[Paper Review] Computing eigenvalues of matrices in a quantum computer

Changpeng Shao|arXiv (Cornell University)|Dec 17, 2019

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper proposes quantum algorithms to compute eigenvalues of diagonalizable matrices—specifically real-eigenvalue matrices and normal matrices—by combining quantum phase estimation, quantum singular value estimation, and quantum linear system solvers. The algorithms output a quantum state encoding eigenvalues and eigenvectors with complexity scaling polylogarithmically in matrix size under sparsity and condition number constraints.

ABSTRACT

Eigenproblem arises in a large number of disciplines of sciences and engineering. Combining quantum algorithms to solve linear differential equations, quantum singular value estimation and quantum phase estimation, we propose quantum algorithms to compute the eigenvalues of diagonalizable matrices of two types: matrices that only have real eigenvalues and normal matrices. The quantum algorithms return a quantum state such that the first register stores eigenvalues and the second register stores the corresponding eigenvectors. For normal matrices, the complexity to obtain this quantum state is dominated by the complexity to perform quantum singular value estimation. For the other case, the complexity is determined by the solving of a linear system of differential equations. Under certain conditions (e.g. sparse), the complexities are polylog on the size $n$ of the given matrix. If we perform measurements, then we can obtain all the eigenvalues classically. If $M$ is $s$ sparse, then the complexity is $\widetilde{O}(sn\|M\|_{\max}\kappa^2/\epsilon^2)$, where $\kappa$ is the condition number of the matrix of eigenvectors.

Motivation & Objective

• To develop efficient quantum algorithms for computing eigenvalues of diagonalizable matrices.
• To address the eigenproblem in quantum computing by leveraging quantum linear system solvers and phase estimation.
• To achieve low computational complexity for sparse matrices by exploiting quantum singular value estimation and differential equation solving.
• To enable classical extraction of all eigenvalues through measurement of the resulting quantum state.
• To establish complexity bounds dependent on sparsity, condition number, and precision.

Proposed method

• The algorithm uses quantum phase estimation to extract eigenvalues from the eigenvectors of the matrix.
• For normal matrices, quantum singular value estimation is the dominant component determining complexity.
• For real-eigenvalue matrices, the method relies on solving a system of linear differential equations using quantum algorithms.
• The output is a quantum state where the first register encodes eigenvalues and the second register encodes corresponding eigenvectors.
• The algorithm assumes the matrix is sparse and uses the condition number and maximum norm to bound error and runtime.
• Precision and accuracy are controlled via the parameter ε, with complexity scaling as Õ(sn∥M∥maxκ²/ε²).

Experimental results

Research questions

• RQ1 Can quantum algorithms efficiently compute eigenvalues of diagonalizable matrices with polylogarithmic complexity in matrix size?
• RQ2 How does the complexity of eigenvalue computation scale with sparsity and condition number in the quantum setting?
• RQ3 What is the role of quantum phase estimation and quantum singular value estimation in solving the matrix eigenproblem?
• RQ4 Can the quantum state encoding eigenvalues and eigenvectors be measured to classically extract all eigenvalues?
• RQ5 What are the runtime bounds for the proposed algorithms under realistic assumptions such as sparsity and bounded condition number?

Key findings

• The algorithm achieves a complexity of Õ(sn∥M∥maxκ²/ε²) for s-sparse matrices, where s is sparsity, ∥M∥max is the maximum absolute entry, κ is the condition number of the eigenvector matrix, and ε is the desired precision.
• For normal matrices, the complexity is dominated by quantum singular value estimation, enabling efficient eigenvalue computation.
• For real-eigenvalue matrices, the complexity is determined by solving a system of linear differential equations using quantum methods.
• The output is a quantum state that encodes eigenvalues in the first register and corresponding eigenvectors in the second register.
• Measurement of the output state allows classical retrieval of all eigenvalues.
• The complexity scales polylogarithmically in the matrix size n under the given assumptions, indicating a significant speedup over classical methods for sparse matrices.