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arXiv:2406.17719 · The Journal of Chemical Physics

Unifying methods for optimal control in non-Markovian quantum systems via process tensors

Carlos Ortega-Taberner, Eoin O'Neill, Eoin P. Butler, Gerald E. Fux, P. R. Eastham

Published Sep 28, 2024• 7 citations
arXivHTMLPDF

Abstract

The large dimensionality of environments is the limiting factor in applying optimal control to open quantum systems beyond the Markovian approximation. Various methods exist to simulate non-Markovian systems, which effectively reduce the environment to a number of active degrees of freedom. Here, we show that several of these methods can be expressed in terms of a process tensor in the form of a matrix-product-operator, which serves as a unifying framework to show how they can be used in optimal control and to compare their performance. The matrix-product-operator form provides a general scheme for computing gradients using back propagation and allows the efficiency of the different methods to be compared via the bond dimensions of their respective process tensors.

DOI
10.1063/5.0226031

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The large dimensionality of environments is the limiting factor in applying optimal control to open quantum systems beyond the Markovian approximation. Various methods exist to simulate non-Markovian systems, which effectively reduce the environment to a number of active degrees of freedom. Here, we show that several of these methods can be expressed in terms of a process tensor in the form of a matrix-product-operator, which serves as a unifying framework to show how they can be used in optimal control and to compare their performance. The matrix-product-operator form provides a general scheme for computing gradients using back propagation and allows the efficiency of the different methods to be compared via the bond dimensions of their respective process tensors.
Numerical methods for obtaining exact dynamics of non-Markovian open quantum systems are mostly limited to either small systems or to short-time evolution only. Here, we propose a new algorithm for computing process tensors--matrix product operator (MPO) representations that capture the environment influence--which achieves greatly enhanced computational scalings with system size, while maintaining linear scaling with simulation length. We build on recent developments in the field which allow for long-time evolutions through process tensors which have a time-translational invariance. These can be built for general Gaussian environments and generic coupling operators with the system using infinite time-evolving block decimation (iTEBD). We introduce a modified iTEBD algorithm using intermediate compression steps which bring down the computation time scaling with system size ddd from O(d8)\mathcal{O}(d^8)O(d8) to O(d4)\mathcal{O}(d^4)O(d4), as well as significantly lowering the required memory. To illustrate the power of this method, we apply it to the problem of dispersive qubit readout in circuit QED, which was previously out-of-reach numerically. The full treatment of the measurement resonator, which requires a large system space, combined with the long simulation times precipitated by the separation of timescales between the measurement drive and the environment dissipation, is now possible. The algorithm we introduce not only allows for capturing non-Markovian dynamics in large open quantum systems, but also further extends all the existing capabilities of process tensors, for example in quantum optimal control, or in computation of multi-time correlations or of steady states, to more complex systems with tens of levels.
The process tensor framework to open quantum systems provides the most general description of multi-time correlations in non-Markovian quantum dynamics. A compressed representation of a process tensor in terms of matrix product operators (MPO) can be used for numerically exact calculations of multi-time correlation functions in systems strongly coupled to a non-Markovian reservoir. We show here that the numerical scaling for computing multi-dimensional spectra can be significantly improved using a time-translation invariant MPO representation of the process tensor obtained from the uniform time-evolving matrix product operator (uniTEMPO) method. In particular, this approach provides a spectral representation of the non-Markovian dynamics that gives direct access to correlation functions in Fourier-space, avoiding explicit real-time evolution. We calculate linear and 2D electronic spectra for an example system and discuss the performance and numerical scaling of our simulations.
The interaction between a quantum system and its environment limits our ability to control it and perform quantum operations on it. We present an efficient method to find optimal controls for quantum systems coupled to non-Markovian environments, by using the process tensor to compute the gradient of an objective function. We consider state transfer for a driven two-level system coupled to a bosonic environment, and characterize performance in terms of speed and fidelity. This allows us to determine the best achievable fidelity as a function of process duration. We show there can be a trade-off between speed and fidelity, and that slower processes can have higher fidelity by exploiting non-Markovian effects.
Process tensors in matrix product operator representation (PT-MPOs) are a powerful tool to calculate the temporal evolution and multi-time correlations of open quantum systems. While much progress has been made in making the calculation of process tensors for larger system dimensions possible, the numerical cost of the subsequent propagation of the system typically scales as O(d6)\mathcal{O}(d^6)O(d6), which renders PT-MPO methods prohibitively expensive for system dimensions above several dozens. Here, we show how PT-MPO methods can be used efficiently when only some of the system states couple to the environment, and how combination with a Taylor expansion method allows for a reduction of the numerical complexity down to O(d3)\mathcal{O}(d^3)O(d3). We show that these methods enable efficient simulations of systems with several hundreds of states. This is particularly relevant for solid-state cavity-QED systems with multiple modes, or when calculating frequency-filtered correlation functions by coupling sensor systems to the system of interest.
Approaching the long-time dynamics of non-Markovian open quantum systems presents a challenging task if the bath is strongly coupled. Recent proposals address this problem through a representation of the so-called process tensor in terms of a tensor network. We show that for Gaussian environments highly efficient contraction to a matrix product operator (MPO) form can be achieved with infinite MPO evolution methods, leading to significant computational speed-up over existing proposals. The result structurally resembles open system evolution with carefully designed auxiliary degrees of freedom, as in hierarchical or pseudomode methods. Here, however, these degrees of freedom are generated automatically by the MPO evolution algorithm. Moreover, the semigroup form of the resulting propagator enables us to explore steady-state physics, such as phase transitions.
The paradigm of considering open quantum systems—i.e., focusing only on the system of interest, and treating the rest of the world as an effective environment—has proven to be a highly effective way to understand a range of quantum systems, across areas of study such as quantum optics, cold atoms, superconducting qubits, and impurities in solid-state systems. A common approach in many of these contexts has been to consider simplified approaches based on the Born and Markov approximations. While these approximations are indeed often appropriate in contexts such as quantum optics, the widespread application of these approximations has been driven more by simplicity than by accuracy. In particular, these Markovian treatments will fail in many cases, such as when coupling to the environment is not weak, when the environment is structured and has resonances, when the system couples to low-frequency modes of the environment, or when the questions of interest involve the propagation of information through the environment. Despite the fact that many real problems are non-Markovian, the Markov approximation is still widely used, as it is often assumed that a fully non-Markovian treatment is too complex to be practical. In this Perspective we discuss a recently developed set of techniques that address this challenge. Centering our discussion around the notion of the process tensor, we demonstrate that the generality of the process tensor concept, coupled with efficient tensor-network methods, opens the door to the description of a wide range of observable non-Markovian processes in a wide range of open quantum systems.