Quantum circuit design for solving linear systems of equations

Abstract

Recently, it is shown that quantum computers can be used for obtaining certain information about the solution of a linear system A x = b exponentially faster than what is possible with classical computation. Here we first review some key aspects of the algorithm from the standpoint of finding its efficient quantum circuit implementation using only elementary quantum operations, which is important for determining the potential usefulness of the algorithm in practical settings. Then we present a small-scale quantum circuit that solves a 2 × 2 linear system. The quantum circuit uses only 4 qubits, implying a tempting possibility for experimental realization. Furthermore, the circuit is numerically simulated and its performance under different circuit parameter settings is demonstrated.

A conceptual framework for quantum accelerated automated design optimization

“…The interested reader is referred to the QLSA primer by Dervovic et al [25] for a description of the different variants of the algorithm and to the research article by Cao et al [32] for a possible quantum circuit design for the QLSA.…”

Quantum circuit design methodology for multiple linear regression

“…The inverse of the Hermitian matrix can be written as , and hence matches . This outcome state, in the standard basis, is component‐wise proportional to the exact solution of the system [17].…”

“…In Section 4, we present an application (in the context of multiple regression) of a modified version of the earlier circuit design by Cao et al [17] which was meant for implementing the HHL algorithm for a linear system on real quantum computers. This circuit requires only 7 qubits and it should be simple enough to experimentally verify it if one gets access to a quantum processor having logic gates with sufficiently low error rates.…”

Bayesian deep learning on a quantum computer

“…with the problem-specific circuit in Ref. [49]. This circuit is much shallower than the full protocol detailed in Ref.…”

“…The reason of choosing the restricted algorithm is that current quantum computers have a small number of qubits, limited qubit-qubit connectivity, and most importantly, short coherence times, which implies that only shallow quantum circuits can be implemented. The restricted algorithm be implemented with a much simpler circuit than the general one, resulting in about 20 gates for the full protocol [49].…”