On the Response of Quantum Linear Systems to Single Photon Input Fields

Linear quantum systems: A tutorial

Preconditioning is the most widely used and effective way for treating ill-conditioned linear systems in the context of classical iterative linear system solvers. We introduce a quantum primitive called fast inversion, which can be used as a preconditioner for solving quantum linear systems. The key idea of fast inversion is to directly block-encode a matrix inverse through a quantum circuit implementing the inversion of eigenvalues via classical arithmetics. We demonstrate the application of preconditioned linear system solvers for computing single-particle Green's functions of quantum many-body systems, which are widely used in quantum physics, chemistry, and materials science. We analyze the complexities in three scenarios: the Hubbard model, the quantum many-body Hamiltonian in the planewave-dual basis, and the Schwinger model. We also provide a method for performing Green's function calculation in second quantization within a fixed particle manifold and note that this approach may be valuable for simulation more broadly. Besides solving linear systems, fast inversion also allows us to develop fast algorithms for computing matrix functions, such as the efficient preparation of Gibbs states. We introduce two efficient approaches for such a task, based on the contour integral formulation and the inverse transform respectively.

In this paper, we propose a measurement nonlinear feedback control scheme to generate Wigner-function negativity in an optical cavity having dynamics described as a linear quantum system. In general, linear optical quantum systems can be easily constructed with reliable devices; therefore, the idea of constructing the entire system with such an optical system and nonlinear feedback is reasonable for generating Wigner-function negativity. However, existing studies have insufficiently examined the realizability or actual implementation of feedback control, which essentially requires fast responses from the sensors and actuators. In order to solve this problem, we consider the realizable feedback control of the optical phase of a pumping beam supplied to a cavity by using electro-optical modulation, which can be utilized as a fast control actuator. Then, we introduce mathematical models of the feedback-controlled system and evaluate its effect on the generation of the Wigner-function negativity by using numerical simulation. Through various numerical simulations, we show that the proposed feedback control can effectively generate the negativity of the Wigner function.

Coherent-classical estimation for linear quantum systems

System identification is a key enabling component for the implementation of new quantum technologies, including quantum control. In this paper we consider a large class of input-output systems, namely linear passive quantum systems, and study the following identifiability question: if the system's Hamiltonian and coupling matrices are unknown, which of these dynamical parameters can be estimated by preparing appropriate input states and performing measurements on the output? The input-output mapping is explicitly given by the transfer function, which contains the maximum information about the system.We show that two minimal systems are indistinguishable (have the same transfer function) if and only if their Hamiltonians and the coupling to the input fields are related by a unitary transformation. Furthermore, we provide a canonical parametrization of the equivalence classes of indistinguishable systems. For models depending on (possibly lower dimensional) unknown parameters, we give a practical identifiability condition which is illustrated on several examples. In particular, we show that systems satisfying a certain Hamiltonian connectivity condition called “infecting”, are completely identifiable.

Sabadini, Walters and others have developed a categorical, machine based theory of concurrency in which there are four essential aspects: a distributive category of data-types; a bicategory Mach whose objects are data types, and whose arrows are input-output machines built from data types; a semantic category (or categories) Sem, suitable to contain the behaviors of machines, and a functor, “behavior”: Mach→Sem. Suitable operations on machines and semantics are found so that the behavior functor preserves these operations. Then, if each machine is decomposable into primitive machines using these operations, the behavior of a general machine is deducible from the behavior of its parts. The theory of non-deterministic finite state automata provides an example of the paradigm and also throws some light on the classical theory of finite state automata. We describe a bicategory whose objects are natural numbers, in which an arrow M: n→p is a finite state automaton with n input states, p output states, and some additional internal states; we require that no transitions begin at output states or end at input states. A machine is represented by an q+n by q+p matrix. The bicategory supports additional operations: non-deterministic choice, parallel interleaving, and feedback. Enough operations are imposed on machines to show that each machine may be obtained from some atomic ones by means of the operations. The semantic category is the (Bloom-Esik) iteration theory Mat (X✶ whose objects are natural numbers and whose arrows from n to p are n×p matrices with entries in the semiring of languages. The behavior functor associates to a machine M: n→p a matrix |M| of languages, one language to each pair of input and output states. Behavior preserves composition, feedback, takes non-deterministic choice to union, and parallel-interleaving to shuffle. Thus, behavior gives a compositional semantics to a primitive notion of concurrent processes.

This paper considers the use of singular perturbation approximations for a class of linear complex quantum systems arising in the area of quantum optics. The class of linear complex quantum systems under consideration arises in systems consisting of interconnections of passive optical components such as cavities, beams splitters and phase shifters. The main result of the paper shows that for a class of singular perturbations of such systems, the approximate reduced dimension quantum system and guaranteed to be physically realizable.

Analysis of quantum linear systems’ response to multi-photon states

Determination of the quantum nature of correlations between two spatially separated systems plays a crucial role in quantum information science. Of particular interest is the questions of if and how these correlations enable quantum information protocols to be more powerful. Here, we report on a distributed quantum computation protocol in which the input and output quantum states are considered to be classically correlated in quantum informatics. Nevertheless, we show that the correlations between the outcomes of the measurements on the output state cannot be efficiently simulated using classical algorithms. Crucially, at the same time, local measurement outcomes can be efficiently simulated on classical computers. We show that the only known classicality criterion violated by the input and output states in our protocol is the one used in quantum optics, namely, phase-space nonclassicality. As a result, we argue that the global phase-space nonclassicality inherent within the output state of our protocol represents true quantum correlations.

We describe the simplest single–photon encoding schemes and introduce the concept of a quantum optical gate. These gates can be used to build up arbitrary entangled states of many initially separate single photons. Experimental realization of such a gate involves the development of nonlinear phase shift elements sensitice at the single quantum level. We report our experimental efforts using linear optical elements and non–classical sources where we are able to demonstrate interference and entanglement of initially separate single–photon pulses.

This paper considers a coherent H <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">infin</sup> control problem for a class of linear quantum systems which can be defined by complex quantum stochastic differential equations in terms of annihilation operators only. For this class of quantum systems, a solution to the H <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">infin</sup> control problem can be obtained in terms of a pair of complex Riccati equations. The paper also considers complex versions of the bounded real lemma, the strict bounded real lemma and the lossless bounded real lemma. For the class of quantum systems under consideration, the question of physical realizability is related to the bounded real and lossless bounded real properties.

In this paper, the synthesis of robust memory modes for linear quantum passive systems in the presence of unknown inputs has been studied, aimed at facilitating secure storage and communication of quantum information. In particular, we can switch on decoherence-free (DF) modes in the storage stage by placing the poles on the imaginary axis via a coherent feedback control scheme, and these memory modes can further be simultaneously made robust against perturbations to the system parameters by minimizing the condition number associated with imaginary poles. The DF modes can also be switched off by tuning the controller parameters to place the poles in the left half of the complex plane in the writing/reading stage. We develop explicit algebraic conditions guiding the design of such a coherent quantum controller, which involves employing an augmented system model to counter the influence of unknown inputs. Examples are provided to illustrate the procedure of synthesizing robust memory modes for linear optical quantum systems.

A new class of non-classical light states has been experimentally generated and their complete phase-space characterization has been achieved by quantum homodyne tomography. Such states are produced by the action of the photon creation operator on a coherent light field and are thus the result of the elementary excitation process of a classical field by a single quantum. Being intermediate between a single-photon Fock state and a coherent one, they offer the unique opportunity to closely follow the smooth evolution between the particle-like and the wave-like behavior of the light field.

Performance of quantum teleportation is typically measured by the average fidelity, an overlap between the input and output states. Along with the first moment, we introduce the second moment of fidelity in continuous-variable (CV) teleportation, i.e., the fidelity deviation as the figure of merit to assess the protocol's efficiency. We show that CV states, both Gaussian and non-Gaussian, can be better characterized by considering both average fidelity and fidelity deviation, which is not possible with only average fidelity. Moreover, we shed light on the performance of the teleportation protocol in two different input scenarios: One is when input states are sampled from constrained uniform distribution while the other one is Gaussian suppression of the input states which again leads to a different classification of CV states according to their performance. The entire analysis is carried out in noiseless and noisy scenarios with noise being incorporated in the measurement and the shared channels. We also report that one type of noise can make the protocol robust against the other one, which leads to a ``constructive effect,'' and identify the noise models which are responsible for decreasing average fidelity and the increment in fidelity deviation.

Quantum information and quantum computing offer a new range of possibilities in terms ofnew computing algorithms, the ability to perform new types of operations and make statementsabout information in terms of what it is physically possible to know about a system.Experiments in quantum information and quantum computing have been advancing significantly,especially over the last decade. In optical quantum computing there are two maintypes of experiments that are performed today. They are experiments involving single photonswith information encoded on the polarisation of the photon and experiments involvingstates with Gaussian noise profiles in their quadratures. There appears to be no way toperform the required operations for universal quantum computing using operations whichtake Gaussian states to Gaussian states. Universal quantum computing using single photonsis possible but uses either enormously large non-linearities or a large number of resources.Also, operations which take Gaussian states to non-Gaussian states are not well understoodin general and can often be very difficult to implement.This thesis presents three sources of optical non-Gaussian states which demonstrate someof the interesting properties of these states. First, a scheme for universal quantum computingusing superpositions of coherent states is presented. Error correction and the fault toleranceproperties of this scheme is also explored and compared with current single photon architectures.These gates require superpositions of coherent states as resources and a relativelysimple scheme is shown for how these states may be generated. Next a scheme for distillingnear ideal Gaussian entanglement using non-Gaussian operations is presented. The loss andnoise conditions under which ideal distillation can be achieved are presented. Finally, a verysimple way of generating mixed non-Gaussian states from two Gaussian states is presented.These states are shown to give rise to counter-intuitive results when measuring their entanglementand when analysing their operational ability in terms of teleportation and quantumkey distribution.