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IEEE Conference on Electrical Performance of Electronic Packaging and Systems (EPEPS)

On the Phase Estimation Amplitudes in the Quantum Matrix Equation Solver

Xinbo Li, Christopher Phillips, Ian Jeffrey, Vladimir Okhmatovski

Published Oct 15, 2023• 0 citations
Publisher

Abstract

The phase estimation amplitude in the quantum matrix equation solver, namely the Harrow/Hassidim/Lloyd (HHL) algorithm is studied in this work. It is the periodic behavior of the amplitude with respect to the eigenvalue estimation error indicates the choice of the simulation time. The impact of the initial states on the quantum phase estimation (QPE) is also provided. This work is expected to be instructive for the formulation of the error analysis in HHL variants.

DOI
10.1109/epeps58208.2023.10314918

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The phase estimation amplitude in the quantum matrix equation solver, namely the Harrow/Hassidim/Lloyd (HHL) algorithm is studied in this work. It is the periodic behavior of the amplitude with respect to the eigenvalue estimation error indicates the choice of the simulation time. The impact of the initial states on the quantum phase estimation (QPE) is also provided. This work is expected to be instructive for the formulation of the error analysis in HHL variants.
We analyze the performance of the Harrow-Hassidim-Lloyd algorithm (HHL algorithm) for solving linear problems and of a variant of this algorithm (HHL variant) commonly encountered in literature. This variant relieves the algorithm of preparing an entangled initial state of an auxiliary register. We prove that the computational error of the variant algorithm does not always converge to zero when the number of qubits is increased, unlike the original HHL algorithm. Both algorithms rely upon two fundamental quantum algorithms, the quantum phase estimation and the amplitude amplification. In particular, the error of the HHL variant oscillates due to the presence of undesired phases in the amplitude to be amplified, while these oscillations are suppressed in the original HHL algorithm. Then, we propose a modification of the HHL variant, by amplifying an amplitude of the state vector that does not exhibit the above destructive interference. We also study the complexity of these algorithms in the light of recent results on simulation of unitaries used in the quantum phase estimation step, and show that the modified algorithm has smaller error and lower complexity than the HHL variant. We supported our findings with numerical simulations.
The Harrow/Hassidim/Lloyd algorithm is a celebrated quantum matrix equation solver. Its Hamiltonian simulation involves a quantum walk process. A newly developed Quantum Walk Unitary HHL (QWU-HHL) leverages the spectral relationship between the quantum walk operator and the system matrix, and uses the quantum walk operator as its unitary in the phase estimation directly, allowing Hamiltonian simulation in the classical HHL to be removed, hence improving its efficiency. Despite the potential of being exponentially faster than classical matrix equation solvers, HHL is not feasible in the noisy intermediate-scale quantum era because its quantum circuit is too deep to preserve an accurate solution. In this work, we investigate the error behavior of QWU-HHL on a 7-qubit quantum system. Instead of executing the entire circuit, the behaviors of its sub-circuits which each contain only an initialization and a single gate are recorded. The error caused by isolated initialization and operation are analyzed.
The Harrow-Hassidim-Lloyd (HHL) quantum algorithm for sampling from the solution of a linear system provides an exponential speed-up over its classical counterpart. The problem of solving a system of linear equations has a wide scope of applications, and thus HHL constitutes an important algorithmic primitive. In these notes, we present the HHL algorithm and its improved versions in detail, including explanations of the constituent sub- routines. More specifically, we discuss various quantum subroutines such as quantum phase estimation and amplitude amplification, as well as the important question of loading data into a quantum computer, via quantum RAM. The improvements to the original algorithm exploit variable-time amplitude amplification as well as a method for implementing linear combinations of unitary operations (LCUs) based on a decomposition of the operators using Fourier and Chebyshev series. Finally, we discuss a linear solver based on the quantum singular value estimation (QSVE) subroutine.
Quantum computing is certainly one of the greatest advances in the computer science field. Thanks to the parallelism and entanglement properties, it has proved to offer several advantages compared to the classical algorithms especially in the great reduction of the processing time. Quantum phase estimation (QPE) is one of the most important algorithms for quantum computing. It is known as the eigenvalue finding module for unitary operators. The Fourier transform is the key to this procedure. It has been researched and used to solve many problems such as the order finding problem, and the factoring problem. It was also applied for quantum sampling algorithms and the calculation of the eigenvalues of unitary matrices. In this paper, we study three important quantum algorithms for machine learning that use the QPE algorithm as a subroutine: the quantum principal components analysis (PCA) for data visualization, the Harrow-Hassidim-Lloyd (HHL) algorithm for solving linear systems, and the quantum singular value thresholding (SVT) for matrix completion in recommender systems. We also discuss the advantages and limits of such algorithms compared to their classical versions. Then we discuss potential ways of amelioration of such algorithms, and end with a proposed approach for further improvement.
We present a classical enhancement to improve the accuracy of the Hybrid variant (Hybrid HHL) of the quantum algorithm for solving linear systems of equations proposed by Harrow, Hassidim, and Lloyd (HHL). We achieve this by using higher precision quantum estimates of the eigenvalues relevant to the linear system, and a new classical step to guide the eigenvalue inversion part of Hybrid HHL. We show that eigenvalue estimates with just two extra bits of precision result in tighter error bounds for our Enhanced Hybrid HHL compared to HHL. Our enhancement reduces the error of Hybrid HHL by an average of 57 percent on an ideal quantum processor for a representative sample of 2x2 systems. On IBM Torino and IonQ Aria-1 hardware, we see that the error of Enhanced Hybrid HHL is on average 13 percent and 20 percent (respectively) less than that of HHL for the same set of systems.