In order to ensure reliable quantum information processing in realistic noisy devices, it is necessary to resort to suitable error correction and avoidance strategies. A key aspect of these techniques is how the relevant information is logically mapped (encoded) in the noise-protected codewords. Here, we study how to engineer Markov dynamics that transfers the desired information from an "upload" subsystem to the target error-correcting or noiseless quantum code, effectively acting as a dissipative quantum encoder. Dissipative encoders offer advantages with respect to standard unitary encoding protocols based on quantum circuits, as they can associate the target states to non-trivial basins of attraction, and thus tolerate more general initializations in the upload qubits. In particular, we show that devising continuous-time dissipative encoders requires the target code to be invariant, making this task more delicate as compared to its discrete-time counterpart. Nonetheless, we show how this is always possible for stabilizer quantum error-correcting codes.
In order to ensure reliable quantum information processing in realistic noisy devices, it is necessary to resort to suitable error correction and avoidance strategies. A key aspect of these techniques is how the relevant information is logically mapped (encoded) in the noise-protected codewords. Here, we study how to engineer Markov dynamics that transfers the desired information from an "upload" subsystem to the target error-correcting or noiseless quantum code, effectively acting as a dissipative quantum encoder. Dissipative encoders offer advantages with respect to standard unitary encoding protocols based on quantum circuits, as they can associate the target states to non-trivial basins of attraction, and thus tolerate more general initializations in the upload qubits. In particular, we show that devising continuous-time dissipative encoders requires the target code to be invariant, making this task more delicate as compared to its discrete-time counterpart. Nonetheless, we show how this is always possible for stabilizer quantum error-correcting codes.
We formalize the problem of dissipative quantum encoding, and explore the advantages of using Markovian evolution to prepare a quantum code in the desired logical space, with emphasis on discrete-time dynamics and the possibility of exact finite-time convergence. In particular, we investigate robustness of the encoding dynamics and their ability to tolerate initialization errors, thanks to the existence of non-trivial basins of attraction. As a key application, we show that for stabilizer quantum codes on qubits, a finite-time dissipative encoder may always be constructed, by using at most a number of quantum maps determined by the number of stabilizer generators. We find that even in situations where the target code lacks gauge degrees of freedom in its subsystem form, dissipative encoders afford nontrivial robustness against initialization errors, thus overcoming a limitation of purely unitary encoding procedures. Our general results are illustrated in a number of relevant examples, including Kitaev's toric code.
Near-term quantum communication protocols suffer inevitably from channel noises, whose alleviation has been mostly attempted with resources such as multiparty entanglement or sophisticated experimental techniques. Generation of multiparty higher dimensional entanglement is not easy. This calls for exploring realistic solutions which are implementable with current devices. Motivated particularly by the difficulty in generation of multiparty entangled states, in this paper, we have investigated error-free information transfer with minimal requirements. For this, we have proposed a new information encoding scheme for communication purposes. The encoding scheme is based on the fact that most noisy channels leave some quantities invariant. Armed with this fact, we encode information in these invariants. These invariants are functions of expectation values of operators. This information passes through the noisy channel unchanged. Pertinently, this approach is not in conflict with other existing error correction schemes. In fact, we have shown how standard quantum error-correcting codes emerge if suitable restrictions are imposed on the choices of logical basis states. As applications, for illustration, we propose a quantum key distribution protocol and an error-immune information transfer protocol.
Codeword-stabilized quantum codes provide a unified approach to constructing quantum error-correcting codes, including both additive and nonadditive quantum codes. Standard codeword-stabilized quantum codes encode quantum information into subspaces. The more general notion of encoding quantum information into a subsystem is known as an operator (or subsystem) quantum error-correcting code. Most operator codes studied to date are based in the usual stabilizer formalism. We introduce operator quantum codes based on the codeword-stabilized quantum code framework. Based on the necessary and sufficient conditions for operator quantum error correction, we derive an error-correction condition for operator codeword-stabilized quantum codes. Based on this condition, the word operators of a operator codeword-stabilized quantum code are constructed from a set of classical binary errors induced by generators of the gauge group. We use this scheme to construct examples of both additive and nonadditive codes that encode quantum information into a subsystem.
We characterize the dynamical behavior of continuous-time, Markovian quantum systems with respect to a subsystem of interest. Markovian dynamics accurately describes a wide class of open quantum systems of relevance to quantum information processing, subsystem encodings offering a general pathway for faithfully representing quantum information. We provide explicit linear-algebraic characterizations of the notion of invariant and noiseless subsystem for Markovian master equations, under different robustness assumptions for model parameter and initial state variations. The stronger concept of an attractive quantum subsystem is introduced, and sufficient existence conditions are identified based on Lyapunov’s stability techniques. As a main control application, we address the potential of output-feedback Markovian control strategies for quantum pure state-stabilization and noiseless-subspace generation. In particular,
This paper proposes an improved quantum coding method based on Markovian dissipative dynamics that is robust against thermal noise. A method for correcting information errors is indispensable for the practical use of quantum information devices, and stabilizer codes for quantum systems have been devised. One of the quantum coding methods, namely, quantum coding based on Markovian dissipative dynamics, has a favorable feature that it does not require strict time control, however it is known that these methods are susceptible to thermal noise. In this paper, we propose a quantum coding method that incorporates a new mechanism to correct the disturbance of the quantum state caused by thermal noise into the quantum coding method using dissipative dynamics. We also analyze the stability of the proposed quantum dynamics in the quantum state corresponding to the target code word. Numerical experiments confirm the effectiveness of the proposed method.
The advance of experimental techniques in circuit quantum electrodynamics over last decadehas made it possible to monitor the evolution of qubits under continuous measurements in\nreal time . This kind of measurements, as compared to the discrete ones, provides a stream\nof information for feedback to alter the system's quantum dynamics continuously in a desired\nmanner, thus allowing broader possibilities. We obtain the information at a cost, the backaction on the states. However, the back-action turns out not to be just something disturbing \nand harmful to the control but instead can be harnessed, together with the control, to steer the system to our targets. \nThis thesis discusses one such application, namely to use measurements to create entanglement from remote qubits under locally optimal control. It is illustrated that by carefully designing symmetric measurement and feedback operators, we can construct efficient Markovian or non-Markovian protocols which generate N-qubit W,\ngeneral Dicke and Greenberger-Horne-Zeilinger states with high fidelity above 94% up to\nN = 100. In some cases, the fidelity even approaches unity. Following that, we describe how\nto take advantages of the continuous stream of information to protect the unitary dynamics of\na quantum system subject to noises. We develop a practical protocol for continuous operation\nof a quantum error correcting code for protection of coherent evolution due to an encoded\nHamiltonian against environmental errors, using the three qubit bit-flip code and bit-flip\nerrors as a canonical example. We demonstrate, for both quantum memory and adiabatic\nHamiltonians, that the protocol, once optimized, can significantly reduce logical errors with\na performance which is close to the theoretical limit of the bit-flip code's capability. In the\nadiabatic Hamiltonian case, we also propose some novel correction operators instead of the\nconventional ones to further reduce the logical errors.