Simulating 2D Effects in Lattice Gauge Theories on a Quantum Computer

Quantum computing is in its greatest upswing, with so-called noisy-intermediate-scale-quantum devices heralding the computational power to be expected in the near future. While the field is progressing toward quantum advantage, quantum computers already have the potential to tackle classically intractable problems. Here, we consider gauge theories describing fundamental-particle interactions. On the way to their full-fledged quantum simulations, the challenge of limited resources on near-term quantum devices has to be overcome. We propose an experimental quantum simulation scheme to study ground-state properties in two-dimensional quantum electrodynamics (2D QED) using existing quantum technology. Our protocols can be adapted to larger lattices and offer the perspective to connect the lattice simulation to low-energy observable quantities, e.g., the hadron spectrum, in the continuum theory. By including both dynamical matter and a nonminimal gauge-field truncation, we provide the novel opportunity to observe 2D effects on present-day quantum hardware. More specifically, we present two variational-quantum-eigensolver- (VQE) based protocols for the study of magnetic field effects and for taking an important first step toward computing the running coupling of QED. For both instances, we include variational quantum circuits for qubit-based hardware. We simulate the proposed VQE experiments classically to calculate the required measurement budget under realistic conditions. While this feasibility analysis is done for trapped ions, our approach can be directly adapted to other platforms. The techniques presented here, combined with advancements in quantum hardware, pave the way for reaching beyond the capabilities of classical simulations.

Popular Summary

The use of quantum computer simulations offers the opportunity to study phenomena in high-energy physics that are extremely hard, if not impossible, to access with classical computers. This concerns the very early universe, the question of why there is more matter than antimatter and, hence, why we exist, and the simulation of experiments in particle colliders. Simple instances of such quantum simulations have already been demonstrated in one-dimensional models. However, going beyond one dimension is a major challenge, due to the appearance of new degrees of freedom, which in turn lead to novel phenomena that are absent in one dimension. In this work, we demonstrate how to overcome this difficulty and outline a strategy that allows for studying higher-dimensional models with existing quantum computers. In particular, we provide a concrete example by studying two-dimensional quantum electrodynamics, the quantum theory of photons and electrons.

The major advantage of our strategy is the possibility of running it on presently available quantum hardware. In our protocols, we reduce the number of degrees of freedom of the considered theory substantially without affecting the physics of the model. In this way, a basic building block of the theory can be simulated with present-day quantum computers with only a small number of qubits. Our work contains a detailed account of the resources that are required to carry out quantum simulations of two-dimensional quantum electrodynamics and we explicitly demonstrate the feasibility of such quantum computations. As an important element, we use a hybrid quantum-classical algorithm to find the energy spectrum of the theory, focusing on the lowest-lying state, i.e., the ground state, which can be utilized to study magnetic field effects and other important phenomena of the model.

Our approach overcomes the present limitations of quantum computing protocols for simulating high-energy physics in higher dimensions. As such, it opens a new and very promising path for quantum simulations, which can push our knowledge of the universe and of the fundamental forces of nature beyond today’s barriers. The present work constitutes a big leap in this direction, with the long-term perspective of unraveling the forces between elementary particles, such as the mysterious interaction between quarks and gluons.

Figure 1

The blueprint for quantum simulations of lattice gauge theories. As described in Sec. 3, we use a VQE to simulate QED. The ingredients to perform the VQE are quantum circuits (Secs. 4b and 4 4c), a classical optimizer (Sec. 3), and a measurement scheme to estimate the energy of a quantum state (Appendix pp2). As described in Sec. 4a, the VQE requires a Hamiltonian ${\hat{H}}_{\mathrm{qubit}}$ cast in terms of qubit operators. ${\hat{H}}_{\mathrm{qubit}}$ is optimized for ion-trap quantum computers (see the figure) but can be used with any platform. In turn, ${\hat{H}}_{\mathrm{qubit}}$ is obtained from an effective Hamiltonian ${\hat{H}}_{\mathrm{eff}}$, which is derived in Sec. 2 from the results of Ref. [28]. As schematically indicated in the “Hamiltonian formulation” box, Ref. [28] uses the Kogut-Susskind [29] formulation of two-dimensional (2D) lattice gauge theories (LGTs), the continuum limits of which correspond to QED. The outcomes of simulated VQEs are presented in Secs. 4b and 4 4c, while the corresponding analytical results are given in Sec. 5. The images of the lattice and the ion-trap quantum computer are from Refs. [30] and [22], respectively.

Figure 2

Conventions for lattice QED in 2D. (a) The ladder system with open boundary conditions. Using Gauss’s law, the number of gauge degrees of freedom can be reduced to one per plaquette (blue ellipses). (b) A single plaquette with matter sites at the vertices and gauge fields on the links. The circles (squares) represent odd (even) sites and black (gray) corresponds to unoccupied (occupied) fermionic fields. The positive field direction is to the right and up. (c) A table showing the mapping between fermionic sites, particles ($e$) and antiparticles ($p$), and spins. (d) The conventions for Gauss’s law. (e) The two gauge-field configurations that minimize the electric field energy for a single plaquette with two particle-antiparticle pairs. As explained in the main text, these configurations are relevant for the 2D effects discussed in Sec. 5a (for more details, see Appendix pp4).

Figure 3

The single periodic boundary plaquette for a pure gauge theory. (a) The representation in terms of the eight gauge fields (blue arrows) that are associated with the links of the lattice. (b) The representation in terms of independent gauge-invariant (i.e., physical) operators called “rotators.” As explained in Ref. [28], the plaquette can be seen as an infinite lattice of plaquettes. Moreover, to explore the ground-state properties, only three independent rotators, ${\hat{R}}_1$, ${\hat{R}}_2$, and ${\hat{R}}_3$ (shown in solid blue), are sufficient for describing the system.

Figure 4

The VQE circuit for preparing the ground state of a plaquette with open boundary conditions. (a) The quantum circuit, with variational parameters ${\theta }_l$ shown for each gate. By identifying symmetries and redundancies via classical simulation, the gates shaded in gray can be eliminated and we set ${\theta }_{16}={\theta }_{19}={\theta }_{21}=\pi /2$, leaving 11 variational parameters. (b) The definitions of the gates used.

Figure 5

The classical simulation of the proposed experiment for observing the dynamical generation of magnetic fields where $\mathrm{\Omega }=5$ and $m=0.1$ (see Sec. 5a), using the circuits given in Fig. 4. The blue and black dots represent data points obtained by variational minimization with a total finite measurement budget of ${10}^7$ measurements for the entire plot. Half of this budget is spent for the black dot alone. The black solid lines are determined via exact diagonalization of the Hamiltonians in Eqs. (13a). (a) The energy of the variational ground state. (b) The plaquette expectation value $⟨◻⟩$ as a function of $g^{-2}$. All dots are calculated using the exact state corresponding to the optimal variational parameters found by the VQE.

Figure 6

VQE circuits for preparing the ground state of a plaquette with periodic boundary conditions. Gate definitions are given in Fig. 4. Gates of the same color in each circuit share a variational parameter after eliminating redundant parameters. The half-shaded gates include an offset of $+\pi /2$ added to the shared parameter, while the gray-shaded gates can be eliminated entirely. The circuit for (a) the electric representation of the Hamiltonian and (b) for the magnetic representation.

Figure 7

The classical simulation of the proposed experiment for observing the running of the coupling (see Sec. 5b) using the circuits given in Fig. 6. The red and blue data points are obtained from variational minimization with a total finite measurement budget of $6\times {10}^5$ measurements for the entire plot. The electric (magnetic) representation is shown in blue (red) and the black solid lines are determined via exact diagonalization of the Hamiltonians in Eqs. (4a) and (5a). (a) The energy of the variational ground state using the electric representation in the region $g^{-2}<1$ and using the magnetic representation in the region $g^{-2}>1$. (b) The plaquette expectation value $⟨◻⟩$ as a function of $g^{-2}$. All dots are calculated using the exact state corresponding to the optimal variational parameters found by the VQE.

Figure 8

(a) The plaquette expectation value as a function of $g^{-2}$ for a single plaquette with mass $m=-50$ and kinetic strength $\mathrm{\Omega }=5$. Regions $1$, $2$, and $3$ correspond to the numbered regions in (b). Exact diagonalization of the truncated model is represented by the black dashed line and perturbation theory is used for the blue, green, and orange lines. The insets are magnifications of parts of regions 1 and 2 (blue and green, respectively), plotting the difference between exact diagonalization and perturbation theory of order reported in the legend to confirm convergence to the full U(1) theory. The difference is not completely vanishing due to the small contribution of the magnetic term, which is ignored in perturbative calculations. (b) A schematic representation of the energy of the ground and first excited states.

Figure 9

The ground-state properties in the nonperturbative regime. (a) The plaquette operator expectation value $⟨◻⟩$ and entanglement entropy $\mathcal{S}({\rho }_g)$ as a function of $g^{-2}$. (b)–(d) The probability by component of the ground state for the indicated $g^{-2}$. At $g^{-2}={10}^{-3}$, the lowest-energy state is the vacuum $|vvvv⟩|0⟩$. For $g^{-2}=1$, the ground state approximates that of the kinetic term. For large $g^{-2}$, the magnetic Hamiltonian in the presence of the kinetic Hamiltonian yields the tensor product between a superposition of matter states and the gauge ground state of the magnetic term. Through (a)–(d), we use $\mathrm{\Omega }=5$ and $m=0.1$ and a gauge-field truncation $l=1$.

Figure 10

The ground-state expectation value of the plaquette operator $◻$ for a pure gauge periodic boundary plaquette (see Fig. 3). Results obtained using the electric and magnetic representation are shown as blue and red lines, respectively. Perturbative calculations are shown as gray dashed lines. The plots correspond to two different truncations $l=1$ and $l=2$, as indicated above. The blue (red) shaded region indicates the values of $g^{-2}$ for which the electric (magnetic) representation achieves the best results.

Figure 11

An example of the measurement protocol explained in the main text and Appendix pp2, applied to the open-boundary plaquette (see Sec. 2b). The circuits shown diagonalize a measurement set ${\hat{R}}_m$ using controlled-not (cnot) gates (a) and $i$swap gates (b) as an entangling operation. The qubit labeling conventions are shown in (c). Applying the circuits before performing local measurements in a VQE protocol amounts to effectively performing measurements in an entangled basis. The table in (d) lists the elements ${\hat{Q}}_k$ in the chosen measurement set and their corresponding diagonalized elements using the cnot circuit and the $i$swap circuit.

Figure 12

A schematic description of our perturbative algorithm when starting from the vacuum. In the first column, we show the bare energies of the corresponding states. Each application of the perturbation ${\hat{H}}_{\mathrm{kin}}$ allows for creating and annihilating particle-antiparticle pairs, as depicted by the tree structure. At each iteration, our algorithm expands the Hilbert space to include new terms contributing to the correction. This allows for efficient use of the resources, ultimately reaching several hundred orders in perturbation theory.