A simple comparative analysis of exact and approximate quantum error correction

We present a comparative analysis of exact and approximate quantum error correction by means of simple unabridged analytical computations. For the sake of clarity, using primitive quantum codes, we study the exact and approximate error correction of the two simplest unital (Pauli errors) and nonunital (non-Pauli errors) noise models, respectively. The similarities and differences between the two scenarios are stressed. In addition, the performances of quantum codes quantified by means of the entanglement fidelity for different recovery schemes are taken into consideration in the approximate case. Finally, the role of self-complementarity in approximate quantum error correction is briefly addressed.Comment: 29 pages, 1 figure, improved v2; accepted for publication in Open Systems and Information Dynamics (2014

Thermodynamic stability criteria for a quantum memory based on stabilizer and subsystem codes

We discuss and review several thermodynamic criteria that have been introduced to characterize the thermal stability of a self-correcting quantum memory. We first examine the use of symmetry-breaking fields in analyzing the properties of self-correcting quantum memories in the thermodynamic limit: we show that the thermal expectation values of all logical operators vanish for any stabilizer and any subsystem code in any spatial dimension. On the positive side, we generalize the results in [R. Alicki et al., arXiv:0811.0033] to obtain a general upper bound on the relaxation rate of a quantum memory at nonzero temperature, assuming that the quantum memory interacts via a Markovian master equation with a thermal bath. This upper bound is applicable to quantum memories based on either stabilizer or subsystem codes.Comment: 23 pages. v2: revised introduction, various additional comments, and a new section on gapped hamiltonian

The Stability of Quantum Concatenated Code Hamiltonians

Protecting quantum information from the detrimental effects of decoherence and lack of precise quantum control is a central challenge that must be overcome if a large robust quantum computer is to be constructed. The traditional approach to achieving this is via active quantum error correction using fault-tolerant techniques. An alternative to this approach is to engineer strongly interacting many-body quantum systems that enact the quantum error correction via the natural dynamics of these systems. Here we present a method for achieving this based on the concept of concatenated quantum error correcting codes. We define a class of Hamiltonians whose ground states are concatenated quantum codes and whose energy landscape naturally causes quantum error correction. We analyze these Hamiltonians for robustness and suggest methods for implementing these highly unnatural Hamiltonians.Comment: 18 pages, small corrections and clarification

Quantum memories based on engineered dissipation

Storing quantum information for long times without disruptions is a major requirement for most quantum information technologies. A very appealing approach is to use self-correcting Hamiltonians, i.e. tailoring local interactions among the qubits such that when the system is weakly coupled to a cold bath the thermalization process takes a long time. Here we propose an alternative but more powerful approach in which the coupling to a bath is engineered, so that dissipation protects the encoded qubit against more general kinds of errors. We show that the method can be implemented locally in four dimensional lattice geometries by means of a toric code, and propose a simple 2D set-up for proof of principle experiments.Comment: 6 +8 pages, 4 figures, Includes minor corrections updated references and aknowledgement

Multipartite entanglement, quantum-error-correcting codes, and entangling power of quantum evolutions

We investigate the average bipartite entanglement, over all possible divisions of a multipartite system, as a useful measure of multipartite entanglement. We expose a connection between such measures and quantum-error-correcting codes by deriving a formula relating the weight distribution of the code to the average entanglement of encoded states. Multipartite entangling power of quantum evolutions is also investigated.Comment: 13 pages, 1 figur