^{1,2}

^{,*}

^{3}

^{,†}

^{4}

^{,‡}

^{3}.

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

Large-scale quantum computers can simulate nonperturbative quantum field theories which are intractable classically

For fermionic fields, relatively efficient mappings to quantum registers are known

In this work, we systematize the proposal of replacing continuous gauge groups

Approximating gauge theories by replacing

The fundamental issue of group discretization can be understood by considering the Wilson gauge action

Both arguments suggest

In lattice calculations, one replaces

In the case of

For the important case of

This name is most common in the mathematical literature

To decrease

The term usually added was the adjoint trace, giving

With these actions, the dimensionless product

Our ultimate goal is to approximate the path integral of group

It is natural to associate every subgroup element

A schematic demonstration of

We then expand the exponential in the path integral and integrate over

Starting with the second order terms computed in Refs.

In the following section, we will calculate Eq.

Through out this work we suppress the argument of

The dimension,

In deriving the decimated action, integrating out the field fluctuations require us to reduce expressions of the form

For

Extending this to non-Abelian groups, e.g.,

In this section, we summarize the derivation of the decimated action order-by-order. Further details can be found in Appendix

It is comforting that at

We now proceed to calculate the second order decimated action while fixing a few typos in

Next, we calculate the case of

We would now like to comment on how the two-plaquette—and general multiplaquette—terms contributes to the

Numerical values of

Example of two plaquettes

It is reasonable to expect these individual reasons to persist at higher orders, suggesting that at a fixed order all multiplaquette terms can be neglected compared to their 1—plaquette counterpart. But can we argue that the multi-plaquette terms generated at order

For the third-order terms of Eq.

Putting together Eqs.

As a demonstration, we simulated Eq.

Average energy per plaquette,

Naively,

At

Taken together, the

With Eq.

For

While superficially the cumulant expansion has appeared as a strong-coupling expansion in

For subgroups of

But what is the origin of this behavior? For simplicity, we can understand this behavior by considering the expansion of

Another feature observed in the

From the behavior observed in

This proxy can be compared to others in the literature, which are collected in Table

Parameters of a discrete subgroups necessary to study the behavior of

Observing the differing

Using our higher order results, one can then gain insight into the effectiveness of the

In this work, we used the cumulant expansion to develop a systematic method for studying and improving lattice actions that replace continuous gauge groups by their discrete subgroups. This is a step in the ongoing trek toward developing accurate and efficient digitization on quantum computers. These decimated actions, through the factor

We further computed the third-order, single-plaquette contribution for the general group. These higher-order terms are necessary for systematizing the decimation procedure of

In order to move beyond pure gauge theory, it will be necessary to consider quark fields. While the computational resources increase substantially for dynamical quarks, an advantage of the discrete subgroup approximations is that many standard lattice field theory techniques such as

Another important step in studying the feasibility of this procedure is to explicitly construct the quantum registers and primitive gates

The authors would like to thank Scott Lawrence, Jesse Stryker, Justin Thaler, and Yukari Yamauchi for helpful comments on this work. Y. J. is grateful for the support of DFG, Grants No. BR 2021/7-2 and No. SFB TRR 257. H. L. is supported by a Department of Energy QuantiSED grant. Fermilab is operated by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the United States Department of Energy. S. Z. is supported by the National Science Foundation CAREER award (Grant No. CCF-1845125).

A useful identity was derived in

At first order, only one integral is needed:

For a given group, the general basis is overcomplete. These leads to simplifications in our derivations for a given group. Here we present the related characters for three groups of relative importance:

For

Another important set of identities are those which relate products of

Applying all the simplifications in Eq.

In this Appendix, we expand upon the derivation of the decimated action. First, for the second-order term in Eq.

For the third-order terms of Eq.