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arXiv:2207.12385 · IEEE Conference on Decision and Control (CDC)

Applying classical control techniques to quantum systems: entanglement versus stability margin and other limitations

Carrie Ann Weidner, S. G. Schirmer, Frank C. Langbein, EDMOND A. JONCKHEERE

Published Dec 6, 2022• 8 citations
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Abstract

Development of robust quantum control has been challenging and there are numerous obstacles to applying classical robust control to quantum system including bilinearity, marginal stability, state preparation errors, nonlinear figures of merit. The requirement of marginal stability, while not satisfied for closed quantum systems, can be satisfied for open quantum systems where Lindbladian behavior leads to non-unitary evolution, and allows for nonzero classical stability margins, but it remains difficult to extract physical insight when classical robust control tools are applied to these systems. We consider a straightforward example of the entanglement between two qubits dissipatively coupled to a lossy cavity and analyze it using the classical stability margin and structured perturbations. We attempt, where possible, to extract physical insight from these analyses. Our aim is to highlight where classical robust control can assist in the analysis of quantum systems and identify areas where more work needs to be done to develop specific methods for quantum robust control.

DOI
10.1109/cdc51059.2022.9992337

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Development of robust quantum control has been challenging and there are numerous obstacles to applying classical robust control to quantum system including bilinearity, marginal stability, state preparation errors, nonlinear figures of merit. The requirement of marginal stability, while not satisfied for closed quantum systems, can be satisfied for open quantum systems where Lindbladian behavior leads to non-unitary evolution, and allows for nonzero classical stability margins, but it remains difficult to extract physical insight when classical robust control tools are applied to these systems. We consider a straightforward example of the entanglement between two qubits dissipatively coupled to a lossy cavity and analyze it using the classical stability margin and structured perturbations. We attempt, where possible, to extract physical insight from these analyses. Our aim is to highlight where classical robust control can assist in the analysis of quantum systems and identify areas where more work needs to be done to develop specific methods for quantum robust control.
Robust performance of control schemes for open quantum systems is investigated under classical uncertainties in the generators of the dynamics and nonclassical uncertainties due to decoherence and initial state preparation errors. A formalism is developed to measure performance based on the transmission of a dynamic perturbation or initial state preparation error to the quantum state error. This makes it possible to apply tools from classical robust control, such as structured singular value analysis. A difficulty arising from the singularity of the closed-loop Bloch equations for the quantum state is overcome by introducing the #-inversion lemma, a specialized version of the matrix inversion lemma. Under some conditions, this guarantees continuity of the structured singular value ats=0s = 0s=0. Additional difficulties occur when symmetry gives rise to multiple open-loop poles, which under symmetry-breaking unfold into single eigenvalues. The concepts are applied to systems subject to pure decoherence and a general dissipative system example of two qubits in a leaky cavity under laser driving fields and spontaneous emission. A nonclassical performance index, steady-state entanglement quantified by the concurrence, a nonlinear function of the system state, is introduced. Simulations confirm a conflict between entanglement, its log-sensitivity and stability margin under decoherence.
Abstract A major obstacle in the way of practical quantum computing is achieving scalable and robust high-fidelity entangling gates. To this end, quantum control has become an essential tool, as it can make the entangling interaction resilient to sources of noise. Nevertheless, it may be difficult to identify an appropriate quantum control technique for a particular need given the breadth of work pertaining to robust entanglement. To this end, we attempt to consolidate the literature by providing a non-exhaustive summary and critical analysis. The quantum control methods are separated into two categories: schemes which extend the robustness to (i) spin or (ii) motional decoherence. We choose to focus on extensions of the σ x ⊗ σ x Mølmer–Sørensen interaction using microwaves and a static magnetic field gradient. Nevertheless, some of the techniques discussed here can be relevant to other trapped ion architectures or physical qubit implementations. Finally, we experimentally realize a proof-of-concept interaction with simultaneous robustness to spin and motional decoherence by combining several quantum control methods presented in this manuscript.
We study the problem of robust performance of quantum systems under structured uncertainties. A specific feature of closed (Hamiltonian) quantum systems is that their poles lie on the imaginary axis and that neither a coherent controller nor physically relevant structured uncertainties can alter this situation. This changes for open systems where decoherence ensures asymptotic stability and creates a unique landscape of pure performance robustness, with the distinctive feature that closed-loop stability is secured by the underlying physics and needs not be enforced. This stability, however, is often detrimental to quantum-enhanced performance, and additive perturbations of the Hamiltonian give rise to dynamic generators that are nonlinear in the perturbed parameters, invalidating classical paradigms to assess robustness to structured perturbations such as singular value analysis. This problem is addressed using a fixed-point iteration approach to determine a maximum perturbation strength δmaxthat ensures that the transfer function remains bounded, ║Tδ║−1for δmax.
Abstract : The proposed work attempted to establish a theoretical framework for the analysis and control of quantum linear systems and their interactions with non-classical quantum fields by developing control theoretic concepts exploiting special features of quantum systems. The research aimed to continue developing a new mathematical framework for robust control of quantum linear systems and extend results to quantum linear systems subject to non-classical quantum fields. The major outcomes of this project are (i) derivation of quantum filtering equations for systems non-classical input states including single photon states, (ii) determination of how linear quantum systems respond to multichannel non-classical states, (iii) linear quantum models for quantum memories and the zero dynamics principle for perfect information transfer, (iv) development of new structured uncertainty methods that ensure robust stability of quantum systems based on nominal linear models, and (v) physical realizability results for finite level quantum systems.
This dissertation is a study of several aspects of discrete time quantum control. After reviewing the necessary background from quantum mechanics and control theory we construct a simple model and solve a control problem on it as a proof of concept for the discrete control formulation developed herein. We apply optimal control to solve a state preparation problem on a physically motivated time discretized single qubit system with simplified dynamics to enable the Bellman equation to be solved both numerically and approximately analytically. However for more than one qubit classical theories of control become problematic, a notable difficulty being the exponential increase in the size of the state space with the number of particles. To study workarounds for this problem we quantify how much information a quantum measurement can extract from the system and then use the results to show that a naive classical technique for controlling a multiparticle system performs poorly when the system has "quantum" properties. We then consider the problem of protecting the entanglement of a bipartite quantum system using discrete open loop control with a technique that seems to have no classical analog. The states of the bipartite system, called X-states, have a special structure which is generalized to states of systems with more than two qubits. We develop the properties of these generalized X-states which are a class of mixed states with interesting multipartite entanglement properties
This dissertation is a study of several aspects of discrete time quantum control. After reviewing the necessary background from quantum mechanics and control theory we construct a simple model and solve a control problem on it as a proof of concept for the discrete control formulation developed herein. We apply optimal control to solve a state preparation problem on a physically motivated time discretized single qubit system with simplified dynamics to enable the Bellman equation to be solved both numerically and approximately analytically. However for more than one qubit classical theories of control become problematic, a notable difficulty being the exponential increase in the size of the state space with the number of particles. To study workarounds for this problem we quantify how much information a quantum measurement can extract from the system and then use the results to show that a naive classical technique for controlling a multiparticle system performs poorly when the system has quantum properties. We then consider the problem of protecting the entanglement of a bipartite quantum system using discrete open loop control with a technique that seems to have no classical analog. The states of the bipartite system, called X-states, have a special structure which is generalized to states of systems with more than two qubits. We develop the properties of these generalized X-states which are a class of mixed states with interesting multipartite entanglement properties