Variational quantum algorithms for many-body systems

Abstract

Variational quantum algorithms (VQAs) incorporate hybrid quantum–classical computation aimed at harnessing the power of noisy intermediate-scale quantum (NISQ) computers to solve challenging computational problems. In this thesis, three main VQAs are presented, each tackling a different facet of many-body physics. The first is the variational quantum eigensolver (VQE), which is designed to determine the ground-state of the extended Fermi-Hubbard model. The VQE was applied to study the ground-state properties of N-component interacting fermions. To this end, an SU(N) fermion-to-qubit encoding was devised, based on an extension of the Jordan–Wigner mapping. The ground-state of the Hubbard model, with different dynamical parameters, was specifically obtained by using a number-conserving parametrised quantum circuit (PQC). The persistent current, having applications in the emergent field of atomtronics, was then investigated and numerically obtained by varying the magnetic flux and adiabatically assisting the VQE. This approach lays out the basis for a current-based quantum simulator of many-body systems that can be implemented on NISQ computers. The second VQA is the variational separability verifier (VSV), which is a novel approach to determining the closest separable state (CSS) of an arbitrary quantum state, with respect to the Hilbert–Schmidt distance (HSD). The performance of the VSV is first assessed by investigating the convergence of the optimisation procedure for GHZ states of up to seven qubits, using both statevector and shot-based simulations. The results indicate that current NISQ devices may be useful in addressing the NP-hard full separability problem using the VSV, due to the shallow quantum circuit imposed by employing the destructive SWAP test to evaluate the HSD. The final VQA was designed for the preparation of thermal states. The preparation of an equilibrium thermal state of a quantum many-body system on NISQ devices is an important task in order to extend the range of applications of quantum computation. Faithful Gibbs state preparation would pave the way to investigate protocols such as thermalisation and out-of-equilibrium thermodynamics, as well as providing useful resources for quantum algorithms, where sampling from Gibbs states constitutes a key subroutine. The novelty of the VQA consists in implementing a PQC acting on two distinct, yet connected, quantum registers. The VQA evaluates the Helmholtz free energy, where the von Neumann entropy is obtained via postprocessing of computational basis measurements on one register, while the Gibbs state is prepared on the other register, via a unitary rotation in the energy basis. Finally, the VQA is benchmarked by preparing Gibbs states of several spin-1/2 models and achieving remarkably high fidelities across a broad range of temperatures in statevector simulations. The performance of the VQA was assessed on IBM quantum computers, showcasing its feasibility on current NISQ devices.