city
Supercomputer/advanced quantum
~10^6 years for n>50 on classical
Key:
Complexity Class: Defines the theoretical difficulty level of the problem based on the type of resources required for a solution.
Big-O Notation: Indicates the rate of growth of the computation time or space as a function of the input size n
Intractable?: If a problem cannot be solved within reasonable time or resources with current technology.
Physical Computability: A qualitative measure of the practical feasibility of computing a solution given current technological limits.
Energy Requirement: Estimated energy consumption, with some values compared to real-world equivalents for context.
Hardware Requirements: Indicates the scale and type of computational infrastructure required.
Time Needed to Compute: Estimated time to compute for large instances of the problem on classical and/or quantum computers, if feasible.
Tractable Problems
These are tractable problems known to be solvable with current technology or feasible computational methods, along with their properties.
Problem
Complexity Class
Big-O Notation
Permutation Space (10^n)
Tractable?
Physical Computability
Energy Requirement
Hardware Requirements
Time Needed to Compute
Integer Factorization (Small Numbers)
P
O(n1.5)
10^6
Yes
Feasible
Low, manageable by typical computers
Classical computer
Minutes to hours
Sorting Algorithms (e.g., Merge Sort)
P
O(nlogn)
10^6
Yes
Feasible
Low to moderate
Classical computer
Milliseconds to seconds
Matrix Multiplication
P
O(n3)
10^9
Yes
Feasible
Low to moderate
Classical computer/GPU
Seconds to minutes for moderate sizes
Discrete Fourier Transform (DFT)
P
O(nlogn)
10^6
Yes
Feasible
Low
Classical computer
Milliseconds
Binary Search
P
O(logn)
10^3
Yes
Feasible
Low
Classical computer
Microseconds
Dijkstra’s Shortest Path Algorithm
P
O(n2)
10^6
Yes
Feasible
Low to moderate
Classical computer
Seconds for moderate graph sizes
Primality Testing (AKS Algorithm)
P
O((logn)6)
10^9
Yes
Feasible
Low to moderate
Classical or quantum computer
Seconds for large primes
Simulating Small Quantum Systems
BQP
O(n3)
10^3
Yes
Feasible on quantum devices
Moderate
Quantum computer
Seconds to hours
Image Compression (e.g., JPEG)
P
O(nlogn)
10^5
Yes
Feasible
Low
Classical computer
Milliseconds
Public Key Encryption (AES-128)
P
O(n3)
10^38
Yes
Feasible
Moderate to high
Classical or quantum computer
Milliseconds to seconds
Shortest Path in a Small Graph
P
O(n2)
10^6
Yes
Feasible
Low
Classical computer
Milliseconds
Pattern Matching in Strings (Knuth-Morris-Pratt)
P
O(n)
10^5
Yes
Feasible
Low
Classical computer
Microseconds
Linear Programming (Simplex)
P
O(n3)
10^6
Yes
Feasible
Moderate
Classical computer
Seconds to minutes
Fourier Transform (FFT)
P
O(nlogn)
10^6
Yes
Feasible
Low to moderate
Classical computer/GPU
Milliseconds
Basic Calculations (e.g., Arithmetic)
P
O(1)
10^1
Yes
Feasible
Very low
Any computing device
Microseconds
Explanation of Tractable Problems
These problems are tractable due to their polynomial or logarithmic growth rates, meaning that as the input size n increases, the required resources grow at a manageable rate, making these problems solvable on typical computing hardware within reasonable time and energy constraints.
Sorting Algorithms – Algorithms like merge sort have a time complexity of O(nlogn) and are feasible on standard computers, running in milliseconds to seconds depending on input size.
Matrix Multiplication – A standard operation in data science and machine learning with a complexity of O(n3), tractable on classical computers or GPUs.
Binary Search – Efficient for ordered lists with O(logn) complexity, taking only microseconds on any device.
Discrete Fourier Transform (DFT) – Common in signal processing, with efficient O(nlogn) algorithms, running in milliseconds to seconds on classical computers.
Dijkstra’s Algorithm – Finds shortest paths in graphs with O(n2)complexity, solvable on a classical computer for moderate-sized graphs.
Primality Testing (AKS) – Used to verify if a number is prime, with polynomial-time complexity O((logn)6) and feasible on classical or quantum computers.
Pattern Matching – Algorithms like Knuth-Morris-Pratt run in linear time O(n) for string matching, widely feasible for string processing.
Linear Programming – The simplex algorithm has a polynomial average case complexity, making it feasible for many practical applications.
Fourier Transform (FFT) – The fast Fourier transform has O(nlogn) complexity and is widely used for rapid signal and data processing.
Public Key Encryption (AES-128) – AES-128 has a polynomial time complexity O(n3)and is solvable quickly, providing practical encryption within seconds.
Integer Factorization (Small Numbers)
Complexity Class: P
Big-O Notation: O(n1.5)
Permutation Space (10^n): 106
Tractable?: Yes
Physical Computability: Feasible
Energy Requirement: Low, manageable by typical computers
Hardware Requirements: Classical computer
Time Needed to Compute: Minutes to hours
Sorting Algorithms (e.g., Merge Sort)
Complexity Class: P
Big-O Notation: O(nlogn)
Permutation Space (10^n): 106
Tractable?: Yes
Physical Computability: Feasible
Energy Requirement: Low to moderate
Hardware Requirements: Classical computer
Time Needed to Compute: Milliseconds to seconds
Matrix Multiplication
Complexity Class: P
Big-O Notation: O(n3)
Permutation Space (10^n): 109
Tractable?: Yes
Physical Computability: Feasible
Energy Requirement: Low to moderate
Hardware Requirements: Classical computer/GPU
Time Needed to Compute: Seconds to minutes for moderate sizes
Discrete Fourier Transform (DFT)
Complexity Class: P
Big-O Notation: O(nlogn)
Permutation Space (10^n): 106
Tractable?: Yes
Physical Computability: Feasible
Energy Requirement: Low
Hardware Requirements: Classical computer
Time Needed to Compute: Milliseconds
Binary Search
Complexity Class: P
Big-O Notation: O(logn)
Permutation Space (10^n): 103
Tractable?: Yes
Physical Computability: Feasible
Energy Requirement: Low
Hardware Requirements: Classical computer
Time Needed to Compute: Microseconds
Dijkstra’s Shortest Path Algorithm
Complexity Class: P
Big-O Notation: O(n2)
Permutation Space (10^n): 106
Tractable?: Yes
Physical Computability: Feasible
Energy Requirement: Low to moderate
Hardware Requirements: Classical computer
Time Needed to Compute: Seconds for moderate graph sizes
Primality Testing (AKS Algorithm)
Complexity Class: P
Big-O Notation: O((logn)6)
Permutation Space (10^n): 109
Tractable?: Yes
Physical Computability: Feasible
Energy Requirement: Low to moderate
Hardware Requirements: Classical or quantum computer
Time Needed to Compute: Seconds for large primes
Simulating Small Quantum Systems
Complexity Class: BQP
Big-O Notation: O(n3)
Permutation Space (10^n): 103
Tractable?: Yes
Physical Computability: Feasible on quantum devices
Energy Requirement: Moderate
Hardware Requirements: Quantum computer
Time Needed to Compute: Seconds to hours
Image Compression (e.g., JPEG)
Complexity Class: P
Big-O Notation: O(nlogn)
Permutation Space (10^n): 105
Tractable?: Yes
Physical Computability: Feasible
Energy Requirement: Low
Hardware Requirements: Classical computer
Time Needed to Compute: Milliseconds
Public Key Encryption (AES-128)
Complexity Class: P
Big-O Notation: O(n3)
Permutation Space (10^n): 1038
Tractable?: Yes
Physical Computability: Feasible
Energy Requirement: Moderate to high
Hardware Requirements: Classical or quantum computer
Time Needed to Compute: Milliseconds to seconds
Shortest Path in a Small Graph
Complexity Class: P
Big-O Notation: O(n2)
Permutation Space (10^n): 106
Tractable?: Yes
Physical Computability: Feasible
Energy Requirement: Low
Hardware Requirements: Classical computer
Time Needed to Compute: Milliseconds
Pattern Matching in Strings (Knuth-Morris-Pratt)
Complexity Class: P
Big-O Notation: O(n)
Permutation Space (10^n): 105
Tractable?: Yes
Physical Computability: Feasible
Energy Requirement: Low
Hardware Requirements: Classical computer
Time Needed to Compute: Microseconds
Linear Programming (Simplex)
Complexity Class: P
Big-O Notation: O(n3)
Permutation Space (10^n): 106
Tractable?: Yes
Physical Computability: Feasible
Energy Requirement: Moderate
Hardware Requirements: Classical computer
Time Needed to Compute: Seconds to minutes
Fourier Transform (FFT)
Complexity Class: P
Big-O Notation: O(nlogn)
Permutation Space (10^n): 106
Tractable?: Yes
Physical Computability: Feasible
Energy Requirement: Low to moderate
Hardware Requirements: Classical computer/GPU
Time Needed to Compute: Milliseconds
Basic Calculations (e.g., Arithmetic)
Complexity Class: P
Big-O Notation: O(1)
Permutation Space (10^n): 101
Tractable?: Yes
Physical Computability: Feasible
Energy Requirement: Very low
Hardware Requirements: Any computing device
Time Needed to Compute: Microseconds
The minimum complexity required to exhaust the computational capacity of the entire universe depends on several factors, primarily the available energy, the information storage capacity, and the physical constraints dictated by the laws of thermodynamics and quantum mechanics. The complexity of problems that would exceed this capacity must involve extremely high computational requirements, typically involving super-exponential growth or non-elementary functions.
To understand the threshold at which computational demands would exhaust the universe’s resources, we can look at key theoretical limits:
Tower of Exponentials and Non-Elementary Growth: For problems with complexity such as O(22n) or non-elementary growth (where the growth rate exceeds any fixed tower of exponentials), the input size n required to reach the universe’s capacity is likely smaller. For example, solving a problem in O(22100) would require far more operations than are computationally feasible within the physical constraints of the universe.
The distinct differences between tractable and intractable problems lie in their computational complexity, required resources (hardware and energy), and the time needed to compute solutions. Here’s a high-level comparison:
Characteristic
Tractable Problems
Intractable Problems
Complexity Class
Polynomial (P), Bounded Quantum Polynomial (BQP)
Exponential (EXP), Nondeterministic Polynomial (NP), unknown
Big-O Notation
Polynomial O(n),O(nlogn),O(n3)
Exponential O(2n),O(2n/2),O(2256), super-polynomial
Permutation Space
Up to 10^10
Often 1025 or higher; can exceed 1077
Physical Computability
Feasible with current technology
Mostly infeasible with current or foreseeable technology
Energy Requirement
Low to moderate
High to extraordinary (e.g., entire Earth’s power)
Hardware Requirements
Classical computers, GPUs, small quantum systems
Large quantum computers, supercomputers, theoretical planetary-scale devices
Time Needed to Compute
Milliseconds to hours
Days to millennia; some exceed the universe's age
Tractable vs Intractable Computability:
Complexity Class and Growth Rate: Tractable problems are typically polynomial-time solvable, with manageable growth rates that make them feasible for real-world applications. Intractable problems, on the other hand, often have exponential or unknown complexity, meaning they grow too rapidly to be solved for large inputs.
Permutation Space and Problem Size: Tractable problems operate within permutation spaces typically under , whereas intractable problems often involve spaces up to 1077 or more. This leads to intractable problems requiring far more computational steps as problem size increases.
Hardware Requirements: Tractable problems can be handled by classical computers or smaller quantum systems. Intractable problems demand hypothetical or advanced quantum computers, supercomputers, or even planetary-scale computing resources, especially for high complexity like O(2256).
Energy and Physical Constraints: Tractable problems generally require low to moderate energy. Intractable problems can demand energy levels that are practically unattainable, such as the power equivalent of an entire city or even Earth’s total energy output, making computation physically impossible on a large scale.
Time to Solve: Tractable problems are solvable in milliseconds to hours. Intractable problems might require timescales far beyond feasible limits—often thousands or millions of years, and in some cases, longer than the universe’s age.
Bibliography
[1] Landauer, R. (1961). Irreversibility and Heat Generation in the Computing Process. IBM Journal of Research and Development.
[2] Bekenstein, J.D. (1981). Universal Upper Bound on the Entropy-to-Energy Ratio for Bounded Systems. Physical Review D, 23(2), 287.
[3] Papadimitriou, C. H. (1994). Computational Complexity. Addison-Wesley. Arora, S., & Barak, B. (2009). Computational Complexity: A Modern Approach. Cambridge University Press.
[4] Shor, P.W. (1994). Algorithms for Quantum Computation: Discrete Logarithms and Factoring. Proceedings of 35th Annual Symposium on Foundations of Computer Science.
[5] Aaronson, S. (2013). Quantum Computing Since Democritus. Cambridge University Press.
Addressing the concept of intractability in computation involves recognizing the inherent limitations imposed by resource requirements and theoretical bounds. Intractable problems are defined by the exponential or even super-exponential growth in computational steps necessary for a solution as the input size grows. For example, a problem requiring operations—a figure approaching the number of atoms in the observable universe—would demand resources that surpass current or even conceivable future technology, demonstrating the challenge posed by such complexity.
The disparity between tractable and intractable problems is often rooted in their respective complexity classes. Problems solvable in polynomial time, denoted P, represent tasks where the computational effort required scales reasonably with input size. In contrast, intractable problems often reside in classes like NP-hard or EXP, where computational requirements grow exponentially, such as , or in higher complexities like double-exponential time . As the input grows, such complexities render even modestly-sized problems unmanageable.
The theoretical upper limits of computation can also be framed in physical terms. Landauer’s Principle asserts a minimum energy cost for erasing information, calculated by kBTln(2), where kB is Boltzmann's constant. At room temperature, each bit operation would require roughly joules. When extrapolated to operations, the energy requirement would approach joules—a significant portion of the universe's energy. Similarly, the Bekenstein Bound limits the information that can be stored within a finite region of space, setting a ceiling at bits for the universe, a constraint particularly relevant for computations with vast input spaces.
While theoretical advances such as quantum computing introduce potential solutions for some types of intractable problems, even these systems face limits. Quantum coherence and error correction present substantial challenges, especially for tasks requiring or more operations. Classical supercomputers, even at exaFLOP speeds, fall short for such demands; calculations of this magnitude would require vastly longer than the age of the universe. Consequently, intractable problems effectively delineate the bounds of feasible computation, underlining the resource constraints imposed by both current technology and the fundamental laws of physics.
Intractable problems are generally those that require computational resources—time, memory, or energy—that grow at a rate which renders them unsolvable for all but the smallest instances. Specifically, intractable problems often belong to complexity classes where resource requirements scale exponentially (e.g., ) or even in double-exponential time (e.g., ), as input size n increases. Problems in these classes rapidly exceed the practical computational limits available, leading to solutions that would require centuries, millennia, or longer—far beyond the lifetime of any computational system or indeed the universe itself.
Moreover, marketing campaigns often push the idea that quantum computers will revolutionize computing across all fields, from cryptography to artificial intelligence to logistics, without limitations. This blanket narrative glosses over the highly specialized nature of quantum computing. For example, while quantum algorithms can indeed solve specific problems more efficiently than classical algorithms, their application is narrowly confined to certain types of operations, such as optimization and searching through unstructured data. Exponential growth in problem complexity, like that required to brute-force possible encryption keys, remains computationally infeasible, even for quantum systems. The naive belief that quantum computers can conquer all computational challenges overlooks their dependency on coherence, error rates, and qubit stability—all of which drastically limit the problems they can realistically address.
This exaggeration is often deliberate. Companies and researchers know that "quantum" carries a powerful allure, especially among non-specialists who associate it with futuristic breakthroughs and unlimited potential. The term becomes a tool for marketing and branding, transforming modest advances in specific computational tasks into broader claims of technological revolution. By framing quantum computing as an impending solution to all hard computational problems, companies can attract funding from venture capitalists, government grants, and private investors eager to back the next big thing. In doing so, they gloss over the significant challenges that quantum computing faces, such as decoherence, error correction, and scaling limitations. These limitations mean that even if we could build highly stable and error-free quantum computers, they would still not be capable of brute-forcing problems involving, say, permutations—a level of complexity reserved for genuinely intractable tasks.
The comparison between classical bits and quantum qubits reveals fundamental differences in information representation and computational potential. While classical bits represent a single state out of possibilities, qubits exploit superposition to embody all possible states simultaneously. This capability challenges traditional computational complexity notions and offers new avenues for algorithm design.
Some cryptographic protocols are built on intractable problems with exponential complexity, such as those with complexity levels reaching possible permutations or beyond. Quantum computers, as powerful as they might become, cannot realistically process an exponentially large number of states due to physical and computational limits. For instance, any brute-force attack requiring 1080 computation is effectively impossible, even with quantum parallelism, as it would exceed the number of atoms in the observable universe.
Quantum computers are bound by the same physical laws as any other system, and energy consumption is a major limiting factor. Landauer’s Principle dictates that erasing even a single bit of information has a minimum energy cost. For quantum systems operating at room temperature, this cost can quickly escalate to astronomical levels when trying to brute-force cryptographic keys. Take an encryption system with possible permutations—a scale frequently involved in cryptographic schemes like RSA. The energy required to process even a fraction of these permutations would far exceed the energy available on Earth. So, while companies may claim their magical quantum devices can break any encryption, basic physics renders that idea ridiculous.
The Bekenstein Bound, a principle from theoretical physics, defines the maximum amount of information that can be stored in a finite region of space. Even the most idealized quantum computer is constrained by this limit, meaning there’s a hard cap on the complexity of problems it can address. Cryptographic problems designed with
Quantum computers are often lauded for their potential to offer exponential speedups. But for problems with exponential complexity—like those in classes such as NEXP or 2-EXP—this speedup is meaningless. A cryptographic problem requiring operations would take far longer than the universe has existed to solve, even with an ideal quantum computer running at full capacity. This reality remains absent from the marketing gloss of quantum companies, who seem to prefer math that stops at the convenient limits of their brochures, rather than extending into the harsh realm of computational feasibility.
Cryptographic systems are designed to be intractable by exploiting exponential complexity. Quantum computers, though capable of factoring small integers or solving niche problems like the discrete logarithm, face an insurmountable task when dealing with truly vast cryptographic keys. For example, a cryptosystem with permutations are immune to brute force attacks, whether from classical or quantum computers. Even with quantum algorithms like Shor’s or Grover’s, the reality is that scaling up to attack large cryptographic systems quickly enters a realm of computational impossibility. Quantum supremacy, in this context, is simply a myth.
This principle establishes a minimum amount of energy required to erase one bit of information, calculated as kBTln(2), where kB is Boltzmann’s constant and T is the temperature in kelvin. At room temperature, each bit flip necessitates roughly joules, and this energy cost accumulates significantly in large-scale computations.As computations become larger and more complex, the energy requirement quickly grows beyond what can be supplied by any practical power source, establishing a fundamental energy barrier to computation.
Proposed by physicist Jacob Bekenstein, the Bekenstein Bound defines the maximum amount of information that can be stored within a given volume of space with a defined amount of energy. For example, the Bekenstein Bound implies that a sphere with the volume and energy equivalent of the observable universe could store a maximum of bits of information.This bound imposes a strict physical limit on how much data any computational system could theoretically process, as it limits the storage and, by extension, the amount of information that can be manipulated.
The Planck scale refers to the smallest measurable length, approximately meters, at which classical concepts of space and time cease to apply. At this scale, quantum gravitational effects dominate, and any attempt to subdivide space further would require energy equivalent to the mass of a black hole.
The distinction between tractable and intractable problems is foundational in computer science, shaping our understanding of what is computationally feasible. Tractable problems can be solved within a reasonable amount of time and resources, typically by algorithms that grow in complexity polynomially with input size (denoted as , where k is a constant). These problems are said to belong to the complexity class P and are often considered efficiently solvable.
(EXP) includes problems that require exponential time to solve, meaning their complexity scales as or higher. Exponential-time problems grow too quickly with input size to be feasible for large instances, as the number of operations doubles with each increment in input.
Definition: Class EXP includes problems solvable in exponential time, where the resources required scale as for some polynomial . Exponential-time algorithms require a number of steps that doubles with each increase in input size.
Landauer’s Principle, formulated by physicist Rolf Landauer in 1961, states that erasing information from a system requires a minimum amount of energy. This minimum energy is due to the second law of thermodynamics, as any information-erasing process results in an increase in entropy. The energy required to erase a single bit of information is kBTln(2), where kB is Boltzmann’s constant ( J/K) and T is the absolute temperature in Kelvin.
Implications for High Complexity Problems: For problems requiring operations or more, the cumulative energy demand would approach or exceed the energy resources of the Earth. Computing beyond this threshold would be constrained by energy availability, making it practically unfeasible to solve certain problems no matter the technological advancements.
Exponential and Super-Exponential Growth: For an exponential complexity problem requiring 2n operations, each increase in n doubles the required steps. For moderately large n, say n=100, this translates to 2100 steps, or roughly operations. If a top supercomputer could hypothetically handle operations per second (1 exaFLOP), it would still take seconds (about 32,000 years) to complete this calculation.
Realistic Limits for Supercomputing: Considering that the age of the universe is approximately years, any problem requiring significantly more than 1018 operations quickly become unfeasible within this lifespan. Real-world constraints in high-complexity fields, such as large-scale physics simulations, are thus limited by what can be computed in timeframes relevant to human life or scientific inquiry.
To compute a problem of operations, the required computational resources, size of the computer, energy needs, and time required would be astronomically large and beyond any feasible technology. Here’s an analysis of what such a computation would entail:
Existing Supercomputers: The current fastest supercomputers, such as Frontier and Fugaku, can perform computations at around FLOPS (floating-point operations per second), or 1 exaFLOP.
Scaling Up: To reach operations, we’d need to run one of these supercomputers continuously for an extraordinarily long time, or build a massively larger system to distribute the workload.
One ExaFLOP Supercomputer: If a single FLOPS supercomputer were used, the time required to complete operations would be: = seconds
Conversion to Years: Converting seconds to years: seconds = years
This is vastly longer than the current age of the universe, estimated at around years. Even with massive scaling, this computation would not be achievable within any practical timeframe using current technology.
Energy per Operation: Assuming highly efficient computing, even at joules per operation (near the theoretical minimum energy per bit flip, according to Landauer’s Principle), the energy required would be operations joules = joules
Comparison to Total Energy: The total energy available in the observable universe is estimated to be around joules. The computation would thus require about joules, a significant fraction of all energy in the universe and practically impossible to harness or supply.
Hypothetical Hardware Requirements: To distribute operations efficiently, a hypothetical computer would need more processing units than there are atoms in the observable universe (around atoms). Each processor would also need to perform around one operation per second continuously, which is unrealistic.
Quantum Parallelism: Even with quantum computing’s potential for parallelism, handling operations would require an astronomical number of qubits, far beyond the capability of any conceivable quantum system.
Decoherence and Error Correction: Quantum systems require error correction to maintain coherence, which scales inefficiently with the number of qubits. Maintaining coherence for the duration needed to complete operations is beyond any realistic quantum computing model.
Requirement for Operations
supercomputers, each running at 1 exaFLOP continuously
years for one exaFLOP machine
joules (significant fraction of universe’s total energy)
A computation requiring operations are far beyond the reach of any conceivable computational technology, classical or quantum. Such a task would require more time, energy, and physical resources than are available in the observable universe, making it fundamentally infeasible. The sheer scale of these requirements demonstrates the practical and theoretical limits of computation as bound by the universe’s resources.
Total Compute Capacity on Earth ( FLOPS)
(thousands of FLOPS)
The potential for quantum computers to impact these areas depends significantly on advancements in qubit coherence, error correction, and algorithm development. While they offer theoretical advantages for specific types of problems (like factoring large integers or simulating quantum systems), practical applications requiring operations or more in fields like climate modeling or neuro-scientific simulations remain out of reach for now.
Problems requiring computations that significantly exceed operations and are currently infeasible include highly detailed simulations of complex systems and cryptanalysis of extremely secure encryption systems. These tasks remain beyond the current scope of both classical supercomputers and quantum computers, pending significant technological advancements in computational power, algorithm efficiency, and quantum technology development.
Scientific Simulations: These include weather modeling, astrophysical simulations, and molecular dynamics, which often require simulating complex interactions over long periods. These could quickly exceed operations, especially when aiming for higher resolution or longer simulation times.
Cryptanalysis: Breaking high-strength encryption, such as RSA with very long keys (e.g., 4096 bits or more), classically requires enormous computational efforts that can surpass operations but are still infeasible with current technology.
While quantum computers hold promise for certain types of problems, their current development stage doesn't allow them to tackle some of the world’s most computationally demanding problems that exceed operations:
Post-Quantum Cryptography: Even though quantum computers could potentially break many of today's encryption standards, algorithms designed to be resistant against quantum attacks (post-quantum cryptography) are designed to withstand quantum capabilities and require computational efforts that could go well beyond operations.
Landauer’s Principle states that each bit operation requires a minimum amount of energy due to information entropy, setting a lower bound on the energy needed for computation. Given this principle, the energy required for operations would be vast, even under ideal efficiency. In practice, it would be impossible to provide sufficient energy within Earth's constraints without fundamentally altering our energy production methods.
Problems requiring operations or more exceed what is practically or theoretically possible to compute, even with hypothetical advances in technology. Physical limits, including power, material availability, time, and fundamental laws of physics, impose barriers that prevent such computations. The enormity of resources required to reach this computational level, whether in terms of energy, space, or material, makes problems of this magnitude fundamentally intractable. In this range, computational problems surpass what can be accomplished within the bounds of the universe’s resources and lifespan.
Permutation Space ( ): An approximation of the number of possible configurations or steps.
Landauer’s Principle states that erasing one bit of information requires a minimum energy of kBTln(2), where kB is Boltzmann's constant and T is the temperature. At room temperature, this is roughly joules per bit.
For the entire observable universe, if we assume its energy availability is on the order of joules, this sets a theoretical maximum limit of around bit operations. However, this would only cover basic logical operations without considering complex algorithms with high computational complexity.
The Bekenstein Bound gives an upper limit on the amount of information that can be contained within a finite region of space with finite energy. For a sphere with the universe’s radius (roughly meters), the Bekenstein Bound is about bits. This storage capacity represents the maximum information that could theoretically be manipulated or stored across the universe.
Exponential and Super-Exponential Complexity: Problems with exponential complexity, such as O(2n), become intractable for large n even with vast resources. The actual threshold where the universe’s computational capacity is exhausted likely lies between and operations, depending on whether classical or quantum resources are used.
Threshold Complexity Class: Problems in the double-exponential class (O(22n)) or non-elementary recursive functions quickly surpass the universe’s computational limits. Approximate Minimum Size: For a problem to exceed the universe’s computational capacity, it would need to involve roughly to operations, corresponding to exponential complexities with inputs around n=300 to n=500 for O(2n) complexity, or smaller input sizes for double-exponential growth like O(22n).
The minimum problem size that exceeds the universe’s computational capacity lies within complexity classes with super-exponential growth, such as double-exponential functions (O(22n)) and non-elementary functions, with operations in the range of to , depending on the specific computational resources and energy constraints.