Simulating gauge theories with variational quantum eigensolvers in superconducting microwave cavities

Jinglei Zhang1,2, Ryan Ferguson1,2, Stefan Kühn3, Jan F. Haase1,2,4, C.M. Wilson1,5, Karl Jansen6, and Christine A. Muschik1,2,7

1Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
2Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
3Computation-based Science and Technology Research Center,The Cyprus Institute, 20 Kavafi Street, 2121 Nicosia, Cyprus
4Institute of Theoretical Physics and IQST, Universität Ulm, Albert-Einstein-Allee 11, D-89069 Ulm, Germany
5Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
6NIC, DESY Zeuthen, Platanenallee 6, 15738 Zeuthen, Germany
7Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada

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Quantum-enhanced computing methods are promising candidates to solve currently intractable problems. We consider here a variational quantum eigensolver (VQE), that delegates costly state preparations and measurements to quantum hardware, while classical optimization techniques guide the quantum hardware to create a desired target state. In this work, we propose a bosonic VQE using superconducting microwave cavities, overcoming the typical restriction of a small Hilbert space when the VQE is qubit based. The considered platform allows for strong nonlinearities between photon modes, which are highly customisable and can be tuned in situ, i.e. during running experiments. Our proposal hence allows for the realization of a wide range of bosonic ansatz states, and is therefore especially useful when simulating models involving degrees of freedom that cannot be simply mapped to qubits, such as gauge theories, that include components which require infinite-dimensional Hilbert spaces. We thus propose to experimentally apply this bosonic VQE to the U(1) Higgs model including a topological term, which in general introduces a sign problem in the model, making it intractable with conventional Monte Carlo methods.

Gauge theories are a fundamental part of modern physics, in particular they constitute the theoretical foundation of the Standard Model, which is the best description we have to date of elementary particles and their interactions, except for gravity. One prominent success of the Standard Model is the Higgs mechanism, which explains how gauge bosons acquire their masses; this was experimentally confirmed by the discovery of the Higgs particle announced in 2013 at CERN. As gauge theories are quantum theories, quantum computers offer an exciting opportunity to understand them more deeply than what we have been able to do so far.

In this work, we propose to use photons in superconducting microwave cavities as a new quantum platform to study gauge theories. While many quantum computing platforms are based on qubits, which have two available states, the photons in a microwave cavity are a higher-dimensional system that can be exploited for the computation. This is particularly relevant because bosonic fields have intrinsically high-dimensional elements, and recent technological developments offer us an excellent level of control and variety of interactions between the microwave photons.

The theory we choose to study is called a U(1) Higgs model with a topological term. This theory contains rich and emblematic physics that we simulate via a hybrid quantum-classical algorithm called a variational quantum eigensolver (VQE). This protocol uses the quantum platform, in our case the microwave cavity, to perform evaluations that are classically hard, and a classical computer to perform a variational optimization that is robust to errors. We show that a VQE is able to compute the lowest-energy state of the model for a range of parameters, allowing us to study different phases of the system that have qualitatively different behaviour.

We discuss in detail and show that the quantum algorithm we developed is experimentally accessible, it studies a gauge theory that would not be accessible with classical methods alone, and it opens up many new possibilities to further develop quantum simulations for gauge theories.

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