Quantum eigenvalue processing

• URL: http://arxiv.org/abs/2401.06240v2
• Date: Fri, 25 Oct 2024 10:13:11 GMT
• Title: Quantum eigenvalue processing
• Authors: Guang Hao Low, Yuan Su,
• Abstract summary: Problems in linear algebra can be solved on a quantum computer by processing eigenvalues of the non-normal input matrices. We present a Quantum EigenValue Transformation (QEVT) framework for applying arbitrary transformations on eigenvalues of block-encoded non-normal operators. We also present a Quantum EigenValue Estimation (QEVE) algorithm for operators with real spectra.
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• Abstract: Many problems in linear algebra -- such as those arising from non-Hermitian physics and differential equations -- can be solved on a quantum computer by processing eigenvalues of the non-normal input matrices. However, the existing Quantum Singular Value Transformation (QSVT) framework is ill-suited to this task, as eigenvalues and singular values are different in general. We present a Quantum EigenValue Transformation (QEVT) framework for applying arbitrary polynomial transformations on eigenvalues of block-encoded non-normal operators, and a related Quantum EigenValue Estimation (QEVE) algorithm for operators with real spectra. QEVT has query complexity to the block encoding nearly recovering that of the QSVT for a Hermitian input, and QEVE achieves the Heisenberg-limited scaling for diagonalizable input matrices. As applications, we develop a linear differential equation solver with strictly linear time query complexity for average-case diagonalizable operators, as well as a ground state preparation algorithm that upgrades previous nearly optimal results for Hermitian Hamiltonians to diagonalizable matrices with real spectra. Underpinning our algorithms is an efficient method to prepare a quantum superposition of Faber polynomials, which generalize the nearly-best uniform approximation properties of Chebyshev polynomials to the complex plane. Of independent interest, we also develop techniques to generate $n$ Fourier coefficients with $\mathbf{O}(\mathrm{polylog}(n))$ gates compared to prior approaches with linear cost.

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