Automated searching for quantum subsystem codes

Quantum error correction allows for faulty quantum systems to behave in an effectively error free manner. One important class of techniques for quantum error correction is the class of quantum subsystem codes, which are relevant both to active quantum error correcting schemes as well as to the design of self-correcting quantum memories. Previous approaches for investigating these codes have focused on applying theoretical analysis to look for interesting codes and to investigate their properties. In this paper we present an alternative approach that uses computational analysis to accomplish the same goals. Specifically, we present an algorithm that computes the optimal quantum subsystem code that can be implemented given an arbitrary set of measurement operators that are tensor products of Pauli operators. We then demonstrate the utility of this algorithm by performing a systematic investigation of the quantum subsystem codes that exist in the setting where the interactions are limited to 2-body interactions between neighbors on lattices derived from the convex uniform tilings of the plane.Comment: 38 pages, 15 figure, 10 tables. The algorithm described in this paper is available as both library and a command line program (including full source code) that can be downloaded from http://github.com/gcross/CodeQuest/downloads. The source code used to apply the algorithm to scan the lattices is available upon request. Please feel free to contact the authors with question

Experiments on joint source-channel fractal image coding with unequal error protection

We propose a joint source-channel coding system for fractal image compression. We allocate the available total bit rate between the source code and a range of error-correcting codes using a Lagrange multiplier optimization technique. The principle of the proposed unequal error protection strategy is to partition the information bits into sensitivity classes and to assign one code from a range of error-correcting codes to each sensitivity class in a nearly optimal way. Experimental results show that joint source-channel coding with fractal image compression is feasible, leads to ef"cient protection strategies, and outperforms previous works in this "eld that only covered channel coding with a "xed source rate

Diversity analysis, code design, and tight error rate lower bound for binary joint network-channel coding

Joint network-channel codes (JNCC) can improve the performance of communication in wireless networks, by combining, at the physical layer, the channel codes and the network code as an overall error-correcting code. JNCC is increasingly proposed as an alternative to a standard layered construction, such as the OSI-model. The main performance metrics for JNCCs are scalability to larger networks and error rate. The diversity order is one of the most important parameters determining the error rate. The literature on JNCC is growing, but a rigorous diversity analysis is lacking, mainly because of the many degrees of freedom in wireless networks, which makes it very hard to prove general statements on the diversity order. In this article, we consider a network with slowly varying fading point-to-point links, where all sources also act as relay and additional non-source relays may be present. We propose a general structure for JNCCs to be applied in such network. In the relay phase, each relay transmits a linear transform of a set of source codewords. Our main contributions are the proposition of an upper and lower bound on the diversity order, a scalable code design and a new lower bound on the word error rate to assess the performance of the network code. The lower bound on the diversity order is only valid for JNCCs where the relays transform only two source codewords. We then validate this analysis with an example which compares the JNCC performance to that of a standard layered construction. Our numerical results suggest that as networks grow, it is difficult to perform significantly better than a standard layered construction, both on a fundamental level, expressed by the outage probability, as on a practical level, expressed by the word error rate

The adversarial joint source-channel problem

This paper introduces the problem of joint source-channel coding in the setup where channel errors are adversarial and the distortion is worst case. Unlike the situation in the case of stochastic source-channel model, the separation principle does not hold in adversarial setup. This surprising observation demonstrates that designing good distortion-correcting codes cannot be done by serially concatenating good covering codes with good error-correcting codes. The problem of the joint code design is addressed and some initial results are offered

Optimal Single-Shot Decoding of Quantum Codes

We discuss single-shot decoding of quantum Calderbank-Shor-Steane codes with faulty syndrome measurements. We state the problem as a joint source-channel coding problem. By adding redundant rows to the code's parity-check matrix we obtain an additional syndrome error correcting code which addresses faulty syndrome measurements. Thereby, the redundant rows are chosen to obtain good syndrome error correcting capabilities while keeping the stabilizer weights low. Optimal joint decoding rules are derived which, though too complex for general codes, can be evaluated for short quantum codes.Comment: 6 pages, 4 figure