Quantum anomaly for benchmarking quantum computing

Abstract

Given the rapid advances in quantum computing hardware, establishing systematic strategies for verifying the correctness of quantum computations has become increasingly important. Exploiting the fact that the axial anomaly in gauge theories is exact to all orders in perturbation theory, we propose the axial anomaly as a nontrivial benchmark for quantum simulations of lattice gauge theories. We simulate anomalous axial-charge production in ${\mathbb Z}_N$ lattice gauge theories on the trapped-ion quantum computer ``Reimei''. After taking the U(1), infinitesimal time, and infinite volume limits, we successfully reproduce the anomaly coefficient within statistical uncertainties, even without error mitigation. Our results demonstrate that the axial anomaly can be simulated on current quantum computers and serves as a verification test of quantum computations.

Figures (4)

• Figure 1: Left: lattice geometry. The two-component fermions $|f_1(x)\rangle$ and $|f_2(x)\rangle$ are defined on sites (black open circles). The gauge fields $|g(x)\rangle$ are defined on links (red filled circles). Periodic boundary conditions are imposed. Right: quantum circuit. The first stage is the initial state preparation implemented by the free fermion creation $C(k)$ and the inverse fermionic Fourier transform ${\rm FFT}_i^\dagger$. The second stage is the time evolution $e^{-\mathrm{i} H_g \delta t}$ with the quantum Fourier transform QFT and its conjugate ${\rm QFT}^\dagger$. The third stage is a projective measurement after the basis transformation $B$ to the measurement basis.
• Figure 2: Axial charge $\langle Q_5 \rangle$ produced by the external gauge field $A_{\rm ext}=\frac{2\pi}{N}$. The data of $e^2 \delta t = 0.1$ are shown. Error bars represent statistical uncertainties due to finite shots.
• Figure 3: Infinitesimal time extrapolation $\delta t \to 0$. Error bars represent fitting uncertainties from Fig. \ref{['figZN']}.
• Figure 4: Infinite volume extrapolation $L \to \infty$. Error bars represent fitting uncertainties from Fig. \ref{['figG']} and the error band represents the uncertainty of the linear extrapolation. The star denotes the correct anomaly coefficient $\frac{1}{\pi}$.