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A Floquet solver for Time Periodic Open Quantum Systems

Fenton Ross Clawson

Published Jan 1, 2024• 1 citation
Publisher

Abstract

Computational physics has been a backbone of experimental and theoretical physics since the 1940’s, when the first nuclear bomb and ballistic simulations were per- formed at Los Alamos Laboratory. Nowadays, however, simulation of open quan- tum systems has been formalized into coding packages and user-friendly functions, such as QuTiP or Qiskit, which allow users to simulate these systems without hav- ing the in-depth knowledge required to build the computational backbone from the ground-up. With the shift in modern quantum physics towards the realization of quantum computation, a strong computational background is still necessary to lend validity to theoretical results and to help explain experimental results. Here, a new solver is considered which fits well into the regime of many modern quantum com- putational experiments, i.e. one which works well for systems which have Hamil- tonians that are repeating in time. This solver, named FLiMESolve (for Floquet- Lindblad Master Equation Solver), is derived from a Floquet interpretation of the Lindblad master equation, is well suited for these time-periodic Hamiltonians, and will be integrated into QuTiP. This solver maintains the benefits of other solvers used in QuTiP, while superseding their usefulness within its own regime. FLiMESolve is able to efficiently and accurately simulate time-periodic systems while being faster and more accurate than other solvers and with the ability to relax the secular ap- proximation inherent to the Lindblad Equation.

DOI
10.33915/etd.12653

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•Sep 3, 2021•0 citations
Computational physics has been a backbone of experimental and theoretical physics since the 1940’s, when the first nuclear bomb and ballistic simulations were per- formed at Los Alamos Laboratory. Nowadays, however, simulation of open quan- tum systems has been formalized into coding packages and user-friendly functions, such as QuTiP or Qiskit, which allow users to simulate these systems without hav- ing the in-depth knowledge required to build the computational backbone from the ground-up. With the shift in modern quantum physics towards the realization of quantum computation, a strong computational background is still necessary to lend validity to theoretical results and to help explain experimental results. Here, a new solver is considered which fits well into the regime of many modern quantum com- putational experiments, i.e. one which works well for systems which have Hamil- tonians that are repeating in time. This solver, named FLiMESolve (for Floquet- Lindblad Master Equation Solver), is derived from a Floquet interpretation of the Lindblad master equation, is well suited for these time-periodic Hamiltonians, and will be integrated into QuTiP. This solver maintains the benefits of other solvers used in QuTiP, while superseding their usefulness within its own regime. FLiMESolve is able to efficiently and accurately simulate time-periodic systems while being faster and more accurate than other solvers and with the ability to relax the secular ap- proximation inherent to the Lindblad Equation.
The dynamics of open quantum systems is commonly described by quantum master equations derived under the assumption of weak system-bath coupling and a separation of timescales between system and bath. When the system is additionally subjected to a periodic driving, the validity of the resulting Floquet master equations is further restricted to regimes of weak or high-frequency driving. Here, we benchmark a set of commonly used Floquet master equations for a model of two locally driven spins coupled to a shared Ohmic reservoir at finite temperature. We systematically probe the accuracy of the equations as a function of the driving parameters, thus identifying limits of their applicability. Dynamical maps predicted by each master equation are compared against numerically exact non-Markovian simulations, tracking the full relaxation dynamics. We find that the accuracy of each master equation closely reflects the assumptions underlying its derivation. For the Floquet-Lindblad equation, errors can be strongly amplified near resonances where the secular approximation breaks down, while approaches that avoid the secular approximation perform better and exhibit a more systematic dependence of the error on driving frequency and amplitude.
For a closed quantum system described by a time-periodic Hamiltonian, Floquet theorem says that the time evolution operator can be written as $U(t,0)\ensuremath{\equiv}P(t){e}^{\frac{\ensuremath{-}i}{\ensuremath{\hbar}}{H}_{F}t}$ with P(t+T)=P(t)P(t+T)=P(t)P(t+T)=P(t), and HF{H}_{F}HF​ is Hermitian and time independent and called the Floquet Hamiltonian. In this work, we extend the Floquet theorem from closed to open systems described by a time-periodic Lindblad master equation with period $T=\frac{2\ensuremath{\pi}}{\ensuremath{\omega}}$. We derive an expansion for the Lindbladian in powers of $\frac{1}{\ensuremath{\omega}}$. Two examples are presented to illustrate the theorem. We find that the results given by the Lindbladian expansion based on the open system Floquet theorem agree well with the exact dynamics in the high-frequency limit. It may find applications in quantum engineering with systems subject to decoherence.
QuTiP, the Quantum Toolbox in Python (Johansson et al., 2012, Johansson et al., 2013), has been at the forefront of open-source quantum software for the past 13 years. It is used as a research, teaching, and industrial tool, and has been downloaded millions of times by users around the world. Here we introduce the latest developments in QuTiP v5, which are set to have a large impact on the future of QuTiP and enable it to be a modern, continuously developed and popular tool for another decade and more. We summarize the code design and fundamental data layer changes as well as efficiency improvements, new solvers, applications to quantum circuits with QuTiP-QIP, and new quantum control tools with QuTiP-QOC. Additional flexibility in the data layer underlying all “quantum objects” in QuTiP allows us to harness the power of state-of-the-art data formats and packages like JAX, CuPy, and more. We explain these new features with a series of both well-known and new examples. The code for these examples is available in a static form on GitHub ( https://github.com/qutip/qutip-paper-v5-examples ) and as continuously updated and documented notebooks in the qutip-tutorials package ( https://github.com/qutip/qutip-tutorials ).
Analogue and digital Hamiltonian simulation represent two distinct approaches to making one quantum system replicate the physical properties of another, with consequences ranging from practical applications to complexity theory and quantum gravity. They are rooted in two distinct regimes of quantum dynamics: time-independent Hamiltonians for the former; fully controllable time-dependent quantum circuits in the latter. Time-periodic Hamiltonians represent an intermediate regime. In this work, we establish a theory of Floquet simulation where we use periodically driven Hamiltonians to synthesise time-independent ones. We show that a set of interactions SSS can Floquet-simulate every Hamiltonian in the Lie algebra Lie(S)\mathrm{Lie}(S)Lie(S), and consequently we give a complete and constructive characterisation of Floquet-universal Hamiltonians that are able to produce any target Hamiltonian. Notably, our construction uses only O(1)O(1)O(1) local interaction strengths and ratios thereof, avoiding the impractical multi-scale local interaction strengths often required in analogue simulation via time-independent Hamiltonians. Instead, the construction requires a range of driving frequencies which scales polynomially with system size for important classes such as kkk-local lattice Hamiltonians. This has immediate complexity-theoretic implications, including numerous BQP-completeness and QMA-hardness results for natural physical properties and quantities studied in Floquet physics.
The characterization of open quantum systems is a central and recurring problem for the development of quantum technologies. For time-independent systems, an (often unique) steady state describes the average physics once all the transient processes have faded out, but interesting quantum properties can emerge at intermediate timescales. Given a Lindblad master equation, these properties are encoded in the spectrum of the Liouvillian whose diagonalization, however, is a challenge even for small-size quantum systems. Here, we propose a new method to efficiently provide the Liouvillian spectral decomposition. We call this method an Arnoldi-Lindblad time evolution, because it exploits the algebraic properties of the Liouvillian superoperator to efficiently construct a basis for the Arnoldi iteration problem. The advantage of our method is double: (i) It provides a faster-than-the-clock method to efficiently obtain the steady state, meaning that it produces the steady state through time evolution shorter than needed for the system to reach stationarity. (ii) It retrieves the low-lying spectral properties of the Liouvillian with a minimal overhead, allowing to determine both which quantum properties emerge and for how long they can be observed in a system. This method is general and model-independent, and lends itself to the study of large systems where the determination of the Liouvillian spectrum can be numerically demanding. Our results can be extended to time evolution with a time-dependent Liouvillian. In particular, our method works for Floquet (i.e., periodically driven) systems, where it allows not only to construct the Floquet map for the slow-decaying processes, but also to retrieve the stroboscopic steady state and the eigenspectrum of the Floquet map. We demonstrate the efficiency of our method on several examples of coupled bosonic systems.