Extended Quantum Computing Patterns

Variational Quantum Eigensolver (VQE)

Aliases: –

Tags:

© Weigold, M.; Barzen, J.; Leymann, F.; and Vietz, D: Patterns For Hybrid Quantum Algorithms. In: Proceedings of the 15th Symposium and Summer School on Service-Oriented Computing (SummerSOC 2021), https://doi.org/10.1007/978-3-030-87568-8_2

Intent

"Approximate the lowest eigenvalue of a matrix" (Weigold et al. 2021)

Context

A hermitian matrix $H$ is given, for which the lowest eigenvalue has to be determined. Since the current NISQ devices cannot provide the resources needed to make use of the quantum phase estimation algorithm, an alternative approach must be used.

Solution

As a first step, $H$ is re-written as a weighted sum of Pauli strings:

$$ H=\sum_{\alpha} h_\alpha P_\alpha $$

Using the iterative structure of the Variational Quantum Algorithm (VQA), a suitable ansatz is chosen for the preparation of trial states. Popular choices include the unitary coupled cluster ansatz (Taube and Bartlett 2006) or an ansatz inspired by hardware (Cerezo et al. 2020). The objective function is then defined as follows:

$$ C(\theta) = \left<\psi(\theta)|H|\psi(\theta) \right> = \sum_{\alpha} h_\alpha \left< \psi(\theta)|P_\alpha|\psi(\theta) \right> $$

i.e., the solution is evaluated by the sum of the expectation values of the Pauli string. The variational principle guarantees that the sum of expectation values is greater than or equal to the smallest eigenvalue which must be approximated. If the termination condition has not been fulfilled by $C(\theta)$, the parameters are further optimized and updated as described in the Variational Quantum Algorithm (VQA).

Result

Since a Variational Quantum Algorithm (VQA) approach is used, the convergence of the overall algorithm depends on the objective function, the ansatz, and the chosen optimization strategy. Based on the outcome of this algorithm (an approximation of the lowest eigenvalue), a follow-up algorithm for finding other eigenvalues of $H$ can be applied (see known uses). If all eigenvalues are known, a principal component analysis can be done to perform a dimension reduction.

Related Patterns

This patterns uses the structure described in the Variational Quantum Algorithm (VQA).

Known Uses

Higgott, Wang and Brierley 2019 extends the initial description of this algorithm Peruzzo et al. 2014 by an approach for finding other eigenvalues of $H$. This pattern is especially important for applications in which the ground state of a quantum chemical system has to be determined and has been experimentally applied for various quantum chemistry systems Cao et al. 2019.

Variational Quantum Eigensolver (PlanQK)