Variational Quantum-Neural Hybrid Eigensolver

The variational quantum eigensolver (VQE) is one of the most representative quantum algorithms in the noisy intermediate-scale quantum (NISQ) era, and is generally speculated to deliver one of the first quantum advantages for the ground-state simulations of some nontrivial Hamiltonians. However, short quantum coherence time and limited availability of quantum hardware resources in the NISQ hardware strongly restrain the capacity and expressiveness of VQEs. In this Letter, we introduce the variational quantum-neural hybrid eigensolver (VQNHE) in which the shallow-circuit quantum Ansatz can be further enhanced by classical post-processing with neural networks. We show that the VQNHE consistently and significantly outperforms the VQE in simulating ground-state energies of quantum spins and molecules given the same amount of quantum resources. More importantly, we demonstrate that, for arbitrary postprocessing neural functions, the VQNHE only incurs a polynomial overhead of processing time and represents the first scalable method to exponentially accelerate the VQE with nonunitary postprocessing that can be efficiently implemented in the NISQ era.

Figure 1

(a) Schematic work flow for the VQNHE. The output state $|\psi ⟩=U(\mathit{\theta })|0⟩$ is attached with a small measurement circuit $V$. Measurement on computational ($Z$) basis is conducted on $VU(\mathit{\theta })|0⟩$ to collect shots of bit string result as $\overrightarrow{s}$. With the zeroth qubit as the star qubit (see main text for details), $0s_{1:n-1}$ and $1{\tilde{s}}_{1:n-1}$ are fed into the classical neural network $f$ with trainable weights $\mathit{\varphi }$. The expectation of $\widehat{H}$ can then be estimated according to Eq. (9). Finally, parameters in both PQC and the neural network are optimized with gradient based optimizer from the expectation result $⟨\widehat{H}⟩$. (b) Measurement protocol for $\widehat{H}=Y_2{Z}_3{X}_4$. The star qubit corresponds to $s_2$ and the $CX$ gate is applied to qubit 4 since $X_4$ is in $\widehat{H}$. Meanwhile, $CZ$ gate is omitted in the hardware level, and its effect is counted by the prefactor $s_3$ in the expression. (c) Measurement protocol for $\widehat{H}=X_1{X}_2{Y}_3$. The star qubit corresponds to $s_1$. $CX$ and $CY$ gates are applied on qubits 2 and 3, respectively, since there are $X_2$ and $Y_3$ in $\widehat{H}$.

Figure 2

Optimized energies on the five-site TFIM model with open boundary condition using either VQNHE (blue) and VQE (red) approaches. Measurement-based results in noiseless, noisy simulators, and real quantum hardware are shown with the error bar. The vertical red and blue lines are the ideal optimized energy values from the VQNHE and VQE, when no quantum noise or measurement uncertainty exists. The blue line also coincides with the exact ground state energy since the ideal VQNHE result has a relative error in the order of ${10}^{-12}$.

Figure 3

LiH dissociation curve. (a) VQNHE optimized energy (purple), VQE converged energy (green), exact energy from full configuration interaction (red), Hartree-Fock (HF) energy (yellow) obtained at different bond distances. (b) Comparison of corresponding energy errors for VQNHE and VQE results. The red dash line is the threshold of chemical accuracy. The cyan vertical line is the bond distance with minimum bonding energy, representing LiH molecule at equilibrium configuration.