Simulation of Quantum Circuits Using the Big-Batch Tensor Network Method

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 $2\times {10}^6$ 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.

Figure 1

The top panel presents a pictorial representation of the 3-dimensional tensor network corresponding to a quantum circuit. The leftmost layer represents the initial state, and the rightmost layer represents the final state, where the blue circles represent measured (closed) qubits, which fix the entries in the final bitstring $\mathit{s}$, and the red circles represent open qubits, where the corresponding entries in $\mathit{s}$ can vary. The yellow plane $\mathcal{C}$ cuts into the tensor network and separates the network into two parts, ${\mathcal{G}}_{\mathrm{head}}$ and ${\mathcal{G}}_{\mathrm{tail}}$, depicted in the bottom left panel. The ${\mathcal{G}}_{\mathrm{head}}$ contains all closed qubits, and ${\mathcal{G}}_{\mathrm{tail}}$ includes all the open qubits. Both are further partitioned into subgraphs hierarchically until the size of the subgraph is smaller than 60. The bottom right panel displays the bottleneck between ${\mathcal{G}}_{\mathrm{head}}$ and ${\mathcal{G}}_{\mathrm{tail}}$ given by $\mathcal{C}$.

Figure 2

Left: Histogram of bitstring probabilities $P_U(\mathit{s})=P_U({\mathit{s}}_1;{\mathit{s}}_2)$ for $2^{21}$ correlated bitstrings obtained from the Sycamore circuit with $n=53$ qubits, $m=20$ cycles, sequence ABCDCDAB, seed 0, with the assignment of partial bitstring ${\mathit{s}}_1$ fixed to $\{0,0,\dots ,0\}$. In the figure, $N=2^{53}$, $p$ denotes the probability of bitstring $P_U(\mathit{s})$. Right: Histogram of probabilities of all $2^{50}$ bitstrings obtained from the Sycamore circuit with $n=50$ qubits, $m=14$ cycles, sequence EFGH. The red line in the figures represent the Porter-Thomas distribution [1].