Variational Phase Estimation with Variational Fast Forwarding

Maria-Andreea Filip1,2, David Muñoz Ramo1, and Nathan Fitzpatrick1

1Quantinuum, 13-15 Hills Road, CB2 1NL, Cambridge, United Kingdom
2Yusuf Hamied Department of Chemistry, University of Cambridge, Cambridge, United Kingdom

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Subspace diagonalisation methods have appeared recently as promising means to access the ground state and some excited states of molecular Hamiltonians by classically diagonalising small matrices, whose elements can be efficiently obtained by a quantum computer. The recently proposed Variational Quantum Phase Estimation (VQPE) algorithm uses a basis of real time-evolved states, for which the energy eigenvalues can be obtained directly from the unitary matrix $U=e^{-iH{\Delta}t}$, which can be computed with cost linear in the number of states used. In this paper, we report a circuit-based implementation of VQPE for arbitrary molecular systems and assess its performance and costs for the $H_2$, $H_3^+$ and $H_6$ molecules. We also propose using Variational Fast Forwarding (VFF) to decrease to quantum depth of time-evolution circuits for use in VQPE. We show that the approximation provides a good basis for Hamiltonian diagonalisation even when its fidelity to the true time evolved states is low. In the high fidelity case, we show that the approximate unitary U can be diagonalised instead, preserving the linear cost of exact VQPE.

One of the promising fields where quantum computers may have impact is quantum chemistry and in particular the problem of Hamiltonian simulation and ground state preparation. Subspace diagonalisation methods are one approach to obtaining the wave function by combining both these techniques. In these approaches, states are generated by repeated application of some operator and the Hamiltonian matrix in this basis is measured using a quantum device. It is then classically diagonalised to give approximate eigenvalues and eigenvectors of the Hamiltonian.

This work is based on the Variational Quantum Phase Estimation (VQPE) algorithm, which uses the time evolution operator to generate basis states, which have a series of mathematically convenient properties. Among these, the eigenfunctions can be computed from the matrix of the time-evolution operator itself, which has a linear number of distinct elements for a uniform time grid. Nevertheless, conventional approaches to expressing the time-evolution operator on a quantum device, such as Trotterised time-evolution, lead to intractably deep quantum circuits for chemistry Hamiltonians.

We combine this method with the Variational Fast Forwarding (VFF) approach, which generates a constant-circuit-dept approximation to the time evolution operator. We show that the method converges well even when the VFF approximation is not extremely accurate. When it is, it can take advantage of the same cost-reduction properties as the original VQPE algorithm, making the algorithm much more amenable to NISQ hardware.

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[1] Francois Jamet, Connor Lenihan, Lachlan P. Lindoy, Abhishek Agarwal, Enrico Fontana, Baptiste Anselme Martin, and Ivan Rungger, "Anderson impurity solver integrating tensor network methods with quantum computing", arXiv:2304.06587, (2023).

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