[Paper Review] Stoquastic simulations of non-stoquastic superconducting flux circuits

Kol-Namer, Harel, Tom Halverson+3 more|arXiv (Cornell University)|Nov 7, 2020

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper demonstrates that non-stoquastic superconducting flux circuits—previously thought to be classically unsimulable due to the quantum Monte Carlo (QMC) sign problem—can be efficiently simulated via direct circuit-level modeling. By treating flux and charge operators in a continuous Hilbert space and applying coordinate transformations, the authors show that sign-problem-free QMC simulations are possible, challenging the assumption that such circuits enable universal adiabatic quantum computation or quantum supremacy.

ABSTRACT

There is a tremendous interest in fabricating superconducting flux circuits that are nonstoquastic -- i.e., have positive off-diagonal matrix elements -- in their qubit representation, as these circuits are thought to be unsimulable by classical approaches due to the presence of a sign problem and thus could play a key role in the demonstration of speedups in quantum annealing protocols. We show, however, that the elimination of the sign problem in these systems is possible by the direct simulation of the flux circuits. Our approach not only obviates the reduction of flux circuits to their qubit representation but also produces results that are more in the spirit of the experimental setup. We discuss the implications of our work, arguing that our findings cast doubt on the conception that superconducting flux circuits represent the correct avenue for universal adiabatic quantum computers.

Motivation & Objective

• To challenge the widely held belief that non-stoquastic superconducting flux circuits are classically unsimulable due to the QMC sign problem.
• To demonstrate that the apparent non-stoquasticity arises from qubit-level approximations, not fundamental circuit-level properties.
• To develop a scalable, sign-problem-free QMC simulation method for superconducting flux circuits by leveraging continuous flux and charge variables.
• To assess whether such circuits can truly support non-stoquastic dynamics necessary for AQC universality and quantum supremacy.
• To explore the implications for future quantum hardware design, particularly in light of the possibility that all non-stoquastic spin models may be low-energy approximations of higher-dimensional stoquastic systems.

Proposed method

• The authors model superconducting flux circuits using continuous flux and charge operators, satisfying canonical commutation relations $[\hat{q}_j, \hat{\phi}_k] = -i\hbar \delta_{jk}$.
• They discretize the flux variables $\hat{\phi}_k$ onto a uniform grid with spacing $\Delta$, transforming the potential energy into a diagonal matrix in the flux basis.
• A coordinate transformation is applied to the charge operators $\hat{q}_k$ to render the kinetic energy term as a Laplacian in the continuous limit, enabling efficient QMC sampling.
• The resulting Hamiltonian is cast in a form amenable to sign-problem-free Quantum Monte Carlo (QMC) simulations, avoiding the sign problem inherent in the qubit representation.
• The method is validated by comparing ODE-based QMC results with exact diagonalization for persistent current and ground state energy across various annealing trajectories.
• The approach is scalable, with computational cost scaling polynomially in system size $n$, assuming constant grid spacing $\Delta$.

Experimental results

Research questions

• RQ1 Can non-stoquastic superconducting flux circuits, which are believed to be unsimulable by classical methods, be efficiently simulated using Quantum Monte Carlo?
• RQ2 Does the apparent non-stoquasticity in qubit-level representations arise from the qubit approximation, rather than from the underlying circuit physics?
• RQ3 Is it possible to construct a sign-problem-free QMC simulation for superconducting flux circuits by working directly in the continuous flux-charge Hilbert space?
• RQ4 What modifications to the Hamiltonian are required for a superconducting circuit to exhibit true non-stoquasticity at the circuit level?
• RQ5 Can all non-stoquastic spin models be understood as low-energy approximations of higher-dimensional stoquastic systems?

Key findings

• The qubit-level representation of superconducting flux circuits appears non-stoquastic and thus classically unsimulable due to the QMC sign problem, but this is an artifact of the qubit approximation.
• When modeled at the circuit level using continuous flux and charge variables, the Hamiltonian becomes stoquastic and amenable to efficient, sign-problem-free QMC simulations.
• The authors achieve agreement between ODE-based QMC and exact diagonalization for persistent current and ground state energy across multiple annealing paths, validating the method.
• The simulation cost scales polynomially in system size $n$, assuming constant grid spacing $\Delta$, which is proportional to the inter-well spacing in flux hyperspace.
• The method remains efficient even for circuits with complex potential landscapes, as long as the charge operators can be transformed into Cartesian form.
• The results suggest that truly non-stoquastic dynamics in superconducting circuits require coupling between flux and charge operators, such as in the bifluxon qubit proposal, rather than standard rf-SQUID configurations.