Efficient Quantum Algorithms for State Measurement and Linear Algebra Applications

TL;DR

This paper introduces a quantum algorithm that efficiently measures local operators and applies to linear algebra problems like solving equations and evaluating thermal states, with logarithmic scaling in system size and accuracy.

Contribution

It presents a novel quantum measurement algorithm that combines state representation and operator decomposition, enabling efficient quantum simulations and linear algebra computations.

Findings

Algorithm scales logarithmically with system size and accuracy

Combines measurement with Hamiltonian evolution for simulation efficiency

Enables efficient evaluation of thermal expectation values

Abstract

We present an algorithm for measurement of $k$-local operators in a quantum state, which scales logarithmically both in the system size and the output accuracy. The key ingredients of the algorithm are a digital representation of the quantum state, and a decomposition of the measurement operator in a basis of operators with known discrete spectra. We then show how this algorithm can be combined with (a) Hamiltonian evolution to make quantum simulations efficient, (b) the Newton-Raphson method based solution of matrix inverse to efficiently solve linear simultaneous equations, and (c) Chebyshev expansion of matrix exponentials to efficiently evaluate thermal expectation values. The general strategy may be useful in solving many other linear algebra problems efficiently.