Simulation of IBM's kicked Ising experiment with Projected Entangled Pair Operator

TL;DR

Confronts the challenge of classically simulating the IBM 127-qubit kicked Ising circuit by recasting the time evolution in the Heisenberg picture as a PEPO-based three-dimensional tensor-network contraction. The approach exploits light-cone structure and intrinsic low-rank contributions from Clifford and near-Clifford gates, avoiding long-range swaps and enabling efficient contraction. It achieves exact results for the shallow 5+1-step circuit via a Clifford expansion and demonstrates that PEPO with small bond dimension $χ$ can outperform state-of-the-art methods, with $χ=2$ matching CPT with $K=10$ and MPO with $χ=1024$, and $χ=184$ delivering near machine-precision; for deeper 20-step circuits, results converge with $χ$ and can be extrapolated to $χ o ∞$ to benchmark against hardware. The findings suggest PEPO as a powerful tool for simulating dynamical properties of quantum many-body systems and near-Clifford quantum algorithms such as QAOA, complemented by an open-source implementation.

Abstract

We perform classical simulations of the 127-qubit kicked Ising model, which was recently emulated using a quantum circuit with error mitigation [Nature 618, 500 (2023)]. Our approach is based on the projected entangled pair operator (PEPO) in the Heisenberg picture. Its main feature is the ability to automatically identify the underlying low-rank and low-entanglement structures in the quantum circuit involving Clifford and near-Clifford gates. We assess our approach using the quantum circuit with 5+1 trotter steps which was previously considered beyond classical verification. We develop a Clifford expansion theory to compute exact expectation values and use them to evaluate algorithms. The results indicate that PEPO significantly outperforms existing methods, including the tensor network with belief propagation, the matrix product operator, and the Clifford perturbation theory, in both efficiency and accuracy. In particular, PEPO with bond dimension $χ=2$ already gives similar accuracy to the CPT with $K=10$ and MPO with bond dimension $χ=1024$. And PEPO with $χ=184$ provides exact results in $3$ seconds using a single CPU. Furthermore, we apply our method to the circuit with 20 Trotter steps. We observe the monotonic and consistent convergence of the results with $χ$, allowing us to estimate the outcome with $χ\to\infty$ through extrapolations. We then compare the extrapolated results to those achieved in quantum hardware and with existing tensor network methods. Additionally, we discuss the potential usefulness of our approach in simulating quantum circuits, especially in scenarios involving near-Clifford circuits and quantum approximate optimization algorithms. Our approach is the first use of PEPO in solving the time evolution problem, and our results suggest it could be a powerful tool for exploring the dynamical properties of quantum many-body systems.

Figures (3)

• Figure 1: Layout of IBM's 127-qubit quantum processor.
• Figure 2: Left: The expectation value of the modified weight-17 stabilizer $\tilde{W}_{17}$ obtained by PEPO, MPS Nature, CPT Chan23, the IBM's quantum hardware with error mitigation (IBM) Nature, and compared against exact results, on the quantum circuit with 5+1 Trotter steps ($5$ Trotter steps with an addition rotation gates, corresponding to Fig.4(a) in Ref. Nature). Right: The absolute errors with respect to the exact results. The computation time of the PEPO method with $\chi=184$ in obtaining a data point is less than $3$ seconds using a single Intel Xeon Gold 6326 CPU.
• Figure 3: Left: Extrapolation of the expectation values of $\langle Z_{62}\rangle$ obtained by our method as a function of the inverse bond dimension $\chi$ on IBM's circuit with $20$ Trotter steps. The solid lines are extrapolated results using a function $b\,e^{-a/\chi}$ with two fitting parameters $a$ and $b$. The largest bond dimension we have calculated is $\chi=256$, which takes about $7$ hours using a single Intel Xeon Gold 6326 CPU to obtain a single data point. Right: The extrapolated value of $\langle Z_{62}\rangle$, PEPO (Extrap.), compared with the results of IBM's quantum hardware with error mitigation (IBM) and other numerical algorithms.