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Digital Pseudoquantum Simulation of $\mathbb{Z}_2$ Gauge Higgs Model

Yiming Ding, Xiaopeng Cui, Yu Shi

Published Aug 31, 2021• 0 citations
OpenAlex

Abstract

We perform a digital pseudoquantum simulation of $\mathbb{Z}_2$ gauge Higgs model on a $3\times 3$ lattice. First we propose the quantum algorithm for the digital quantum simulation, based on Trotter decomposition, quantum adiabatic algorithm and its circuit realization. Then we classically demonstrate it in a GPU simulator, obtaining useful results, which indicate the topological properties of deconfined phase and clarify the phase diagram. Especially, our work suggests that the tricitical point, where the two critical lines of second-order transitions meet, lies on the critical line of the first-order transition rather than its end.

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Digital Pseudoquantum Simulation of Z2\mathbb{Z}_2Z2​ Gauge Higgs Model
We perform a digital pseudoquantum simulation of Z2\mathbb{Z}_2Z2​ gauge Higgs model on a 3×33\times 33×3 lattice. First we propose the quantum algorithm for the digital quantum simulation, based on Trotter decomposition, quantum adiabatic algorithm and its circuit realization. Then we classically demonstrate it in a GPU simulator, obtaining useful results, which indicate the topological properties of deconfined phase and clarify the phase diagram. Especially, our work suggests that the tricitical point, where the two critical lines of second-order transitions meet, lies on the critical line of the first-order transition rather than its end.
We present a quantum algorithm for digital quantum simulation of the Z2{\mathbb{Z}}_{2}Z2​ gauge--Higgs model on a $3\ifmmode\times\else\texttimes\fi{}3$ lattice, which is based on Trotter decomposition, the quantum adiabatic algorithm, and its circuit realization. Then, we perform a classical demonstration, dubbed a pseudoquantum simulation, on a GPU simulator. We obtain useful results on this model, which suggest the topological properties of the deconfined phase and help to clarify the phase diagram. It is suggested that the tricitical point, where the second-order critical lines of deconfinement-confinement transition and of deconfinement-Higgs transition meet, seems to be on the first-order critical line of confinement-Higgs transition, at a point other than the end of this critical line.
A bstract Gauge theory is the framework of the Standard Model of particle physics and is also important in condensed matter physics. As its major non-perturbative approach, lattice gauge theory is traditionally implemented using Monte Carlo simulation, consequently it usually suffers such problems as the Fermion sign problem and the lack of real-time dynamics. Hopefully they can be avoided by using quantum simulation, which simulates quantum systems by using controllable true quantum processes. The field of quantum simulation is under rapid development. Here we present a circuit-based digital scheme of quantum simulation of quantum ℤ 2 lattice gauge theory in 2 + 1 and 3 + 1 dimensions, using quantum adiabatic algorithms implemented in terms of universal quantum gates. Our algorithm generalizes the Trotter and symmetric decompositions to the case that the Hamiltonian varies at each step in the decomposition. Furthermore, we carry through a complete demonstration of this scheme in classical GPU simulator, and obtain key features of quantum ℤ 2 lattice gauge theory, including quantum phase transitions, topological properties, gauge invariance and duality. Hereby dubbed pseudoquantum simulation, classical demonstration of quantum simulation in state-of-art fast computers not only facilitates the development of schemes and algorithms of real quantum simulation, but also represents a new approach of practical computation.
Trotter decomposition is the basis of the digital quantum simulation. Asymmetric and symmetric decompositions are used in our GPU demonstration of the digital adiabatic quantum simulations of (2[Formula: see text]+[Formula: see text]1)-dimensional quantum [Formula: see text] lattice gauge theory. The actual errors in Trotter decompositions are investigated as functions of the coupling parameter and the number of Trotter substeps in each step of the variation of coupling parameter. The relative error of energy is shown to be equal to the Trotter error usually defined in terms of the evolution operators. They are much smaller than the order-of-magnitude estimation. The error in the symmetric decomposition is much smaller than that in the asymmetric decomposition. The features of the Trotter errors obtained here are useful in the experimental implementation of digital quantum simulation and its numerical demonstration.
To investigate gauge theories with near-term quantum computers warrants exploration of nontraditional quantum simulators to find resource-efficient simulation protocols and ultimately access exotic features of different field theories, including unexplored regimes of the QCD phase diagram. In this work, using superconducting transmon qutrit processors, we formulate and implement a pulse-based, three-level, hybrid analog-digital simulation protocol of the (1+1) dimensional Abelian Higgs model (AHM) on two sites. Alongside this approach, we experimentally realize a gate-based implementation of the same model. Using the natural mapping of the three-level truncation of the transmon Hilbert space to the spin-1 truncated AHM, we observe real time dynamics of AHM field observables, which are analogous to electric field operators, with both protocols. For the analog-digital protocol, we engineer a Floquet simulation with a combination of local analog drives, driven modification of the natural interaction Hamiltonian of the two transmons, and dynamical decoupling pulses. For the digital protocol, we use a state-of-the-art qutrit processor to implement a Trotterized simulation of the model incorporating advanced error mitigation techniques. We further discuss the scalability of the two approaches, and their potential to be extended to the simulation of other model Hamiltonians. Our experiments demonstrate a viable platform for future studies of spin-1 and SU(3) based gauge theory models on current and near-term transmon qutrit processors.
Two-dimensional Z2\mathbb{Z}_2Z2​ lattice gauge theory on a near-term quantum simulator: variational quantum optimization, confinement, and topological order
We propose an implementation of a two-dimensional Z2\mathbb{Z}_2Z2​ lattice gauge theory model on a shallow quantum circuit, involving a number of single and two-qubits gates comparable to what can be achieved with present-day and near-future technologies. The ground state preparation is numerically analyzed on a small lattice with a variational quantum algorithm, which requires a small number of parameters to reach high fidelities and can be efficiently scaled up on larger systems. Despite the reduced size of the lattice we consider, a transition between confined and deconfined regimes can be detected by measuring expectation values of Wilson loop operators or the topological entropy. Moreover, if periodic boundary conditions are implemented, the same optimal solution is transferable among all four different topological sectors, without any need for further optimization on the variational parameters. Our work shows that variational quantum algorithms provide a useful technique to be added in the growing toolbox for digital simulations of lattice gauge theories.
We construct a mapping between the two-dimensional toric code model in external magnetic fields, hz{h}_{z}hz​ and hx{h}_{x}hx​, and the three-dimensional classical Ising system with plaquette interactions, which is equivalent to the three-dimensional Z2{\mathbb{Z}}_{2}Z2​ gauge Higgs model with anisotropy between the imaginary time and spatial directions. The isotropic limit of the latter model was studied using Monte Carlo simulations on large (up to 603{60}^{3}603) lattices in order to determine the stability of the topological phase against generic magnetic field perturbations and to resolve fine details of the phase diagram. We find that the topological phase is bounded by second-order transition lines, which merge into a first-order line at what appears to be a multicritical point arising from the competition between the Higgs and confinement transitions in the Z2{\mathbb{Z}}_{2}Z2​ gauge system. An effective field theory for this type of multicritical point (if one actually exists) is not known. Our results have potential applications to frustrated magnets, quantum computation, lattice gauge models in particle physics, and critical phenomena.