Since Turing, it's been generally accepted that computing systems have two types of limitations: algorithmic (some problems are fundamentally unsolvable) and engineering (the hardware is imperfect, noisy, and breaks). Quantum computing has added a third type— complexity : some problems are solvable, but require resources that grow faster than any polynomial. This supplement introduces a fourth type of restriction, previously not described in the literature: The fundamental limit of quantum computing (FQC) is a limit on the precision of any quantum computing device, which arises not from imperfections in the implementation, but from the discrete structure of spacetime at the Planck scale. This limit cannot be overcome by any technological advancement. It is not a consequence of decoherence , thermal noise, or imperfect gates. It is built into the geometry of reality—just as the speed of light is built into the structure of spacetime and cannot be overcome by any acceleration.Unlike collapse-model-based limits on quantum hardware, the Fundamental Limit of Quantum Computation is proposed as an ontological bound arising from the discrete structure of spacetime itself.
Since Turing, it's been generally accepted that computing systems have two types of limitations: algorithmic (some problems are fundamentally unsolvable) and engineering (the hardware is imperfect, noisy, and breaks). Quantum computing has added a third type— complexity : some problems are solvable, but require resources that grow faster than any polynomial. This supplement introduces a fourth type of restriction, previously not described in the literature: The fundamental limit of quantum computing (FQC) is a limit on the precision of any quantum computing device, which arises not from imperfections in the implementation, but from the discrete structure of spacetime at the Planck scale. This limit cannot be overcome by any technological advancement. It is not a consequence of decoherence , thermal noise, or imperfect gates. It is built into the geometry of reality—just as the speed of light is built into the structure of spacetime and cannot be overcome by any acceleration.Unlike collapse-model-based limits on quantum hardware, the Fundamental Limit of Quantum Computation is proposed as an ontological bound arising from the discrete structure of spacetime itself.
**Building on our earlier structural analysis of gate-based quantum computing,** recent advances in quantum technologies have further sharpened a long-standing tension between continuous quantum dynamics and discrete logical computation. Despite decades of progress in qubit coherence, fault-tolerant coding, and hybrid NISQ architectures, current quantum devices remain fundamentally constrained by a structural mismatch between fluctuating quantum states and the stability requirements of digital logic.This work extends and unifies that analysis by providing a geometry-based account of these limitations and situating quantum computing within a broader comparative framework of next-generation computation paradigms. The contributions of this work are threefold:• Historical and technical analysis: We review the evolution from Feynman’s proposal to Shor’s algorithm and modern fault-tolerance theory, highlighting how the community gradually shifted toward NISQ-based noisetolerant strategies.• Structural diagnosis of quantum limits: Building on the Protiformframework, we show that quantum systems inherently realize continuous fluctuation manifolds, whereas logical computation requires discrete convergencesurfaces. Measurement is therefore a forced collapse between incompatible geometries, making full fault-tolerant digital logic rendering practicalfault-tolerant scaling prohibitive—not merely technologically difficult.• Comparative evaluation of post-von Neumann architectures: Using a set of structural criteria (digital stability, noise resilience, scalability,native gate realizability, generality, and materials feasibility), we evaluatemajor candidates—quantum, photonic, neuromorphic, analog solvers, andtopological computing. The analysis shows that topological computing isthe only architecture satisfying all structural requirements, owingto its intrinsic protection of logical degrees of freedom by global invariants.Taken together, these results suggest that the fundamental limitation of quantum computing is not technological but structural. Protiform provides a unifiedexplanation for this boundary, while also identifying topological computing as thestructurally consistent path beyond the von Neumann paradigm.
The rapid evolution of computers in the half century since their invention has resulted in dramatically smaller and faster computers. However, from a computational point of view, all these computers look alike; for example, they are built out of simple logic gates. A fundamental thesis of computer science---the modern form of the Church--Turing thesis---asserts that this is inevitable in a deep sense. Any computer can be simulated with at most a polynomial factor slowdown by a probabilistic Turing machine. Quantum computation poses the first credible challenge to this thesis. It goes back to a suggestion by Feynman [4], who pointed out that there appears to be no efficient way of simulating a quantum mechanical system on a computer, and suggested that, perhaps, a computer based on quantum physical principles might be able to carry out the simulation efficiently. Two formal models for quantum computers---the quantum Turing machine [2] and quantum computational networks [3]---were defined by Deutsch. The first three papers in this issue describe efficient quantum algorithms for computational tasks that we do not know how to solve classically. In "Quantum Complexity Theory," Bernstein and Vazirani give the first formal evidence that quantum computers violate the modern form of the Church--Turing thesis. They show that a certain problem---the recursive Fourier sampling problem---can be solved in polynomial time on a quantum Turing machine, but relative to an oracle, requires superpolynomial time on a classical probabilistic Turing machine. Simon, in the paper "On the Power of Quantum Computation" introduces a fundamental projection technique and uses it to design an efficient quantum algorithm to determine whether a certain type of function is 2-1 or 1-1. He further shows that, relative to an oracle, this problem requires exponential time on a classical probabilistic Turing machine. In the paper "Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer," Shor gives remarkable polynomial time quantum algorithms for two of the most famous problems in computer science: factoring and discrete log. Since the computational hardness of these problems is the basis of several famous cryptosystems, Shor's paper very dramatically underlines the power of quantum computers. To understand the computational power of quantum computers, it is helpful to consider a quantum mechanical system of n particles, each of which can be in one of two states, labeled ∣0⟩ and ∣1⟩. If this were a classical system, then its instantaneous state could be described by n bits. However, in quantum physics, the system is allowed to be in a linear superposition of configurations, and indeed the instantaneous state of the system is described by a unit vector in the 2n dimensional vector space, whose basis vectors correspond to all the2n classical configurations. Therefore, to describe the instantaneous state of the system, we must specify 2n complex numbers. Nature must update 2n complex numbers at each instant to evolve the system in time. This is an extraordinary amount of effort, since even for n = 200, 2n is larger than estimates of the number of elementary particles in the visible universe. Nonetheless, there are limits to the power of quantum computers. In "Strengths and Weaknesses of Quantum Computing," Bennett, Bernstein, Brassard, and Vazirani show that, relative to a random oracle, with probability 1, the class NP cannot be solved on a quantum Turing machine in time o(2n/2). This bound is tight, since recent work of Grover [5] has shown how to accept any language in NP in time O(2n/2) on a quantum Turing machine. Quantum computers are necessarily time reversible. Indeed, Bennett's work [1] on reversible computation inspired early work on quantum computation that preceded Feynman's paper [4]. The reversibility requirement makes it quite complex to implement even basic computational primitives such as looping or composition. In "Quantum Complexity Theory," Bernstein and Vazirani show how to implement quantum programming primitives and give a construction for an efficient universal quantum Turing machine. The structure of the universal quantum Turing machine is quite simple: it consists of a deterministic Turing machine with a single "quantum coin flip." In "Quantum Computability," Adleman, DeMarrais, and Huang greatly simplify this further by showing that a very simple type of coin flip is sufficient---a rotation by an angle θ such that sinθ=3/5. Making quantum computers robust against noise and decoherence is an important and challenging problem. In "Stabilization of Quantum Computations by Symmetrization," Barenco, Berthiaume, Deutsch, Ekert, Jozsa, and Macchiavello show how to use the quantum watchdog effect to stabilize a quantum computation against noise. Their method is based on running several copies of the quantum computer in parallel and projecting its state into the symmetric subspace at frequent intervals. They show that the quantum watchdog effect results in the suppression of errors that lie outside the symmetric subspace. Quantum computation touches upon the foundations of both computer science and quantum physics. It is not unlikely that the issues raised by quantum computation will stimulate further research into the foundations of quantum physics. I wish to express my gratitude to several people who made this special section possible. Oded Goldreich acted as editor for two of the papers in the issue and dealt with them with his characteristic efficiency and judgment. The editorial staff at SIAM, most notably Lisa Dougherty, Beth Gallagher, Deidre Wunderlich, and Sam Young, were extremely helpful, patient, and resourceful. Finally, I would like to thank a number of referees whose careful and timely reviews were critical to putting together this issue. Umesh Vazirani
Quantum computing represents a revolutionary leap in computational power, leveraging the principles of quantum mechanics to process information in fundamentally new ways. Unlike classical computers, which use bits to represent data as 0s or 1s, quantum computers use quantum bits or qubits, enabling the phenomena of superposition and entanglement to perform complex calculations more efficiently (Nielsen & Chuang, 2010). This article explores the basic principles of quantum computing, the different models and algorithms that have been developed, and the current state of quantum hardware. The power of quantum computing lies in its ability to solve certain types of problems exponentially faster than classical computers. For instance, Shor's algorithm can factorize large integers in polynomial time, posing a significant threat to classical encryption methods (Shor, 1997). Similarly, Grover's algorithm provides a quadratic speedup for unstructured search problems, showcasing the potential for dramatic improvements in database search capabilities (Grover, 1996). These algorithms exemplify how quantum computing can revolutionize fields that rely on complex computations. Quantum hardware, however, is still in its nascent stage, with significant challenges to be overcome. Current quantum computers, such as those developed by IBM and Google, have demonstrated quantum supremacy but are limited by issues of qubit coherence and error rates (Arute et al., 2019). Quantum error correction and the development of more stable qubit technologies are critical areas of ongoing research (Preskill, 2018). Despite these challenges, recent advancements in quantum hardware and algorithms continue to drive the field forward, promising transformative impacts on industries such as cryptography, drug discovery, and artificial intelligence. As quantum computing continues to advance, it holds the promise of solving problems that are currently intractable for classical computers, paving the way for unprecedented technological and scientific advancements (Shor, 1997; Grover, 1996). This article provides a comprehensive overview of the current landscape of quantum computing, discussing its foundational principles, key algorithms, hardware challenges, and potential applications. By understanding these aspects, we can better appreciate the profound implications of quantum computing for the future of technology and science.
Quantum computation was proposed as an alternate computing process in the late-20th century, when classical computing models faced limitation in solving quantum mechanical problems. Major breakthroughs came in the 1990s, when quantum algorithms for integer factoring and black-box searching were invented that are faster than the best classical algorithms known even today. In this thesis, we look at quantum circuits with limited depth and size and analyse their power and limitations.
Computer Science is about the study of computational problems, many of which are motivated by real-world applications. We are interested as much in an actual method to solve a problem as in its unsolvability. In computational complexity theory we separate out the actual problem from the computing apparatus (e.g. a Turing machine, an AND-OR circuit) and analyse the hardness of the computation, due both to the intrinsic complexity of the problem and to the underlying model.
In this thesis, we consider quantum circuits built with a universal set of gates and show that non-constant depth is required to compute a particular function (parity) and other similar functions. The parity function, extensively studied previously, was shown to take non-constant depth for classical circuits too, thus providing an evidence of an inherent hardness in the function itself. On the other hand we show that the quantum circuit model is rich enough to build a universal circuit (one circuit that can simulate all circuits) that has roughly the same depth or size as the simulated circuit. We also consider a fault model in which some of the inputs to a gate might not be connected to the gate's output. We ask if we can detect the faulty inputs by testing the gate on different inputs. Our goal is to minimise the number of tests. We show that this is very hard for general gates but for certain gates it can be provably easier to detect faulty inputs. We even show that quantum algorithms can provably outperform classical algorithms for these gates. These results give us a sense of wherein lies the power of quantum computation and quantum circuits in particular.
This paper presents a comprehensive review of quantum computing and the technological foundations of quantum processors. Quantum computing represents a transformative computational paradigm that leverages the fundamental principles of quantum mechanics—such as superposition, entanglement, and quantum interference—to solve complex problems that are intractable for classical computers. At the core of this paradigm lies the quantum processor, a specialized hardware system designed to initialize, control, and measure quantum bits (qubits) through precise gate operations and coherent interactions. The survey examines the essential theoretical concepts underlying quantum computation and provides a structured comparison of leading quantum processor platforms. Prominent architectures, including superconducting circuits, trapped-ion systems, and photonic implementations, are analyzed in terms of their physical mechanisms, scalability, coherence properties, and engineering challenges. Each platform offers distinct advantages and faces specific technical limitations related to noise, error correction, and fabrication complexity. In addition to reviewing hardware developments, this study highlights key obstacles in advancing quantum processors toward large-scale, fault-tolerant systems. Issues such as qubit stability, error mitigation, thermal management, and system integration are discussed to present a realistic perspective on current progress. Performance indicators including gate fidelity, coherence time, connectivity, and scalability are used to compare architectural approaches. The objective of this paper is to provide a clear and organized synthesis of recent advancements in quantum computing hardware, thereby offering researchers and practitioners a foundational reference to support continued innovation and development in this rapidly evolving field.
The paper proposes a new category of limitation on quantum computation distinct from decoherence, control error, and algorithmic complexity arising from the discreteness of an underlying periodic gauge-graph model of spacetime, and derives it explicitly as εF ≳ 2π/L from single-valuedness of the U(1) holonomy on the graph. We quantify the bound against the phase precision required by Shor's algorithm across several RSA key sizes, showing the gap is too large for any near-term cryptographic relevance, and we identify the physically meaningful, testable content as a saturation of residual phase error with hardware scale rather than the underlying cosmological number itself. What we believe is new here is the derivation: rather than asserting a fundamental limit by analogy with other Planck-scale arguments, we obtain it as a direct structural consequence of a specific lattice model and give a falsifiable laboratory protocol (Sec. V) for testing it on existing superconducting processors.