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arXiv:2108.02201

Testing Quantum Mechanics using Noisy Quantum Computers

Kevin Slagle

Published Aug 4, 2021• 2 citations
arXivHTMLPDF

Abstract

We outline a proposal to test quantum mechanics in the high-complexity regime using noisy intermediate-scale quantum (NISQ) devices. The procedure involves simulating a non-Clifford random circuit, followed by its inverse, and then checking that the resulting state is the same as the initial state. We are motivated by the hypothesis that quantum mechanics is not fundamental, but instead emerges from a theory with less computational power, such as classical mechanics. This emergent quantum mechanics (EmQM) hypothesis makes the prediction that quantum computers will not be capable of sufficiently complex quantum computations. We show that quantum mechanics predicts that the fidelity of our procedure decays exponentially with circuit depth (due to noise in NISQ devices), while EmQM predicts that the fidelity will decay significantly more rapidly for sufficiently deep circuits, which is the experimental signature that we propose to search for. We estimate rough bounds for when possible signals of EmQM should be expected. Furthermore, we find that highly informative experiments should require only thousands qubits.

DOI
10.48550/arxiv.2108.02201

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We outline a proposal to test quantum mechanics in the high-complexity regime using noisy intermediate-scale quantum (NISQ) devices. The procedure involves simulating a non-Clifford random circuit, followed by its inverse, and then checking that the resulting state is the same as the initial state. We are motivated by the hypothesis that quantum mechanics is not fundamental, but instead emerges from a theory with less computational power, such as classical mechanics. This emergent quantum mechanics (EmQM) hypothesis makes the prediction that quantum computers will not be capable of sufficiently complex quantum computations. We show that quantum mechanics predicts that the fidelity of our procedure decays exponentially with circuit depth (due to noise in NISQ devices), while EmQM predicts that the fidelity will decay significantly more rapidly for sufficiently deep circuits, which is the experimental signature that we propose to search for. We estimate rough bounds for when possible signals of EmQM should be expected. Furthermore, we find that highly informative experiments should require only thousands qubits.
Noisy Intermediate-Scale Quantum (NISQ) devices fail to produce outputs with sufficient fidelity for deep circuits with many gates today. Such devices suffer from read-out, multi-qubit gate and crosstalk noise combined with short decoherence times limiting circuit depth. This work develops a methodology to generate shorter circuits with fewer multi-qubit gates whose unitary transformations approximate the original reference one. It explores the benefit of such generated approximations under NISQ devices. Experimental results with Grover's algorithm, multiple-control Toffoli gates, and the Transverse Field Ising Model show that such approximate circuits produce higher fidelity results than longer, theoretically precise circuits on NISQ devices, especially when the reference circuits have many CNOT gates to begin with. With this ability to fine-tune circuits, it is demonstrated that quantum computations can be performed for more complex problems on today's devices than was feasible before, sometimes even with a gain in overall precision by up to 60%.
Noise is the defining feature of the NISQ era, but it remains unclear if noisy quantum devices are capable of quantum speedups. Quantum supremacy experiments have been a major step forward, but gaps remain between the theory behind these experiments and their actual implementations. In this work we initiate the study of the complexity of quantum random circuit sampling experiments with realistic amounts of noise. Actual quantum supremacy experiments have high levels of uncorrected noise and exponentially decaying fidelities. It is natural to ask if there is any signal of exponential complexity in these highly noisy devices. Surprisingly, we show that it remains hard to compute the output probabilities of noisy random quantum circuits without error correction. More formally, so long as the noise rate of the device is below the error detection threshold, we show it is #P-hard to compute the output probabilities of random circuits with a constant rate of noise per gate. This hardness persists even though these probabilities are exponentially close to uniform. Therefore the small deviations away from uniformity are hard to compute, formalizing an important intuition behind Google's supremacy claim. Interestingly these hardness results also have implications for the complexity of experiments in a low-noise setting. The issue here is that prior hardness results for computing output proba-bilities of random circuits are not robust enough to imprecision to connect with the Stockmeyer argument for hardness of sampling from circuits with constant fidelity. We exponentially improve the robustness of prior results to imprecision, both in the cases of Random Circuit Sampling and BosonSampling. In the latter case we bring the proven hardness within a constant factor in the exponent of the robustness required for hardness of sampling for the first time. We then show that our results are in tension with one another - the high-noise result implies the low-noise result is essentially optimal, even with generalizations of our techniques.
Understanding the computational power of noisy intermediate-scale quantum (NISQ) devices is of both fundamental and practical importance to quantum information science. Here, we address the question of whether error-uncorrected noisy quantum computers can provide computational advantage over classical computers. Specifically, we study noisy random circuit sampling in one dimension (or 1D noisy RCS) as a simple model for exploring the effects of noise on the computational power of a noisy quantum device. In particular, we simulate the real-time dynamics of 1D noisy random quantum circuits via matrix product operators (MPOs) and characterize the computational power of the 1D noisy quantum system by using a metric we call MPO entanglement entropy. The latter metric is chosen because it determines the cost of classical MPO simulation. We numerically demonstrate that for the two-qubit gate error rates we considered, there exists a characteristic system size above which adding more qubits does not bring about an exponential growth of the cost of classical MPO simulation of 1D noisy systems. Specifically, we show that above the characteristic system size, there is an optimal circuit depth, independent of the system size, where the MPO entanglement entropy is maximized. Most importantly, the maximum achievable MPO entanglement entropy is bounded by a constant that depends only on the gate error rate, not on the system size. We also provide a heuristic analysis to get the scaling of the maximum achievable MPO entanglement entropy as a function of the gate error rate. The obtained scaling suggests that although the cost of MPO simulation does not increase exponentially in the system size above a certain characteristic system size, it does increase exponentially as the gate error rate decreases, possibly making classical simulation practically not feasible even with state-of-the-art supercomputers.
Noisy intermediate-scale quantum (NISQ) devices offer unique platforms to test and evaluate the behavior of quantum computing. However, validating circuits on NISQ devices is difficult due to fluctuations in the underlying noise sources and other nonreproducible behaviors that generate computational errors. Here we present a test-driven approach that decomposes a noisy, application-specific circuit into a series of bootstrapped experiments on a NISQ device. By characterizing individual subcircuits, we generate a composite noise model for the original quantum circuit. We demonstrate this approach to model applications of Greenberger-Horne-Zeilinger(GHZ)-state preparation and the Bernstein-Vazirani algorithm on a family of superconducting transmon devices. We measure the model accuracy using the total variation distance between predicted and experimental results, and we demonstrate that the composite model works well across multiple circuit instances. Our approach is shown to be computationally efficient and offers a trade-off in model complexity that can be tailored to the desired predictive accuracy.
The viability of algorithms on Noisy Intermediate-Scale Quantum (NISQ) computers is limited by operational errors and decoherence. To optimize execution, it is crucial to understand how circuit complexity (depth and number of gates), statistical sampling (shots), and result fidelity interact. This work investigates that relationship through large-scale simulations of 100,000 random circuits under a physically motivated noise model with temporal decoherence (T1/T2) and gate errors. The analysis shows a strong negative correlation between circuit complexity and Hellinger fidelity, revealing a “fidelity ceiling” where increasing shots reduces statistical error but cannot overcome noise. For complex circuits, the number of gates is identified as a stronger predictor of error than circuit depth.
Noisy Intermediate-Scale Quantum computers are expected to be available this year. It is proposed to exploit such a device for decision making under uncertainty. The probabilistic character of quantum mechanics reflects this uncertainty. Concomitantly, the noise may add to it. The approach is standard in the sense that Bayes decision rule is used to decide on the basis of maximum expected reward. The novelty is to model the various action profiles and the development of `nature' as unitary transformations on a set of qubits. Measurement eventually yields samples of classical binary random variables in which the reward function has to be expressed. In order to achieve sufficiently low variances for reliable decision making more runs of such a quantum algorithm are necessary. Some simple examples have been worked out to elucidate the idea. Here the calculations are still analytically feasible. Presently lacking an operating quantum device, the QX simulator of Quantum Inspire has been used to generate the necessary samples for comparison and demonstration. First obtained results are promising and point at a possible useful application for noisy intermediate-scale quantum computers.