Quantum eigenvalue processing (QIP 2024)


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.

Jan 18, 2024
Taipei International Convention Center, No. 1, Section 5, Xinyi Road, Taipei City 11049, Taiwan
Yuan Su
Yuan Su
Senior Researcher

I work on quantum algorithms for simulating Hamiltonian dynamics. I am particularly interested in the design, analysis, implementation, and application of quantum simulation.