Neural-network variational quantum algorithm for simulating many-body dynamics

We propose a neural-network variational quantum algorithm to simulate the time evolution of quantum many-body systems. Based on a modified restricted Boltzmann machine (RBM) wave function ansatz, the proposed algorithm can be efficiently implemented in near-term quantum computers with low measurement cost. Using a qubit recycling strategy, only one ancilla qubit is required to represent all the hidden spins in an RBM architecture. The variational algorithm is extended to open quantum systems by employing a stochastic Schrödinger equation approach. Numerical simulations of spin-lattice models demonstrate that our algorithm is capable of capturing the dynamics of closed and open quantum many-body systems with high accuracy without suffering from the vanishing gradient (or “barren plateau”) issue for the considered system sizes.

Figure 1

Quantum circuit for preparing the unitary-coupled restricted Boltzmann machine (uRBM) state with the qubit recycling scheme described in Eq. (4). All qubits are initialized in $|0\rangle$ state. The single rotations are governed by the relations in Eq. (5). The $j\mathrm{th}$ entangling block implements the $exp\left(\mathrm{i}{\sum }_i{W}_{ij}^I{\widehat{v}}_i^z{\widehat{h}}_j^z\right)$ operator, and its explicit form is given in Appendix pp1. After each entangling block, the ancilla qubit representing the $j\mathrm{th}$ hidden spin is projected onto $|+\rangle$ state before being recycled.

Figure 2

Time evolution induced by quantum quench. Results from the unitary-coupled restricted Boltzmann machine (uRBM) algorithm (symbols) are compared with exact calculations (solids lines). (a) and (b) Dynamics of transverse polarization and its correlation in a one-dimensional (1D) Ising model. (c) and (d) Dynamics of magnetization and transverse polarization correlations in a 1D Heisenberg model in a global field.

Figure 3

Dynamics of a dissipative one-dimensional (1D) Ising model obtained from exact numerical solution of Eq. (12) (solid lines) and from the hybrid unitary-coupled restricted Boltzmann machine (uRBM) algorithm (symbols). The simulation results from the hybrid uRBM algorithm are obtained from averaging over 10 000 pure state trajectories, and the error bars are smaller than the symbols

Figure 4

Quantum circuits for the coupling terms (a) $exp\left(\mathrm{i}{\sum }_i{W}_{ij}^I{\widehat{v}}_i^z{\widehat{h}}_j^z\right)$ and (b) $exp\left({\sum }_i{W}_{ij}^R{\widehat{v}}_i^z{\widehat{h}}_j^z\right)$ between $i\mathrm{th}$ visible and $j\mathrm{th}$ hidden spins.

Figure 5

Imaginary time evolution of (a) one-dimensional (1D) Ising model and (b) Heisenberg models. The solid lines are the exact ground-state energy. The dashed black lines represent the imaginary time evolution using the variational unitary-coupled restricted Boltzmann machine (uRBM) algorithm.

Figure 6

Time evolution in a two-dimensional triangular antiferromagnetic Ising lattice induced by quantum quench. (a) Configuration of the triangular lattice with periodic boundary condition. (b) and (c) Results from the unitary-coupled restricted Boltzmann machine (uRBM) algorithm (symbols) are compared with exact calculations (solids lines) for dynamics of transverse polarization and its correlation.

Figure 7

Autocorrelation of a two-dimensional triangular antiferromagnetic Ising model as a function of Monte Carlo step number for $O={\sigma }_{(0,0)}^z{\sigma }_{(L/2,L/2)}^z$. $y$ aixs is scaled by $\langle O(\tau =0)O(\tau =0)\rangle$ to ensure that maximum value is unity.

Figure 8

Noisy simulations of a 6-spin one-dimensional (1D) Ising lattice induced by quantum quench. To account for the imperfections of actual quantum devices, random Gaussian noises with standard deviation $\delta$ are added to the covariance matrix $A$ and the force vector $f$. Results from noisy uRBM simulations (dashed lines) are compared with exact calculations (black solid lines) for dynamics of transverse polarization and its correlation.

Figure 9

The norms of $f$ and $A^{-1}f$ as a function of system size for (a) and (b) a one-dimensional (1D) transverse field Ising model and (c) and (d) a 1D Heisenberg model in the longitudinal field. The norms are normalized by the number of variational parameters. The blue and red lines denote the average and minimum from 100 random initializations, respectively.