Quantum techniques for eigenvalue problems

Changpeng Shao, Jin-Peng Liu

Many eigenvalue problems arising in practice are often of the generalized form A x = λ B x . One particularly important case is symmetric, namely A , B are Hermitian and B is positive definite. The standard algorithm for solving this class of eigenvalue problems is to reduce them to Hermitian eigenvalue problems. For a quantum computer, quantum phase estimation is a useful technique to solve Hermitian eigenvalue problems. In this work, we propose a new quantum algorithm for symmetric generalized eigenvalue problems using ordinary differential equations. The algorithm has lower complexity than the standard one based on quantum phase estimation. Moreover, it works for a wider case than symmetric: B is invertible, B − 1 A is diagonalizable and all the eigenvalues are real.

Jin‐Min Liang, Shu‐Qian Shen, Ming Li +1 more

Selman Hershfield

Jeffrey B. Parker, Ilon Joseph

Quantum phase estimation provides a path to quantum computation of solutions to Hermitian eigenvalue problems $Hv=\ensuremath{\lambda}v$, such as those occurring in quantum chemistry. It is natural to ask whether the same technique can be applied to generalized eigenvalue problems $Av=\ensuremath{\lambda}Bv$, which arise in many areas of science and engineering. We answer this question affirmatively. A restricted class of generalized eigenvalue problems could be solved as efficiently as standard eigenvalue problems. A paradigmatic example is provided by Sturm-Liouville problems. Another example comes from linear ideal magnetohydrodynamics, where phase estimation could be used to determine the stability of magnetically confined plasmas in fusion reactors.

Laura Clinton, Johannes Bausch, Joel Klassen +1 more

Abstract In this work we investigate a binned version of quantum phase estimation (QPE) set out by Somma (2019 New J. Phys. 21 123025) and known as the quantum eigenvalue estimation problem (QEEP). Specifically, we determine whether the circuit decomposition techniques we set out in previous work, Clinton et al (2021 Nat. Commun. 12 1–10), can improve the performance of QEEP in the noisy intermediate scale quantum (NISQ) regime. To this end we adopt a physically motivated abstraction of NISQ device capabilities as in Clinton et al (2021 Nat. Commun. 12 1–10). Within this framework, we find that our techniques reduce the threshold at which it becomes possible to perform the minimum two-bin instance of this algorithm by an order of magnitude. This is for the specific example of a two dimensional spin Fermi-Hubbard model. For example, we estimate that the depolarizing single qubit error rate required to implement a minimum two bin example of QEEP—with a 5 × 5 Fermi-Hubbard model and up to a precision of 10 % —can be reduced from 10−7 to 10−5. We explore possible modifications to this protocol and propose an application, which we dub randomized quantum eigenvalue estimation problem (rQEEP). rQEEP outputs estimates on the fraction of eigenvalues which lie within randomly chosen bins and upper bounds the total deviation of these estimates from the true values. One use case we envision for this algorithm is resolving density of states features of local Hamiltonians.

Martin Wilner