Average-Case Quantum Advantage with Shallow Circuits

Gall, François Le

doi:10.48550/arxiv.1810.12792

Cited by 2 publications

(4 citation statements)

References 9 publications

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“…Two other works obtained concurrently and independently from ours establish directly related, but strictly incomparable, results. In [Gal18] Le Gall obtains an average-case hardness result that is very similar to our Theorem 1.2, with a concrete constant c ′ = 1/2 that is likely better than the one that we achieve here. Le Gall's proof is based on an ingenious construction using the framework of graph states; although some aspects are similar in spirit to ours (such as the use of parallel repetition to amplify the soundness guarantees) the proof rests on rather different intuition.…”

Section: Introductionsupporting

confidence: 68%

Trading locality for time: certifiable randomness from low-depth circuits

Coudron¹,

Stark²,

Vidick³

2018

Preprint

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The generation of certifiable randomness is the most fundamental information-theoretic task that meaningfully separates quantum devices from their classical counterparts. We propose a protocol for exponential certified randomness expansion using a single quantum device. The protocol calls for the device to implement a simple quantum circuit of constant depth on a 2D lattice of qubits. The output of the circuit can be verified classically in linear time, and is guaranteed to contain a polynomial number of certified random bits assuming that the device used to generate the output operated using a (classical or quantum) circuit of sub-logarithmic depth. This assumption contrasts with the locality assumption used for randomness certification based on Bell inequality violation and more recent proposals for randomness certification based on computational assumptions. Furthermore, to demonstrate randomness generation it is sufficient for a device to sample from the ideal output distribution within constant statistical distance.Our procedure is inspired by recent work of Bravyi et al. (Science 2018), who introduced a relational problem that can be solved by a constant-depth quantum circuit, but provably cannot be solved by any classical circuit of sub-logarithmic depth. We develop the discovery of Bravyi et al. into a framework for robust randomness expansion. Our results leads to a new proposal for a demonstrated quantum advantage that has some advantages compared to existing proposals. First, our proposal does not rest on any complexity-theoretic conjectures, but relies on the physical assumption that the adversarial device being tested implements a circuit of sub-logarithmic depth. Second, success on our task can be easily verified in classical linear time. Finally, our task is more noise-tolerant than most other existing proposals that can only tolerate multiplicative error, or require additional conjectures from complexity theory; in contrast, we are able to allow a small constant additive error in total variation distance between the sampled and ideal distributions.

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Section: Introductionsupporting

confidence: 68%

Trading locality for time: certifiable randomness from low-depth circuits

Coudron¹,

Stark²,

Vidick³

2018

Preprint

View full text Add to dashboard Cite

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“…This separation also holds in the average-case setting when the classical circuit only needs to solve a few instances of the problem that are drawn randomly from a suitable distribution [13, Supplementary Material]. Similar proofs of quantum advantage with associated average-case hardness results for classical circuits have been obtained more recently in [14,15], see also [16]. In this work we extend these results in two distinct ways.…”

Section: Contentssupporting

confidence: 61%

“…Acknowledgements 53 References 54 (i) the problem can be solved with certainty by a constant-depth quantum circuit composed of geometrically local gates on a 2D grid of qubits, while (ii) any classical probabilistic circuit which solves the problem with success probability at least 7/8 must have depth growing logarithmically with the input size.This separation also holds in the average-case setting when the classical circuit only needs to solve a few instances of the problem that are drawn randomly from a suitable distribution [13, Supplementary Material]. Similar proofs of quantum advantage with associated average-case hardness results for classical circuits have been obtained more recently in [14,15], see also [16]. In this work we extend these results in two distinct ways.…”

supporting

confidence: 61%

“…To this end, we construct new relation problems: these incorporate fault-tolerance mechanisms allowing for a solution by noisy constantdepth quantum circuits. We emphasize that these constructions do not proceed by "amplification" of the classical threshold (or the "soundness") towards 1 (as considered in [14,15]). Indeed, as our relational problems involve an extensive number of output bits such amplification techniques do not appear to be suitable for this purpose.…”

Section: Noisy Quantum Circuits Versus Noiseless Classical Circuitsmentioning

confidence: 99%

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Quantum Advantage with Noisy Shallow Circuits in 3D

Bravyi

Gosset

Koenig

et al. 2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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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. It may also be more practical for future implementations. 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 constantdepth 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. To this end, we provide a protocol for single-shot logical state preparation in the surface code which may be of independent interest. CONTENTS

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Average-Case Quantum Advantage with Shallow Circuits

Cited by 2 publications

References 9 publications

Trading locality for time: certifiable randomness from low-depth circuits

Trading locality for time: certifiable randomness from low-depth circuits

Quantum Advantage with Noisy Shallow Circuits in 3D

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