Applying quantum computing to simulations of Quantum Field Theory

Numerical integration of the path integral in lattice-field theory is usually done by the Markov Chain Monte Carlo method which regards the Boltzmann weight as a probability. Therefore, it encounters a problem when the integrand is negative or complex and highly oscillating because then the sampling method becomes much less efficient. This problem is known as the sign problem and physically, it happens when we have topological terms, chemical potentials, or when we want to compute real time dynamics. Perhaps we could avoid the sign problem entirely by switching to the Hamiltonian formalism where the sign problem is absent from the beginning however then we have to deal with regularising an infinite dimensional Hilbert space and then computing matrix mechanics operations in a huge vector space whose dimension grows exponential in the volume of the system. Quantum computers may aid in this regard as the Hamiltonian formalism is the most common framework to compute static and dynamical observables as the unitary time evolution can be naturally implemented on a quantum device. Therefore, it is reasonable to expect that quantum computers could do this job in the not so distant future.

To progress the field towards this goal, research has to be done on both the algorithm front as well as the theoretical front. Work on the theoretical front may involve fundamentally new formulations of problems within the framework of quantum field theory which can more efficiently encode the infinite-dimensional Hilbert space of quantum field theories into a finite-dimensional one. For example, different approaches may result in different rates of convergence to the predictions of the infinite-dimensional theory, as well as different resource requirements in hardware. More research is needed to determine the pros and cons of different formulations. On the algorithmic research side there is a need for novel algorithms that maximize efficiency in encodings of degrees of freedom to qubits, as well as algorithms with tight and rigorous error bounds. Furthermore, it is a non-trivial task to create protocols that efficiently prepare initial states and that measure outcomes relevant for a range of observables. Furthermore, it is a non-trivial task to create protocols that efficiently prepare initial states and that measure outcomes relevant for a range of observables. Again thorough benchmarking of such algorithms is important. Possible projects could therefore aim to build on one or more of these tasks.

Quantum computers are therefore expected to extend the domain of problem classes in simulations of quantum field theory to include theories with topological terms, chemical potentials and also calculations of real time dynamics. Therefore possible range of applications in high energy physics problems include computations of full scattering processes, simulation of high density systems such as collective neutrino oscillations in core-collapse supernova, non-equilibrium dynamics in high-energy particle collisions and also as a tool for exploring quantum gravity theories. Aside from high energy physics the study of how to simulate quantum field theories may prove to be useful in areas outside of particle physics such as in condensed matter where again QFT's play a pivotal role. Therefore effective quantum simulations of quantum field theories could lead to numerous practical and theoretical advancements in various fields across physics.