Efficient application of the factorized form of the unitary coupled-cluster ansatz for the variational quantum eigensolver algorithm by using linear combination of unitaries

TL;DR

This paper introduces a more efficient method for implementing the factorized unitary coupled-cluster ansatz in the variational quantum eigensolver, significantly reducing the complexity of applying high-rank operators on quantum computers.

Contribution

It exploits SU(2) symmetry and linear combination of unitaries to reduce the scaling complexity of applying rank-n operators from exponential to cubic in the variational quantum eigensolver.

Findings

Reduces application complexity of rank-n operators to O(n^3)

Uses n+2 ancilla qubits for preparing operators

Employs O(n) CNOT gates in the select scheme

Abstract

The variational quantum eigensolver is one of the most promising algorithms for near-term quantum computers. It has the potential to solve quantum chemistry problems involving strongly correlated electrons, which are otherwise difficult to solve on classical computers. The variational eigenstate is constructed from a number of factorized unitary coupled-cluster terms applied onto an initial (single-reference) state. Current algorithms for applying one of these operators to a quantum state require a number of operations that scales exponentially with the rank of the operator. We exploit a hidden SU($2$) symmetry to allow us to employ the linear combination of unitaries approach, Our \textsc{Prepare} subroutine uses $n+2$ ancilla qubits for a rank-$n$ operator. Our \textsc{Select}($\hat U$) scheme uses $\mathcal{O}(n)$ \textsc{Cnot} gates. This results in an full algorithm that scales like the cube of the rank of the operator $n^3$, a significant reduction in complexity for rank five or higher operators. This approach, when combined with other algorithms for lower-rank operators (when compared to the standard implementation, will make the factorized form of the unitary coupled-cluster approach much more efficient to implement on all types of quantum computers.