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The interplay between quantum foundations and circuit complexity

Michael Oliveira

Published Feb 1, 2025• 0 citations
OpenAlex

Abstract

Quantum computers were initially proposed to surpass the limitations of classical computers in simulating physical systems. Soon after, the field of quantum computing advanced through landmark discoveries of quantum algorithms, with some conjectured to provide exponential advantages over classical counterparts. These include Shor’s factoring algorithm, which has profound implications in cryptography. Additionally, moderate yet significant polynomial separations have further increased interest in the field. However, despite this progress, unconditional proofs confirming quantum superiority on the theoretical side remain elusive. At the same time, large, noise-resilient quantum computations on the experimental side, which could challenge the extended Church-Turing thesis by potentially breaking classical computational barriers, have yet to be achieved. Meanwhile, our understanding of the differences between classical and quantum mechanics has deepened. We now possess rigorous theoretical tools, such as Bell inequalities and nonlocal games, to describe behaviors unique to these theories. Moreover, the main loopholes for observing non-classical correlations have been closed experimentally, showing that correlations in the physical world exceed those predicted by classical mechanics. This thesis investigates computational complexity differences between quantum and classical models by studying shallow-depth circuits. The primary focus lies in identifying, elucidating, and proving computational quantum advantages. Additionally, the analysis delves into the bijection between these circuits and foundational non-local game constructions, exploring how quantum-classical deviations in multi-party correlation settings translate to computational advantages. A key question addressed is how to synthesize constant-depth quantum circuits for various problems to leverage quantum advantages. To this end, circuit synthesis techniques tailored to measurement-based quantum computation are developed, enabling simultaneously optimal quantum strategies for certain nonlocal games based on a particular bijection between this model and XOR non-local games. Building on this work, we confront a pivotal challenge: If a candidate quantum circuit potentially provides a computational advantage, how can we prove that no classical circuit surpasses it? This work presents new unconditional quantum-classical separations within shallow-depth settings, distinguishing for the first time constant-depth quantum circuits with one- and two-qubit gates from biased polynomial threshold circuits, which incorporate neural-network-type functionalities. These separations extend to prime-dimensional quantum systems, and we conjecture to have identified Razborov-Smolensky-type problems uniquely solvable by specific prime dimensions. The thesis also differentiates constant-depth quantum circuits with multi-qubit Toffoli gates from their classical counterparts, emphasizing the power of parallel quantum computation. Finally, verifying shallow-depth quantum computations is challenging due to conjectured exponential separations from classical computations, limiting correctness verification through simulation. While specific quantum algorithms within BQP have been addressed, we explore whether a quantum computer can reliably execute any instance from a fixed class of quantum circuits. Moving beyond heuristic-based solutions, we propose a benchmarking protocol that certifies, with exponentially high confidence, the execution of arbitrary circuits of specified dimensions, widths, and depths.

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On provable quantum advantages in combinatorial optimization and distribution learning

Pirnay, Niklas

This dissertation investigates the fundamental computational capabilities of quantum algorithms with respect to combinatorial optimization and distribution learning. We establish rigorous quantum advantages over classical algorithms for such problems under both widely accepted cryptographic assumptions and unconditionally. First, we demonstrate a super-polynomial quantum advantage for approximating solutions to combinatorial optimization problems, explicitly constructing advantage-bearing instances of Integer Linear Programming. Our proofs build upon approximation hardness results that instantiate a connection between the representation-independent hardness of learning and approximation hardness, as captured by Occam’s razor within computational learning theory. We show that the instances we construct can be efficiently approximated by quantum algorithms utilizing Shor’s factoring algorithm, yet remain classically intractable unless standard cryptographic assumptions fail. Second, we fill a previously open gap by proving a super-polynomial quantum advantage for learning the density function of probability distributions, complementing prior results established for generative distribution learning. We show that the hardness of density modeling can be established under weaker cryptographic assumptions involving pseudorandom functions. The quantum advantage for density modeling fundamentally arises from Shor’s algorithm for computing discrete logarithms. Moreover, we explicitly articulate known connections between Boolean function learning hardness and distribution learning hardness using established constructions. Third, addressing practical constraints of near-term quantum devices, we establish an unconditional quantum advantage for generative distribution learning using shallow, constant-depth quantum circuits. We find that shallow quantum circuits pose a superior class of hypotheses compared to classical counterparts. This result underscores the potential of near-term quantum hardware by highlighting the unique computational capabilities enabled by quantum entanglement and non-local correlations, offered even by constrained devices. Collectively, these contributions deepen our understanding of quantum computational advantages for combinatorial optimization and distribution learning, clarifying their foundations and pointing toward promising avenues for future exploration.

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Quantum computers were initially proposed to surpass the limitations of classical computers in simulating physical systems. Soon after, the field of quantum computing advanced through landmark discoveries of quantum algorithms, with some conjectured to provide exponential advantages over classical counterparts. These include Shor’s factoring algorithm, which has profound implications in cryptography. Additionally, moderate yet significant polynomial separations have further increased interest in the field. However, despite this progress, unconditional proofs confirming quantum superiority on the theoretical side remain elusive. At the same time, large, noise-resilient quantum computations on the experimental side, which could challenge the extended Church-Turing thesis by potentially breaking classical computational barriers, have yet to be achieved. Meanwhile, our understanding of the differences between classical and quantum mechanics has deepened. We now possess rigorous theoretical tools, such as Bell inequalities and nonlocal games, to describe behaviors unique to these theories. Moreover, the main loopholes for observing non-classical correlations have been closed experimentally, showing that correlations in the physical world exceed those predicted by classical mechanics. This thesis investigates computational complexity differences between quantum and classical models by studying shallow-depth circuits. The primary focus lies in identifying, elucidating, and proving computational quantum advantages. Additionally, the analysis delves into the bijection between these circuits and foundational non-local game constructions, exploring how quantum-classical deviations in multi-party correlation settings translate to computational advantages. A key question addressed is how to synthesize constant-depth quantum circuits for various problems to leverage quantum advantages. To this end, circuit synthesis techniques tailored to measurement-based quantum computation are developed, enabling simultaneously optimal quantum strategies for certain nonlocal games based on a particular bijection between this model and XOR non-local games. Building on this work, we confront a pivotal challenge: If a candidate quantum circuit potentially provides a computational advantage, how can we prove that no classical circuit surpasses it? This work presents new unconditional quantum-classical separations within shallow-depth settings, distinguishing for the first time constant-depth quantum circuits with one- and two-qubit gates from biased polynomial threshold circuits, which incorporate neural-network-type functionalities. These separations extend to prime-dimensional quantum systems, and we conjecture to have identified Razborov-Smolensky-type problems uniquely solvable by specific prime dimensions. The thesis also differentiates constant-depth quantum circuits with multi-qubit Toffoli gates from their classical counterparts, emphasizing the power of parallel quantum computation. Finally, verifying shallow-depth quantum computations is challenging due to conjectured exponential separations from classical computations, limiting correctness verification through simulation. While specific quantum algorithms within BQP have been addressed, we explore whether a quantum computer can reliably execute any instance from a fixed class of quantum circuits. Moving beyond heuristic-based solutions, we propose a benchmarking protocol that certifies, with exponentially high confidence, the execution of arbitrary circuits of specified dimensions, widths, and depths.
The rapid evolution of quantum devices fuels concerted efforts to experimentally establish quantum advantage over classical computing. Many demonstrations of quantum advantage, however, rely on computational assumptions and face verification challenges. Furthermore, steady advances in classical algorithms and machine learning make the issue of provable, practically demonstrable quantum advantage a moving target. In this work, we unconditionally demonstrate that parallel quantum computation can exhibit greater computational power than previously recognized. We prove that polynomial-size biased threshold circuits of constant depth-which model neural networks with tunable expressivity-fail to solve certain problems solvable by small constant-depth quantum circuits with local gates, for values of the bias that allow quantifiably large computational power. Additionally, we identify a family of problems that are solvable in constant depth by a universal quantum computer over prime-dimensional qudits with bounded connectivity, but remain hard for polynomial-size biased threshold circuits. We thereby bridge the foundational theory of non-local games in higher dimensions with computational advantage on emerging devices operating on a wide range of physical platforms. Finally, we show that these quantum advantages are robust to noise across all prime qudit dimensions with all-to-all connectivity, enhancing their practical appeal.
Circuit depth is a central resource in complexity theory. While bounded-depth classical circuits admit well-understood hierarchy theorems, the internal structure of constant-depth quantum computation remains comparatively unexplored. We prove an explicit depth hierarchy theorem for QNC0\mathsf{QNC}^0QNC0. For each d≥12d\ge 12d≥12, we construct a family of two-round interactive problems on which no depth-(d−1)(d-1)(d−1) quantum circuit can achieve near-perfect success, regardless of gate set, circuit size, or ancillary qubits. In contrast, we prove that our construction admits realizations by simple bounded fan-in quantum circuits of depth larger than ddd by a small constant factor. Moreover, all bounded fan-in classical circuits of sublogarithmic depth (in the input size) fail to achieve perfect success on these tasks for every ddd, yielding a hierarchy of problems that show unconditional quantum advantage of QNC0\mathsf{QNC}^0QNC0 over NC0\mathsf{NC}^0NC0. A key obstacle is the scarcity of lower bound techniques for quantum circuits. To address this, we develop methods to analyze how depth affects a circuit's ability to realize nonlocal correlations amongst its output qubits in a fine-grained manner. Our approach exploits the correspondence between constraint systems and nonlocal games, translating group-theoretic constructions into rigid operator-valued constraint systems and then into non-local games. In particular, we construct constraint systems whose unique faithful operator-valued solutions require every perfect strategy, and every near-perfect strategy to a fixed precision, to implement multi-controlled phase operations. This reduces to a nonlocal unitary-synthesis problem, yielding depth lower bounds for both shallow quantum and classical circuits. These results show that increasing depth strictly increases computational power within QNC0\mathsf{QNC}^0QNC0, establishing a genuinely quantum hierarchy.
Abstract The rapid evolution of quantum devices fuels concerted efforts to experimentally establish quantum advantage over classical computing. Many demonstrations of quantum advantage, however, rely on computational assumptions and face verification challenges. Furthermore, steady advances in classical algorithms and machine learning make the issue of provable, practically demonstrable quantum advantage a moving target. In this work, we unconditionally demonstrate that parallel quantum computation can exhibit greater computational power than previously recognized. We prove that polynomial-size biased threshold circuits of constant depth—which model neural networks with tunable expressivity—fail to solve certain problems solvable by small constant-depth quantum circuits with local gates, for values of the bias that allow quantifiably large computational power. Additionally, we identify a family of problems that are solvable in constant depth by a universal quantum computer over prime-dimensional qudits with bounded connectivity, but remain hard for polynomial-size biased threshold circuits. We thereby bridge the foundational theory of non-local games in higher dimensions with computational advantage on emerging devices operating on a wide range of physical platforms. Finally, we show that these quantum advantages are robust to noise across all prime qudit dimensions with all-to-all connectivity, enhancing their practical appeal.
The relevance of shallow-depth quantum circuits has recently increased, mainly due to their applicability to near-term devices. In this context, one of the main goals of quantum circuit complexity is to find problems that can be solved by quantum shallow circuits but require more computational resources classically. Our first contribution in this work is to prove new separations between classical and quantum constant-depth circuits. Firstly, we show a separation between constant-depth quantum circuits with quantum advice QNC0/qpoly\mathsf {QNC}^0/\mathsf {qpoly} QNC0/qpoly , and AC0[p]\mathsf {AC}^0[p] AC0[p] , which is the class of classical constant-depth circuits with unbounded-fan in and MODp\mathsf {MOD}_{p} MODp​ gates. Additionally, we show a separation between QAC0\mathsf {QAC}^0 QAC0 , the circuit class containing Toffoli gates with unbounded control, and AC0[p]\mathsf {AC}^0[p] AC0[p] , when QAC0\mathsf {QAC}^0 QAC0 is augmented with additional mid-circuit measurements and classical fanout. This establishes the first such separation for a shallow-depth quantum class that does not involve quantum fanout gates, while relying solely on finite quantum gate sets. Equivalently, this yields a separation between AC0[p]\mathsf {AC}^0[p] AC0[p] and [QNC0,AC0]2[\mathsf {QNC}^0, \mathsf {AC}^0]^2 [QNC0,AC0]2 , i.e., shallow quantum circuits interleaved with simple classical computation. Secondly, we consider QNC0\mathsf {QNC}^0 QNC0 circuits with infinite-size gate sets. We show that these circuits, along with quantum prime modular gates or classical prime modular gates in combination with classical fanout, can implement threshold gates, showing that QNC0[p]=QTC0\mathsf {QNC}^0[p]=\mathsf {QTC}^0 QNC0[p]=QTC0 . Finally, we also show that in the infinite-size gate set case, these quantum circuit classes for higher-dimensional Hilbert spaces do not offer any advantage to standard qubit implementations.
Quantum effects can enhance information-processing capabilities and speed up the solution of certain computational problems. Whether a quantum advantage can be rigorously proven in some setting or demonstrated experimentally using near-term devices is the subject of active debate. We show that parallel quantum algorithms running in a constant time period are strictly more powerful than their classical counterparts; they are provably better at solving certain linear algebra problems associated with binary quadratic forms. Our work gives an unconditional proof of a computational quantum advantage and simultaneously pinpoints its origin: It is a consequence of quantum nonlocality. The proposed quantum algorithm is a suitable candidate for near-future experimental realizations, as it requires only constant-depth quantum circuits with nearest-neighbor gates on a two-dimensional grid of qubits (quantum bits).
This dissertation investigates the fundamental computational capabilities of quantum algorithms with respect to combinatorial optimization and distribution learning. We establish rigorous quantum advantages over classical algorithms for such problems under both widely accepted cryptographic assumptions and unconditionally. First, we demonstrate a super-polynomial quantum advantage for approximating solutions to combinatorial optimization problems, explicitly constructing advantage-bearing instances of Integer Linear Programming. Our proofs build upon approximation hardness results that instantiate a connection between the representation-independent hardness of learning and approximation hardness, as captured by Occam’s razor within computational learning theory. We show that the instances we construct can be efficiently approximated by quantum algorithms utilizing Shor’s factoring algorithm, yet remain classically intractable unless standard cryptographic assumptions fail. Second, we fill a previously open gap by proving a super-polynomial quantum advantage for learning the density function of probability distributions, complementing prior results established for generative distribution learning. We show that the hardness of density modeling can be established under weaker cryptographic assumptions involving pseudorandom functions. The quantum advantage for density modeling fundamentally arises from Shor’s algorithm for computing discrete logarithms. Moreover, we explicitly articulate known connections between Boolean function learning hardness and distribution learning hardness using established constructions. Third, addressing practical constraints of near-term quantum devices, we establish an unconditional quantum advantage for generative distribution learning using shallow, constant-depth quantum circuits. We find that shallow quantum circuits pose a superior class of hypotheses compared to classical counterparts. This result underscores the potential of near-term quantum hardware by highlighting the unique computational capabilities enabled by quantum entanglement and non-local correlations, offered even by constrained devices. Collectively, these contributions deepen our understanding of quantum computational advantages for combinatorial optimization and distribution learning, clarifying their foundations and pointing toward promising avenues for future exploration.