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arXiv:2009.00247 · International Journal of Modern Physics B

Trotter errors in digital adiabatic quantum simulation of quantum ℤ2 lattice gauge theory

Xiaopeng Cui, Yu Shi

Published Aug 19, 2020• 3 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1142/s0217979220502926

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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.
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.
A higher-order Suzuki-Trotter decomposition or Trotterization can be exploited to mitigate the Trotter error in digital quantum simulation. This work revisits the second-order symmetric Trotterization in terms of the Trotter error, where quantum many-body spin dynamics of the transverse-field Ising model is simulated. While the work presents a pedagogical way to exploit a real quantum computer, the effectiveness of the symmetric Trotterization is evaluated in a prototype superconducting quantum device on IBM Quantum Experience. It turns out that the symmetric Trotterization does not provide higher accuracy than the first-order Trotterization in the testbed using the transverse-field Ising model. The result indicates that apart from the quantum errors, such as logical gate error and readout error, the use of a higher-order Trotterization should be circumspect, and the Trotter error would play an insignificant role in particular applications in an early stage of realized noisy intermediate-scale quantum (NISQ) devices.
The simulation of adiabatic evolution has deep connections with adiabatic quantum computation, the quantum approximate optimization algorithm, and adiabatic state preparation. Here we address the error analysis problem in quantum simulation of adiabatic process using Trotter formulas. We show that with additional conditions, the circuit depth can be linear in simulation time TTT. The improvement comes from the observation that the fidelity error here can't be estimated by the norm distance between evolution operators. This phenomenon is termed the robustness of discretization in digital adiabatic simulation. It can be explained in three steps, from analytical and numerical evidence: (1) The fidelity error should be estimated by applying adiabatic theorem on the effective Hamiltonian instead. (2) Because of the specialty of Riemann-Lebesgue lemma, most adiabatic process is naturally robust against discretization. (3) As the Trotter step gets larger, the spectral gap of effective Hamiltonian tends to close, which results in the failure of digital adiabatic simulation.
The Trotter decomposition is a basic approach to Hamiltonian simulation (digital quantum simulation). The first-order Trotter decomposition is the simplest one, whose deviations from target dynamics are of the first order of a small coefficient in terms of the infidelity. In this paper, we consider the first-order Trotter decomposition in the dynamical-invariant basis. By using a state-dependent inequality, we point out that deviations of this decomposition are of the second order of a small coefficient. Moreover, we also show that this decomposition includes a useful example, i.e., digital implementation of shortcuts to adiabaticity by counterdiabatic driving.
The simulation of adiabatic evolution has deep connections with Adiabatic Quantum Computation, the Quantum Approximate Optimization Algorithm and adiabatic state preparation. Here we address the error analysis problem in quantum simulation of adiabatic process using Trotter formulas. We show that with additional conditions, the circuit depth can be linear in simulation time TTT. The improvement comes from the observation that the fidelity error here can't be estimated by the norm distance between evolution operators. This phenomenon is termed as the robustness of discretization in digital adiabatic simulation. It can be explained in three steps, from analytical and numerical evidence: 1. the fidelity error should be estimated by applying adiabatic theorem on the effective Hamiltonian instead. 2. because of the specialty of Riemann-Lebesgue lemma, most adiabatic process is naturally robust against discretization. 3. as the Trotter step gets larger, the spectral gap of effective Hamiltonian tends to close, which results in the failure of digital adiabatic simulation.
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.