On the average-case complexity of learning output distributions of quantum circuits

• URL: http://arxiv.org/abs/2305.05765v2
• Date: Thu, 09 Oct 2025 13:44:03 GMT
• Title: On the average-case complexity of learning output distributions of quantum circuits
• Authors: Alexander Nietner, Marios Ioannou, Ryan Sweke, Richard Kueng, Jens Eisert, Marcel Hinsche, Jonas Haferkamp,
• Abstract summary: We show that learning the output distributions of brickwork random quantum circuits is average-case hard in the statistical query model.<n>This learning model is widely used as an abstract computational model for most generic learning algorithms.
• Score: 33.76498647184212
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• Abstract: In this work, we show that learning the output distributions of brickwork random quantum circuits is average-case hard in the statistical query model. This learning model is widely used as an abstract computational model for most generic learning algorithms. In particular, for brickwork random quantum circuits on $n$ qubits of depth $d$, we show three main results: - At super logarithmic circuit depth $d=\omega(\log(n))$, any learning algorithm requires super polynomially many queries to achieve a constant probability of success over the randomly drawn instance. - There exists a $d=O(n)$, such that any learning algorithm requires $\Omega(2^n)$ queries to achieve a $O(2^{-n})$ probability of success over the randomly drawn instance. - At infinite circuit depth $d\to\infty$, any learning algorithm requires $2^{2^{\Omega(n)}}$ many queries to achieve a $2^{-2^{\Omega(n)}}$ probability of success over the randomly drawn instance. As an auxiliary result of independent interest, we show that the output distribution of a brickwork random quantum circuit is constantly far from any fixed distribution in total variation distance with probability $1-O(2^{-n})$, which confirms a variant of a conjecture by Aaronson and Chen.

Related papers

• Incompressibility and spectral gaps of random circuits [2.359282907257055]
  reversible and quantum circuits form random walks on the alternating group $mathrmAlt (2n)$ and unitary group $mathrmSU (2n)$, respectively.<n>We prove that the gap for random reversible circuits is $Omega(n-3)$ for all $tgeq 1$, and the gap for random quantum circuits is $Omega(n-3)$ for $t leq Theta(2n/2)$.
  arXiv  Detail & Related papers  (2024-06-11T17:23:16Z)
• Classical simulation of peaked shallow quantum circuits [2.6089354079273512]
  We describe an algorithm with quasipolynomial runtime $nO(logn)$ that samples from the output distribution of a peaked constant-depth circuit. Our algorithms can be used to estimate output probabilities of shallow circuits to within a given inverse-polynomial additive error.
  arXiv  Detail & Related papers  (2023-09-15T14:01:13Z)
• Near-Optimal Bounds for Learning Gaussian Halfspaces with Random Classification Noise [50.64137465792738]
  We show that any efficient SQ algorithm for the problem requires sample complexity at least $Omega(d1/2/(maxp, epsilon)2)$. Our lower bound suggests that this quadratic dependence on $1/epsilon$ is inherent for efficient algorithms.
  arXiv  Detail & Related papers  (2023-07-13T18:59:28Z)
• Mind the gap: Achieving a super-Grover quantum speedup by jumping to the end [114.3957763744719]
  We present a quantum algorithm that has rigorous runtime guarantees for several families of binary optimization problems. We show that the algorithm finds the optimal solution in time $O*(2(0.5-c)n)$ for an $n$-independent constant $c$. We also show that for a large fraction of random instances from the $k$-spin model and for any fully satisfiable or slightly frustrated $k$-CSP formula, statement (a) is the case.
  arXiv  Detail & Related papers  (2022-12-03T02:45:23Z)
• A single $T$-gate makes distribution learning hard [56.045224655472865]
  This work provides an extensive characterization of the learnability of the output distributions of local quantum circuits. We show that for a wide variety of the most practically relevant learning algorithms -- including hybrid-quantum classical algorithms -- even the generative modelling problem associated with depth $d=omega(log(n))$ Clifford circuits is hard.
  arXiv  Detail & Related papers  (2022-07-07T08:04:15Z)
• Quantum Resources Required to Block-Encode a Matrix of Classical Data [56.508135743727934]
  We provide circuit-level implementations and resource estimates for several methods of block-encoding a dense $Ntimes N$ matrix of classical data to precision $epsilon$. We examine resource tradeoffs between the different approaches and explore implementations of two separate models of quantum random access memory (QRAM) Our results go beyond simple query complexity and provide a clear picture into the resource costs when large amounts of classical data are assumed to be accessible to quantum algorithms.
  arXiv  Detail & Related papers  (2022-06-07T18:00:01Z)
• Random quantum circuits transform local noise into global white noise [118.18170052022323]
  We study the distribution over measurement outcomes of noisy random quantum circuits in the low-fidelity regime. For local noise that is sufficiently weak and unital, correlations (measured by the linear cross-entropy benchmark) between the output distribution $p_textnoisy$ of a generic noisy circuit instance shrink exponentially. If the noise is incoherent, the output distribution approaches the uniform distribution $p_textunif$ at precisely the same rate.
  arXiv  Detail & Related papers  (2021-11-29T19:26:28Z)
• Quantum supremacy and hardness of estimating output probabilities of quantum circuits [0.0]
  We prove under the theoretical complexity of the non-concentration hierarchy that approximating the output probabilities to within $2-Omega(nlogn)$ is hard. We show that the hardness results extend to any open neighborhood of an arbitrary (fixed) circuit including trivial circuit with identity gates.
  arXiv  Detail & Related papers  (2021-02-03T09:20:32Z)
• Quantum learning algorithms imply circuit lower bounds [7.970954821067043]
  We establish the first general connection between the design of quantum algorithms and circuit lower bounds. Our proof builds on several works in learning theory, pseudorandomness, and computational complexity.
  arXiv  Detail & Related papers  (2020-12-03T14:03:20Z)
• The Quantum Supremacy Tsirelson Inequality [0.22843885788439797]
  For a quantum circuit $C$ on $n$ qubits and a sample $z in 0,1n$, the benchmark involves computing $|langle z|C|0n rangle|2$. We show that for any $varepsilon ge frac1mathrmpoly(n)$, outputting a sample $z$ is the optimal 1-query for $|langle z|C|0nrangle|2$ on average.
  arXiv  Detail & Related papers  (2020-08-20T01:04:32Z)
• FANOK: Knockoffs in Linear Time [73.5154025911318]
  We describe a series of algorithms that efficiently implement Gaussian model-X knockoffs to control the false discovery rate on large scale feature selection problems. We test our methods on problems with $p$ as large as $500,000$.
  arXiv  Detail & Related papers  (2020-06-15T21:55:34Z)
• Spoofing Linear Cross-Entropy Benchmarking in Shallow Quantum Circuits [8.401078947103475]
  Google's noisy quantum simulation of a 53 qubit circuit $C$ achieved a fidelity value of $(2.24pm0.21)times10-3$. Our results can be considered as an evidence that fooling the linear XEB test might be easier than achieving a full simulation of quantum circuit.
  arXiv  Detail & Related papers  (2020-05-05T18:01:48Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.