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arXiv:2005.14221 · Physical review. D/Physical review. D.

Gluon field digitization via group space decimation for quantum computers

Ji Yao, Henry Lamm, Shuchen Zhu

Published Dec 21, 2020• 70 citations
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Abstract

Efficient digitization is required for quantum simulations of gauge theories. Schemes based on discrete subgroups use fewer qubits at the cost of systematic errors. We systematize this approach by deriving a single plaquette action for approximating general continuous gauge groups through integrating out field fluctuations. This provides insight into the effectiveness of these approximations, and how they could be improved. We accompany the scheme by simulations of pure gauge over the largest discrete subgroup of $SU(3)$ up to the third order.

DOI
10.1103/physrevd.102.114513

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Efficient digitization is required for quantum simulations of gauge theories. Schemes based on discrete subgroups use fewer qubits at the cost of systematic errors. We systematize this approach by deriving a single plaquette action for approximating general continuous gauge groups through integrating out field fluctuations. This provides insight into the effectiveness of these approximations, and how they could be improved. We accompany the scheme by simulations of pure gauge over the largest discrete subgroup of SU(3)SU(3)SU(3) up to the third order.
Efficient digitization is required for quantum simulations of gauge theories. Schemes based on discrete subgroups use a smaller, fixed number of qubits at the cost of systematic errors. We systematize this approach by deriving the single plaquette action through matching the continuous group action to that of a discrete one via group character expansions modulo the field fluctuation contributions. We accompany this scheme by simulations of pure gauge over the largest discrete crystal-like subgroup of SU(3)SU(3)SU(3) up to the fifth order in the coupling constant.
Simulations of gauge theories on quantum computers require the digitization of continuous field variables. Digitization schemes that use the minimum amount of qubits are desirable. We present a practical scheme for digitizing SU(3)SU(3)SU(3) gauge theories via its discrete subgroup S(1080)S(1080)S(1080). The S(1080)S(1080)S(1080) standard Wilson action cannot be used since a phase transition occurs as the coupling is decreased, well before the scaling regime. We propose a modified action that allows simulations in the scaling window and carry out classical Monte Carlo calculations down to lattice spacings of order $a\ensuremath{\approx}0.08\text{ }\text{ }\mathrm{fm}$. We compute a set of observables with subpercent precision at multiple lattice spacings and show that the continuum extrapolated value agrees with the full SU(3)SU(3)SU(3) results. This suggests that this digitization scheme provides sufficient precision for noisy intermediate-scale quantum era QCD simulations.
Quantum computers can circumvent the numerical sign problem in gauge theories at finite density or in real time. Quantum simulations of gauge theories require a finite-dimensional representation of continuous gauge fields. Replacing a continuous gauge group by a finite subgroup can substantially reduce the required quantum resources, but introduces digitization errors that must be controlled in the Hamiltonian, or continuous-time, limit. Previous studies, using the isotropic Euclidean lattices showed that the freezing transition of the discrete subgroup can make it a bad approximation for the continuous group at large Euclidean couplings. Here, we study the digitization of U(1) by its Z(NNN) subgroups in 2+1 dimensions using anisotropic Euclidean lattices. We derive the trajectories of the spatial and temporal gauge couplings along which the Hamiltonian limit is approached at fixed Hamiltonian coupling. While the temporal coupling exhibits power-law scaling in the continuous U(1) theory, it grows only logarithmically for finite Z(NNN). Using classical lattice simulations and exact diagonalization, we verify that these trajectories reproduce the corresponding Hamiltonian theories. We find that the freezing transition persists in the Hamiltonian limit of discrete gauge groups and that finite-NNN theories can differ substantially from U(1) even outside the frozen regime, in contrast to the behavior on isotropic Euclidean lattices, where for small couplings, the discrete group provides a very accurate approximation of the continuous group. Our results provide a classical benchmark for quantifying the systematic errors due to gauge-field digitization in quantum simulations.
With the long term perspective of using quantum computers and tensor networks for lattice gauge theory simulations, an efficient method of digitizing gauge group elements is needed. We thus present our results for a handful of discretization approaches for the non-trivial example of SU(2), such as its finite subgroups, as well as different classes of finite subsets. We focus our attention on a freezing transition observed towards weak couplings. A generalized version of the Fibonacci spiral appears to be particularly efficient and close to optimal.
With the long term perspective of using quantum computers and tensor networks for lattice gauge theory simulations, an efficient method of digitizing gauge group elements is needed. We thus present our results for a handful of discretization approaches for the non-trivial example of SU(2), such as its finite subgroups, as well as different classes of finite subsets. We focus our attention on a freezing transition observed towards weak couplings. A generalized version of the Fibonacci spiral appears to be particularly efficient and close to optimal.
We describe the simulation of dihedral gauge theories on digital quantum computers. The nonabelian discrete gauge group DND_NDN​ -- the dihedral group -- serves as an approximation to U(1)×Z2U(1)\times\mathbb{Z}_2U(1)×Z2​ lattice gauge theory. In order to carry out such a lattice simulation, we detail the construction of efficient quantum circuits to realize basic primitives including the nonabelian Fourier transform over DND_NDN​, the trace operation, and the group multiplication and inversion operations. For each case the required quantum resources scale linearly or as low-degree polynomials in n=log⁡Nn=\log Nn=logN. We experimentally benchmark our gates on the Rigetti Aspen-9 quantum processor for the case of D4D_4D4​. The fidelity of all D4D_4D4​ gates was found to exceed 80%80\%80%.