Variational quantum simulation of many-body dissipative dynamics on a superconducting quantum processor

TL;DR

This work addresses the challenge of simulating non-unitary, dissipative many-body dynamics on quantum hardware by introducing a variational quantum simulation (VQS) framework built atop the linear combination of Hamiltonian simulation (LCHS). The non-Hermitian evolution is recast as a discretized sum of unitaries, and a parameterized quantum circuit is trained in a hybrid quantum-classical loop using a fidelity-based loss plus an optional penalty term, with a Hadamard-test simplification that keeps circuit depth independent of the total simulation time. The authors demonstrate the approach on a superconducting processor (Wukong) for two open-system models: a dissipative transverse Ising chain and an interacting Hatano-Nelson model, observing consistent dynamics and, in the latter, many-body non-Hermitian skin effects and dynamic symmetry. The results establish that VQAs can realistically capture dissipative quantum phenomena on NISQ devices and offer a pathway toward larger-scale simulations of open quantum systems with near-term hardware. The methodology combines rigorous decomposition, efficient circuit design, and hardware-aware optimization to push the boundary of what is experimentally accessible in open quantum-system physics.

Abstract

Open quantum systems host a wide range of intriguing phenomena, yet their simulation on well-controlled quantum devices is challenging, owing to the exponential growth of the Hilbert space and the inherently non-unitary nature of the dynamics. Here we propose and experimentally demonstrate a variational quantum algorithm capable of scalable simulation of non-unitary many-body dissipative dynamics. The algorithm builds on the framework of linear combination of Hamiltonian simulation, which converts non-unitary dynamics into a weighted sum of unitary evolutions. With the further introduction of a simplified quantum circuit for loss-function evaluation, our scheme is suitable for near-term quantum hardware, with the circuit depth independent of the simulation time. We illustrate our scheme by simulating the collective dynamics of a dissipative transverse Ising model, as well as an interacting Hatano-Nelson model, on the superconducting quantum processor Wukong. Our work underlines the capability of noisy intermediate-scale quantum devices in simulating dissipative many-body dynamics and represents a step forward in exploiting their potential for solving outstanding physical problems.

Figures (7)

• Figure 1: Workflow of the variational quantum simulation algorithm. (a) Schematic overview. The objective is to determine a sequence of parameters $\{\bm{\theta}_m\}$ such that $U(\bm{\theta}_m)\ket{0} = \ket{\psi_m}$. We first use a VQA to obtain $\bm{\theta}_0$ by maximizing the fidelity between $U(\bm{\theta})\ket{0}$ and the initial state $\ket{\psi_0}$. The VQS procedure is then applied to generate the parameters step-by-step. (b) The VQS framework at step $m$. The initial parameters $\bm{\theta}$ are input to a quantum processor. Given the LCHS decomposition $\sum_k c_k U_k$ and a set of observables $\{O_l\}$, we apply the Hadamard test to obtain $X_{m,k}$, and perform direct measurements to evaluate $\langle O_l \rangle_{\operatorname{mea}}$. These quantities are sent to a classical computer to evaluate the loss function and its gradient information. Parameters are optimized in such a hybrid quantum-classical loop until convergence is reached. We then obtain $\bm{\theta}_{m+1}$, and the time-evolved state at the next time step. (c) Quantum circuit for the Hadamard test to evaluate $\Re[X_{m,k}]$. (d) Schematic overview of the simplification procedure. The circuit can be simplified in pairs. Further details of the VQS algorithm and the simplification procedure can be found in the Methods section.
• Figure 2: Simulating results of the dissipative Ising model. Panels (a)–(d) correspond to $g_{\mathrm{i}} = 0$, $0.5$, $1.0$, and $2.0$, respectively. The solid lines represent results from numerical simulations, and the dots are experimental data. For each data point in panels (b)–(d), the parameterized circuit is executed 50 times, with each execution sampled by $2\times 10^{4}$ shots. Among these, the result exhibiting the minimal bias is selected for presentation.
• Figure 3: Simulation results of the interacting HN model. (a) Theoretical predictions of $\langle n_j\rangle$ for $H_{\text{p}}(t_R,t_L)$, showing progressive localization of fermions toward the right edge. (b) Simulation results for $H_{\text{p}}(t_L,t_R)$ using our VQS algorithm, showing excellent agreement with the theoretical results in (a). (c) Experimental evolution of the occupations for the two edge sites $q_8$ and $q_9$ for $H_{\text{p}}(t_R,t_L)$. Each data point is obtained from $2\times 10^4$ shots. The experimental results match the theoretical predictions throughout the time evolution. (d)(e)(f) Simulations results for $H_{\text{p}}(t_L,t_R)$.
• Figure 4: Circuit simplification for the Hadamard test with a two-qubit PQC as one example. (a) The example of the two-qubit PQC with two single-qubit gates and a two-qubit gate. (b) Circuit for $C_1U(\bm{\theta})\cdot C_0U(\bm{\theta}_m)$. (c) Detailed quantum circuit for (b) after we rearrange the quantum gates according to Eq. \ref{['eq:arrange']}, which has 4 two-qubit gates and 2 three-qubit gates. Quantum gates in yellow and green dashed boxes can be simplified similarly to Eqs. \ref{['eq:simsingle']} and \ref{['eq:simdouble']}, respectively. (d) The simplified quantum circuit. Compared to (c), the circuit depth is reduced, and there are no three-qubit gates.
• Figure S1: Benchmarking of the LCHS method for different choices of $K$ and $\delta k$. Higher accuracy is achieved with larger $K$ and smaller $\delta k$. Notably, $\{K=40, \delta k=1\}$ yields better performance than $\{K=80, \delta k=2\}$ with equal resource cost.