Stabilizer Formalism for Operator Quantum Error Correction

Operator quantum error correction is a recently developed theory that provides a generalized framework for active error correction and passive error avoiding schemes. In this paper, we describe these codes in the stabilizer formalism of standard quantum error correction theory. This is achieved by adding a "gauge" group to the standard stabilizer definition of a code that defines an equivalence class between encoded states. Gauge transformations leave the encoded information unchanged; their effect is absorbed by virtual gauge qubits that do not carry useful information. We illustrate the construction by identifying a gauge symmetry in Shor's 9-qubit code that allows us to remove 4 of its 8 stabilizer generators, leading to a simpler decoding procedure and a wider class of logical operations without affecting its essential properties. This opens the path to possible improvements of the error threshold of fault-tolerant quantum computing.Comment: Corrected claim based on exhaustive searc

Fast Decoders for Topological Quantum Codes

We present a family of algorithms, combining real-space renormalization methods and belief propagation, to estimate the free energy of a topologically ordered system in the presence of defects. Such an algorithm is needed to preserve the quantum information stored in the ground space of a topologically ordered system and to decode topological error-correcting codes. For a system of linear size L, our algorithm runs in time log L compared to L^6 needed for the minimum-weight perfect matching algorithm previously used in this context and achieves a higher depolarizing error threshold.Comment: 4 pages, 4 figure

Tensor Networks and Quantum Error Correction

We establish several relations between quantum error correction (QEC) and tensor network (TN) methods of quantum many-body physics. We exhibit correspondences between well-known families of QEC codes and TNs, and demonstrate a formal equivalence between decoding a QEC code and contracting a TN. We build on this equivalence to propose a new family of quantum codes and decoding algorithms that generalize and improve upon quantum polar codes and successive cancellation decoding in a natural way.Comment: Accepted in Phys. Rev. Lett. 8 pages, 9 figure

Coarse grained belief propagation for simulation of interacting quantum systems at all temperatures

We continue our numerical study of quantum belief propagation initiated in [Phys. Rev. A, 77 (2008), p. 052318]. We demonstrate how the method can be expressed in terms of an effective thermal potential that materializes when the system presents quantum correlations, but is insensitive to classical correlations. The thermal potential provides an efficient means to assess the precision of belief propagation on graphs with no loops. We illustrate these concepts using the one-dimensional quantum Ising model and compare our results with exact solutions. We also use the method to study the transverse field quantum Ising spin glass for which we obtain a phase diagram that is largely in agreement with the one obtained in [arXiv:0706.4391] using a different approach. Finally, we introduce the coarse grained belief propagation (CGBP) algorithm to improve belief propagation at low temperatures. This method combines the reliability of belief propagation at high temperatures with the ability of entanglement renormalization to efficiently describe low energy subspaces of quantum systems with local interactions. With CGBP, thermodynamic properties of quantum systems can be calculated with a high degree of accuracy at all temperatures.Comment: updated references and acknowledgement

An NPZ Model with State-Dependent Delay due to Size-Structure in Juvenile Zooplankton

The study of planktonic ecosystems is important as they make up the bottom trophic levels of aquatic food webs. We study a closed Nutrient-Phytoplankton-Zooplankton (NPZ) model that includes size structure in the juvenile zooplankton. The closed nature of the system allows the formulation of a conservation law of biomass that governs the system. The model consists of a system of nonlinear ordinary differential equation coupled to a partial differential equation. We are able to transform this system into a system of delay differential equations where the delay is of threshold type and is state-dependent. The system of delay differential equations can be further transformed into one with fixed delay. Using the different forms of the model we perform a qualitative analysis of the solutions, which includes studying existence and uniqueness, positivity and boundedness, local and global stability, and conditions for extinction. Key parameters that are explored are the total biomass in the system and the maturity level at which the juvenile zooplankton reach maturity. Numerical simulations are also performed to verify our analytical results

Relational time for systems of oscillators

Using an elementary example based on two simple harmonic oscillators, we show how a relational time may be defined that leads to an approximate Schrodinger dynamics for subsystems, with corrections leading to an intrinsic decoherence in the energy eigenstates of the subsystem.Comment: Contribution to the Int. J. of Quant. Info. issue dedicated to the memory of Asher Pere

Hybridization of Magnetism and Piezoelectricity for an Energy Scavenger based on Temporal Variation of Temperature

Autonomous microsystems are confronted today to a major challenge : the one of energy supply. Energy scavenging, i.e. collecting energy from the ambient environment has been developed to answer this problematic. Various sources have already been successfully used (solar, vibration). This article presents temporal variations of temperature as a new source of exploitable energy. A brief review will take place at the beginning, exposing the different approaches used in the past. Then we will focus our attention on hybridization of magnetism and piezoelectricity. A new kind of thermal generator is proposed and a preliminary model is exposed. Conclusions will be drawn on the suitability of this prototype and the improvements that are needed to increase its potential.Comment: Submitted on behalf of EDA Publishing Association (http://irevues.inist.fr/handle/2042/16838