Using results from quantum filtering theory and methods from classical control theory, we derive an optimal control strategy for an open two-level system (a qubit in interaction with the electromagnetic field) controlled by a laser. The aim is to optimally choose the laser's amplitude and phase in order to drive the system into a desired state. The Bellman equations are obtained for the case of diffusive and counting measurements for vacuum field states. A full exact solution of the optimal control problem is given for a system with simpler, linear, dynamics. These linear dynamics can be obtained physically by considering a two-level atom in a strongly driven, heavily damped, optical cavity.
We exploit the separation of the filtering and control aspects of quantum feedback control to consider the optimal control as a classical stochastic problem on the space of quantum states. We derive the corresponding Hamilton-Jacobi-Bellman equations using the elementary arguments of classical control theory and show that this is equivalent, in the Stratonovich calculus, to a stochastic Hamilton-Pontryagin setup. We show that, for cost functionals that are linear in the state, the theory yields the traditional Bellman equations treated so far in quantum feedback.
We show that the stochastic Schrodinger equation for the filtered state of a system, with linear free dynamics, undergoing continual non-demolition measurement or either position or momentum, or both together, can be solved explicitly within a class of Gaussian states which we call extended coherent states. The asymptotic limit yields a class of relaxed states which we describe explicitly. Bellman's principle is then applied directly to optimal feedback control of such dynamical systems and the Hamilton Jacobi Bellman equation for the minimum cost is derived. The situation of quadratic perform…
We consider two-qubit undergoing local dissipation and subject to local driving. We then determine the optimal Markovian feedback action to preserve initial entanglement as well as to create stationary entanglement with the help of an XY interaction Hamiltonian. Such feedback actions are worked out in a way not depending on the initial two-qubit state, whence called universal.
Recently two papers [K. Jacobs, Phys. Rev. A {\bf 67}, 030301(R) (2003); H. M. Wiseman and J. F. Ralph, New J. Physics {\bf 8}, 90 (2006)] have derived control strategies for rapid purification of qubits, optimized with respect to various goals. In the former paper the proof of optimality was not mathematically rigorous, while the latter gave only heuristic arguments for optimality. In this paper we provide rigorous proofs of optimality in all cases, by applying simple concepts from optimal control theory, including Bellman equations and verification theorems.
Quantum mechanical systems exhibit an inherently probabilistic nature upon measurement. Using a quantum noise model to describe the stochastic evolution of the open quantum system and working in parallel with classical indeterministic control theory, we present the theory of nonlinear optimal quantum feedback control. The resulting quantum Bellman equation is then applied to the explicitly solvable quantum linear-quadratic-Gaussian (LQG) problem which emphasizes many similarities with the corresponding classical control problem.
We study a protocol for two-qubit state guidance that does not rely on feedback mechanisms. In our scheme, entanglement can be concentrated by arranging the interactions of the qubits with a continuous variable ancilla. By properly post-selecting the outcomes of repeated measurements of the state of the ancilla, the qubit state is driven to have a desired amount of purity and entanglement. We stress the primary role played by the first iterations of the protocol. Inefficiencies in the detection operations can be fully taken into account.
This paper reconsiders the claimed rapidity of a scheme for the purification of the quantum state of a qubit, proposed recently in Jacobs 2003 Phys. Rev. A67 030301(R). The qubit starts in a completely mixed state, and information is obtained by a continuous measurement. Jacobs' rapid purification protocol uses Hamiltonian feedback control to maximise the average purity of the qubit for a given time, with a factor of two increase in the purification rate over the no-feedback protocol. However, by re-examining the latter approach, we show that it mininises the average time taken for a qubit to …
The state of a continuously monitored qubit evolves stochastically, exhibiting competition between coherent Hamiltonian dynamics and diffusive partial collapse dynamics that follow the measurement record. We couple these distinct types of dynamics together by linearly feeding the collected record for dispersive energy measurements directly back into a coherent Rabi drive amplitude. Such feedback turns the competition cooperative, and effectively stabilizes the qubit state near a target state. We derive the conditions for obtaining such dispersive state stabilization and verify the stabilizatio…
It is known that any two-outcome quantum measurement can be decomposed into a continuous stochastic process using a feedback loop. In this article, we characterize which of these decompositions are possible when each iteration of the feedback loop consists of a weak measurement caused by an interaction with a probe system. We restrict ourselves to the case when the probe is a qubit and the interaction Hamiltonian between the probe and system is constant. We find that even given the ability to perform arbitrary unitary pulses throughout the continuous decomposition, only generalized measurement…
We propose a class of preparation schemes for orbital angular momentum and polarisation qubits carried by single photons or classical states of light based on coherent feedback control by an ancillary degree of freedom of light. The preparation methods use linear optics and include the transcription of an arbitrary polarisation state onto a two-level OAM system (swap) for arbitrary OAM values plus/minus "l" within a light beam, i.e. without spatial interferometer. The preparations can be carried out with unit efficiency independent from the potentially unknown initial state of the system. More…
Quantum feedback control is a technology which can be used to drive a quantum system into a predetermined eigenstate. In this article, sufficient conditions for the experiment parameters of a quantum feedback control process of a homodyne QND measurement are given to guarantee feedback control of a spin-1/2 quantum system in case of imperfect detection efficiency. For the case of pure states and perfect detection efficiency, time scales of feedback control processes are calculated.
Readout of the state of a superconducting qubit by homodyne detection of the output signal from a dispersively coupled microwave resonator is a common technique in circuit quantum electrodynamics, and is often claimed to be quantum non-demolition (QND) up to the same order of approximation as in the dispersive approximation. However, in this work we show that only in the limit of infinite measurement time is this protocol QND, as the formation of a dressed coherent state in the qubit-cavity system applies an effective rotation to the qubit state. We show how this rotation can be corrected by a…
We study the correction of errors intervening in two-qubit dissipating into their own environments. This is done by resorting to local feedback actions with the aim of preserving as much as possible the initial amount of entanglement. Optimal control is found by first gaining insights from the subsystem purity and then by numerical analysis on the concurrence. This is tantamount to a double optimization, on the actuation and on the measurement processes. Repeated feedback action is also investigated, thus paving the way for a continuous time formulation and solution of the problem.
We study the quantum-jump-based feedback control on the entanglement shared between two qubits with one of them subject to decoherence, while the other qubit is under the control. This situation is very relevant to a quantum system consisting of nuclear and electron spins in solid states. The possibility to prolong the coherence time of the dissipative qubit is also explored. Numerical simulations show that the quantum-jump-based feedback control can improve the entanglement between the qubits and prolong the coherence time for the qubit subject directly to decoherence.
The purpose of this paper is to study the dynamics of a quantum coherent feedback network composed of two two-level systems (qubits) driven by two counter-propagating photons, one in each input channel. The coherent feedback network enhances the nonlinear photon-photon interaction inside the feedback loop. By means of quantum stochastic calculus and the input-output framework, the analytic form of the steady-state output two-photon state is derived. Based on the analytic form, the applications on the Hong-Ou-Mandel (HOM) interferometer and marginally stable single-photon devices using this coh…
We investigate the entanglement dynamics of two two-level emitters (qubits) mediated by a semiinfinite, one-dimensional (1D) photonic waveguide. The coupling of each qubit to the waveguide is chiral, which depends on the propagation direction of light. The finite end of the waveguide is terminated by a perfect mirror, such that coherent quantum feedback is introduced to the system. We show that the chirally generated entanglement between the qubits can be preserved by controlling the time delay of the feedback. Moreover, when the time delay is negligible, the qubit-qubit reduced system evolves…
Feedback control of quantum systems via continuous measurement involves complex nonlinear dynamics. Except in very special cases, even for a single qubit optimal feedback protocols are unknown. Not even do intuitive candidates exist for choosing the measurement basis, which is the primary non-trivial ingredient in the feedback control of a qubit. Here we present a series of arguments that suggest a particular form for the optimal protocol for a broad class of noise sources in the regime of good control. This regime is defined as that in which the control is strong enough to keep the system clo…
Superconducting qubits are a promising platform for building a larger-scale quantum processor capable of solving otherwise intractable problems. In order for the processor to reach practical viability, the gate errors need to be further suppressed and remain stable for extended periods of time. With recent advances in qubit control, both single- and two-qubit gate fidelities are now in many cases limited by the coherence times of the qubits. Here we experimentally employ closed-loop feedback to stabilize the frequency fluctuations of a superconducting transmon qubit, thereby increasing its coh…
We present a general approach to measurement-based quantum feedback that employs proportional and quantum state-based (PaQS) feedback components to obtain locally optimal protocols. To demonstrate the power of the method, we first show that it reproduces many known feedback protocols, and then apply it to generation of multipartite entanglement with an emphasis on remote entanglement, which requires spatially local feedback Hamiltonians. The symmetry of both measurement and feedback operators is found to be essential for construction of effective protocols. We show that under perfect measureme…
We compare the performance of continuous coherent feedback, implemented using an ideal single-qubit controller, to that of continuous measurement-based feedback for the task of controlling the state of a single qubit. Here the basic dynamical resource is the ability to couple the system to a traveling-wave field (for example, a transmission line) via a system observable, and the fundamental limitation is the maximum rate that is available for this coupling. We focus on the question of the best achievable control given ideal controllers. To obtain a fair comparison we acknowledge that the ampli…
We propose to use a cryogenic nonlinear resonator for the projective readout, classical memory, and feedback for a superconducting qubit. This approach sidesteps many of the inefficiencies inherent in two-way communication between temperature stages in typical systems with room temperature controllers, and avoids increasing the cryogenic heat load. This controller may find a broad range of uses in multi-qubit systems, but here we analyze two specific demonstrative cases in single qubit-control. In the first case, the nonlinear controller is used to initialize the qubit in a definite eigenstate…
When one performs a continuous measurement, whether on a classical or quantum system, the measurement provides a certain average rate at which one becomes certain about the state of the system. For a quantum system this is an average rate at which the system is projected onto a pure state. We show that for a standard kind of continuous measurement, for a qubit this rate may be increased by applying unitary operations during the measurement (that is, by using Hamiltonian feedback), in contrast to the equivalent measurement on a classical bit, where reversible operations cannot be used to enhanc…
We consider the use of feedback control during a measurement to increase the rate at which a single qubit is purified, and more generally the rate at which near-pure states may be prepared. We derive the optimal bang-bang algorithm for rapid state preparation from an initially completely mixed state when the measurement basis is unrestricted, and evaluate its performance numerically. We also consider briefly the case in which the measurement basis is fixed with respect to the state to be prepared, and describe the qualitative structure of the optimal bang-bang algorithm.
By using an ancilla qubit as a mediator, two distant qubits can undergo a non-local entangling unitary operation. This is desirable for when attempting to scale up or distribute quantum computation by combining fixed static local sets of qubits with ballistic mediators. Using a model driven by measurements on the ancilla, it is possible to generate a maximally entangling CZ gate while only having access to a less entangling gate between the pair qubits and the ancilla. However this results in a stochastic process of generating control phase rotation gates where the expected time for success do…
Classical feedback is defined here as the knowledge by the transmitter of the quantum state of the qubit received by the receiver. Such classical feedback doubles capacities of certain memoryless quantum channels without preexisting entanglement between transmitter and receiver. The increase in capacity, which is absent on classical memoryless channels, occurs because we can transform an entangled qubit pair into any other entangled state by applying a unitary operator to only one of the qubits.
A quantum system may be purified, i.e., projected into a pure state, faster if one applies feedback operations during the measurement process. However existing results suggest that such an enhancement is only possible when the measurement efficiency exceeds 0.5, which is difficult to achieve experimentally. We address the task of finding the global optimal feedback control for purifying a single qubit in the presence of measurement inefficiency. We use the Bloch vector length, a more physical and practical quantity than purity, to assess the quality of the state, and employ a backward iteratio…
The protection of quantum states is challenging for non-orthogonal states especially in the presence of noises. The recent research breakthrough shows that this difficulty can be overcome by feedback control with weak measurements. However, the state-protection schemes proposed recently work optimally only for special quantum states. In this paper, {by applying different weak measurements, we extend the idea of the state-protection scheme to protect general states.} We calculate numerically the optimal parameters and discuss the performance of the scheme. Comparison between this extended schem…
We have analyzed theoretically the operation of the Bayesian quantum feedback of a solid-state qubit, designed to maintain perfect coherent oscillations in the qubit for arbitrarily long time. In particular, we have studied the feedback efficiency in presence of dephasing environment and detector nonideality. Also, we have analyzed the effect of qubit parameter deviations and studied the quantum feedback control of an energy-asymmetric qubit.
Quantum feedback is a technique for measuring a qubit and applying appropriate feedback depending on the measurement results. Here, we propose a new on-chip quantum feedback method where the measurement-result information is not taken from the chip to the outside of a dilution refrigerator. This can be done by using a selective qubit-energy shift induced by measurement apparatus. We demonstrate on-chip quantum feedback and succeed in the rapid initialization of a qubit by flipping the qubit state only when we detect the ground state of the qubit. The feedback loop of our quantum feedback metho…
Measurement-based feedback control is central in quantum computing and precise quantum control. Here we realize a fast and flexible field-programmable-gate-array-based feedback control in a superconducting Xmon qubit system. The latency of room-temperature electronics is custom optimized to be as short as 140 ns. Projective measurement of a signal qubit produces a feedback tag to actuate a conditional pulse gate to the qubit. In a feed-forward process, the measurement-based feedback tag is brought to a different target qubit for a conditional control. In a two-qubit experiment, the feedback an…
We demonstrate feedback control of a superconducting transmon qubit using discrete, projective measurement and conditional coherent driving. Feedback realizes a fast and deterministic qubit reset to a target state with 2.4% error averaged over input superposition states, and cooling of the transmon from 16% spurious excitation to 3%. This closed-loop qubit control is necessary for measurement-based protocols such as quantum error correction and teleportation.
We have studied theoretically the basic operation of a quantum feedback loop designed to maintain a desired phase of quantum coherent oscillations in a single solid-state qubit. The degree of oscillations synchronization with external harmonic signal is calculated as a function of feedback strength, taking into account available bandwidth and coupling to environment. The feedback can efficiently suppress the dephasing of oscillations if the qubit coupling to the detector is stronger than coupling to environment.
It was shown by Ahn, Wiseman, and Milburn [PRA {\bf 67}, 052310 (2003)] that feedback control could be used as a quantum error correction process for errors induced by weak continuous measurement, given one perfectly measured error channel per qubit. Here we point out that this method can be easily extended to an arbitrary number of error channels per qubit. We show that the feedback protocols generated by our method encode $n-2$ logical qubits in $n$ physical qubits, thus requiring just one more physical qubit than in the previous case.
In the context of a charge qubit under continuous monitoring by a single electron transistor, we propose an unraveling of the generalized quantum Markovian master equation into an ensemble of individual quantum trajectories for stochastic point process. A suboptimal feedback algorism is implemented into individual quantum trajectories to protect a desired pure state. Coherent oscillations of the charge qubit could be maintained in principle for an arbitrarily long time in case of sufficient feedback strength. The effectiveness of the feedback control is also reflected in the detector's noise s…
Quantum measurements not only extract information from a system but also alter its state. Although the outcome of the measurement is probabilistic, the backaction imparted on the measured system is accurately described by quantum theory. Therefore, quantum measurements can be exploited for manipulating quantum systems without the need for control fields. We demonstrate measurement-only state manipulation on a nuclear spin qubit in diamond by adaptive partial measurements. We implement the partial measurement via tunable correlation with an electron ancilla qubit and subsequent ancilla readout.…
We propose the manipulation of an isolated qubit by a simple instantaneous closed-loop feedback scheme in which a time-dependent electronic detector current is directly back-coupled into qubit parameters. As specific detector model we employ a capacitively coupled single-electron transistor. We demonstrate the stabilization of pure delocalized qubit states above a critical detector-qubit coupling. This electronic purification is independent of the initial qubit state and is accomplished after few electron jumps through the detector. Our simple scheme can be used for the efficient and robust in…
In solid-state circuit QED system, we extend the previous study of generating and stabilizing two-qubit Bell state [Phys. Rev. A 82, 032335 (2010)], to three-qubit GHZ state. In dispersive regime, we employ the homodyne joint readout for multiple qubits to infer the state for further processing, and in particular use it to stabilize the state directly by means of an alternate-flip-interrupted Zeno (AFIZ) scheme. Moreover, the state-of-the-art feedback action based on the filtered current enables not only a deterministic generation of the pre-GHZ state in the initial stage, but also a fast reco…
The informational approach to continuous quantum measurement is derived from POVM formalism for a mesoscopic scattering detector measuring a charge qubit. Quantum Bayesian equations for the qubit density matrix are derived, and cast into the form of a stochastic conformal map. Measurement statistics are derived for kicked quantum nondemolition measurements, combined with conditional unitary operations. These results are applied to derive a feedback protocol to produce an arbitrary pure state after a weak measurement, as well as to investigate how an initially mixed state becomes purified with …
We propose an experiment on quantum feedback control of a solid-state qubit, which is almost within the reach of the present-day technology. Similar to the earlier proposal, the feedback loop is used to maintain the coherent (Rabi) oscillations in a qubit for an arbitrary long time; however, this is done in a significantly simpler way, which requires much smaller bandwidth of the control circuitry. The main idea is to use the quadrature components of the noisy detector current to monitor approximately the phase of qubit oscillations. The price for simplicity is a less-than-ideal operation: the…
Quantum computing architectures rely on classical electronics for control and readout. Employing classical electronics in a feedback loop with the quantum system allows to stabilize states, correct errors and to realize specific feedforward-based quantum computing and communication schemes such as deterministic quantum teleportation. These feedback and feedforward operations are required to be fast compared to the coherence time of the quantum system to minimize the probability of errors. We present a field programmable gate array (FPGA) based digital signal processing system capable of real-t…
A measurement-based quantum feedback protocol is developed for spin state initialization in a gate-defined double quantum dot spin qubit coupled to a superconducting resonator. The protocol improves qubit state initialization as it is able to robustly prepare the spin in shorter time and reach a higher fidelity, which can be pre-set. Being able to pre-set the fidelity aimed at is a highly desired feature enabling qubit initialization to be more deterministic. The protocol developed herein is also effective at high temperatures, which is critical for the current efforts towards scaling up the n…
Various schemes have been proposed to overcome the drawback of the decoherence on quantum-enhanced parameter estimation. Here we suggest an alternative method, quantum feedback, to enhance the parameter precision of optimal quantum estimation of a dissipative qubit by investigating its dynamics of quantum Fisher information. We find that compared with the case without feedback, the quantum Fisher information of the dissipative qubit in the case of feedback has a large maximum value in time evolution and a smaller decay rate in the long time.
Fault tolerant quantum computing relies on the ability to detect and correct errors, which in quantum error correction codes is typically achieved by projectively measuring multi-qubit parity operators and by conditioning operations on the observed error syndromes. Here, we experimentally demonstrate the use of an ancillary qubit to repeatedly measure the $ZZ$ and $XX$ parity operators of two data qubits and to thereby project their joint state into the respective parity subspaces. By applying feedback operations conditioned on the outcomes of individual parity measurements, we demonstrate the…
Repeated closed-loop control operations acting as piecewise-constant Liouville superoperators conditioned on the outcomes of regularly performed measurements may effectively be described by a fixed-point iteration for the density matrix. Even when all Liouville superoperators point to the completely mixed state, feedback of the measurement result may lead to a pure state, which can be interpreted as selective dampening of undesired states. Using a microscopic model, we exemplify this for a single qubit, which can be purified in an arbitrary single-qubit state by tuning the measurement directio…
Persistent control of a transmon qubit is performed by a feedback protocol based on continuous heterodyne measurement of its fluorescence. By driving the qubit and cavity with microwave signals whose amplitudes depend linearly on the instantaneous values of the quadratures of the measured fluorescence field, we show that it is possible to stabilize permanently the qubit in any targeted state. Using a Josephson mixer as a phase-preserving amplifier, it was possible to reach a total measurement efficiency $η$=35%, leading to a maximum of 59% of excitation and 44% of coherence for the stabilized …
We show that for qubits and qutrits it is always possible to perfectly recover quantum coherence by performing a measurement only on the environment, whereas for dimension d>3 there are situations where recovery is impossible, even with complete access to the environment. For qubits, the minimal amount of classical information to be extracted from the environment equals the entropy exchange.
Quantum error correction will be essential for realizing the full potential of large-scale quantum information processing devices. Fundamental to its experimental realization is the repetitive detection of errors via projective measurements of quantum correlations among qubits, and correction using conditional feedback. Performing these tasks repeatedly requires a system in which measurement and feedback decision times are short compared to qubit coherence times, where the measurement reproduces faithfully the desired projection, and for which the measurement process has no detrimental effect …
This chapter covers the development of feedback control of superconducting qubits using projective measurement and a discrete set of conditional actions, here referred to as digital feedback. We begin with an overview of the applications of digital feedback in quantum computing. We then introduce an implementation of high-fidelity projective measurement of superconducting qubits. This development lays the ground for closed-loop control based on the binary measurement result. A first application of digital feedback control is fast and deterministic qubit reset, allowing the repeated initializat…
We find that feedback control may induce "pseudo" nonlinear dynamics in a damped harmonic oscillator, whose centroid trajectory in the phase space behaves like a classical nonlinear system. Thus, similar to nonlinear amplifiers (e.g., rf-driven Josephson junctions), feedback control on the harmonic oscillator can induce nonlinear bifurcation, which can be used to amplify small signals and further to measure quantum states of qubits. Using the circuit QED systems as an example, we show how to apply our method to measure superconducting charge qubits.
We derive the equations of motion describing the feedback control of quantum systems in the regime of "good control", in which the control is sufficient to keep the system close to the desired state. One can view this regime as the quantum equivalent of the "linearized" regime for feedback control of classical nonlinear systems. Strikingly, while the dynamics of a single qubit in this regime is indeed linear, that of all larger systems remains nonlinear, in contrast to the classical case. As a first application of these equations, we determine the steady-state performance of feedback protocols…
Circuit cavity quantum electrodynamics (QED) is proving to be a powerful platform to implement quantum feedback control schemes due to the ability to control superconducting qubits and microwaves in a circuit. Here, we present a simple and promising quantum feedback control scheme for deterministic generation and stabilization of a three-qubit $W$ state in the superconducting circuit QED system. The control scheme is based on continuous joint Zeno measurements of multiple qubits in a dispersive regime, which enables us not only to infer the state of the qubits for further information processin…
