Quantum eigenvalue processing

TL;DR

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.

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.

Figures (3)

• Figure 1: A diagrammatic illustration of quantum eigenvalue processing and its applications. See \ref{['tab:nonnormal']} for a summary of common treatments of non-normality of the input matrix and query complexity.
• Figure 2: Illustration of eigenvalue enclosing regions and the corresponding nearly best uniform polynomial approximations. Subfigure (a) represents the real interval $[-1,1]$, for which Chebyshev expansion provides a nearly best approximation. Subfigure (b) represents the unit disk, for which Taylor expansion provides a nearly best approximation. Subfigures (c) and (d) represent more general regions in the complex plane, for which Faber expansion provides a nearly best approximation. Subfigure (c) shows a unit semidisk on the left half-plane, with Faber expansion generated by the Elliott's conformal map Coleman1987FaberCircularSectors, whereas Subfigure (d) is a smooth deformation of (c).
• Figure 3: Illustration of the unit disk $\mathcal{D}$, the target region $\mathcal{E}$, and the exterior Riemann mappings $\mathbf{\Psi}$, $\mathbf{\Phi}$ associated with the definition of Faber polynomials.

Theorems & Definitions (85)

• Lemma 1: Spectral norm bound for block matrices
• proof
• Proposition 2: Chebyshev expansion of exponential function Low2016HamSim
• Proposition 3: Chebyshev expansion of error function Low17Wan22
• Lemma 4: Riesz inequality king_2009
• Lemma 5: Carleson-Hunt theorem Reyna2002 and TaoBlog
• Proposition 6: Eigendecomposition
• Proposition 7: Jordan form decomposition
• Proposition 8: Spectral decomposition
• Proposition 9: Singular value decomposition