A scalable quantum-neural hybrid variational algorithm for ground state estimation

TL;DR

The unitary variational quantum-neural hybrid eigensolver (U-VQNHE), which improves upon the original VQNHE by enforcing unitary neural transformations, and retains improved accuracy and stability over standard variational quantum eigensolvers.

Abstract

We propose the unitary variational quantum-neural hybrid eigensolver (U-VQNHE), which improves upon the original VQNHE by enforcing unitary neural transformations. The non-unitary nature of VQNHE causes normalization issues and divergence of the loss function during training, leading to exponential scaling of measurement overhead with qubit number. U-VQNHE resolves these issues, significantly reduces required measurements, and retains improved accuracy and stability over standard variational quantum eigensolvers.

Figures (4)

• Figure 1: Flowchart of the VQNHE algorithm. The round box indicates the quantum part of the algorithm. The parameters $\theta$ of the quantum circuit are trained before the parameters $\phi$ of the neural network, whose process is not shown in the figure.
• Figure 2: VQNHE implementation of a 7-site TFIM with 7 qubits. (a) Training of the neural network in VQNHE. The vertical axis shows the loss function—the expectation value of the Hamiltonian—on a log scale. The quantum circuit simulation uses the Qiskit sampler with 500 shots per circuit. The inset highlights the region between the lowest energy from the bare VQE (i.e., VQE without a neural network) and the exact ground state energy; values below this are invalid. (b) Output of the trained neural network after 200 epochs. The horizontal axis shows bit strings (in decimal), and the vertical axis shows neural network outputs. Blue lines mark bit strings that were sampled (observed) from the ansatz, and red lines mark bit strings that were never observed. Dots mark the network output values. While most values lie near $10^{-6}$ (shown by black dots), extreme values (above $10^{-2}$, shown in red) appear. The neural network assigns extremely large output values to some of those unobserved (red) strings, effectively contributing only to the numerator of the loss (since they are absent in the denominator).
• Figure 3: VQNHE results for different numbers of shots, where all bit strings are measured from the ansatz circuit. Yellow markers represent individual VQNHE trials with shots ranging from 500 to 50,000. The parameters of the PQC are fixed throughout. Ideally, values should fall between the two dotted lines, the upper of which represents the exact VQE value and the lower represents exact ground state energy. However, without large number of shots, VQNHE significantly deviates from this appropriate range. Error bars indicate the standard deviation, $\sigma([\langle \hat{H} \rangle - \langle \hat{H} \rangle_m])$, computed using $p_m(s;P)$ and $p_a(s)$ from the exact VQNHE and the neural network values, averaged over trials at each shot count.
• Figure 4: Comparison between the training performance of VQNHE and U-VQNHE. (a) Result for a 5-site TFIM using a single-layer hardware-efficient ansatz and 100 shots. (b) Result for a 12-site TFIM using a two-layer ansatz and 5000 shots. In both cases, the region between the exact ground state energy and the VQE result is shown in white, while all other regions are shaded in gray. Solid lines represent results from shot-based simulators, and dashed lines are from statevector simulators. (c) U-VQNHE results of 5-site TFIM for different numbers of shots compared to VQNHE. Unlike VQNHE, U-VQNHE does not fall below the exact ground state by a significant amount, showing more stable results even with a small number of shots. The number of shots used was greater than $2^n$ to highlight this stability.