Quantum expectation value estimation by doubling the number of qubits

TL;DR

This paper evaluates a qubit-doubling method for estimating expectation values of Hamiltonians in quantum chemistry, showing it reduces measurements for moderate precision, thus potentially improving efficiency in quantum algorithms.

Contribution

It assesses the efficiency of a two-copy Bell basis measurement method for expectation value estimation, demonstrating its advantages over traditional sampling in certain regimes.

Findings

Fewer measurements needed for tens of milli-Hartree precision

Effective for molecular Hamiltonians up to 12 qubits

Potentially useful for applications with moderate precision requirements

Abstract

Expectation value estimation is ubiquitous in quantum algorithms. The expectation value of a Hamiltonian, which is essential in various practical applications, is often estimated by measuring a large number of Pauli strings on quantum computers and performing classical post-processing. In the case of $n$-qubit molecular Hamiltonians in quantum chemistry calculations, it is necessary to evaluate $O(n^4)$ Pauli strings, requiring a large number of measurements for accurate estimation. To reduce the measurement cost, we assess an existing idea that uses two copies of an $n$-qubit quantum state of interest and coherently measures them in the Bell basis, which enables the simultaneous estimation of the absolute values of expectation values of all the $n$-qubit Pauli strings. We numerically investigate the efficiency of energy estimation for molecular Hamiltonians of up to 12 qubits. The results show that, when the target precision is no smaller than tens of milli-Hartree, this method requires fewer measurements than conventional sampling methods. This suggests that the method may be useful for many applications that rely on expectation value estimation of Hamiltonians and other observables as well when moderate precision is sufficient.