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 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 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 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.
Efficient simulation of quantum circuits has become indispensable with the rapid development of quantum hardware. The primary simulation methods are based on state vectors and tensor networks. As the number of qubits and quantum gates grows larger in current quantum devices, traditional state-vector-based quantum circuit simulation methods prove inadequate due to the overwhelming size of the Hilbert space and extensive entanglement. Consequently, brutal force tensor network simulation algorithms become the only viable solution in such scenarios. The two main challenges faced in tensor network simulation algorithms are optimal contraction path finding and efficient execution on modern computing devices, with the latter determining the actual efficiency. In this study, we investigate the optimization of such tensor network simulations on modern GPUs and propose general optimization strategies from two aspects: computational efficiency and accuracy. First, we propose to transform critical Einstein summation operations into GEMM operations, leveraging the specific features of tensor network simulations to amplify the efficiency of GPUs. Second, by analyzing the data characteristics of quantum circuits, we employ extended precision to ensure the accuracy of simulation results and mixed precision to fully exploit the potential of GPUs, resulting in faster and more precise simulations. Our numerical experiments demonstrate that our approach can achieve a 3.96× reduction in verification time for random quantum circuit samples in the 18-cycle case of Sycamore, with sustained performance exceeding 21 TFLOPS on one A100. This method can be easily extended to the 20-cycle case, maintaining the same performance, accelerating by 12.5× compared to the state-of-the-art CPU-based results and 4.48–6.78× compared to the state-of-the-art GPU-based results reported in the literature. Furthermore, the strategies presented in this article hold general applicability for accelerating problems that can be tackled by contracting tensor networks such as graphical models and combinatorial optimizations.
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.
Here we present a flexible tensor network based simulator for quantum circuits on different topologies, including the Google Bristlecone QPU. Our simulator can compute both exact amplitudes, a task essential for the verification of the quantum hardware, as well as low-fidelity amplitudes to mimic Noisy Intermediate-Scale Quantum (NISQ) devices. While our simulator can be used to compute amplitudes of arbitrary quantum circuits, we focus on random quantum circuits (RQCs) [Boixo et al., Nature Physics 14] in the range of sizes expected for supremacy experiments. Our simulator enables the simulation of sampling on quantum circuits that were out of reach for previous approaches. For instance, our simulator is able to output single amplitudes with depth 1+32+1 for the full Bristlecone QPU in less than (f⋅4200) hours on a single core, where 0<f≤1 is the target fidelity, on 2×20-core Intel Xeon Gold 6148 processors (Skylake). We also estimate that computing 106 amplitudes (with fidelity 0.50\%) needed to sample from the full Bristlecone QPU with depth (1+32+1) would require about 3.5 days using the NASA Pleiades and Electra supercomputers combined. In addition, we discuss the hardness of the classical simulation of RQCs, as well as give evidence for the higher complexity in the simulation of Bristlecone topology as compared to other two-dimensional grids with the same number of qubits. Our analysis is supported by extensive simulations on NASA HPC clusters Pleiades and Electra. For the most computationally demanding simulation we had, namely the simulation of a 60-qubit sub-lattice of Bristlecone, the two HPC clusters combined reached a peak of 20 PFLOPS (single precision), that is 64% of their maximum achievable performance. To date, this numerical computation is the largest in terms of sustained PFLOPS and number of nodes utilized ever run on NASA HPC clusters.
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.