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arXiv:2112.01657 · PRX Quantum

Limitations of Linear Cross-Entropy as a Measure for Quantum Advantage

Xun Gao, Marcin Kalinowski, Chi-Ning Chou, Mikhail D. Lukin, Boaz Barak, Soonwon Choi

Published Feb 29, 2024• 54 citations
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Abstract

Demonstrating quantum advantage requires experimental implementation of a computational task that is hard to achieve using state-of-the-art classical systems. One approach is to perform sampling from a probability distribution associated with a certain class of highly entangled many-body wave functions. It has been suggested that such a quantum advantage can be certified with the (XEB). We critically examine this notion. First, we consider a “benign” setting, where an honest implementation of a noisy quantum circuit is assumed, and characterize the conditions under which the XEB approximates the of quantum dynamics. Second, we assume an “adversarial” setting, where all possible classical algorithms are considered for comparisons, and show that achieving relatively high XEB values does not imply faithful simulation of quantum dynamics. Specifically, we present an efficient classical algorithm that achieves high XEB values, namely 5–12% of those obtained in the state-of-the-art experiments, within just a few seconds using a single GPU machine. This is made possible by identifying and exploiting several vulnerabilities of the XEB, which allows us to achieve high XEB values without simulating a full quantum circuit. Remarkably, our algorithm features with the system size than a noisy quantum device for commonly studied random circuit ensembles in various architecture. We quantitatively explain the success of our algorithm and the limitations of the XEB by using a theoretical framework, in which the dynamics of the average XEB and fidelity are mapped to classical statistical mechanics models. Using this framework, we illustrate the relation between the XEB and the fidelity for quantum circuits in various architectures, with different choices of gate sets, and in the presence of noise. Taken together, our results demonstrate that XEB’s utility as a proxy for fidelity hinges on several conditions, which should be independently checked in the benign setting, but cannot be assumed in the general adversarial setting. Therefore, the XEB on its own has a limited utility as a benchmark for quantum advantage. We briefly discuss potential ways to overcome these limitations. Published by the American Physical Society 2024

DOI
10.1103/prxquantum.5.010334

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The linear cross-entropy benchmark (Linear XEB) has been used as a test for procedures simulating quantum circuits. Given a quantum circuit C with n inputs and outputs and purported simulator whose output is distributed according to a distribution p over {0,1}ⁿ, the linear XEB fidelity of the simulator is ℱ_C(p) = 2ⁿ 𝔼_{x ∼ p} q_C(x) -1, where q_C(x) is the probability that x is output from the distribution C |0ⁿ⟩. A trivial simulator (e.g., the uniform distribution) satisfies ℱ_C(p) = 0, while Google’s noisy quantum simulation of a 53-qubit circuit C achieved a fidelity value of (2.24 ±0.21)×10^{-3} (Arute et. al., Nature'19). In this work we give a classical randomized algorithm that for a given circuit C of depth d with Haar random 2-qubit gates achieves in expectation a fidelity value of Ω(n/L⋅15^{-d}) in running time poly(n,2^L). Here L is the size of the light cone of C: the maximum number of input bits that each output bit depends on. In particular, we obtain a polynomial-time algorithm that achieves large fidelity of ω(1) for depth O(√{log n}) two-dimensional circuits. This is the first such result for two dimensional circuits of super-constant depth. Our results can be considered as an evidence that fooling the linear XEB test might be easier than achieving a full simulation of the quantum circuit.

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Demonstrating quantum advantage requires experimental implementation of a computational task that is hard to achieve using state-of-the-art classical systems. One approach is to perform sampling from a probability distribution associated with a certain class of highly entangled many-body wave functions. It has been suggested that such a quantum advantage can be certified with the (XEB). We critically examine this notion. First, we consider a “benign” setting, where an honest implementation of a noisy quantum circuit is assumed, and characterize the conditions under which the XEB approximates the of quantum dynamics. Second, we assume an “adversarial” setting, where all possible classical algorithms are considered for comparisons, and show that achieving relatively high XEB values does not imply faithful simulation of quantum dynamics. Specifically, we present an efficient classical algorithm that achieves high XEB values, namely 5–12% of those obtained in the state-of-the-art experiments, within just a few seconds using a single GPU machine. This is made possible by identifying and exploiting several vulnerabilities of the XEB, which allows us to achieve high XEB values without simulating a full quantum circuit. Remarkably, our algorithm features with the system size than a noisy quantum device for commonly studied random circuit ensembles in various architecture. We quantitatively explain the success of our algorithm and the limitations of the XEB by using a theoretical framework, in which the dynamics of the average XEB and fidelity are mapped to classical statistical mechanics models. Using this framework, we illustrate the relation between the XEB and the fidelity for quantum circuits in various architectures, with different choices of gate sets, and in the presence of noise. Taken together, our results demonstrate that XEB’s utility as a proxy for fidelity hinges on several conditions, which should be independently checked in the benign setting, but cannot be assumed in the general adversarial setting. Therefore, the XEB on its own has a limited utility as a benchmark for quantum advantage. We briefly discuss potential ways to overcome these limitations. Published by the American Physical Society 2024
Demonstrations of quantum computational advantage and benchmarks of quantum processors via quantum random circuit sampling are based on evaluating the linear cross-entropy benchmark (XEB). A key question in the theory of XEB is whether it approximates the fidelity of the quantum state preparation. Previous works have shown that the XEB generically approximates the fidelity in a regime where the noise rate per qudit ε\varepsilonε satisfies εN≪1\varepsilon N \ll 1εN≪1 for a system of NNN qudits and that this approximation breaks down at large noise rates. Here, we show that the breakdown of XEB as a fidelity proxy occurs as a sharp phase transition at a critical value of εN\varepsilon NεN that depends on the circuit architecture and properties of the two-qubit gates, including in particular their entangling power. We study the phase transition using a mapping of average two-copy quantities to statistical mechanics models in random quantum circuit architectures with full or one-dimensional connectivity. We explain the phase transition behavior in terms of spectral properties of the transfer matrix of the statistical mechanics model and identify two-qubit gate sets that exhibit the largest noise robustness.
Validation of quantum advantage claims in the context of Gaussian boson sampling (GBS) currently relies on providing evidence that the experimental samples genuinely follow their corresponding ground truth, i.e., the theoretical model of the experiment that includes all the possible losses that the experimenters can account for. This approach to verification has an important drawback: it is necessary to assume that the ground truth distributions are computationally hard to sample, that is, that they are sufficiently close to the distribution of the ideal, lossless experiment, for which there is evidence that sampling, either exactly or approximately, is a computationally hard task. This assumption, which cannot be easily confirmed, opens the door to classical algorithms that exploit the noise in the ground truth to efficiently simulate the experiments, thus undermining any quantum advantage claim. In this work, we argue that one can avoid this issue by validating GBS implementations using their corresponding ideal distributions directly. We explain how to use a modified version of the linear cross-entropy, a quantity that we call the LXE score, to find reference values that help us assess how close a given GBS implementation is to its corresponding ideal model. Finally, we analytically compute the score that would be obtained by a lossless GBS implementation. Published by the American Physical Society 2024
With the advent of quantum processors exceeding 100 qubits and the high engineering complexities involved, there is a need for holistically benchmarking the processor to have quality assurance. Linear cross-entropy benchmarking (XEB) has been used extensively for systems with 50 or more qubits but is fundamentally limited in scale due to the exponentially large computational resources required for classical simulation. In this work we propose conducting linear XEB with random Clifford circuits of constant to logarithmic depth, a scheme we call Clifford XEB. Since Clifford circuits can be simulated in polynomial time, Clifford XEB can be scaled to much larger systems. To validate this claim, we run numerical simulations for the classes of Clifford circuits we propose with noise and observe exponential decays. When noise levels are low, the decay rates are well correlated with the noise of each cycle assuming a multiplicative error accumulation, i.e., where the fidelities of individual gates multiply. We perform simulations of systems up to 1225 qubits, where the classical processing task can be easily dealt with by a workstation. Furthermore, using the theoretical guarantees in Chen et al. [PRX Quantum 3, 030320 (2022)], we prove that Clifford XEB with our proposed Clifford circuits must yield exponential decays under a general error model for sufficiently low errors. Our theoretical results explain some of the phenomena observed in the simulations and shed light on the behavior of general linear XEB experiments.
The computational advantage of noisy quantum computers has been demonstrated by sampling the bitstrings of quantum random circuits. An important issue is how the performance of quantum devices could be quantified in the so-called "supremacy regime". The standard approach is through the linear cross entropy benchmark (XEB), where the theoretical value of the probability is required for each bitstring. However, the computational cost of XEB grows exponentially. So far, random circuits of the 53-qubit Sycamore chip were verified up to 10 cycles of gates only; the XEB fidelities of deeper circuits were approximated with simplified circuits instead. Here we present a multi-tensor contraction algorithm for speeding up the calculations of XEB for quantum circuits, where the computational cost can be significantly reduced through some form of memoization. As a demonstration, we analyzed the experimental data of the 53-qubit Sycamore chip and obtained the exact values of the corresponding XEB fidelities up to 16 cycles using only moderate computing resources (few GPUs). If the algorithm was implemented on the Summit supercomputer, we estimate that for the supremacy (20 cycles) circuits, it would only cost 7.5 days, which is several orders of magnitude lower than previously estimated in the literature.
Random circuit sampling experiments, the leading demonstrations of quantum computational advantage, are tested with the linear cross-entropy benchmark (XEB), which scores outcomes by their ideal probabilities. We study what a high XEB score certifies about an untrusted quantum device. In the quantum query model, we show that the honest sampler's ideal score acts as a Tsirelson bound for efficient devices and certified randomness. First, exceeding the ideal score by a constant requires Ω(N1/3)Ω(N^{1/3})Ω(N1/3) queries for the Haar and Fourier oracle ensembles with NNN outcomes, and this bound is tight; this proves an earlier conjecture for Haar-random states and extends it to Fourier sampling. Second, for Fourier sampling and for Haar-random states accessed through the canonical state-preparation oracle, the nnn-bit output of a device that makes polynomially many queries and scores within o(1)o(1)o(1) of the ideal score has n−O(log⁡n)n-O(\log n)n−O(logn) bits of smooth min-entropy, even against an adversary who is entangled with the device and later learns the oracle; this is nearly optimal. As a corollary, sampling the collision distribution, which is proportional to the square of the ideal distribution, has query complexity Θ(N1/3)Θ(N^{1/3})Θ(N1/3), and a coherent version of the same algorithm approximately prepares quantum Hadamard products of states with uniform amplitude magnitudes. Both lower bounds rest on progress measures that bound the excess over the ideal score and increase only slightly with each query. For Fourier sampling the measure is the norm of the coherent amplitudes for removing pairs in a purification of the random function, combining the polynomial method with the compressed oracle technique; for Haar-random states it is the square root of the average level in an expansion in Dirichlet orthogonal polynomials, controlled by an exact identity for the effect of one observation.
The linear cross-entropy benchmark (Linear XEB) has been used as a test for procedures simulating quantum circuits. Given a quantum circuit C with n inputs and outputs and purported simulator whose output is distributed according to a distribution p over {0,1}ⁿ, the linear XEB fidelity of the simulator is ℱ_C(p) = 2ⁿ 𝔼_{x ∼ p} q_C(x) -1, where q_C(x) is the probability that x is output from the distribution C |0ⁿ⟩. A trivial simulator (e.g., the uniform distribution) satisfies ℱ_C(p) = 0, while Google’s noisy quantum simulation of a 53-qubit circuit C achieved a fidelity value of (2.24 ±0.21)×10^{-3} (Arute et. al., Nature'19). In this work we give a classical randomized algorithm that for a given circuit C of depth d with Haar random 2-qubit gates achieves in expectation a fidelity value of Ω(n/L⋅15^{-d}) in running time poly(n,2^L). Here L is the size of the light cone of C: the maximum number of input bits that each output bit depends on. In particular, we obtain a polynomial-time algorithm that achieves large fidelity of ω(1) for depth O(√{log n}) two-dimensional circuits. This is the first such result for two dimensional circuits of super-constant depth. Our results can be considered as an evidence that fooling the linear XEB test might be easier than achieving a full simulation of the quantum circuit.