Quantum Error Mitigation Simulates General Non-Hermitian Dynamics

Abstract

While non-Hermitian Hamiltonians enable exotic dynamical phenomena, implementing their nonunitary time evolution on near-term quantum devices remains challenging. We propose a hardware-friendly protocol that simulates non-Hermitian dynamics without continuous monitoring. Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) evolution via classical Gaussian white-noise averaging and to subsequently cancel the quantum-jump contribution at the level of the measured observable using stochastic quantum error mitigation (QEM). The scheme requires no ancillas or controlled time-evolution, while the mitigation layer uses only single-qubit operations. We validate the method through numerical simulations of a model with asymmetric hopping, interaction, and disorder. Our work provides a programmable and ancilla-free framework investigating exotic dynamics that are not completely-positive and trace-preserving using QEM.

Figures (3)

• Figure 1: Overview of the proposed protocol for realizing non-Hermitian dynamics via a noise-averaged GKSL evolution and sQEM. (a) Conventional setup: the desired non-Hermitian (no-jump) dynamics is obtained by postselecting no-jump events using a detector. (b) Proposed protocol: in each run, we alternately apply the coherent evolution generated by $H_{\mathrm{Re}}$ and the stochastically driven unitary evolution generated by $H_{\mathrm{I}}(k)\coloneqq \sum_{\ell \in E}\xi_{k,\ell} H_{\mathrm{I},\ell}$, where $\boldsymbol{\xi}_k=\{\xi_{k,\ell}\}_{\ell \in E}$ denotes the integrated white-noise increment at step $k$. The noise-averaged evolution reproduces the auxiliary GKSL dynamics. Using the decomposition of the GKSL generator into a non-Hermitian (no-jump) part and a quantum-jump part, we implement sQEM within each trajectory to cancel the jump contribution, thereby realizing the target non-Hermitian Hamiltonian dynamics at the level of normalized observables. Here, $\Delta t$ is the simulation time step and $\delta t$ is the time step in the sQEM cancellation map.
• Figure 2: Two-qubit benchmark: normalized expectation values (a) $\langle \sigma^z_1\rangle_{\mathrm{norm}}(t)$ and (b) $\langle \sigma^z_0\sigma^z_1\rangle_{\mathrm{norm}}(t)$ starting from $|\psi_0\rangle=|+\rangle^{\otimes 2}$. The solid blue line and the dashed orange lines show the exact Non-Hermitian Hamiltonian and GKSL time evolutions, respectively. Green circles and red squares with error bars show Monte Carlo estimates from our protocol with and without sQEM. We use parameters $J=1.0$, $g=0.1$, $U=2.0$, $\gamma=1.0$, $(h_0,h_1)=(-0.8071,0.3890)$, $\Delta t=10^{-3}$ , and $t\in\{0,0.05,\ldots,1.50\}$. We use $B=8000$ independent runs, each averaging over $M_{\mathrm{num}}=2000$ stochastic trajectories (total $BM_{\mathrm{num}}=1.6\times 10^7$ trajectories). Error bars show jackknife standard errors obtained from resampling over the $B$ independent runs.
• Figure 3: Four-qubit benchmark: (a) edge imbalance $\Delta n_{\mathrm{edge}}(t)=\langle n_{3}(t)\rangle_{\mathrm{norm}}-\langle n_{0}(t)\rangle_{\mathrm{norm}}$ and (b) site occupations $\langle n_i(t)\rangle_{\mathrm{norm}}$ ($i=0,1,2,3$) for weak disorder ($h_{\mathrm{amp}}=0.1$) and strong disorder ($h_{\mathrm{amp}}=8.0$), where the initial state is $|0110\rangle$. Solid orange and dashed purple lines show the exact target non-Hermitian dynamics (asym, $g=0.1$) for $h_{\mathrm{amp}}=0.1$ and $h_{\mathrm{amp}}=8.0$, respectively. Dotted green and dash-dotted brown lines show the corresponding symmetric-hopping model (symm, $g=0$). Blue circles ($h_{\mathrm{amp}}=0.1$) and red squares ($h_{\mathrm{amp}}=8.0$) show Monte Carlo estimates from our protocol. Parameters: $J=1.0$, $g=0.1$, $U=1.0$, $\gamma=1.0$, $\Delta t=1.0\times 10^{-3}$, and $t\in\{0,0.05,\ldots,0.80\}$. We use $B=210240$ independent runs, each obtained by averaging over $M_{\mathrm{num}}=2000$ stochastic trajectories (total $BM_{\mathrm{num}}=4.2048\times 10^8$ trajectories). Error bars show jackknife standard errors over the $B$ runs.

Theorems & Definitions (44)

• Theorem 1: RMSE of the normalized sQEM ratio estimator in the presence of trace-decreasing basis operation channels
• proof
• Lemma 1: Duhamel formula (variation of constants)
• proof
• Lemma 2: Telescoping bound for products of unitaries
• proof
• Lemma 3: Gaussian absolute moments
• proof
• Lemma 4: Bias--variance decomposition and a sufficient condition
• proof