Unitary Feedback Research Articles

Measurement and Feedback Driven Entanglement Transition in the Probabilistic Control of Chaos.

We uncover a dynamical entanglement transition in a monitored quantum system that is heralded by a local order parameter. Classically, chaotic systems can be stochastically controlled onto unstable periodic orbits and exhibit controlled and uncontrolled phases as a function of the rate at which the control is applied. We show that such control transitions persist in open quantum systems where control is implemented with local measurements and unitary feedback. Starting from a simple classical model with a known control transition, we define a quantum model that exhibits a diffusive transition between a chaotic volume-law entangled phase and a disentangled controlled phase. Unlike other entanglement transitions in monitored quantum circuits, this transition can also be probed by correlation functions without resolving individual quantum trajectories.

Forward Action to Stabilize multiple Time-Delays MIMO Systems

In this paper, the problem of closed-loop stability of linear time-invariant systems with multiple time-delays is studied. We prove that stability of closed-loop configurations of such systems with unitary feedback can be guaranteed by means of a forward control action. The case of single-input single-output (SISO) systems is first considered, and then, an extension to the more general case of multi-input multi-output (MIMO) systems with multiple time-delays is dealt with. Numerical examples illustrating the theoretical results are also discussed.

Feedback‐Assisted Quantum Search by Continuous‐Time Quantum Walks

AbstractThe quantum search of a target node on a cycle graph by means of a quantum walk assisted by continuous measurement and feedback are addressed. Unlike previous spatial search approaches, where the oracle is described as a projector on the target state, a dynamical oracle implemented through a feedback Hamiltonian is considered. The idea is based on continuously monitoring the position of the quantum walker on the graph and then applying a unitary feedback operation based on the information obtained from measurement. The feedback changes the couplings between the nodes and it is optimized at each time via a numerical procedure. The stochastic trajectories describing the evolution for graphs of dimensions up to are numerically simulated, and the performance of the protocol is quantified via the average fidelity between the state of the walker and the target node. Different constraints on the control strategy are discussed. For unbounded controls, the protocol is able to quickly localize the walker on the target node. How the performance is lowered by posing an upper bound on the control couplings is then discussed. Finally, it is shown how a digital feedback protocol seems in general as efficient as the continuous bounded one.

Error correction of quantum system dynamics via measurement–feedback control

In this paper, we generalize the application of a recently proposed quantum control technique called self-fulfilling prophesy (Uys et al 2018 Phys. Rev. A 97 060102) to achieve high-fidelity target quantum dynamics which are disturbed by unwanted external couplings, based on sequential unsharp measurements and tailored unitary feedback operations. The unsharp measurements are carried out periodically during system evolution, while the feedback operations are well-designed based on the eigenstates of the density matrices of the expected (without external couplings) dynamical states and their corresponding post-measurement states. By means of quantum state estimation, the mechanism of the measurement–feedback scheme is explained generally. For illustrative examples, we show that the dynamical trajectory errors caused by both constant couplings and time-varying couplings are substantially reduced in typical two-level and multi-level systems, i.e. the high-fidelity quantum dynamics are achieved. Furthermore, we discuss the influence of external coupling strength and measurement strength on system dynamics. Crucially, the scheme is universal and can be applied to any dimension systems. Thus, it may find applications in quantum information processing.

Time-varying feedback matrices in feedback delay networks and their application in artificial reverberation

This paper introduces a time-variant reverberation algorithm as an extension of the feedback delay network (FDN). By modulating the feedback matrix nearly continuously over time, a complex pattern of concurrent amplitude modulations of the feedback paths evolves. Due to its complexity, the modulation produces less likely perceivable artifacts and the time-variation helps to increase the liveliness of the reverberation tail. A listening test, which has been conducted, confirms that the perceived quality of the reverberation tail can be enhanced by the feedback matrix modulation. In contrast to the prior art time-varying allpass FDNs, it is shown that unitary feedback matrix modulation is guaranteed to be stable. Analytical constraints on the pole locations of the FDN help to describe the modulation effect in depth. Further, techniques and conditions for continuous feedback matrix modulation are presented.

Global versus local optimality in feedback-controlled qubit purification: new insights from minimizing Rényi entropies

It was first shown by Jacobs, in 2003, that the process of qubit state purification by continuous measurement of one observable can be enhanced, on average, by unitary feedback control. Here, we quantify this by the reduction in any one of the family of Rényi entropies , with , at some terminal time, revealing the rich structure of stochastic quantum control even for this simple problem. We generalize Jacobs’ original argument, which was for the (unique) impurity measure with a linear evolution map under his protocol, by replacing linearity with convexity, thereby making it applicable to Rényi entropies for α in a finite interval about one. Even with this generalization, Jacobs’ argument fails to identify the surprising fact, which we prove by Bellmanʼs principle of dynamic programming, that his protocol is globally optimal for all Rényi entropies whose decrease is locally maximized by that protocol. Also surprisingly, even though there is a range of Rényi entropies whose decrease is always locally maximized by the null-control protocol, that null-control protocol cannot be shown to be globally optimal in any instance. These results highlight the non-intuitive relation between local and global optimality in stochastic quantum control.

Quantum effects improve the energy efficiency of feedback control.

The laws of thermodynamics apply equally well to quantum systems as to classical systems, and because of this, quantum effects do not change the fundamental thermodynamic efficiency of isothermal refrigerators or engines. We show that, despite this fact, quantum mechanics permits measurement-based feedback control protocols that are more thermodynamically efficient than their classical counterparts. As part of our analysis, we perform a detailed accounting of the thermodynamics of unitary feedback control and elucidate the sources of inefficiency in measurement-based and coherent feedback.

Differential Evolution for Tuning Fractional Order Controllers Approximated by Particle Swarm Optimization

AbstractFractional-Order Controllers (FOC) extend PID controllers by using non-integer order integral and derivative actions. They have many advantages even if the tuning methods are still at their infancy. Hence we apply Differential Evolution (DE) optimization to tune parameters of FOC in a unitary feedback linear control system of a first order plana with a lag plus an integrator. The FOC combines two differintegrators. Closed-loop specifications are peak value and settling time of the step response. To determine the controller parameters, DE minimizes the sum of the squared real and imaginary parts of the characteristic equation, expressed in terms of the desired pair of closed-loop dominant poles and of the unknown parameters. Then, the Particle Swarm Optimization with constriction factor approximates the irrational operators of the controller with rational transfer functions that are of low order, stable, minimum-phase, with zeros interlacing poles. Frequency and step responses show the efficiency of the proposed method.

Feedback Stabilization of Isospectral Control Systems on Complex Flag Manifolds: Application to Quantum Ensembles

In an N-level quantum mechanical system, the problem of unitary feedback stabilization of mixed density operators to periodic orbits admits a natural Lyapunov-based time-varying feedback design. A global description of the domain of attraction of the closed-loop system can be provided based on a ``root-space''-like structure of the space of density operators. This convex set foliates as a complex flag manifold where each leaf is identified with the coadjoint orbit of the eigenvalues of the density operator. The converging conditions are time-independent but depend from the topology of the flag manifold: it is shown that the closed loop must have a number of equilibria at least equal to the Euler characteristic of the manifold, thus imposing obstructions of topological nature to global stabilizability.

CONTROL BY (STATE–MODULATED) INTERCONNECTION OF PORT–HAMILTONIAN SYSTEMS

It is well known that the dynamics of many physical processes can be suitably described by Port–Hamiltonian (PH) models. In this paper we consider the Passivity–Based Control (PBC) technique of Control by Interconnection (CbI), where the controller is another PH system connected to the plant to add up their energy functions. We propose two extensions to this method, first, we exploit the non–uniqueness of the PH representation of the system to generate new cyclo–passive outputs. Applying CbI through these new port variables overcomes the so–called dissipation obstacle. Second, when the plant state variables are measurable, we show that the conditions for applicability of the method can be relaxed replacing the simple unitary feedback by a state–modulated interconnection. A central contribution of the paper is the proof that the conditions for energy shaping via CbI are equivalent to those imposed in Interconnection and Damping Assignment PBC, providing in this way a nice geometric interpretation to this successful controller design technique.

Evolution of a qubit under the influence of a succession of weak measurements with unitary feedback

We investigate the evolution of a single qubit subject to a continuous unitary dynamics and an additional interrupting influence which occurs periodically. One may imagine a dynamically evolving closed quantum system which becomes open at certain times. The interrupting influence is represented by an operation, which is assumed to equivalently describe a non-selective unsharp measurement. It may be decomposed into a positive operator, which in case of a measurement represents the pure measurement part, followed by an unitary back-action operator. Equations of motion for the state evolution are derived in the form of difference equations. It is shown that the 'free' Hamiltonian is completed by an averaged Hamiltonian, which goes back to the back-action. The positive operator specifies a decoherence rate and results in a decoherence term. The continuum limit to a master equation is performed. The selective evolution is discussed and correcting higher order terms are worked out in an Appendix.