Improving Hamiltonian encodings with the Gray code

TL;DR

This paper introduces a Gray code-based encoding for quantum Hamiltonian simulation that improves resource efficiency, reduces circuit depth, and decreases variance in VQE solutions compared to traditional methods, enabling larger system simulations on current hardware.

Contribution

The paper presents a novel Gray code encoding for Hamiltonians that enhances quantum resource utilization and performance over standard encodings in variational algorithms.

Findings

Smaller variance in VQE energy estimates with Gray code encoding

Reduced circuit depth and gate count for time evolution simulations

Enables larger problem sizes on current quantum hardware

Abstract

Due to the limitations of present-day quantum hardware, it is especially critical to design algorithms that make the best possible use of available resources. When simulating quantum many-body systems on a quantum computer, straightforward encodings that transform many-body Hamiltonians into qubit Hamiltonians use $N$ of the available basis states of an $N$-qubit system, whereas $2^N$ are in theory available. We explore an efficient encoding that uses the entire set of basis states, where terms in the Hamiltonian are mapped to qubit operators with a Hamiltonian that acts on the basis states in Gray code order. This encoding is applied to the commonly-studied problem of finding the ground state energy of a deuteron with a simulated variational quantum eigensolver (VQE). It is compared to a standard "one-hot" encoding, and various trade-offs that arise are analyzed. The energy distribution of VQE solutions has smaller variance than the one obtained by the one-hot encoding even in the presence of simulated hardware noise, despite an increase in the number of measurements. The reduced number of qubits and a shorter-depth variational ansatz enables the encoding of larger problems on current-generation machines. This encoding also demonstrates improvements for simulating time evolution of the same system, producing circuits for the evolution operators with reduced depth and roughly half the number of gates compared to a one-hot encoding.