Qubit-Efficient Randomized Quantum Algorithms for Linear Algebra

Abstract

We propose a class of randomized quantum algorithms for the task of sampling from matrix functions, without the use of quantum block encodings or any other coherent oracle access to the matrix elements. As such, our use of qubits is purely algorithmic, and no additional qubits are required for quantum data structures. For N × N Hermitian matrices, the space cost is log(N ) + 1 qubits and depending on the structure of the matrices, the gate complexity can be comparable to state-of-the-art methods that use quantum data structures of up to size O(N 2 ), when considering equivalent end-to-end problems. Within our framework, we present a quantum linear system solver that allows one to sample properties of the solution vector, as well as an algorithm for sampling properties of ground states of Hamiltonians. As a concrete application, we combine these two sub-routines to present a scheme for calculating Green's functions of quantum many-body systems.

Computing Ground State Properties with Early Fault-Tolerant Quantum Computers

“…trading time resources for space resources). All of these features make our algorithm a great candidate for realizing quantum advantage for industrially-relevant problems on early fault-tolerant quantum computers [33,25,15,14,39,47,40,48].…”

A parallel quantum eigensolver for quantum machine learning

“…Some variational quantum algorithms (VQAs) are also proposed to explore quantum advantages on Noisy Intermediate-Scale Quantum (NISQ) devices, such as variational quantum eigensolver and variational quantum linear solver [15,[31][32][33][34][35][36][37]. There are also some works that utilize linear combination of unitaries [38] or block encodings to solve linear algebra problems [39][40][41]. Recently, quantum resonance transition (QRT) that merely needs two ancillary qubits and achieves squared acceleration over QPE has been proposed to solve the eigenproblem of physical systems [42].…”