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.
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.
The computational advantage of noisy quantum computers have 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 regime. The standard approach is through the linear cross entropy (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 was 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 of quantum circuits, where the computational cost can be significantly reduced through a recursive manner with 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 20-cycles supremacy circuits, it would only cost 7.5 days, which is several orders of magnitudes lower than previously estimated in the literature.
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.
Random circuit sampling has become a popular means for demonstrating the superiority of quantum computers over classical supercomputers. While quantum chips are evolving rapidly, classical sampling algorithms are also getting better and better. The major challenge is to generate bitstrings exhibiting an XEB fidelity above that of the quantum chips. Here we present a classical sampling algorithm for producing the probability distribution of any given random quantum circuit, where the fidelity can be rigorously bounded. Specifically, our algorithm performs rejection sampling after the introduced very recently multi-tensor contraction algorithm. We show that the fidelity can be controlled by partially contracting the dominant paths in the tensor network and by adjusting the number of batches used in the rejection sampling. As a demonstration, we classically produced 1 million samples with the fidelity bounded by 0.2%, based on the 20-cycle circuit of the Sycamore 53-qubit quantum chip. Though this task was initially estimated to take 10,000 years on the Summit supercomputer, it took about 14.5 days using our algorithm on a relatively small cluster with 32 GPUs (Tesla V100 16GB). Furthermore, we estimate that for the Zuchongzhi 56-qubit 20-cycle circuit one can produce 1M samples with fidelity 0.066% using the Selene supercomputer with 4480 GPUs (Tesla A100 80GB) in about 4 days.
It is believed that random quantum circuits are difficult to simulate classically. These have been used to demonstrate quantum supremacy: the execution of a computational task on a quantum computer that is infeasible for any classical computer. The task underlying the assertion of quantum supremacy by Arute et al. (Nature, 574, 505--510 (2019)) was initially estimated to require Summit, the world's most powerful supercomputer today, approximately 10,000 years. The same task was performed on the Sycamore quantum processor in only 200 seconds. In this work, we present a tensor network-based classical simulation algorithm. Using a Summit-comparable cluster, we estimate that our simulator can perform this task in less than 20 days. On moderately-sized instances, we reduce the runtime from years to minutes, running several times faster than Sycamore itself. These estimates are based on explicit simulations of parallel subtasks, and leave no room for hidden costs. The simulator's key ingredient is identifying and optimizing the "stem" of the computation: a sequence of pairwise tensor contractions that dominates the computational cost. This orders-of-magnitude reduction in classical simulation time, together with proposals for further significant improvements, indicates that achieving quantum supremacy may require a period of continuing quantum hardware developments without an unequivocal first demonstration.
The quantum supremacy experiment, such as Google Sycamore [F. Arute et al., Nature (London) 574, 505 (2019).NATUAS0028-083610.1038/s41586-019-1666-5], poses a great challenge for classical verification due to the exponentially increasing compute cost. Using a new-generation Sunway supercomputer within 8.5 d, we provide a direct verification by computing 3×10^{6} exact amplitudes for the experimentally generated bitstrings, obtaining a cross-entropy benchmarking fidelity of 0.191% (the estimated value is 0.224%). The leap of simulation capability is built on a multiple-amplitude tensor network contraction algorithm which systematically exploits the "classical advantage" (the inherent "store-and-compute" operation mode of von Neumann machines) of current supercomputers, and a fused tensor network contraction algorithm which drastically increases the compute efficiency on heterogeneous architectures. Our method has a far-reaching impact in solving quantum many-body problems, statistical problems, as well as combinatorial optimization problems.
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.