Sparse quantum state preparation with improved Toffoli cost

TL;DR

This paper introduces an improved method for preparing sparse quantum states on $n$ qubits, significantly reducing Toffoli gate costs by optimizing the isometry implementation, especially for states with real coefficients.

Contribution

It presents a new efficient algorithm for constructing isometries that reduces the Toffoli gate count in sparse state preparation compared to previous methods.

Findings

Toffoli cost is roughly 2s for large n, improving over prior approaches.

Numerical benchmarks show the cost is closer to s for random states.

Optimizations are proposed for states with real coefficients, reducing overall complexity.

Abstract

The preparation of quantum states is one of the most fundamental tasks in quantum computing, and a key primitive in many quantum algorithms. Of particular interest to areas such as quantum simulation and linear-system solvers are sparse quantum states, which contain only a small number $s$ of non-zero computational basis states compared to a generic state. In this work, we present an approach that prepares $s$-sparse states on $n$ qubits, reducing the number of Toffoli gates required compared to prior art. We work in the established framework of first preparing a dense state on a $\lceil{\log(s)}\rceil$-qubit sub-register, and then mapping this state to the target state via an isometry, with the latter step dominating the cost of the full algorithm. The speed-up is achieved by designing an efficient algorithm for finding and implementing the isometry. The worst-case Toffoli cost of our isometry circuit, which may be viewed as a batched version of an approach by Malvetti et al., is essentially $2s$ for sufficiently large values of $n$, yielding roughly a $\log(s)/2$ improvement factor over the state-of-the-art. In numerical benchmarks on randomly chosen states, the cost is closer to $s$. With the improved isometry circuit, we examine the dense-state preparation step and present ways to optimize the joint cost of both steps, particularly in the case of target states with purely real coefficients, by outsourcing some sub-tasks from the dense-state preparation to the isometry.