Two-Unitary Decomposition Algorithm and Open Quantum System Simulation

TL;DR

This paper introduces a quantum two-unitary decomposition algorithm that efficiently simulates non-unitary operators in open quantum systems, reducing complexity and resource requirements compared to traditional methods.

Contribution

The proposed TUD algorithm decomposes non-unitary operators into two unitaries using quantum singular value transformation, avoiding costly classical SVD and enabling efficient implementation.

Findings

Reduces the complexity of simulating non-unitary operators.

Requires only a single call to the state preparation oracle per unitary.

Potential applications in quantum linear algebra and machine learning.

Abstract

Simulating general quantum processes that describe realistic interactions of quantum systems following a non-unitary evolution is challenging for conventional quantum computers that directly implement unitary gates. We analyze complexities for promising methods such as the Sz.-Nagy dilation and linear combination of unitaries that can simulate open systems by the probabilistic realization of non-unitary operators, requiring multiple calls to both the encoding and state preparation oracles. We propose a quantum two-unitary decomposition (TUD) algorithm to decompose a $d$-dimensional operator $A$ with non-zero singular values as $A=(U_1+U_2)/2$ using the quantum singular value transformation algorithm, avoiding classically expensive singular value decomposition (SVD) with an $O(d^3)$ overhead in time. The two unitaries can be deterministically implemented, thus requiring only a single call to the state preparation oracle for each. The calls to the encoding oracle can also be reduced significantly at the expense of an acceptable error in measurements. Since the TUD method can be used to implement non-unitary operators as only two unitaries, it also has potential applications in linear algebra and quantum machine learning.