Barren-plateau free variational quantum simulation of Z2 lattice gauge theories

TL;DR

This paper addresses the challenge of simulating ground states and static string breaking in lattice gauge theories by employing a variational quantum eigensolver (VQE) on a $\mathbb{Z}_2$ lattice gauge theory defined on a two-leg ladder with Kogut-Susskind fermions. It introduces two complementary ansätze: a gauge-invariant (GI) Hamiltonian-variational circuit and a hardware-efficient MBQC-inspired ZZ circuit, demonstrating that gauge-invariant ground states can be reached without explicit Gauss-law penalties and that gradient scalability avoids barren plateaus in practice. Tensor-network methods verify the VQE results and reveal cases where TNs get trapped in local minima, illustrating the VQE’s potential to access regions difficult for classical methods. Static string breaking is observed both in simulations and on IBM hardware, with the static potential $V(d)$ showing linear growth up to a critical distance followed by breaking via pair creation. The work suggests that VQEs are a promising route for studying $\mathbb{Z}_2$ LGTs and motivates extending these techniques to other gauge groups, while highlighting initialization and symmetry constraints as key factors for success on near-term devices.

Abstract

In this work, we design a variational quantum eigensolver (VQE) suitable for investigating ground states and static string breaking in a $\mathbb{Z}_2$ lattice gauge theory (LGT). We consider a two-leg ladder lattice coupled to Kogut-Susskind staggered fermions and verify the results of the VQE simulations using tensor network methods. We find that for varying Hamiltonian parameter regimes and in the presence of external charges, the VQE is able to arrive at the gauge-invariant ground state without explicitly enforcing gauge invariance through penalty terms. Additionally, experiments showing string breaking are performed on IBM's quantum platform. Thus, VQEs are seen to be a promising tool for $\mathbb{Z}_2$ LGTs, and could pave the way for studies of other gauge groups. We find that the scaling of gradients with the number of qubits is favorable for avoiding barren plateaus. At the same time, it is not clear how to efficiently simulate the LGT using classical methods. Furthermore, strategies that avoid barren plateaus arise naturally as features of LGTs, such as choosing the initialization by setting the Gauss law sector and restricting the Hilbert space to the gauge-invariant subspace.

Figures (9)

• Figure 1: Features of LGTs on the two-leg ladder lattice. The LGT specifies matter sites and links between them pertaining to the underlying gauge field. The arrows on the links show the orientation of the lattice. We have labeled $\hat{E}_x$ and $\hat{E}_y$ the orientations of the electric field and the ring labeled $\hat{P}$ indicates the magnetic plaquette term. For the vertex $\bm l$, we have indicated the notation for the adjacent link and matter site that is one unit vector $\hat{x}$ across. The red arrows show how we linearly arrange the qubits to perform the Jordan-Wigner transformation: we count sites from the leftmost lower corner, then traverse upwards, then rightwards along the lattice. The alternating colors of the matter sites illustrate the coupling to staggered fermions where the sign alternates. On this diagram we have also included an example string configuration, through the solid arrow connecting two charges that have been placed. The overlapping circles show the change in sign, indicating the placed charges.
• Figure 2: The ansatz circuits considered in this work. These are circuits for the simplest example of one plaquette. The red gates set the initial state, while the green gates are parametrized, and form the repeating layers. Here we use the shorthand $X\rightarrow R_X(\theta)$, so each green gate has associated one variable angle. a) shows the gauge-invariant (GI) ansatz and b) the hardware-efficient, MBQC-inspired (ZZ) ansatz.
• Figure 3: Scaling of the variance with system size across ansätze and numbers of layers. We consider the GI and ZZ ansatz, across 1, 2, and 3 layers, averaged over 100 samples. As the system size is increased, we see a favorable scaling in the variance, so that gradients remain non-vanishing when the number of qubits is beyond classical simulation.
• Figure 4: Average fidelity of Gauss law operator $G_l$ across each site during the optimization. For the ZZ ansatz we plot how the fidelity varies for the Hamiltonian with $J=m=\mu=1$ over the course of the optimization using SLSQP. In dark green, we initialize in the gauge-invariant subspace by setting all the initial angles to $\pi$, whereas for light green this choice is random. When the optimization begins in this subspace, in a few iterations, the VQE explores states outside of the space to then return, arriving on the state that minimizes the cost function. When the optimization begins outside of this subspace, the VQE nonetheless converges upon the state within the physical subspace.
• Figure 5: Scaling of the dimension of the DLA. For increasing number of qubits of the GI and ZZ ansatz and the one- and two-dimensional LGT. Here we include an exponential and polynomial fit for the one-dimensional cases. For the GI ansatz the exponential fitting is given by $\dim(\mathfrak g) \approx 3.49e-1 2^{1.50n}$, where $n$ is the number of qubits, and the mean squared error (MSE) $10.56$, while the polynomial fit is $\dim(\mathfrak g) \approx 5.71e-7 n^{10.33}$ with MSE $8550.98$. Whereas for the ZZ ansatz the exponential fit is $\dim(\mathfrak g) \approx 5.47e-1 2^{1.98n}$ with MSE $1.63$, and then the polynomial fit is $\dim(\mathfrak g) \approx 1.04e-3 n^{8.17}$ with MSE $258.65$. Up to $7$ qubits, the one- and two-dimensional cases are the same, as the 8th qubit introduces a plaquette term. With this additional term, the DLA is larger than for the one-dimensional case.