Algorithms for perturbative analysis and simulation of quantum dynamics

Abstract

We develop general purpose algorithms for computing and utilizing both the Dyson series and Magnus expansion, with the goal of facilitating numerical perturbative studies of quantum dynamics. To enable broad applications to models with multiple parameters, we phrase our algorithms in terms of multivariable sensitivity analysis, for either the solution or the time-averaged generator of the evolution over a fixed time-interval. These tools simultaneously compute a collection of terms up to arbitrary order, and are general in the sense that the model can depend on the parameters in an arbitrary time-dependent way. We implement the algorithms in the open source software package \qiskitdynamics{}, utilizing the JAX array library to enable just-in-time compilation, automatic differentiation, and GPU execution of all computations. Using a model of a single transmon, we demonstrate how to use these tools to approximate fidelity in a region of model parameter space, as well as construct perturbative robust control objectives. We also derive and implement Dyson and Magnus-based variations of the recently introduced Dysolve algorithm [Shillito et al., Physical Review Research, 3(3):033266] for simulating linear matrix differential equations. We show how the pre-computation step can be phrased as a multivariable expansion computation problem with fewer terms than in the original method. When simulating a two-transmon entangling gate on a GPU, we find the Dyson and Magnus-based solvers provide a speedup over traditional ODE solvers, ranging from roughly $2\times$ to $4\times$ for a solution and $10\times$ to $60\times$ for a gradient, depending on solution accuracy.

Accepted version: Speedup results updated after rerunning simulations with updated packages, modified figures, and expanded discussions

References in corpus (7)

• QuTiP 2: A Python framework for the dynamics of open quantum systems
• The Magnus expansion and some of its applications
• Fault-Tolerant Quantum Dynamical Decoupling
• How to Enhance Dephasing Time in Superconducting Qubits
• Analytic control methods for high fidelity unitary operations in a weakly nonlinear oscillator
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Cited by in corpus (8)

• Dissipative Dynamics of Graph-State Stabilizers with Superconducting Qubits
• PANSATZ: Pulse-based Ansatz for Variational Quantum Algorithms
• Efficient Lindblad synthesis for noise model construction
• Fast microwave-driven two-qubit gates between fluxonium qubits with a transmon coupler
• GPU-accelerated Effective Hamiltonian Calculator
• SuperGrad: a differentiable simulator for superconducting processors
• Engineering Precise and Robust Effective Hamiltonians
• Decoherence, Perturbations and Symmetry in Lindblad Dynamics -- Implications for Diffractive Dissociation