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arXiv:2106.01398

Worldline Path Integrals for Gauge Fields and Quantum Computing

Feng, Yuan, Michael McGuigan

Published Jun 2, 2021• 0 citations
arXivHTMLPDF

Abstract

We study different aspects the worldline path integrals with gauge fields using quantum computing. We use the Variational Quantum Eigensolver (VQE) and Evolution of Hamiltonian (EOH) quantum algorithms and IBM QISKit to perform our computations. We apply these methods to the path integral of a particle moving in a Abelian and non-Abelian background gauge field associated with a constant magnetic field and the field of a chromo-magnetic field. In all cases we found excellent agreement with the classical computation. We also discuss the insertion of vertex operators into the worldline path integrals to study scattering and show how to represent them using unitary operators and quantum gates on near term quantum computers.

DOI
10.48550/arxiv.2106.01398

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We study different aspects the worldline path integrals with gauge fields using quantum computing. We use the Variational Quantum Eigensolver (VQE) and Evolution of Hamiltonian (EOH) quantum algorithms and IBM QISKit to perform our computations. We apply these methods to the path integral of a particle moving in a Abelian and non-Abelian background gauge field associated with a constant magnetic field and the field of a chromo-magnetic field. In all cases we found excellent agreement with the classical computation. We also discuss the insertion of vertex operators into the worldline path integrals to study scattering and show how to represent them using unitary operators and quantum gates on near term quantum computers.
We ran quantum simulations on lattices of interacting fermions in a background magnetic field [1]. We created discretized Hamiltonian matrices for each system and pre-processed them so that they could be read by the quantum simulator implemented in IBM's Qiskit package [2]. The simulator uses the variational quantum eigensolver (VQE) algorithm [3] to approximate the ground state energy of the input Hamiltonian, which is then compared to the classically-calculated exact value. This classical evaluation is possible because the systems studied are still within the grasp of classical computers. We also investigated ways to simplify the system before the pre-processing step. We found that in almost all cases these simplifications made our quantum calculations converge to better answers using simpler starting conditions. We are exploring other ways to interface with IBM's quantum computing software so that we no longer need to explicitly define the Hamiltonian matrices for our systems, a problem which will itself soon become intractable.
Quantum computers are promising candidates to radically expand the domain of computational science by their increased computing power and more effective algorithms. In particular, quantum computing could have a tremendous impact on the field of quantum cosmology. The goal of quantum cosmology is to describe the evolution of the universe through the Wheeler-DeWitt equation or path integral methods without having to first formulate a full theory of quantum gravity. The quantum computer provides an advantage in this endeavor because it can perform path integrals directly in Lorentzian space and does not require constructing contour integrations in Euclidean gravity. Also, quantum computers can provide advantages in systems with fermions which are difficult to simulate on classical computers. In this study, we first employed classical computational methods to analyze a Friedmann-Robertson-Walker mini-superspace with a scalar field and visualize the calculated wavefunction of the universe for a variety of different values of the curvature of the universe and the cosmological constant. We then used IBM's Quantum Information Science Kit Python library and the variational quantum eigensolver algorithm to study the same systems on a quantum computer. The framework was extended to the world line approach to quantum field theory.
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We study the Z3 gauge theory with fermions on the quantum computer using the Variational Quantum Eigensolver (VQE) algorithm with IBM QISKit software. Using up to 9 qubits we are able to obtain accurate results for the ground state energy. Introducing nonzero chemical potential we are able to determine the Equation of State (EOS) for finite density on the quantum computer. We discuss possible realizations of quantum advantage for this system over classical computers with regards to finite density simulations and the fermion sign problem.