Limited-Magnitude Error-Correcting Gray Codes for Rank Modulation

We construct Gray codes over permutations for the rank-modulation scheme, which are also capable of correcting errors under the infinity-metric. These errors model limited-magnitude or spike errors, for which only single-error-detecting Gray codes are currently known. Surprisingly, the error-correcting codes we construct achieve a better asymptotic rate than that of presently known constructions not having the Gray property, and exceed the Gilbert-Varshamov bound. Additionally, we present efficient ranking and unranking procedures, as well as a decoding procedure that runs in linear time. Finally, we also apply our methods to solve an outstanding issue with error-detecting rank-modulation Gray codes (snake-in-the-box codes) under a different metric, the Kendall $\tau$-metric, in the group of permutations over an even number of elements $S_{2n}$, where we provide asymptotically optimal codes.Comment: Revised version for journal submission. Additional results include more tight auxiliary constructions, a decoding shcema, ranking/unranking procedures, and application to snake-in-the-box codes under the Kendall tau-metri

Experimental implementation of bit commitment in the noisy-storage model

Fundamental primitives such as bit commitment and oblivious transfer serve as building blocks for many other two-party protocols. Hence, the secure implementation of such primitives are important in modern cryptography. In this work, we present a bit commitment protocol which is secure as long as the attacker's quantum memory device is imperfect. The latter assumption is known as the noisy-storage model. We experimentally executed this protocol by performing measurements on polarization-entangled photon pairs. Our work includes a full security analysis, accounting for all experimental error rates and finite size effects. This demonstrates the feasibility of two-party protocols in this model using real-world quantum devices. Finally, we provide a general analysis of our bit commitment protocol for a range of experimental parameters.Comment: 21 pages (7 main text +14 appendix), 6+3 figures. New version changed author's name from Huei Ying Nelly Ng to Nelly Huei Ying Ng, for consistency with other publication

Fault Secure Encoder and Decoder for NanoMemory Applications

Memory cells have been protected from soft errors for more than a decade; due to the increase in soft error rate in logic circuits, the encoder and decoder circuitry around the memory blocks have become susceptible to soft errors as well and must also be protected. We introduce a new approach to design fault-secure encoder and decoder circuitry for memory designs. The key novel contribution of this paper is identifying and defining a new class of error-correcting codes whose redundancy makes the design of fault-secure detectors (FSD) particularly simple. We further quantify the importance of protecting encoder and decoder circuitry against transient errors, illustrating a scenario where the system failure rate (FIT) is dominated by the failure rate of the encoder and decoder. We prove that Euclidean geometry low-density parity-check (EG-LDPC) codes have the fault-secure detector capability. Using some of the smaller EG-LDPC codes, we can tolerate bit or nanowire defect rates of 10% and fault rates of 10^(-18) upsets/device/cycle, achieving a FIT rate at or below one for the entire memory system and a memory density of 10^(11) bit/cm^2 with nanowire pitch of 10 nm for memory blocks of 10 Mb or larger. Larger EG-LDPC codes can achieve even higher reliability and lower area overhead

Quantum Locally Testable Codes

We initiate the study of quantum Locally Testable Codes (qLTCs). We provide a definition together with a simplification, denoted sLTCs, for the special case of stabilizer codes, together with some basic results using those definitions. The most crucial parameter of such codes is their soundness, $R(\delta)$, namely, the probability that a randomly chosen constraint is violated as a function of the distance of a word from the code ($\delta$, the relative distance from the code, is called the proximity). We then proceed to study limitations on qLTCs. In our first main result we prove a surprising, inherently quantum, property of sLTCs: for small values of proximity, the better the small-set expansion of the interaction graph of the constraints, the less sound the qLTC becomes. This phenomenon, which can be attributed to monogamy of entanglement, stands in sharp contrast to the classical setting. The complementary, more intuitive, result also holds: an upper bound on the soundness when the code is defined on poor small-set expanders (a bound which turns out to be far more difficult to show in the quantum case). Together we arrive at a quantum upper-bound on the soundness of stabilizer qLTCs set on any graph, which does not hold in the classical case. Many open questions are raised regarding what possible parameters are achievable for qLTCs. In the appendix we also define a quantum analogue of PCPs of proximity (PCPPs) and point out that the result of Ben-Sasson et. al. by which PCPPs imply LTCs with related parameters, carries over to the sLTCs. This creates a first link between qLTCs and quantum PCPs.Comment: Some of the results presented here appeared in an initial form in our quant-ph submission arXiv:1301.3407. This is a much extended and improved version. 30 pages, no figure

Spin glass reflection of the decoding transition for quantum error correcting codes

We study the decoding transition for quantum error correcting codes with the help of a mapping to random-bond Wegner spin models. Families of quantum low density parity-check (LDPC) codes with a finite decoding threshold lead to both known models (e.g., random bond Ising and random plaquette $\Z2$ gauge models) as well as unexplored earlier generally non-local disordered spin models with non-trivial phase diagrams. The decoding transition corresponds to a transition from the ordered phase by proliferation of extended defects which generalize the notion of domain walls to non-local spin models. In recently discovered quantum LDPC code families with finite rates the number of distinct classes of such extended defects is exponentially large, corresponding to extensive ground state entropy of these codes. Here, the transition can be driven by the entropy of the extended defects, a mechanism distinct from that in the local spin models where the number of defect types (domain walls) is always finite.Comment: 15 pages, 2 figure