Non-zero noise extrapolation: accurately simulating noisy quantum circuits with tensor networks

TL;DR

The paper addresses the challenge of accurately simulating noisy quantum circuits with tensor networks in the low-noise regime, where entanglement limits fidelity. It introduces non-zero noise extrapolation (NZNE), which artificially adds controllable noise to improve tensor-network emulations and then extrapolates results back to the desired low-noise level. Across benchmarks on TFIM, Fermi-Hubbard, XY, and large 60-qubit systems, NZNE significantly improves observable accuracy over direct high-fidelity emulations and remains scalable to larger qubit counts. The approach leverages a VMPO representation for density matrices, defines a fidelity-based extrapolation workflow, and outlines extensions to open systems and higher dimensions, offering a practical path toward reliable noise-aware quantum simulations on near-term hardware.

Abstract

Understanding the effects of noise on quantum computations is fundamental to the development of quantum hardware and quantum algorithms. Simulation tools are essential for quantitatively modelling these effects, yet unless artificial restrictions are placed on the circuit or noise model, accurately modelling noisy quantum computations is an extremely challenging task due to unfavourable scaling of required computational resources. Tensor network methods offer a viable solution for simulating computations that generate limited entanglement or that have noise models which yield low gate fidelities. However, in the most interesting regime of entangling circuits (with high gate fidelities) relevant for error correction and mitigation tensor network simulations often achieve poor accuracy. In this work we develop and numerically test a method for significantly improving the accuracy of tensor network simulations of noisy quantum circuits in the low-noise (i.e. high gate-fidelity) regime. Our method comes with the advantages that it (i) allows for the simulation of quantum circuits under generic types of noise model, (ii) is especially tailored to the low-noise regime, and (iii) retains the benefits of tensor network scaling, enabling efficient simulations of large numbers of qubits. We build upon the observations that adding extra noise to a quantum circuit makes it easier to simulate with tensor networks, and that the results can later be reliably extrapolated back to the low-noise regime of interest. These observations form the basis for a novel emulation technique that we call non-zero noise extrapolation, in analogy to the quantum error mitigation technique of zero-noise extrapolation.

Figures (16)

• Figure 1: Left: Average (over 10 runs) of the entropy of the quantum state after each layer random quantum circuits on 16 qubits. Each layer of the circuit consists of Haar-random single-qubit gates followed by controlled-$Z$ gates on all neighbouring pairs of qubits with 1D connectivity. For $\lambda > 0$, the entanglement entropy is the MPO entanglement entropy of the VMPO representing the state. For $\lambda = 0$ the entanglement entropy is the 'usual' entanglement entropy of the pure state (represented by a matrix product state). For both, the entropy is measured at a cut across the middle 2 qubits. Right: Emulation fidelity $\mathcal{F}_\lambda$ for various noise strengths $\lambda$ of the depolarizing noise model whist keeping the bond dimension fixed (to 1500). The circuit is a random 2D quantum circuit on 25 qubits, consisting of 30 layers of: Haar-random single-qubit gates on all qubits followed by controlled-$Z$ gates on all neighbouring pairs of qubits. All circuits were applied to the all-zeros initial (pure) state.
• Figure 2: Demonstration of the method for the observable $YX_{30,31}$ on a $60$-qubit instance of the $XY$ model (see main text or Section \ref{['app:xym']} for details of the circuit). First we perform fidelity extrapolation to obtain estimates of $\overline{\langle YX \rangle}_{30, 31}$ at different noise strengths $\lambda$. For each $\lambda$ the fidelity extrapolation is obtained from 4 tensor network emulations with different bond dimensions $D \leq 300$. (a). Any poorly extrapolated results are removed ((a) red line). The remaining points are used for non-zero noise extrapolation, which consists of a (weighted) straight line fit applied to the logs of the absolute values of the expectation values (b). The weighted fit puts more emphasis on high-fidelity data points close to the target noise strength. The method is most useful when the emulation fidelity at the target noise strength is low, which was the case here.
• Figure 3: $\langle Z_iZ_j \rangle$ after $10$ Trotter steps of a $2 \times 7$ instance of the TFIM with depolarising noise at $\lambda^*=0.01$. A single emulation with $D_{\max}=32$ (and $\mathcal{F}_{\lambda^*}^{\max}=0.39$) yields an average absolute error of $0.0590$ compared to the exact values. Non-zero noise extrapolation with $D\leq32$ yields an average absolute error of $0.0093$.
• Figure 4: $\langle Z_iZ_j \rangle$ after $10$ Trotter steps of a $2 \times 7$ instance of the TFIM with cat noise at $\lambda^*=0.0005$. A single emulation with $D_{\max}=32$ (and $\mathcal{F}_{\lambda^*}^{\max}=0.29$ ) yields an average absolute error of $0.0696$ compared to the exact values. Non-zero noise extrapolation with $D\leq32$ yields an average absolute error of $0.0085$.
• Figure 5: Relative error in the energy per site $\langle \bar{E}_{\text{TFIM}} \rangle$ against emulation fidelity $\mathcal{F}_{\lambda^*}^{\max}$ at the target noise strength, for a circuit implementing 10 Trotter steps of time evolution of a $2 \times 7$ (14 qubit) instance of the TFIM under (a) depolarizing noise at $\lambda^*=0.01$ with exact value $\langle \bar{E}_{\text{TFIM}} \rangle_{\lambda^*} = -1.833$ and (b) cat noise at $\lambda^*=0.0005$ with exact value $\langle \bar{E}_{\text{TFIM}} \rangle_{\lambda^*} = -1.710$, for both a single $D_{\max}$ emulation at the target noise strength (blue line) and non-zero noise extrapolation using only $D \leq D_{\max}$ emulations (green line). Emulations were run with $D_{\max} \in \{8, \dots, 512\}$ in order to obtain different emulation fidelities between 0 and 1.