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A theory of time-delayed coherent quantum feedback is developed. More specifically, we consider a quantum system coupled to a bosonic reservoir creating a unidirectional feedback loop. It is shown that the dynamics can be mapped onto a fictitious series of cascaded quantum systems, where the system is driven by past versions of itself. The derivation of this model relies on a tensor network representation of the system-reservoir time-propagator. For concreteness, this general theory is applied to a driven two-level atom scattering into a coherent feedback loop. We demonstrate how delay effects can qualitatively change the dynamics of the atom, and how quantum control can be implemented in the presence of time-delays. Introduction.-Delayed autonomous feedback, where a signal is directly fed back to a system after a controllable time-delay, is an important control tool for classical systems [1][2][3]. It is highly attractive as a tool for stabilizing non-equilibrium states of fast dynamical systems, where avoiding any time-costly signal-processing is crucial. Such stabilization is of great experimental and technological relevance [4][5][6]. In particular, delayed autonomous feedback has been used to stabilize the high frequency dynamics of optical systems and high speed electrical circuits [7, 8].Autonomous feedback is also receiving substantial and growing interest for controlling quantum systems [9][10][11][12][13][14][15][16]. Because of the relatively short coherence time and fast dynamics of quantum systems, very fast feedback control possible with autonomous feedback is highly desirable. In addition, any measurement of the feedback signal will necessarily destroy its quantum character, making a fully quantum mechanical feedback loop that preserves coherence attractive from a fundamental point of view. Compelling evidence that this type of coherent feedback can outperform any measurement-based counterpart for important quantum information processing tasks has been given [17,18].A natural way of implementing coherent feedback control loops is by coupling remote quantum systems via waveguides [19][20][21][22]. Time-delays are unavoidable in practice in such setups and are likely to become important if current experiments are scaled up to larger and more complex networks [23][24][25]. Despite of this, relatively little theoretical research has been done on delay effects for coherent quantum feedback. A major obstacle is the lack of tractable and general theoretical models for treating the highly non-Markovian dynamics induced by this type of feedback. The theoretical difficulty lies in the quantum correlations between the control target system and the in-loop quantum field: The field cannot simply be traced out, and one has to deal with a highly entangled quantum state over a continuum of degrees of freedom.Previous investigations have typically been limited to

Quantum annealing aims at solving combinatorial optimization problems mapped to Ising interactions between quantum spins. Here, with the objective of developing a noise-resilient annealer, we propose a paradigm for quantum annealing with a scalable network of two-photon-driven Kerr-nonlinear resonators. Each resonator encodes an Ising spin in a robust degenerate subspace formed by two coherent states of opposite phases. A fully connected optimization problem is mapped to local fields driving the resonators, which are connected with only local four-body interactions. We describe an adiabatic annealing protocol in this system and analyse its performance in the presence of photon loss. Numerical simulations indicate substantial resilience to this noise channel, leading to a high success probability for quantum annealing. Finally, we propose a realistic circuit QED implementation of this promising platform for implementing a large-scale quantum Ising machine.

Bosonic rotation codes, introduced here, are a broad class of bosonic error-correcting codes based on phase-space rotation symmetry. We present a universal quantum computing scheme applicable to a subset of this class-number-phase codes-which includes the well-known cat and binomial codes, among many others. The entangling gate in our scheme is code-agnostic and can be used to interface different rotation-symmetric encodings. In addition to a universal set of operations, we propose a teleportation-based error correction scheme that allows recoveries to be tracked entirely in software. Focusing on cat and binomial codes as examples, we compute average gate fidelities for error correction under simultaneous loss and dephasing noise and show numerically that the error-correction scheme is close to optimal for error-free ancillae and ideal measurements. Finally, we present a scheme for fault-tolerant, universal quantum computing based on concatenation of number-phase codes and Bacon-Shor subsystem codes.I.

We realize an open version of the Dicke model by coupling two hyperfine ground states using two cavity-assisted Raman transitions. The interaction due to only one of the couplings is described by the Tavis-Cummings model and we observe a normal mode splitting in the transmission around the dispersively shifted cavity. With both couplings present the dynamics are described by the Dicke model and we measure the onset of superradiant scattering into the cavity above a critical coupling strength.

Superconducting circuits rank among some of the most interesting architectures for the implementation of quantum information processing devices. The recently proposed 0-π qubit (Brooks et al 2013 Phys. Rev. A 87 52306) promises increased protection from spontaneous relaxation and dephasing. In this paper we present a detailed theoretical study of the coherence properties of the 0-π device, investigate relevant decoherence channels, and show estimates for achievable coherence times in multiple parameter regimes. In our analysis, we include disorder in circuit parameters, which results in the coupling of the qubit to a low-energy, spurious harmonic mode. We analyze the effects of such coupling on decoherence, in particular dephasing due to photon shot noise, and outline how such a noise channel can be mitigated by appropriate parameter choices. In the end we find that the 0-π qubit performs well and may become an attractive candidate for the implementation of the next-generation superconducting devices for uses in quantum computing and information.Conceptually, the 0-π circuit exhibits a rudimentary form of topological protection that combines exponential suppression of noise-induced transitions (dissipation) with exponential suppression of dephasing, see figure 1. The former is achieved by engineering qubit states with disjoint support, the latter by rendering qubit states (nearly) degenerate and exponentially suppressing the sensitivity of the corresponding energies to lowfrequency environmental noise.The circuit underlying the 0-π qubit consists of four nodes connected by a pair of linear inductors, a pair of capacitors, and a pair of Josephson junctions as shown in figure 2. Two issues pose challenges to the implementation of the 0-π design: first, to achieve the desired regime it is necessary to simultaneously realize large superinductances, large shunting capacitors, and high junction charging energies (very low stray capacitances); second, circuit elements should ideally be pairwise identical (no disorder in circuit element parameters) in order to prevent coupling of the qubit to a spurious circuit mode [14], which we will refer to as the ζ-mode 6 .While notable increases in accessible inductance values by means of junction-array based superinductances may partially address the first issue [15][16][17][18][19], some amount of circuit parameter disorder and hence residual coupling to the ζ-mode is unavoidable. In the present work, we theoretically assess the coherence properties of 0-π devices, ones that are possible to realize with todayʼs state-of-the art fabrication techniques, as well as those that will require technological advances. Specifically, we present calculations of relevant decoherence rates resulting from the qubitʼs coupling to known noise sources, including both intrinsic sources, such as flux, charge and critical current noise, which couple directly to the qubitʼs degree of freedom, as well as noise mediated by the coupling to the spurious ζ-mode. We concentrate our study on three representa...

We investigate the quantum dynamics of a single transmon qubit coupled to surface acoustic waves (SAWs) via two distant connection points. Since the acoustic speed is five orders of magnitude slower than the speed of light, the travelling time between the two connection points needs to be taken into account. Therefore, we treat the transmon qubit as a giant atom with a deterministic time delay. We find that the spontaneous emission of the system, formed by the giant atom and the SAWs between its connection points, initially decays polynomially in the form of pulses instead of a continuous exponential decay behaviour, as would be the case for a small atom. We obtain exact analytical results for the scattering properties of the giant atom up to two-phonon processes by using a diagrammatic approach. We find that two peaks appear in the inelastic (incoherent) power spectrum of the giant atom, a phenomenon which does not exist for a small atom. The time delay also gives rise to novel features in the reflectance, transmittance, and second-order correlation functions of the system. Furthermore, we find the short-time dynamics of the giant atom for arbitrary drive strength by a numerically exact method for open quantum systems with a finite-time-delay feedback loop.

We propose a quantum simulation of a two-level atom coupled to a single mode of the electromagnetic field in the ultrastrong coupling regime based upon resonant Raman transitions in an atom interacting with a high finesse optical cavity mode. We show by numerical simulation the possibility of realizing the scheme with a single rubidium atom, in which two hyperfine ground states make up the effective two-level system, and for cavity QED parameters that should be achievable with, for example, microtoroidal whispering-gallery-mode resonators. Our system also enables simulation of a generalized model in which a nonlinear coupling between the atomic inversion and the cavity photon number occurs on an equal footing with the (ultrastrong) dipole coupling and can give rise to critical-type behavior even at the single-atom level. Our model takes account of dissipation, and we pay particular attention to observables that would be readily observable in the output from the system.