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arXiv:2109.14114 · Physical Review A

Block Lanczos method for excited states on a quantum computer

Thomas E. Baker

Published Jul 8, 2024• 6 citations
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Abstract

The method of quantum Lanczos recursion is extended to solve for multiple excitations on the quantum computer. While quantum Lanczos recursion is, in principle, capable of obtaining excitations, the extension to a block Lanczos routine can resolve degeneracies with better precision and only costs O ( d 2 ) for d excitations on top of the previously introduced quantum Lanczos recursion method. Extension to non-Hermitian operators is also discussed. Published by the American Physical Society 2024

DOI
10.1103/physreva.110.012420

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The method of quantum Lanczos recursion is extended to solve for multiple excitations on the quantum computer. While quantum Lanczos recursion is, in principle, capable of obtaining excitations, the extension to a block Lanczos routine can resolve degeneracies with better precision and only costs O ( d 2 ) for d excitations on top of the previously introduced quantum Lanczos recursion method. Extension to non-Hermitian operators is also discussed. Published by the American Physical Society 2024
A state-preserving quantum counting algorithm is used to obtain coefficients of a Lanczos recursion from a single ground-state wave function on the quantum computer. This is used to compute the continued fraction representation of an interacting Green's function for use in condensed matter, particle physics, and other areas. The wave function does not need to be reprepared at each iteration. The quantum algorithm represents an exponential reduction in memory over known classical methods. An extension of the method to determining the ground state is also discussed.
The recursion method, which solves coupled Heisenberg equations in a Lanczos operator basis, has recently emerged as a powerful nonperturbative tool for computing dynamical correlation functions in strongly correlated two- and three-dimensional quantum many-body systems. Motivated by this success, we investigate whether the method can be extended to expectation values of observables following a quantum quench. We find that such an extension encounters an obstacle absent in the computation of dynamical correlation functions. The latter are fully determined by the Lanczos coefficients b_n b n , which in generic systems exhibit universal behavior, enabling reliable extrapolation from the first few dozens of explicitly computed coefficients. In contrast, quench dynamics additionally requires “quench coefficients” c_n c n , defined as overlaps of Lanczos basis operators with the initial state. We show that, unlike the Lanczos coefficients, the quench coefficients exhibit no universal structure and cannot be reliably extrapolated, thereby limiting the time up to which the method yields accurate results. The behavior of quench coefficients is highly state-dependent, ranging from decaying to irregular or even growing sequences; typically, the less regular the sequence c_n c n , the shorter the accessible timescale. Nevertheless, for favorable initial states, the method remains competitive with state-of-the-art approaches. Moreover, its symbolic implementation allows a single computation to be reused across different Hamiltonian parameters and initial states, making it particularly advantageous in studies requiring extensive scans over Hamiltonian parameters or initial states.
We present an algorithm that uses block encoding on a quantum computer to exactly construct a Krylov space, which can be used as the basis for the Lanczos method to estimate extremal eigenvalues of Hamiltonians. While the classical Lanczos method has exponential cost in the system size to represent the Krylov states for quantum systems, our efficient quantum algorithm achieves this in polynomial time and memory. The construction presented is exact in the sense that the resulting Krylov space is identical to that of the Lanczos method, so the only approximation with respect to the exact method is due to finite sample noise. This is possible because, unlike previous quantum Krylov methods, our algorithm does not require simulating real or imaginary time evolution. We provide an explicit error bound for the resulting ground state energy estimate in the presence of noise. For our method to be successful efficiently, the only requirement on the input problem is that the overlap of the initial state with the true ground state must be Ω ( 1 / poly ( n ) ) for n qubits.
Quantum subspace-diagonalization methods, particularly Quantum Krylov Diagonalization (QKD), provide a promising route for computing low-energy spectra of quantum many-body Hamiltonians. However, existing quantum Krylov approaches rely on non-orthogonal Krylov bases, requiring overlap-matrix regularization that limits numerical stability and accuracy. In this work, we introduce an Orthogonal Quantum Krylov Diagonalization (OQKD) framework that reformulates the classical Lanczos recursion at the operator level, enabling an orthogonal quantum implementation of Krylov-subspace diagonalization. By expressing Lanczos vectors as polynomial transformations of the Hamiltonian, OQKD reproduces the orthogonality, tridiagonal structure, and convergence behavior of the classical Lanczos algorithm thus eliminating the need for overlap-matrix regularization. We further show that the required Lanczos polynomials can be implemented using block encoding and Generalized Quantum Signal Processing with the same asymptotic query complexity as Chebyshev-based QKD methods. Numerical simulations of the J1J_1J1​--J2J_2J2​ Heisenberg model confirm the classical Lanczos convergence and numerical stability of the proposed method, while the measurement-complexity scaling is established analytically. Building upon the OQKD framework, we then introduce a restarted state-preparation protocol that replaces a single high-degree polynomial transformation with a sequence of fixed low-degree transformations, maintaining an affordable block encoding success probability while retaining comparable convergence. These results establish OQKD as an orthogonal quantum analog of the classical Lanczos algorithm and identify the restarted protocol as a promising state-preparation strategy for Quantum Phase Estimation.
We provide a very efficient procedure for obtaining the excited states of a quantum operator H, in any arbitrary chosen energy range, independently from the knowledge of the states at lower energies. Our procedure consists in determining, within the Lanczos algorithm, the ground state of the auxiliary operator A=(H-Et{\mathit{E}}_{\mathit{t}}Et​)2{)}^{2})2, and hence the eigenvalue of H nearest in energy to Et{\mathit{E}}_{\mathit{t}}Et​, where Et{\mathit{E}}_{\mathit{t}}Et​ is any chosen trial energy in the energy range of interest. We show that a variational method exploiting diagonalization of 2\ifmmode\times\else\texttimes\fi{}2 Lanczos matrices, combined with a two-pass Lanczos procedure of relatively small number of iterations, produces eigenvalues of H within any desired accuracy, as well as the corresponding eigenfunctions. We discuss in particular the T\ensuremath{\bigotimes}\ensuremath{\tau} Jahn-Teller model, as a realistic prototype of a quantum system with a very large number of degrees of freedom.
The recursion method of Haydock, Heine and Kelly is a powerful tool for calculating diagonal matrix elements of the resolvent of quantum-mechanical Hamiltonian operators by elegantly expressing them in terms of continued fractions. In this paper we extend the recursion method to off-diagonal matrix elements of general (possibly non-Hermitian) operators and apply it to the simulation of molecular optical absorption and photoemission spectra within time-dependent density-functional and many-body perturbation theories, respectively. This method is demonstrated with a couple of applications to the optical absorption and photoemission spectra of the caffeine molecule.