Variational quantum-neural hybrid imaginary time evolution

TL;DR

This work addresses the challenge of simulating imaginary-time evolution on quantum computers by augmenting variational imaginary-time evolution (VITE) with a neural-network–generated non-unitary operator. The proposed method, VQNHITE, forms a more expressive trial state $|\tilde{\varphi}(\bm{\theta},\bm{\phi})\rangle=\hat{f}(\bm{\phi})\hat{U}(\bm{\theta})|\bar{0}\rangle$ and employs a two-stage procedure: an initial optimization at a small imaginary time $\beta=\delta\beta$ to stabilize parameters, followed by joint updates for general $\beta$ using extended McLachlan-based equations and Hadamard-test circuits. Numerical simulations on Heisenberg spin chains show that VQNHITE achieves higher fidelity to the exact ITE state than VITE, across both nearest-neighbor and all-to-all connectivities. This demonstrates that NN-assisted non-unitary layers can significantly improve variational imaginary-time simulations on NISQ hardware, albeit with increased sampling costs and optimization complexity.

Abstract

Numerous methodologies have been proposed to implement imaginary time evolution (ITE) on quantum computers. Among these, variational ITE (VITE) methods for noisy intermediate-scale quantum (NISQ) computers have attracted much attention, which uses parametrized quantum circuits to mimic non-unitary dynamics. Although widely studied, conventional variational quantum algorithms including face challenges in achieving high accuracy due to their strong dependence on the choice of ansatz quantum circuits. Recently, the variational quantum-neural hybrid eigensolver (VQNHE), which combines the neural network (NN) with a variational quantum eigensolver, has been proposed. This approach enhances the expressive power of variational states and improves the estimation of expectation values. Motivated by this idea, we explore the hybridization of VITE with a NN-based non-unitary operator. In this study, we propose a method named variational quantum-neural hybrid ITE (VQNHITE). By combining the NN and parameterized quantum circuit, our proposal enhances the expressive power compared to conventional approaches, enabling more accurate tracking of imaginary-time dynamics. In addition, to mitigate the instability arising from randomly initialized NN parameters, we introduce an initial-parameter optimization procedure at a small imaginary-time step, which stabilizes the subsequent variational evolution. We tested our approach with numerical simulations on Heisenberg spin chains under both nearest-neighbor and all-to-all circuit connectivities. The results demonstrate that VQNHITE consistently achieves higher fidelity with the exact ITE state compared to VITE.

Figures (5)

• Figure 1: Quantum circuits for calculating (a) $\Re(e^{i\phi}\bra{\bar{0}}\mathcal{\hat{U}}_{j,p}^\dag \mathcal{\hat{U}}_{k,q}\ket{\bar{0}})$ and (b) $\Re(e^{i\phi}\bra{\bar{0}} \mathcal{\hat{U}}_{j,k}^\dag \hat{P}_j \hat{U}\ket{\bar{0}})$, which are required to obtain $M_{j,k}$ in Eq. (\ref{['eq:M']}) and $C_{j}$ in Eq. (\ref{['eq:V']}). The upper (lower) horizontal line represents the ancillary qubit (the system qubits). The initial state is prepared as $(\ket{0}+e^{i \phi}\ket{1})/\sqrt{2}$. $X$ and $H$ denote the Pauli $\hat{\sigma}^x$ gate and the Hadamard gate, respectively, and $U_j$ ($j=1,\cdots, N_{\mathrm{ p}}$) is a parametrized unitary gate constituting the variational circuit. The expectation values of the $Z$-measurement on the ancillary qubit yield (a) $\Re(e^{i\phi}\bra{\bar{0}}\mathcal{\hat{U}}_{j,p}^\dag \mathcal{\hat{U}}_{k,q}\ket{\bar{0}})$ and (b) $\Re(e^{i\phi}\bra{\bar{0}} \mathcal{\hat{U}}_{j,k}^\dag \hat{P}_j \hat{U}\ket{\bar{0}})$.
• Figure 2: The ansätz circuit for $6$ qubits. Panels (a) and (b) show circuits containing only single-qubit rotation gates together with (a) nearest-neighbor interactions or (b) all-to-all interactions. $R^Y_{\theta_i}$ denotes a rotation along the $Y$-axis, and $\theta_i$ represents the parameter of the $i$th rotation angle.
• Figure 3: Fidelity between the variational states obtained by our VQNHITE (blue markers) or the VITE (orange square markers) and the exact imaginary-time–evolved state, plotted as a function of the imaginary time $\beta$. The horizontal axis corresponds to $\beta\in[0.1,6]$ with increments $\Delta\beta=0.1$. Results are shown for the nearest-neighbor interactions ansatz in Fig. \ref{['fig:ansatz']} (a). We consider (a) $N=6$ and (b) $N=8$, set $J=-1$, and randomly choose $h_j\sim\mathrm{Unif}[-1,1]$. Each data point represents the mean fidelity over $100$ samples. The fidelities for VITE and VQNHITE are defined as $F(\beta)=|\langle\psi(\beta)\vert\varphi(\bm{\theta})\rangle|^2$ and $F(\beta)=|\langle\psi(\beta)\vert\tilde{\varphi}(\bm{\theta,\phi})\rangle|^2$, respectively.
• Figure 4: Fidelity between the variational states obtained by our VQNHITE (blue markers) or the VITE (orange square markers) and the exact imaginary-time–evolved state, plotted as a function of the imaginary time $\beta$. The horizontal axis corresponds to $\beta\in[0.1,6]$ with increments $\Delta\beta=0.1$. Results are shown for the all-to-all interactions ansatz in Fig. \ref{['fig:ansatz']}(a). We consider (a) $N=6$ and (b) $N=8$, set $J=-1$, and randomly choose $h_j\sim\mathrm{Unif}[-1,1]$. Each data point represents the mean fidelity over $100$ samples. The fidelities for VITE and VQNHITE are defined as $F(\beta)=|\langle\psi(\beta)\vert\varphi(\bm{\theta})\rangle|^2$ and $F(\beta)=|\langle\psi(\beta)\vert\tilde{\varphi}(\bm{\theta,\phi})\rangle|^2$, respectively.
• Figure 5: Quantum circuits for calculating: (a)-1 $e^{i\phi_{j,k}}\bra{\varphi(\bm{\theta})}\ket{s}\bra{s} \mathcal{\hat{U}}_{j,k} \ket{\bar{0}}$, without the measurement circuit $V$; (a)-2 $\Re(\bra{\bar{0}}\partial_{\theta_j}\hat{U}^{\dagger}\hat{f}(\bm{\phi})\partial_{\phi_k}\hat{f}(\bm{\phi})\ket{\varphi(\bm{\theta})})$ with the measurement circuit $V$; (b) $\Re\left(e^{i\phi_{j,l_1,k,l_2}}\bra{\bar{0}}\mathcal{\hat{U}}_{j,l_1}^{\dag}\ket{s}\bra{s} \mathcal{\hat{U}}_{k,l_2} \ket{\bar{0}}\right)$; (c) $\braket{\psi(\delta\beta)}{\tilde{\varphi}(\bm{\theta}, \bm{\phi})}$ and $\partial \theta_k\braket{\psi(\delta\beta)}{\tilde{\varphi}(\bm{\theta}, \bm{\phi})}$; (d) $\partial \theta_j\braket{\psi(\delta\beta)}{\tilde{\varphi}(\bm{\theta}, \bm{\phi})}$. To measure the real part in (c) and (d), the ancillary qubit is measured in the $X$ basis using the Hadamard gate. To measure the imaginary part, the phase gate $S^{\dagger}$ is inserted before the Hadamard gate so that the measurement basis becomes $Y$ since $S^{\dagger} Z S=Y$. The upper (lower) horizontal line represents the ancillary qubit (the system qubits).