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Journal of Physics A Mathematical and General

Unitary integration with operator splitting for weakly dissipative systems

Brad A. Shadwick, Walter F. Buell

Published May 24, 2001• 8 citations
Publisher

Abstract

Unitary integration is a numerical method that preserves the structure of the quantum Liouville equation by evolving the density via unitary transformations. Unitary integrators preserve the kinematic invariants C(sub j) = trp(sup j), j = 1,..., n to all orders in the time step. Here we extend unitary integration to weakly dissipative systems. We apply the technique of operator splitting, using a unitary integrator for the Hamiltonian evolution and a conventional integrator for the dissipative piece. In this way, we guarantee that all dissipation and decoherence (variation of the C(sub j)) is due to the new non-Hamiltonian terms and not to any numerical artifacts. We illustrate the method with examples.

DOI
10.1088/0305-4470/34/22/315

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Unitary integration is a numerical method that preserves the structure of the quantum Liouville equation by evolving the density via unitary transformations. Unitary integrators preserve the kinematic invariants C(sub j) = trp(sup j), j = 1,..., n to all orders in the time step. Here we extend unitary integration to weakly dissipative systems. We apply the technique of operator splitting, using a unitary integrator for the Hamiltonian evolution and a conventional integrator for the dissipative piece. In this way, we guarantee that all dissipation and decoherence (variation of the C(sub j)) is due to the new non-Hamiltonian terms and not to any numerical artifacts. We illustrate the method with examples.
The quantum Liouville equation for an nnn-level atomic system driven by external fields has a nontrivial kinematic structure; the quantities $\mathrm{tr}{\ensuremath{\rho}}^{j}$, j=1,2,…,nj\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1,2,\dots{},nj=1,2,…,n remain constant in time, independent of the Hamiltonian. These invariants are physically significant; the qualitative character of the solution depends on their existence. A generic numerical method will not, in general, preserve these invariants. We present a numerical technique which evolves the density matrix via unitary transformations thus exactly preserving these invariants to all orders in the time step.
Dissipation and irreversibility are central to many physical systems, yet they lead to non-unitary dynamics that are challenging to realise on quantum processors. High-order operator splitting is an attractive approach for simulating unitary dynamics, yet conventional product formulas introduce negative time steps at high orders that are ill-conditioned for dissipative dynamics. We show how block encodings of complex-coefficient product formulas can be constructed by a sequence of simple Hamiltonian evolutions in real and imaginary time with high-order accuracy. The unitary substages use positive real coefficients, while the dissipative substages use complex coefficients with positive real parts; the real parts preserve the contractive evolution, and the imaginary parts are additional unitary evolutions. We use known formulas of orders 4 and 6 and give a new symmetric formula of order 8 in this restricted class. We apply the approach to the classical problem of lossy mechanical wave propagation in statevector simulations and on a trapped-ion quantum processor. Orders 4 and 6 require comparable CNOT gates to orders 1 and 2 at large error tolerances, but quickly become more efficient as the tolerance is reduced. On the quantum hardware, a single step of order 4 achieves lower error than order 1 and comparable error to order 2, while order 6 is noise-limited. Our results show that operator splitting can combine practical circuit constructions with high-order convergence for non-unitary dissipative dynamics.
Dissipation and decoherence, and the evolution from pure to mixed states in quantum physics, are handled through master equations for the density matrix. By embedding elements of this matrix in a higher-dimensional Liouville-Bloch equation, the methods of unitary integration are adapted to solve for the density matrix as a function of time, including the nonunitary effects of dissipation and decoherence. The input requires only solutions of classical, initial value time-dependent equations. Results are illustrated for a damped, driven two-level system.
Splitting methods constitute a widely used class of numerical integrators for ordinary and partial differential equations, particularly well suited to problems that can be decomposed into simpler subproblems. High-order splitting schemes are available that achieve high accuracy while preserving key qualitative properties of the underlying dynamical system, and are successfully used across a broad range of fields. In this work, we present a systematic analysis of both local and global errors arising from arbitrary splitting methods applied to unitary problems. Two complementary types of error estimates are derived. The first is expressed in terms of operator norms, while the second is formulated using norms of commutators and can, under suitable assumptions, be extended to certain classes of unbounded operators. Special attention is devoted to the case where only two operators are involved. The theoretical results are illustrated by deriving explicit error bounds for some representative schemes.
We discuss a new analytical approach to real-time evolution in quantum many-body systems. Our approach extends the framework of continuous unitary transformations such that it amounts to a novel solution method for the Heisenberg equations of motion for an operator. It is our purpose to illustrate the accuracy of this approach by studying dissipative quantum systems on all timescales. In particular, we obtain results for non-equilibrium correlation functions for general initial conditions. We illustrate our ideas for the exactly solvable dissipative oscillator and, as a non-trivial model, for the dissipative two-state system.
If an open quantum system is initially uncorrelated from its environment, then its dynamics can be written in terms of a Lindblad-form master equation. The master equation is divided into a unitary piece, represented by an effective Hamiltonian, and a dissipative piece, represented by a hermiticity-preserving superoperator; however, the division of open system dynamics into unitary and dissipative pieces is non-unique. For finite-dimensional quantum systems, we resolve this non-uniqueness by specifying a norm on the space of dissipative superoperators and defining the canonical Hamiltonian to be the one whose dissipator is minimal. We show that the canonical Hamiltonian thus defined is equivalent to the Hamiltonian initially defined by Lindblad, and that it is uniqueley specified by requiring the dissipator's jump operators to be traceless. For a system weakly coupled to its environment, we give a recursive formula for computing the canonical effective Hamiltonian to arbitrary orders in perturbation theory, which we can think of as a perturbative scheme for renormalizing the system's bare Hamiltonian.