Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2102.06833

Interactive quantum advantage with noisy, shallow Clifford circuits

Daniel Grier, Nathan Ju, Luke Schaeffer

Published Feb 13, 2021• 1 citation
arXivHTMLPDF

Abstract

Recent work by Bravyi et al. constructs a relation problem that a noisy constant-depth quantum circuit (QNC$^0$) can solve with near certainty (probability $1 - o(1)$), but that any bounded fan-in constant-depth classical circuit (NC$^0$) fails with some constant probability. We show that this robustness to noise can be achieved in the other low-depth quantum/classical circuit separations in this area. In particular, we show a general strategy for adding noise tolerance to the interactive protocols of Grier and Schaeffer. As a consequence, we obtain an unconditional separation between noisy QNC$^0$ circuits and AC$^0[p]$ circuits for all primes $p \geq 2$, and a conditional separation between noisy QNC$^0$ circuits and log-space classical machines under a plausible complexity-theoretic conjecture. A key component of this reduction is showing average-case hardness for the classical simulation tasks -- that is, showing that a classical simulation of the quantum interactive task is still powerful even if it is allowed to err with constant probability over a uniformly random input. We show that is true even for quantum tasks which are $\oplus$L-hard to simulate. To do this, we borrow techniques from randomized encodings used in cryptography.

DOI
10.48550/arxiv.2102.06833

Related papers

Interactive shallow Clifford circuits: Quantum advantage against NC¹ and beyond

Daniel Grier, Luke Schaeffer

Recent work of Bravyi et al. and follow-up work by Bene Watts et al. demonstrates a quantum advantage for shallow circuits: constant-depth quantum circuits can perform a task which constant-depth classical (i.e., 0) circuits cannot. Their results have the advantage that the quantum circuit is fairly practical, and their proofs are free of hardness assumptions (e.g., factoring is classically hard, etc.). Unfortunately, constant-depth classical circuits are too weak to yield a convincing real-world demonstration of quantum advantage. We attempt to hold on to the advantages of the above results, while increasing the power of the classical model. Our main result is a two-round interactive task which is solved by a constant-depth quantum circuit (using only Clifford gates, between neighboring qubits of a 2D grid, with Pauli measurements), but such that any classical solution would necessarily solve -hard problems. This implies a more powerful class of constant-depth classical circuits (e.g., 0[p] for any prime p) unconditionally cannot perform the task. Furthermore, under standard complexity-theoretic conjectures, log-depth circuits and log-space Turing machines cannot perform the task either. Using the same techniques, we prove hardness results for weaker complexity classes under more restrictive circuit topologies. Specifically, we give 0 interactive tasks on 2 × n and 1 × n grids which require classical simulations of power 1 and 0[6], respectively. Moreover, these hardness results are robust to a small constant fraction of error in the classical simulation.

arXiv:1911.02555•ACM Symposium on Theory of Computing (STOC)•Jun 6, 2020•14 citations

3D-local noisy shallow quantum circuits defeat unbounded fan-in classical circuits

Libor Caha, Xavier Coiteux-Roy, Robert König

Abstract Although quantum computers are believed to be more powerful than classical ones, a convincing experimental demonstration of this fact remains elusive. Proposed schemes either rely on unproven complexity-theoretic hardness assumptions, and/or require universal, fault-tolerant scalable quantum computers to implement. This has motivated the study of restricted models of computation. Constant-depth quantum circuits are known to be more powerful than unbounded fan-in classical ( $${{\mathsf{AC}}}^{0}$$ AC 0 -)circuits. Here we ask if this advantage persists in the presence of noise and under locality constraints. We present a computational problem for which every instance can be solved with near-certainty, despite noise, by a constant-depth quantum circuit with local operations in 3D. In contrast, every $${{\mathsf{AC}}}^{0}$$ AC 0 -circuit of size smaller than a certain (sub)exponential fails with near-certainty on a uniformly random instance. This constitutes a proposal with built-in fault-tolerance to experimentally observe the strongest known complexity-theoretic separation between classical and quantum computation.

Nature Communications•Aug 11, 2026•0 citations

Quantum Advantage with Noisy Shallow Circuits in 3D

Sergey B. Bravyi, David Gosset, Robert König +1 more

Prior work has shown that there exists a relation problem which can be solved with certainty by a constant-depth quantum circuit composed of geometrically local gates in two dimensions, but cannot be solved with high probability by any classical constant depth circuit composed of bounded fan-in gates. Here we provide two extensions of this result. Firstly, we show that a separation in computational power persists even when the constant-depth quantum circuit is restricted to geometrically local gates in one dimension. The corresponding quantum algorithm is the simplest we know of which achieves a quantum advantage of this type. Our second, main result, is that a separation persists even if the shallow quantum circuit is corrupted by noise. We construct a relation problem which can be solved with near certainty using a noisy constant-depth quantum circuit composed of geometrically local gates in three dimensions, provided the noise rate is below a certain constant threshold value. On the other hand, the problem cannot be solved with high probability by a noise-free classical circuit of constant depth. A key component of the proof is a quantum error-correcting code which admits constant-depth logical Clifford gates and single-shot logical state preparation. We show that the surface code meets these criteria.

Annual Symposium on Foundations of Computer Science•Nov 1, 2019•7 citations

Parallel classical simulation of noisy shallow circuits: no quantum advantage in 1D

Robert Koenig, Marco Tomamichel

We consider quantum circuits consisting of $d$ layers of nearest-neighbor two-qubit gates acting on $n$ qubits arranged on a line, where every qubit is independently depolarized with a constant probability before each layer. We describe a randomized parallel algorithm which samples from the output distribution of any such circuit to within total variation error $δ$, with parallel runtime $2^{O(d)}\log\log(n/δ)$ and $n 2^{O(d)}$ elementary real-arithmetic operations. Without noise, the same approach gives an exact sampler with parallel runtime $O(\log n)$. For constant depth, we further show that the input/output behavior of the noisy quantum circuit is reproduced up to a constant error by a randomized $\mathsf{AC}^0$-circuit, that is, a Boolean circuit of polynomial size and constant depth with unbounded fan-in AND and OR gates and NOT gates. Consequently, every relation problem solved by a noisy constant-depth quantum circuit in one dimension is also solved, with essentially the same success probability, by a randomized $\mathsf{AC}^0$-circuit. This rules out, for noisy circuits in one dimension, the unconditional quantum advantage established for noisy shallow circuits in two and three dimensions, where polynomial-size classical circuits over the same gate set require depth $Ω(\log n/\log\log n)$. Our algorithm exploits the fact that depolarizing noise cuts a one-dimensional circuit into independent pieces of logarithmic width, each of which can be sampled exactly in parallel.

quant-pharXiv:2609.39448•Oct 1, 2026•0 citations

Unconditional Quantum Advantage for Sampling with Shallow Circuits

Adam Bene Watts, Natalie Parham

Recent work by Bravyi, Gosset, and Koenig showed that there exists a search problem that a constant-depth quantum circuit can solve, but that any constant-depth classical circuit with bounded fan-in cannot. They also pose the question: Can we achieve a similar proof of separation for an input-independent sampling task? In this paper, we show that the answer to this question is yes when the number of random input bits given to the classical circuit is bounded.We introduce a distribution D n over { 0 , 1 } n and construct a constant-depth uniform quantum circuit family { C n } n such that C n samples from a distribution close to D n in total variation distance. For any δ < 1 we also prove, unconditionally, that any classical circuit with bounded fan-in gates that takes as input k n + n δ i.i.d. Bernouli random variables with entropy 1 / k and produces output close to D n in total variation distance has depth Ω ( log ⁡ log ⁡ n ) . This gives an unconditional proof that constant-depth quantum circuits can sample from distributions that can't be reproduced by constant-depth bounded fan-in classical circuits, even up to additive error. We also show a similar separation between constant-depth quantum circuits with advice and classical circuits with bounded fan-in and fan-out, but access to an unbounded number of i.i.d random inputs.The distribution D n and classical circuit lower bounds are inspired by work of Viola, in which he shows a different (but related) distribution cannot be sampled from approximately by constant-depth bounded fan-in classical circuits.

arXiv:2301.00995•Quantum•Aug 12, 2026•2 citations

Worst-case depth hierarchy for shallow quantum circuits

Min-Hsiu Hsieh, Michael de Oliveira, Sathyawageeswar Subramanian +1 more

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 $\mathsf{QNC}^0$. For each $d\ge 12$, we construct a family of two-round interactive problems on which no depth-$(d-1)$ 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 $d$ 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 $d$, yielding a hierarchy of problems that show unconditional quantum advantage of $\mathsf{QNC}^0$ over $\mathsf{NC}^0$. 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 $\mathsf{QNC}^0$, establishing a genuinely quantum hierarchy.

quant-pharXiv:2606.16425•Jun 16, 2026•0 citations
Recent work by Bravyi et al. constructs a relation problem that a noisy constant-depth quantum circuit (QNC0^00) can solve with near certainty (probability 1−o(1)1 - o(1)1−o(1)), but that any bounded fan-in constant-depth classical circuit (NC0^00) fails with some constant probability. We show that this robustness to noise can be achieved in the other low-depth quantum/classical circuit separations in this area. In particular, we show a general strategy for adding noise tolerance to the interactive protocols of Grier and Schaeffer. As a consequence, we obtain an unconditional separation between noisy QNC0^00 circuits and AC0[p]^0[p]0[p] circuits for all primes p≥2p \geq 2p≥2, and a conditional separation between noisy QNC0^00 circuits and log-space classical machines under a plausible complexity-theoretic conjecture. A key component of this reduction is showing average-case hardness for the classical simulation tasks -- that is, showing that a classical simulation of the quantum interactive task is still powerful even if it is allowed to err with constant probability over a uniformly random input. We show that is true even for quantum tasks which are ⊕\oplus⊕L-hard to simulate. To do this, we borrow techniques from randomized encodings used in cryptography.
Recent work of Bravyi et al. and follow-up work by Bene Watts et al. demonstrates a quantum advantage for shallow circuits: constant-depth quantum circuits can perform a task which constant-depth classical (i.e., 0) circuits cannot. Their results have the advantage that the quantum circuit is fairly practical, and their proofs are free of hardness assumptions (e.g., factoring is classically hard, etc.). Unfortunately, constant-depth classical circuits are too weak to yield a convincing real-world demonstration of quantum advantage. We attempt to hold on to the advantages of the above results, while increasing the power of the classical model. Our main result is a two-round interactive task which is solved by a constant-depth quantum circuit (using only Clifford gates, between neighboring qubits of a 2D grid, with Pauli measurements), but such that any classical solution would necessarily solve -hard problems. This implies a more powerful class of constant-depth classical circuits (e.g., 0[p] for any prime p) unconditionally cannot perform the task. Furthermore, under standard complexity-theoretic conjectures, log-depth circuits and log-space Turing machines cannot perform the task either. Using the same techniques, we prove hardness results for weaker complexity classes under more restrictive circuit topologies. Specifically, we give 0 interactive tasks on 2 × n and 1 × n grids which require classical simulations of power 1 and 0[6], respectively. Moreover, these hardness results are robust to a small constant fraction of error in the classical simulation.
Abstract Although quantum computers are believed to be more powerful than classical ones, a convincing experimental demonstration of this fact remains elusive. Proposed schemes either rely on unproven complexity-theoretic hardness assumptions, and/or require universal, fault-tolerant scalable quantum computers to implement. This has motivated the study of restricted models of computation. Constant-depth quantum circuits are known to be more powerful than unbounded fan-in classical ( AC0{{\mathsf{AC}}}^{0}AC0 AC 0 -)circuits. Here we ask if this advantage persists in the presence of noise and under locality constraints. We present a computational problem for which every instance can be solved with near-certainty, despite noise, by a constant-depth quantum circuit with local operations in 3D. In contrast, every AC0{{\mathsf{AC}}}^{0}AC0 AC 0 -circuit of size smaller than a certain (sub)exponential fails with near-certainty on a uniformly random instance. This constitutes a proposal with built-in fault-tolerance to experimentally observe the strongest known complexity-theoretic separation between classical and quantum computation.
Prior work has shown that there exists a relation problem which can be solved with certainty by a constant-depth quantum circuit composed of geometrically local gates in two dimensions, but cannot be solved with high probability by any classical constant depth circuit composed of bounded fan-in gates. Here we provide two extensions of this result. Firstly, we show that a separation in computational power persists even when the constant-depth quantum circuit is restricted to geometrically local gates in one dimension. The corresponding quantum algorithm is the simplest we know of which achieves a quantum advantage of this type. Our second, main result, is that a separation persists even if the shallow quantum circuit is corrupted by noise. We construct a relation problem which can be solved with near certainty using a noisy constant-depth quantum circuit composed of geometrically local gates in three dimensions, provided the noise rate is below a certain constant threshold value. On the other hand, the problem cannot be solved with high probability by a noise-free classical circuit of constant depth. A key component of the proof is a quantum error-correcting code which admits constant-depth logical Clifford gates and single-shot logical state preparation. We show that the surface code meets these criteria.
We consider quantum circuits consisting of ddd layers of nearest-neighbor two-qubit gates acting on nnn qubits arranged on a line, where every qubit is independently depolarized with a constant probability before each layer. We describe a randomized parallel algorithm which samples from the output distribution of any such circuit to within total variation error δδδ, with parallel runtime 2O(d)log⁡log⁡(n/δ)2^{O(d)}\log\log(n/δ)2O(d)loglog(n/δ) and n2O(d)n 2^{O(d)}n2O(d) elementary real-arithmetic operations. Without noise, the same approach gives an exact sampler with parallel runtime O(log⁡n)O(\log n)O(logn). For constant depth, we further show that the input/output behavior of the noisy quantum circuit is reproduced up to a constant error by a randomized AC0\mathsf{AC}^0AC0-circuit, that is, a Boolean circuit of polynomial size and constant depth with unbounded fan-in AND and OR gates and NOT gates. Consequently, every relation problem solved by a noisy constant-depth quantum circuit in one dimension is also solved, with essentially the same success probability, by a randomized AC0\mathsf{AC}^0AC0-circuit. This rules out, for noisy circuits in one dimension, the unconditional quantum advantage established for noisy shallow circuits in two and three dimensions, where polynomial-size classical circuits over the same gate set require depth Ω(log⁡n/log⁡log⁡n)Ω(\log n/\log\log n)Ω(logn/loglogn). Our algorithm exploits the fact that depolarizing noise cuts a one-dimensional circuit into independent pieces of logarithmic width, each of which can be sampled exactly in parallel.
Recent work by Bravyi, Gosset, and Koenig showed that there exists a search problem that a constant-depth quantum circuit can solve, but that any constant-depth classical circuit with bounded fan-in cannot. They also pose the question: Can we achieve a similar proof of separation for an input-independent sampling task? In this paper, we show that the answer to this question is yes when the number of random input bits given to the classical circuit is bounded.We introduce a distribution D n over { 0 , 1 } n and construct a constant-depth uniform quantum circuit family { C n } n such that C n samples from a distribution close to D n in total variation distance. For any δ < 1 we also prove, unconditionally, that any classical circuit with bounded fan-in gates that takes as input k n + n δ i.i.d. Bernouli random variables with entropy 1 / k and produces output close to D n in total variation distance has depth Ω ( log ⁡ log ⁡ n ) . This gives an unconditional proof that constant-depth quantum circuits can sample from distributions that can't be reproduced by constant-depth bounded fan-in classical circuits, even up to additive error. We also show a similar separation between constant-depth quantum circuits with advice and classical circuits with bounded fan-in and fan-out, but access to an unbounded number of i.i.d random inputs.The distribution D n and classical circuit lower bounds are inspired by work of Viola, in which he shows a different (but related) distribution cannot be sampled from approximately by constant-depth bounded fan-in classical circuits.
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.