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.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.

arXiv:2002.08497Physical Review AAug 25, 202038 citations

Quantum Phase Estimation

Hiu Yung Wong

Springer eBooksJan 1, 20226 citations

Bayesian phase difference estimation: a general quantum algorithm for the direct calculation of energy gaps

Kenji Sugisaki, Chikako Sakai, Kazuo Toyota +3 more

Quantum computers can perform full configuration interaction (full-CI) calculations by utilising the quantum phase estimation (QPE) algorithms including Bayesian phase estimation (BPE) and iterative quantum phase estimation (IQPE). In these quantum algorithms, the time evolution of wave functions for atoms and molecules is simulated conditionally with an ancillary qubit as the control, which make implementation to real quantum devices difficult. Also, most of the problems in chemistry discuss energy differences between two electronic states rather than total energies themselves, and thus direct calculations of energy gaps are promising for future applications of quantum computers to real chemistry problems. In the race of finding efficient quantum algorithms to solve quantum chemistry problems, we test a Bayesian phase difference estimation (BPDE) algorithm, which is a general algorithm to calculate the difference of two eigenphases of unitary operators in the several cases of the direct calculations of energy gaps between two electronic states on quantum computers, including vertical ionisation energies, singlet-triplet energy gaps, and vertical excitation energies. In the BPDE algorithm, state preparation is carried out conditionally on the ancillary qubit, and the time evolution of the wave functions in superposition of two electronic states are executed unconditionally. Based on our test, we conclude that BPDE is capable of computing the energy gap with an accuracy similar to BPE without controlled-time evolution simulations and with the smaller number of iterations in Bayesian optimisations.Quantum computers can perform full configuration interaction (full-CI) calculations by utilising the quantum phase estimation (QPE) algorithms including Bayesian phase estimation (BPE) and iterative quantum phase estimation (IQPE). In these quantum algorithms, the time evolution of wave functions for atoms and molecules is simulated conditionally with an ancillary qubit as the control, which make implementation to real quantum devices difficult. Also, most of the problems in chemistry discuss energy differences between two electronic states rather than total energies themselves, and thus direct calculations of energy gaps are promising for future applications of quantum computers to real chemistry problems. In the race of finding efficient quantum algorithms to solve quantum chemistry problems, we test a Bayesian phase difference estimation (BPDE) algorithm, which is a general algorithm to calculate the difference of two eigenphases of unitary operators in the several cases of the direct calculations of energy gaps between two electronic states on quantum computers, including vertical ionisation energies, singlet-triplet energy gaps, and vertical excitation energies. In the BPDE algorithm, state preparation is carried out conditionally on the ancillary qubit, and the time evolution of the wave functions in superposition of two electronic states are executed unconditionally. Based on our test, we conclude that BPDE is capable of computing the energy gap with an accuracy similar to BPE without controlled-time evolution simulations and with the smaller number of iterations in Bayesian optimisations.

Physical Chemistry Chemical PhysicsJan 1, 202129 citations

Quantum Phase Estimation

Hiu Yung Wong

Springer eBooksJun 21, 20230 citations

Quantum Phase Estimation

Osama M. Raisuddin, Suvranu De

Springer eBooksNov 14, 20250 citations

Accelerated Variational Quantum Eigensolver

Daochen Wang, Oscar Higgott, STEPHEN D. BRIERLEY

The problem of finding the ground state energy of a Hamiltonian using a quantum computer is currently solved using either the quantum phase estimation (QPE) or variational quantum eigensolver (VQE) algorithms. For precision $\ensuremath{\epsilon}$, QPE requires $O(1)$ repetitions of circuits with depth $O(1/\ensuremath{\epsilon})$, whereas each expectation estimation subroutine within VQE requires $O(1/{\ensuremath{\epsilon}}^{2})$ samples from circuits with depth $O(1)$ . We propose a generalized VQE algorithm that interpolates between these two regimes via a free parameter $\ensuremath{\alpha}\ensuremath{\in}[0,1]$, which can exploit quantum coherence over a circuit depth of $O(1/{\ensuremath{\epsilon}}^{\ensuremath{\alpha}})$ to reduce the number of samples to $O(1/{\ensuremath{\epsilon}}^{2(1\ensuremath{-}\ensuremath{\alpha})})$. Along the way, we give a new routine for expectation estimation under limited quantum resources that is of independent interest.The problem of finding the ground state energy of a Hamiltonian using a quantum computer is currently solved using either the quantum phase estimation (QPE) or variational quantum eigensolver (VQE) algorithms. For precision $\ensuremath{\epsilon}$, QPE requires $O(1)$ repetitions of circuits with depth $O(1/\ensuremath{\epsilon})$, whereas each expectation estimation subroutine within VQE requires $O(1/{\ensuremath{\epsilon}}^{2})$ samples from circuits with depth $O(1)$. We propose a generalized VQE algorithm that interpolates between these two regimes via a free parameter $\ensuremath{\alpha}\ensuremath{\in}[0,1]$, which can exploit quantum coherence over a circuit depth of $O(1/{\ensuremath{\epsilon}}^{\ensuremath{\alpha}})$ to reduce the number of samples to $O(1/{\ensuremath{\epsilon}}^{2(1\ensuremath{-}\ensuremath{\alpha})})$. Along the way, we give a new routine for expectation estimation under limited quantum resources that is of independent interest.

arXiv:1802.00171Physical Review LettersApr 12, 2019241 citations

O(1)

repetitions of circuits with depth $O(1/\ensuremath{\epsilon})$, whereas each expectation estimation subroutine within VQE requires $O(1/{\ensuremath{\epsilon}}^{2})$ samples from circuits with depth

O(1)

. We propose a generalized VQE algorithm that interpolates between these two regimes via a free parameter $\ensuremath{\alpha}\ensuremath{\in}[0,1]$, which can exploit quantum coherence over a circuit depth of $O(1/{\ensuremath{\epsilon}}^{\ensuremath{\alpha}})$ to reduce the number of samples to $O(1/{\ensuremath{\epsilon}}^{2(1\ensuremath{-}\ensuremath{\alpha})})$. Along the way, we give a new routine for expectation estimation under limited quantum resources that is of independent interest.