Generalized Quantum Singular Value Transformation

TL;DR

This paper introduces two generalizations of quantum singular value transformation that enable the use of complex polynomials and indefinite parity, broadening the applicability and efficiency of quantum algorithms involving matrix transformations.

Contribution

It proposes the generalized quantum singular value and eigenvalue transformations, allowing complex polynomials and indefinite parity, with new techniques for block encoding and matrix multiplication.

Findings

Higher expressivity of polynomials can lead to advantages in quantum algorithms.

Faster computation of phase factors is possible for non-unitary matrices.

New block encoding techniques reduce circuit complexity and resource requirements.

Abstract

The quantum singular value transformation has revolutionised quantum algorithms. By applying a polynomial to an arbitrary matrix, it provides a unifying picture of quantum algorithms. However, polynomials are restricted to definite parity and real coefficients, and finding the circuit (the phase factors) has proven difficult in practice. Recent work has removed these restrictions and enabled faster computation of phase factors, yet only for unitary matrices. Here we propose two generalisations. The generalised quantum singular value transformation allows complex polynomials for arbitrary matrices. For Hermitian matrices, we propose the generalised quantum eigenvalue transformation that even allows polynomials of indefinite parity. While we find that the polynomial might have to be downscaled compared to the quantum singular value transformation, the higher expressivity of polynomials and faster computation of phase factors can sometimes result in advantages. The results are achieved with various block encoding (or projected unitary encoding) techniques, including qubitisation, Hermitianisation, and multiplication. We show how to multiply block-encoded matrices with only one extra qubit, and introduce measure-early multiplication to further avoid the extra qubit and decrease average circuit length.