The first achievement of quantum supremacy has been claimed recently by Google for the random quantum circuit benchmark with 53 superconducting qubits. Here, we analyze the randomness of Google's quantum random-bit sampling. The heat maps of Google's random bit-strings show stripe patterns at specific qubits in contrast to the Haar-measure or classical random-bit strings. Google's data contains more bit 0 than bit 1, i.e., about 2.8\% difference, and fail to pass the NIST random number tests, while the Haar-measure or classical random-bit samples pass. Their difference is also illustrated by the Marchenko-Pastur distribution and the Girko circular law of random matrices of random bit-strings. The calculation of the Wasserstein distances shows that Google's random bit-strings are farther away from the Haar-measure random bit-strings than the classical random bit-strings. Our results imply that random matrices and the Wasserstein distance could be new tools for analyzing the performance of quantum computers.
The first achievement of quantum supremacy has been claimed recently by Google for the random quantum circuit benchmark with 53 superconducting qubits. Here, we analyze the randomness of Google's quantum random-bit sampling. The heat maps of Google's random bit-strings show stripe patterns at specific qubits in contrast to the Haar-measure or classical random-bit strings. Google's data contains more bit 0 than bit 1, i.e., about 2.8% difference, and fail to pass the NIST random number tests, while the Haar-measure or classical random-bit samples pass. Their difference is also illustrated by the Marchenko-Pastur distribution and the Girko circular law of random matrices of random bit-strings. The calculation of the Wasserstein distances shows that Google's random bit-strings are farther away from the Haar-measure random bit-strings than the classical random bit-strings. Our results imply that random matrices and the Wasserstein distance could be new tools for analyzing the performance of quantum computers.
Quantum supremacy has been recently reported for random circuit sampling on the Sycamore processor with 53 qubits. Here, we analyze the statistical properties of bit strings sampled from random quantum circuits. In contrast to classical random bit strings, bit strings sampled from Sycamore random circuits give rise to heat maps with stripe patterns at specific qubits, have more bit 1 than 0, and do not pass the NIST random number tests. The difference between the Sycamore bit strings and classical random bit strings is also demonstrated by the Marchenko-Pastur distribution and the Girko circular law of random matrices. The calculation of Wasserstein distances shows that the Sycamore bit strings are farther from bit strings sampled from Haar-measure random quantum circuits than classical random bit strings. Our results show that random matrices and Wasserstein distances could be used to analyze the performance of quantum computers.
Random quantum circuit sampling, a task to sample bit strings from a random quantum circuit, is considered a suitable benchmark task to demonstrate the outperformance of quantum computers even with noisy qubits. Recently, random quantum circuit sampling was performed on the Sycamore quantum processor with 53 qubits [Nature (London) 574, 505 (2019)] and on the Zuchongzhi quantum processor with 56 qubits [Phys. Rev. Lett. 127, 180501 (2021)]. Here, we analyze and compare the statistical properties of the outputs of the random quantum circuit sampling by the Sycamore and Zuchongzhi processors. Using the Marchenko-Pastur law of random matrices of bit strings and the Wasssertein distances between bit strings, we find that the statistical properties of Sycamore bit strings are quite different from those of Zuchongzhi bit strings, while both processors score similar values of linear cross-entropy fidelity for random circuit sampling. Some bit strings sampled by the Zuchongzhi processor pass the NIST random number tests while both Sycamore and Zuchongzhi processors show similar patterns in the heat maps of bit strings. Zuchongzhi bit strings are much closer to classical uniform random bits than those of Sycamore. It is shown that the statistical properties of bit strings of both random quantum circuits change little as the depth of the random quantum circuits increases. Our findings raise a question about the computational reliability of noisy quantum processors because two quantum processors with similar noise levels and similar qubit structures produced statistically different outputs for the same random quantum circuit sampling.
Random circuit sampling, the task to sample bit strings from a random unitary operator, has been performed to demonstrate quantum advantage on the Sycamore quantum processor with 53 qubits and on the Zuchongzhi quantum processor with 56 and 61 qubits. Recently, it has been claimed that classical computers using tensor network simulation could catch on current noisy quantum processors for random circuit sampling. While the linear cross entropy benchmark fidelity is used to certify all these claims, it may not capture in detail statistical properties of outputs. Here, we compare the bit strings sampled from classical computers using tensor network simulation by Pan et al. [Phys. Rev. Lett. 129, 090502 (2022)] and by Kalachev et al. [arXiv:2112.15083 (2021)] and from the Sycamore and Zuchongzhi quantum processors. It is shown that all Kalachev et al.'s samples pass the NIST random number tests. The heat maps of bit strings show that Pan et al.'s and Kalachev et al.'s samples are quite different from the Sycamore or Zuzhongzhi samples. The analysis with the Marchenko-Pastur distribution and the Wasssertein distances demonstrates that Kalachev et al.'s samples are statistically close to the Sycamore samples than Pan et al.'s while the three datasets have similar values of the linear cross entropy fidelity. Our finding implies that further study is needed to certify or beat the claims of quantum advantage for random circuit sampling.
Random circuit sampling, the task of sampling bit strings from a random unitary operator, has been implemented to demonstrate quantum advantage on the Sycamore quantum processor with 53 qubits and on the Zuchongzhi quantum processor with 56 and 61 qubits. Recently, it was claimed that classical computers using tensor network simulation could catch on to current noisy quantum processors for random circuit sampling. While the linear cross-entropy benchmark fidelity was used to certify all these claims, it may not capture statistical properties of outputs in detail. Here, we compare the bit strings sampled from classical computers using tensor network simulation by Pan et al. [F. Pan, K. Chen, and P. Zhang, Phys. Rev. Lett. 129, 090502 (2022)] and by Kalachev et al. [G. Kalachev, P. Panteleev, P. Zhou, and M.-H. Yung, arXiv:2112.15083] with the bit strings from the Sycamore quantum processor. It is shown that all of Kalachev et al.'s samples passed the NIST random number tests. The heat maps of bit strings show that Pan et al.'s and Kalachev et al.'s samples are quite different from the Sycamore or Zuchongzhi samples. The analysis with the Marchenko-Pastur distribution and the Wasssertein distances demonstrates that Kalachev et al.'s samples are statistically closer to the Sycamore samples than Pan et al.'s while the three datasets have similar values for the linear cross-entropy fidelity. Our finding implies that further study is needed to certify or beat the claims of quantum advantage using random circuit sampling.
As Moore's law reaches its limits, quantum computers are emerging with the promise of dramatically outperforming classical computers. We have witnessed the advent of quantum processors with over 50 quantum bits (qubits), which are expected to be beyond the reach of classical simulation. Quantum supremacy is the event at which the old Extended Church-Turing Thesis is overturned: A quantum computer performs a task that is practically impossible for any classical (super)computer. The demonstration requires both a solid theoretical guarantee and an experimental realization. The lead candidate is Random Circuit Sampling (RCS), which is the task of sampling from the output distribution of random quantum circuits. Google recently announced a 53−qubit experimental demonstration of RCS. Soon after, classical algorithms appeared that challenge the supremacy of random circuits by estimating their outputs. How hard is it to classically simulate the output of random quantum circuits? We prove that estimating the output probabilities of random quantum circuits is formidably hard (#P-Hard) for any classical computer. This makes RCS the strongest candidate for demonstrating quantum supremacy relative to all other proposals. The robustness to the estimation error that we prove may serve as a new hardness criterion for the performance of classical algorithms. To achieve this, we introduce the Cayley path interpolation between any two gates of a quantum computation and convolve recent advances in quantum complexity and information with probability and random matrices. Furthermore, we apply algebraic geometry to generalize the well-known Berlekamp-Welch algorithm that is widely used in coding theory and cryptography. Our results imply that there is an exponential hardness barrier for the classical simulation of most quantum circuits.
We propose an application for near-term quantum devices: namely, generating cryptographically certified random bits, to use (for example) in proof-of-stake cryptocurrencies. Our protocol repurposes the existing "quantum supremacy" experiments, based on random circuit sampling, that Google and USTC have successfully carried out starting in 2019. We show that, whenever the outputs of these experiments pass the now-standard Linear Cross-Entropy Benchmark (LXEB), under plausible hardness assumptions they necessarily contain Ω(n) min-entropy, where n is the number of qubits. To achieve a net gain in randomness, we use a small random seed to produce pseudorandom challenge circuits. In response to the challenge circuits, the quantum computer generates output strings that, after verification, can then be fed into a randomness extractor to produce certified nearly-uniform bits -- thereby "bootstrapping" from pseudorandomness to genuine randomness. We prove our protocol sound in two senses: (i) under a hardness assumption called Long List Quantum Supremacy Verification, which we justify in the random oracle model, and (ii) unconditionally in the random oracle model against an eavesdropper who could share arbitrary entanglement with the device. (Note that our protocol's output is unpredictable even to a computationally unbounded adversary who can see the random oracle.) Currently, the central drawback of our protocol is the exponential cost of verification, which in practice will limit its implementation to at most n∼60 qubits, a regime where attacks are expensive but not impossible. Modulo that drawback, our protocol appears to be the only practical application of quantum computing that both requires a QC and is physically realizable today.