In this paper, we propose a general procedure, based on a step scaling approach, to compute the running coupling as a function of a physical scale. We then consider a matching between MC and quantum computing, through suitable observables, such as the plaquette expectation value, as we employ in this work. This matching is carried out in a regime of , where both methods are reliable. MC can, in principle, be used to obtain the physical value of the lattice spacing from large-volume calculations. The so obtained physical scale can then be transferred to the quantum computing analysis. The step scaling function has been considered, e.g., in O (3) sigma model in (1 + 1) dimensions in refs. 28,29,30,31,32. Here we focus on implementing and testing the feasibility of the method in compact U (1) pure gauge theory at a fixed 3 × 3 lattice volume as an initial demonstration, and propose a follow-up extension to matter fields. The latter, however, goes beyond the scope of the present work. The inclusion of matter fields will lead to a nontrivial β-function, rendering the system physically meaningful. The proposed procedure can be directly generalized to (2 + 1)-dimensional QED but also to non-Abelian lattice gauge theories, and eventually to QCD. Furthermore, to study the continuum limit, large-scale computations are required. This would, in principle, be possible with access to quantum devices with a large number of qubits33. Moreover, this matching procedure is independent of the way we get the quantum state to compute the expectation value of the physical observables relevant to this work. For example, the quantum simulation could be performed on cold atom quantum simulators or trapped ions quantum simulators34.
We consider a lattice discretization of the U (1) LGT using Kogut-Susskind staggered fermions. A naive discretization of the fermionic degrees of freedom leads to the so-called doubling problem, i.e., an incorrect continuum limit of the theory. In the staggered formulation, the spinor components are distributed on different lattice sites to avoid this problem. We present the full Hamiltonian, including matter fields, for completeness, even though it is not used in the present work. The Hamiltonian reads
where is the electric energy, the magnetic energy contribution, the fermionic mass term and the kinetic term for the fermions. The electric energy is given by
where is the dimensionless electric field operator that acts on the link emanating from the lattice site with the coordinates r = (r, r) in direction μ ∈ {x, y}. The bare coupling g determines the strength of the interaction, playing a pivotal role throughout the work. The second term in , the magnetic interaction, reads
with a the lattice spacing and the so-called plaquette operator consisting of a product of the operators acting on the links of a plaquette of the lattice (with the subscripts notation r + x ≡ (r + 1, r) or r + y ≡ (r, r + 1)). The unitary operators are related to the discretized vector field as
They represent the gauge connection between the fermionic fields, and we choose to work with a compact formulation where is restricted to (0, 2π). The lattice vector field is the canonical conjugate variable to the electric field, hence one finds for the commutation relations between and
The fermionic mass term is given by
where m is the bare lattice fermion mass and a and a single-component fermionic field residing on site r, since we start from a continuum formulation with two-component Dirac spinors (see Supplementary Note 4 for details). The kinetic term corresponds to a correlated fermion hopping between two lattice sites while simultaneously changing the electric field on the link in between,
Note that here we consider a different kinetic Hamiltonian compared to a previous work, by including an additional phase factor and which corresponds to the original Kogut-Susskind formulation. From now on, we set the a = 1, unless stated otherwise. The physically relevant subspace of gauge invariant states is given by those that fulfill Gauss's law at each site r, which reads
In the above expression, the operators
correspond to the dynamical charges, and Q represent static charges. The static charges will be particularly relevant for the computation of the static potential in the Section "Step scaling approach". Since in this paper, we are focusing on a U (1) pure gauge theory, we will study only the Hamiltonian .
We remark that instead of working on the full Hilbert space and enforcing Gauss's law a posteriori, in this work we impose it beforehand and work on a gauge invariant subspace.
The electric field values on a gauge link are unbounded, which leads to infinite infinite-dimensional Hilbert space for the gauge degrees of freedom. Therefore, for a numerical implementation of the Hamiltonian, the gauge degrees of freedom have to be truncated to a finite dimension. In ref. , the continuous U(1) group is discretized, in the electric basis, to , where l introduces a truncation and dictates the dimensionality of the Hilbert space. The discretized gauge fields are constrained to integer values within the range [-l, l], resulting in a total Hilbert space dimension of (2l + 1), where N denotes the number of gauge fields in the system. The eigenstates of the electric field operator, , form a basis for the link degrees of freedom (see e.g., Section VI C of ref. ),
The link operators () act as a raising (lowering) operator on the electric field eigenstates,
The link operators have the following form,
With this truncation, unitarity is lost but is recovered in the l → ∞ limit. The commutation relations between the electric field and link operators from Eqs. (5) and (6) are preserved even for the truncated operators. Other approaches for the definition of the gauge field operators have been considered in refs. and with qudits. The errors introduced by finite truncation l have been studied in refs. .
In Supplementary Note 3, we will explore an alternative representation of the Hamiltonian known as the magnetic basis or dual basis, recently explored in refs. . This approach becomes relevant when the coupling constant g decreases and the magnetic term of the Hamiltonian becomes dominant. In this regime, the electric basis cannot provide a good approximation of the system with small values of l. However, by exploiting the discrete Fourier transform, we can obtain a diagonal expression for the plaquette terms, thus reducing the resources needed for the calculations. With the magnetic formulation introduced in ref. , the group under consideration changes to , where J serves as an additional parameter dictating the discretization. The dimensionality of the Hilbert space remains defined by the truncation parameter l. An alternative formulation for the magnetic basis implementation has been explored in ref. . We remark that the Hamiltonian formalism can be extended to non-Abelian gauge groups, like SU(2) (see e.g., refs. ). For this case, the discretization in the magnetic basis has been investigated more recently in refs. . Further approaches can be found e.g., in refs. .
The step scaling approach is a computational method employed for the determination of the running coupling, introduced in ref. and used also for instance in refs. . For a general description based on the Schrödinger functional approach see refs. . Let us assume we define a running, renormalized coupling α(r) at a physical scale r(g). We then define the step scaling function σ in the continuum from
which can be understood as an integrated form of the β-function of the theory. Starting from α(r), we then apply the function in Eq. (14). The step then is repeated, going to α(sr) and subsequent values, by creating the steps in Fig. 1a.
This method can be iterated up to arbitrary N + 1 steps, obtaining α(sr) and thus getting the running coupling as a function of the physical scale. The goal of this work is to compute the step scaling function non-perturbatively on the lattice, by starting with some distance r in lattice units and with a bare coupling g. The lattice spacing is encoded in the coupling which is an implicit function of a in physical units. We fix two scales, r and r ≡ s ⋅ r in lattice units and compute the renormalized coupling at a fixed value of g, i.e., α(r, g) and α(sr, g), which corresponds to σ(α(r, g)) on the lattice. We tune g, finding the value where α(r, g) = α(sr, g). The corresponding sequence of steps can be illustrated in Fig. 1b.
In this paper, we consider a lattice calculation and then we convert the lattice distances into physical ones with an artificial value of a in physical units and r = ar (see the Section "Towards defining a physical scale"). Where, as mentioned in the introduction, the numerical value of the lattice spacing can be obtained in principle with large-volume Monte Carlo computations. Since a in physical units is a function of the bare coupling, by decreasing g, we change the physical distance to smaller values. In this way, we get the running of the coupling as a function of the physical scale. Results of the application of this method are discussed in the Section "Towards defining a physical scale". We use the static force, F(r, g), as the physical quantity of interest, focusing in particular on the dimensionless quantity rF(r, g), with g the bare coupling at which the force is computed and r the distance between two static charges. The calculation of the static force involves the application of a discrete derivative, approximated as , where V(r) denotes the static potential between two static charges separated by r. This potential is, for short distances, proportional to a logarithmic Coulomb term on the lattice. Thus, rF(r, g) can be related eventually to the renormalized coupling. For the analysis of the step scaling, we need two values of the static force,
Therefore, it is necessary to involve three distances, namely r, r, and r, in the calculation of the step scaling function. Note that in this paper, we introduce the step scaling method for a pure gauge U(1) theory. Once we will include matter fields, we will have a non-trivial running coupling.
In this section, we describe the matching between the Variational Quantum Eigensolver (VQE) approach and Markov Chain Monte Carlo (MCMC). For our study, we consider the pure gauge case, i.e., the theory without fermionic fields, on a 3 × 3 lattice with periodic boundary conditions (PBC) (see Fig. 2a for an illustration). The quantity we analyze is the expectation value of the plaquette operator,
where is the number of plaquettes in the lattice.
We focus on bare couplings in the interval corresponding to 0.8 ≤ β = 1/g ≤ 2.6, selected to be in a regime accessible with MCMC methods. We analyze the convergence behavior of the results with exact diagonalization (ED) with respect to the truncation parameter l, as illustrated by the lines in the upper panel of Fig. 2b. The results from ED show that with increasing values of l we observe convergence, and for the range of couplings chosen here l = 3 is sufficient to reliably determine the plaquette expectation value, as the results for l = 3 and l = 4 are essentially indistinguishable. For a truncation l = 2, we see slight deviations for larger values of β. While for l = 1 deviations towards larger values of β are noticeable, the data still qualitatively reproduce the behavior observed for larger values of l. For the VQE approach, we have developed a quantum circuit with an entanglement structure connecting all gauge fields. The resources required for running the circuit are shown in Table 1.
Due to the limited resources available on current quantum devices, we focus on truncation l = 1 for a proof of principle. In order to benchmark the approach, we classically simulate the VQE, assuming a noise-free quantum device. For the optimization, we employ the SLSQP classical optimizer and infinite shots, i.e., number of measurements, setting as the first goal only the expressivity of the quantum circuit. After applying Gauss's law, only 10 of the 18 links remain dynamical, thus, we need 20 qubits for the computation. As illustrated in Fig. 2b, the top panel shows the VQE results for this truncation, indicated by circles, along with the relative error with respect to the exact values in the bottom panel. These results are in line with the plaquette curve from ED.
For the future, a closer examination of the entanglement structure could help to improve the accuracy of the data and optimize the scalability of the gate number. Such exploration should focus on three main purposes: to improve our understanding of the interplay between circuit and lattice structure, to extend the results to higher truncation while preserving the depths of the circuits and to prepare for analysis on quantum hardware platforms.
Part of the authors are involved in a MC calculation of the Hamiltonian limit of the theory analyzed in this work, namely the continuum limit in the time direction at fixed spatial lattice spacing, within the Lagrangian formalism. Preliminary account of the latter can be found in ref. . This procedure returns a β-value for which we know the corresponding bare coupling value of the discretized theory. Thus, the spatial lattice spacing is identical up to lattice artefacts. At this value of the coupling, we can then perform large-volume Monte Carlo simulations and set the physical scale. Since this analysis was still ongoing at the time of writing, in this work we adopt the preliminary value β = 1.4, corresponding to the vertical line in Fig. 2b.
In this section, we illustrate the methodology for the pure gauge case on a 3 × 3 lattice with Open Boundary Conditions (OBC). Two static charges of opposite values are placed on two sites, as in Fig. 3. The choice of boundary conditions allows us to obtain more distinct lattice distances than the periodic case. We focus on two sets of distances to generalize any findings regarding the coupling behavior. We have tested all the combinations of the five possible distances for two static charges on a pure gauge lattice, , and have chosen the two combinations with more points in the step scaling procedure below a certain threshold for the bare coupling, i.e., β ≤ 10. Note that with a system size of 3 × 3 sites and a range of βs considered here, we always work at distances below the confinement scale, and the correlation length is given by the small system size.
In the following analysis, a variational quantum algorithm is used to calculate the static potential at different distances, and the results are compared with those derived from exact diagonalization. The data presented here are computed with a combination of two classical optimizers: we performed a first minimization with NFT, which gave us fidelities of up to ~95% (noise-free simulations with shots). As the coupling decreases, higher precision in the VQE results becomes necessary. Consequently, we have used the final optimal parameters as a starting point for a new optimization with COBYLA and a larger number of shots . This aspect is crucial for our objectives, as the values of the static forces in the weak coupling regime are almost equivalent. A better understanding of the entanglement structure may be helpful for increasing the precision with fewer shots. We will consider a more in-depth analysis of this in future work. In Table 2, we show the resource estimation for three values of the truncation parameter l.
In this section, we focus on illustrating the step scaling method using a variational approach and exact diagonalization as a reference, for the configuration with charges placed as in Fig. 3a, c, d and relative distances on the lattice r = 1, and, respectively.
We show the step scaling procedure starting from a weak coupling regime. As explained in the Section "Running coupling and step scaling", we repeat the steps until we reach a certain value of the bare coupling. In Fig. 4, we follow the step scaling procedure starting from β = 10 and moving to the left with a variational Ansatz (up(down)ward-pointing triangles) and exact results (empty circles/squares). One can see that the precision required for small couplings increases because of the small values required in the step scaling function. Considering only the electric basis becomes more difficult, as the superposition within the ground state expands significantly towards weaker couplings. It is thus advisable to start at large β-values with the magnetic basis and monitor the convergence of results throughout the process towards smaller β-values.
Here, we discuss the variational results for the step scaling method, starting from the value of the bare coupling where we have a matching with MC, see the Section "Matching strategy for 3 × 3 PBC system," and continuing towards a weaker regime. We first illustrate the procedure with a fixed truncation, l = 1, and then discuss higher truncations, involving also a magnetic representation. Starting from β = 1.4, we compute and . Next, using the result of the static force at a distance r, the bare coupling g is adjusted to a reduced value until a new is obtained. This step scaling process is then repeated using a similar approach as in the previous paragraph, but aimed at the weak coupling regime. The comparison between variational results (denoted by up/downward-pointing triangles) and exact results (denoted by empty circles/squares) is illustrated in Fig. 5a.
To give physical meaning to the data, it is essential to increase the truncation parameter l applied to the gauge operators until independent solutions are obtained. Given the limited resources, we adopt a strategy involving an interplay between electric and magnetic basis. Starting from the electric formulation, we progressively increase l until convergence is achieved within the desired range, 1.4 ≤ β ≤ 10. Once a reference value is established, we restrict ourselves to l = 7, which will result in a total of 16 qubits, which seems feasible on current quantum hardware. Next, following the procedure described in Supplementary Note 3, we find the value of the bare coupling where the accuracy of the electric basis is not sufficient anymore (i.e., exceeding a relative error of ϵ ≥ 0.01). At this point, we move on to the magnetic basis with l = 3 and discretization J = 200, parameters that give us reliable results. Initially, we conducted tests with l = 7 also for the magnetic basis, maintaining an equal number of qubits for each register as for the electric one. The outcomes proved to be comparable to those obtained with l = 3. Consequently, we can decrease the computational resources required while preserving a high level of accuracy in the solutions. Fig. 5b illustrates the step-scaling method employing the technique described, with exact diagonalization. Similarly, we proceed through the weak coupling regime by increasing β and constructing the steps accordingly.
In this section, the analysis is repeated for a new set of distances,, and , Fig. 3b-d. Here, we solely explore the step scaling starting from β. The results are then combined in the the Section "Towards defining a physical scale" with the previous set of distances, in order to show the dependence of rF(r, g) in terms of a physical scale.
The step scaling procedure is illustrated in Fig. 6a in the fixed bare coupling interval. In this case, four steps are observed within the range 1.4 ≤ β ≤ 10.
We apply the same technique with higher truncations for this set of distances, Fig. 6b. Also in such a case, we consider l = 7 for the electric basis and l = 3, J = 200 for the magnetic, obtaining a total of five pairs of points.
We are not aware of any real experiment which can be described by the effective (2 + 1)-dimensional compact pure gauge theory considered in this paper. Therefore, we cannot extract the physical value of the lattice spacing with large-volume MC calculations. See Section V C in ref. for the illustration of the principle to determine the value of the lattice spacing. Thus, for the sake of demonstrating our method, we consider an artificial value for the lattice spacing, e.g., a = 0.1 fm, and we use the data in the previous sections to identify the physical value for the scales. With two sets of distances, we have two scale factors s to connect r and r, (r = s ⋅ r), i.e., and . We then combine the results in a single plot.
Let us first consider the set r = 1, , . Our aim is to start with β and invert the sequence by changing the scale by s and include the physical value of the lattice spacing, a = 0.1 fm. At β ≡ β we have,
Then, we go to the next value of the bare coupling, where we have,
The procedure iterates through multiple steps, and eventually, the static force values can be written in terms of a physical scale, as depicted in Fig. 7, with data from Figs. 5a and 6a. Note, for example, that Eqs. (17b) and (18a), correspond to the same physical scale (rightmost full circle and second rightmost full downward-pointing triangle).
In compact pure gauge U(1) theory, the β-function of the dimensionful coupling is trivial and therefore there is no renormalization of the coupling. Consequently, there is, in principle, no scale dependence. Nevertheless, in Fig. 7, we observe a non-trivial behavior of the dimensionless quantity rF(r, g) as a function of the physical distance. However, since the results are at non-zero lattice spacing, we cannot control the uncertainties from a non-zero a and, currently, we cannot compare directly with the continuum perturbation theory.
Note that, when including matter fields, the β-function becomes nontrivial, see again ref. .
We can replicate the procedure using the outcomes from the variational quantum algorithm (again Figs. 5a and 6a), as depicted in Fig. 8. Despite fluctuations in the results, attributed in part to the finite number of shots and the limited convergence of the optimization, the data effectively captures the dependence of the coupling as a function of the physical distance.
The procedure is repeated also for the analysis with electric and magnetic basis, using the data from Figs. 5b and 6b and combining them in Fig. 9.
As a final remark, without going into details, we mention a possible strategy to reach the continuum limit for the discussed analysis. The idea is to keep the physical distance between two static charges fixed while changing the lattice size and, correspondingly, the distance of two static charges on the lattice. The step scaling function parameter s is defined by the fixed ratio (s = r/r). One computes until the same value as in smaller lattice size is found and then applies the step scaling function. The results are studied as a function of . The process is repeated for more steps and the data are subject to a fit, reaching thus the limit . Carrying out this process will be considered in future work.