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Quantum 2.0 Conference and Exhibition

Preparation of the SU(3) Lattice Yang-Mills Vacuum with Variational Quantum Methods

Anthony N. Ciavarella, Ivan A. Chernyshev

Published Jan 1, 2022• 2 citations
Publisher

Abstract

Studying QCD and other gauge theories on quantum hardware requires the preparation of physical states. In this work, VQE is applied to pure SU(3) lattice Yang-Mills on a single plaquette and one dimensional plaquette chains. Ansatz states for plaquette chains are constructed in a scalable manner from smaller systems using a stitching procedure.

DOI
10.1364/quantum.2022.qm3c.5

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Studying QCD and other gauge theories on quantum hardware requires the preparation of physical states. In this work, VQE is applied to pure SU(3) lattice Yang-Mills on a single plaquette and one dimensional plaquette chains. Ansatz states for plaquette chains are constructed in a scalable manner from smaller systems using a stitching procedure.
Studying QCD and other gauge theories on quantum hardware requires the preparation of physically interesting states. The variational quantum eigensolver provides a way of performing vacuum state preparation on quantum hardware. In this work, variational quantum eigensolver is applied to pure SU(3) lattice Yang-Mills on a single plaquette and one dimensional plaquette chains. Bayesian optimization and gradient descent were investigated for performing the classical optimization. Ansatz states for plaquette chains are constructed in a scalable manner from smaller systems using domain decomposition and a stitching procedure analogous to the density matrix renormalization group. Small examples are performed on IBM's superconducting Manila processor.
Quantum simulations of the dynamics of QCD have been limited by the complexities of mapping the continuous gauge fields onto quantum computers. By parametrizing the gauge invariant Hilbert space in terms of plaquette degrees of freedom, we show how the Hilbert space and interactions can be expanded in inverse powers of N_{c}. At leading order in this expansion, the Hamiltonian simplifies dramatically, both in the required size of the Hilbert space as well as the type of interactions involved. Adding a truncation of the resulting Hilbert space in terms of local energy states we give explicit constructions that allow simple representations of SU(3) gauge fields on qubits and qutrits. This formulation allows a simulation of the real time dynamics of a SU(3) lattice gauge theory on a 5×5 and 8×8 lattice on ibm_torino with a CNOT depth of 113.
Computing the vacuum and energy spectrum in non-Abelian, interacting lattice gauge theories remains an open challenge, in part because approximating the continuum limit requires large lattices and huge Hilbert spaces. To address this difficulty with near-term quantum computing devices, we adapt the variational quantum eigensolver to non-Abelian gauge theories. We outline scaling advantages when using a spin-network basis to simulate the gauge-invariant Hilbert space and develop a systematic state preparation ansatz that creates gauge-invariant excitations while alleviating the barren plateau problem. We illustrate our method in the context of SU(2) Yang-Mills theory by testing it on a minimal toy model consisting of a single vertex in 3+1 dimensions. In this toy model, simulations allow us to investigate the impact of noise expected in current quantum devices.
We present an overview of a program to understand the low-energy physics of quantum Yang-Mills theory from a quantum-information perspective. Our setting is that of the Hamiltonian formulation of pure Yang-Mills theory in the temporal gauge on the lattice. Firstly, inspired by recent constructions for Z/2Z\mathbb{Z}/2\mathbb{Z}Z/2Z lattice gauge theory, in particular, Kitaev's toric code, we describe the gauge-invariant sector of Hilbert space by introducing a primitive quantum gate: the quantum parallel transporter. We then develop a non-Abelian generalization of Laplace interpolation to present an ansatz for the ground state of pure Yang-Mills theory which interpolates between the weak- and strong-coupling renormalization group fixed points. The resulting state acquires the structure of a tensor network, namely, a multiscale entanglement renormalization ansatz, and allows for the efficient computation of local observables and Wilson loops. Various refinements of the tensor network are discussed leading to several generalizations. Finally, the continuum limit of our ansatz as the lattice regulator is removed is then described. This paper is intended as an abstract for an ongoing program: there are still many open problems.
We develop a quantum algorithmic framework for the efficient simulation of Yang--Mills theories, including the SU(3)\mathrm{SU}(3)SU(3) gauge theory in Quantum Chromodynamics (QCD). The framework uses maximal-tree gauge in terms of gauge field variables that removes all local gauge redundancies. In the resulting gauge-fixed formulation and digitization in the field-amplitude basis, we show that Hamiltonian time evolution admits an efficient implementation based on quantum singular value transformation (QSVT). We derive upper bounds on the total number of qubits and gate complexity, finding polynomial scaling with the inverse simulation precision 1/εs1/\varepsilon_s1/εs​, lattice volume V\mathcal{V}V, gauge coupling ggg, and target energy scale EEE. Our results provide a rigorous complexity-theoretic demonstration that non-Abelian Yang--Mills theories can be simulated efficiently on quantum computers, paving the way toward first-principles quantum simulations of non-perturbative QCD dynamics.