Abstract
Variational quantum eigensolvers (VQEs) are a promising class of quantum algorithms for preparing approximate ground states in near-term quantum devices. Minimizing the error in such an approximation requires designing ansatzes using physical considerations that target the studied system. One such consideration is size-extensivity, meaning that the ground state quantum correlations are to be compactly represented in the ansatz. On digital quantum computers, however, the size-extensive ansatzes usually require expansion via Trotter-Suzuki methods. These introduce additional costs and errors to the approximation. In this work, we present a diagrammatic scheme for the digital VQE ansatzes, which is size-extensive but does not rely on Trotterization. We start by designing a family of digital ansatzes that explore the entire Hilbert space with the minimum number of free parameters. We then demonstrate how one may compress an arbitrary digital ansatz, by enforcing symmetry constraints of the target system, or by using them as parent ansatzes for a hierarchy of increasingly long but increasingly accurate sub-ansatzes. We apply a perturbative analysis and develop a diagrammatic formalism that ensures the size-extensivity of generated hierarchies. We test our methods on a short spin chain, finding good convergence to the ground state in the paramagnetic and the ferromagnetic phase of the transverse-field Ising model.
Highlights
Despite promises of exponential speedups, quantum algorithms require optimization to achieve an advantage over their classical counterparts on state of the art supercomputers for problems of interest
We start by designing a class of Variational quantum eigensolvers (VQEs) ansatzes, based on the stabilizer formalism in quantum error correction, which provably tightly span the entire Hilbert space of Nq qubits
A variational quantum eigensolver (VQE) is an algorithm executed on a quantum register that aims to approximate the minimum eigenvalue E0 of a target Hamiltonian H on C2Nq by finding low energy states |ψ ∈ C2Nq variationally
Summary
Despite promises of exponential speedups, quantum algorithms require optimization to achieve an advantage over their classical counterparts on state of the art supercomputers for problems of interest. As the manifold of obtainable states for a given VQE will only ever be an exponentially small region in the larger Hilbert space, optimizing VQE design is critical to obtain good approximations of the system’s ground state [4, 5, 6] This has spurred much recent work in optimizing VQEs based on the unitary coupled cluster expansion [4, 5, 7], or on the quantum approximate optimization algorithm [8, 9]. We develop a Trotterization-free diagrammatic method to generate size-extensive VQEs. We start by designing a class of VQE ansatzes, based on the stabilizer formalism in quantum error correction, which provably tightly span the entire Hilbert space of Nq qubits. We find that strictly following the perturbative approach is beneficial in the weak-coupling regime, but restricting the ansatz to lowest-order gives better convergence in the strong-coupling regime — even though such ansatzes are seemingly less-informed about the strong-coupling physics
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