We propose a general tensor network method for simulating quantum circuits. The method is massively more efficient in computing a large number of correlated bitstring amplitudes and probabilities than existing methods. As an application, we study the sampling problem of Google's Sycamore circuits, which are believed to be beyond the reach of classical supercomputers and have been used to demonstrate quantum supremacy. Using our method, employing a small computational cluster containing 60 graphical processing units (GPUs), we have generated one million correlated bitstrings with some entries fixed, from the Sycamore circuit with 53 qubits and 20 cycles, with linear cross-entropy benchmark (XEB) fidelity equals 0.739, which is much higher than those in Google's quantum supremacy experiments.
We propose a general tensor network method for simulating quantum circuits. The method is massively more efficient in computing a large number of correlated bitstring amplitudes and probabilities than existing methods. As an application, we study the sampling problem of Google's Sycamore circuits, which are believed to be beyond the reach of classical supercomputers and have been used to demonstrate quantum supremacy. Using our method, employing a small computational cluster containing 60 graphical processing units (GPUs), we have generated one million correlated bitstrings with some entries fixed, from the Sycamore circuit with 53 qubits and 20 cycles, with linear cross-entropy benchmark (XEB) fidelity equals 0.739, which is much higher than those in Google's quantum supremacy experiments.
We propose a tensor network approach to compute amplitudes and probabilities for a large number of correlated bitstrings in the final state of a quantum circuit. As an application, we study Google's Sycamore circuits, which are believed to be beyond the reach of classical supercomputers and have been used to demonstrate quantum supremacy. By employing a small computational cluster containing 60 graphical processing units (GPUs), we compute exact amplitudes and probabilities of 2×10^{6} correlated bitstrings with some entries fixed (which span a subspace of the output probability distribution) for the Sycamore circuit with 53 qubits and 20 cycles. The obtained results verify the Porter-Thomas distribution of the large and deep quantum circuits of Google, provide datasets and benchmarks for developing approximate simulation methods, and can be used for spoofing the linear cross entropy benchmark of quantum supremacy. Then we extend the proposed big-batch method to a full-amplitude simulation approach that is more efficient than the existing Schrödinger method on shallow circuits and the Schrödinger-Feynman method in general, enabling us to obtain the state vector of Google's simplifiable circuit with n=43 qubits and m=14 cycles using only one GPU. We also manage to obtain the state vector for Google's simplifiable circuits with n=50 qubits and m=14 cycles using a small GPU cluster, breaking the previous record on the number of qubits in full-amplitude simulations. Our method is general in computing bitstring probabilities for a broad class of quantum circuits and can find applications in the verification of quantum computers. We anticipate that our method will pave the way for combining tensor network-based classical computations and near-term quantum computations for solving challenging problems in the real world.
We study the problem of generating independent samples from the output distribution of Google's Sycamore quantum circuits with a target fidelity, which is believed to be beyond the reach of classical supercomputers and has been used to demonstrate quantum supremacy. We propose a method to classically solve this problem by contracting the corresponding tensor network just once, and is massively more efficient than existing methods in generating a large number of uncorrelated samples with a target fidelity. For the Sycamore quantum supremacy circuit with 53 qubits and 20 cycles, we have generated 1×10^{6} uncorrelated bitstrings s which are sampled from a distribution P[over ^](s)=|ψ[over ^](s)|^{2}, where the approximate state ψ[over ^] has fidelity F≈0.0037. The whole computation has cost about 15 h on a computational cluster with 512 GPUs. The obtained 1×10^{6} samples, the contraction code and contraction order are made public. If our algorithm could be implemented with high efficiency on a modern supercomputer with ExaFLOPS performance, we estimate that ideally, the simulation would cost a few dozens of seconds, which is faster than Google's quantum hardware.
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.
We present a framework for effectively simulating the execution of quantum circuits originally designed to demonstrate quantum supremacy using accessible high-performance computing (HPC) infrastructure. Building on prior CPU-only approaches, our pipeline combines a single NVIDIA A100 GPU for quantum state construction, followed by N parallel CPU jobs that perform distributed measurement sampling. We validate the fidelity by simulating the 53-qubit, 14-cycle Sycamore circuit and achieving a linear cross-entropy benchmarking (XEB) score of 0.549, exceeding the published XEB score of 0.002 from Google's reference data. We then evaluate execution time performance with the more complex 53-qubit, 20-cycle circuit, completing the full 2.5 million-shot workload over 100 CPU jobs in 01:15:36, representing a 6.95 x 10^7 speedup compared to Google's original classical estimate. Further, we show that if 1,000 CPU jobs were employed, the estimated duration would be approximately 00:17:35, only 12 minutes slower than the time taken by the original QPU-based experiment. These results illustrate that 'quantum supremacy' is not fixed and continues to be a moving target. In addition, hybrid classical-quantum strategies may provide broader near-term quantum utility than once thought.
Quantum Computational Superiority boasts rapid computation and high energy efficiency. Despite recent advances in classical algorithms aimed at refuting the milestone claim of Google's sycamore, challenges remain in generating uncorrelated samples of random quantum circuits. In this paper, we present a groundbreaking large-scale system technology that leverages optimization on global, node, and device levels to achieve unprecedented scalability for tensor networks. This enables the handling of large-scale tensor networks with memory capacities reaching tens of terabytes, surpassing memory space constraints on a single node. Our techniques enable accommodating large-scale tensor networks with up to tens of terabytes of memory, reaching up to 2304 GPUs with a peak computing power of 561 PFLOPS half-precision. Notably, we have achieved a time-to-solution of 14.22 seconds with energy consumption of 2.39 kWh which achieved fidelity of 0.002 and our most remarkable result is a time-to-solution of 17.18 seconds, with energy consumption of only 0.29 kWh which achieved a XEB of 0.002 after post-processing, outperforming Google's quantum processor Sycamore in both speed and energy efficiency, which recorded 600 seconds and 4.3 kWh, respectively.
A major milestone in the era of noisy intermediate scale quantum computers is \textit{quantum supremacy} [Nature \textbf{574}, 505 (2019)] claimed on the Sycamore quantum processor of 53 qubits, which can perform a random circuit sampling task within 200 seconds while the same task is estimated to require a runtime of 10,000 years on Summit. This record has been renewed with two recent experiments on the Zuchongzhi 2.0 (56 qubits) and Zuchongzhi 2.1 (60 qubits) quantum processors. On the other front of quantum supremacy comparison, there has also been continuous improvements on both the classical simulation algorithm as well as the underlying hardware. And a fair justification of the computational advantages for those quantum supremacy experiments would require to practically simulate the same problems on current top supercomputers, which is still in lack. Here we report the full-scale simulations of these problems on new generation Sunway supercomputer, based on a customized tensor network contraction algorithm. Our benchmark shows that the most challenging sampling task performed on Sycamore can be accomplished within 1 week, thus collapsing the quantum supremacy claim of Sycamore. Additionally, we show that the XEB fidelities of the \textit{quantum supremacy circuits} with up to 14 cycles can be verified in minutes, which also provides strong consistency check for quantum supremacy experiments. Our results redefine quantum supremacy baseline using the new generation Sunway supercomputer.