Structured Error Recovery for Codeword-Stabilized Quantum Codes

Codeword stabilized (CWS) codes are, in general, non-additive quantum codes that can correct errors by an exhaustive search of different error patterns, similar to the way that we decode classical non-linear codes. For an n-qubit quantum code correcting errors on up to t qubits, this brute-force approach consecutively tests different errors of weight t or less, and employs a separate n-qubit measurement in each test. In this paper, we suggest an error grouping technique that allows to simultaneously test large groups of errors in a single measurement. This structured error recovery technique exponentially reduces the number of measurements by about 3^t times. While it still leaves exponentially many measurements for a generic CWS code, the technique is equivalent to syndrome-based recovery for the special case of additive CWS codes.Comment: 13 pages, 9 eps figure

On optimal quantum codes

We present families of quantum error-correcting codes which are optimal in the sense that the minimum distance is maximal. These maximum distance separable (MDS) codes are defined over q-dimensional quantum systems, where q is an arbitrary prime power. It is shown that codes with parameters [[n,n-2d+2,d]]_q exist for all 3 <= n <= q and 1 <= d <= n/2+1. We also present quantum MDS codes with parameters [[q^2,q^2-2d+2,d]]_q for 1 <= d <= q which additionally give rise to shortened codes [[q^2-s,q^2-2d+2-s,d]]_q for some s.Comment: Accepted for publication in the International Journal of Quantum Informatio

Low-complexity quantum codes designed via codeword-stabilized framework

We consider design of the quantum stabilizer codes via a two-step, low-complexity approach based on the framework of codeword-stabilized (CWS) codes. In this framework, each quantum CWS code can be specified by a graph and a binary code. For codes that can be obtained from a given graph, we give several upper bounds on the distance of a generic (additive or non-additive) CWS code, and the lower Gilbert-Varshamov bound for the existence of additive CWS codes. We also consider additive cyclic CWS codes and show that these codes correspond to a previously unexplored class of single-generator cyclic stabilizer codes. We present several families of simple stabilizer codes with relatively good parameters.Comment: 12 pages, 3 figures, 1 tabl

Minimal-memory realization of pearl-necklace encoders of general quantum convolutional codes

Quantum convolutional codes, like their classical counterparts, promise to offer higher error correction performance than block codes of equivalent encoding complexity, and are expected to find important applications in reliable quantum communication where a continuous stream of qubits is transmitted. Grassl and Roetteler devised an algorithm to encode a quantum convolutional code with a "pearl-necklace encoder." Despite their theoretical significance as a neat way of representing quantum convolutional codes, they are not well-suited to practical realization. In fact, there is no straightforward way to implement any given pearl-necklace structure. This paper closes the gap between theoretical representation and practical implementation. In our previous work, we presented an efficient algorithm for finding a minimal-memory realization of a pearl-necklace encoder for Calderbank-Shor-Steane (CSS) convolutional codes. This work extends our previous work and presents an algorithm for turning a pearl-necklace encoder for a general (non-CSS) quantum convolutional code into a realizable quantum convolutional encoder. We show that a minimal-memory realization depends on the commutativity relations between the gate strings in the pearl-necklace encoder. We find a realization by means of a weighted graph which details the non-commutative paths through the pearl-necklace. The weight of the longest path in this graph is equal to the minimal amount of memory needed to implement the encoder. The algorithm has a polynomial-time complexity in the number of gate strings in the pearl-necklace encoder.Comment: 16 pages, 5 figures; extends paper arXiv:1004.5179v

Efficient Quantum Circuits for Non-Qubit Quantum Error-Correcting Codes

We present two methods for the construction of quantum circuits for quantum error-correcting codes (QECC). The underlying quantum systems are tensor products of subsystems (qudits) of equal dimension which is a prime power. For a QECC encoding k qudits into n qudits, the resulting quantum circuit has O(n(n-k)) gates. The running time of the classical algorithm to compute the quantum circuit is O(n(n-k)^2).Comment: 18 pages, submitted to special issue of IJFC

Quantum generalized Reed-Solomon codes: Unified framework for quantum MDS codes

We construct a new family of quantum MDS codes from classical generalized Reed-Solomon codes and derive the necessary and sufficient condition under which these quantum codes exist. We also give code bounds and show how to construct them analytically. We find that existing quantum MDS codes can be unified under these codes in the sense that when a quantum MDS code exists, then a quantum code of this type with the same parameters also exists. Thus as far as is known at present, they are the most important family of quantum MDS codes.Comment: 9 pages, no figure

Codeword stabilized quantum codes: algorithm and structure

The codeword stabilized ("CWS") quantum codes formalism presents a unifying approach to both additive and nonadditive quantum error-correcting codes (arXiv:0708.1021). This formalism reduces the problem of constructing such quantum codes to finding a binary classical code correcting an error pattern induced by a graph state. Finding such a classical code can be very difficult. Here, we consider an algorithm which maps the search for CWS codes to a problem of identifying maximum cliques in a graph. While solving this problem is in general very hard, we prove three structure theorems which reduce the search space, specifying certain admissible and optimal ((n,K,d)) additive codes. In particular, we find there does not exist any ((7,3,3)) CWS code though the linear programming bound does not rule it out. The complexity of the CWS search algorithm is compared with the contrasting method introduced by Aggarwal and Calderbank (arXiv:cs/0610159).Comment: 11 pages, 1 figur

SIC~POVMs and Clifford groups in prime dimensions

We show that in prime dimensions not equal to three, each group covariant symmetric informationally complete positive operator valued measure (SIC~POVM) is covariant with respect to a unique Heisenberg--Weyl (HW) group. Moreover, the symmetry group of the SIC~POVM is a subgroup of the Clifford group. Hence, two SIC~POVMs covariant with respect to the HW group are unitarily or antiunitarily equivalent if and only if they are on the same orbit of the extended Clifford group. In dimension three, each group covariant SIC~POVM may be covariant with respect to three or nine HW groups, and the symmetry group of the SIC~POVM is a subgroup of at least one of the Clifford groups of these HW groups respectively. There may exist two or three orbits of equivalent SIC~POVMs for each group covariant SIC~POVM, depending on the order of its symmetry group. We then establish a complete equivalence relation among group covariant SIC~POVMs in dimension three, and classify inequivalent ones according to the geometric phases associated with fiducial vectors. Finally, we uncover additional SIC~POVMs by regrouping of the fiducial vectors from different SIC~POVMs which may or may not be on the same orbit of the extended Clifford group.Comment: 30 pages, 1 figure, section 4 revised and extended, published in J. Phys. A: Math. Theor. 43, 305305 (2010

Radiative effects of convection in the tropical Pacific

The radiative effects of tropical clouds at the tropopause and the ocean surface have been estimated by using in situ measurements from the Central Equatorial Pacific Experiment (CEPEX). The effect of clouds is distinguished from the radiative effects of the surrounding atmosphere by calculating the shortwave and longwave cloud forcing. These terms give the reduction in insolation and the increase in absorption of terrestrial thermal emission associated with clouds. At the tropopause the shortwave and longwave cloud forcing are nearly equal and opposite, even on daily timescales. Therefore the net effect of an ensemble of convective clouds is small compared to other radiative terms in the surface-tropospheric heat budget. This confirms the statistical cancellation of cloud forcing observed in Earth radiation budget measurements from satellites. At the surface the net effect of clouds is to reduce the radiant energy absorbed by the ocean. Under deep convective clouds the diurnally averaged reduction exceeds 150 W m(-2). The divergence of flux in the cloudy atmosphere can be estimated from the difference in cloud forcing at the surface and tropopause. The CEPEX observations show that the atmospheric cloud forcing is nearly equal and opposite to the surface forcing. Based upon the frequency of convection, the atmospheric forcing approaches 100 W m(-2) when the surface temperature is 303 K. The cloud forcing is closely related to the frequency of convective cloud systems. This relation is used in conjunction with cloud population statistics derived from satellite to calculate the change in surface cloud forcing with sea surface temperature. The net radiative cooling of the surface by clouds increases at a rate of 20 W m(-2)K(-1)during the CEPEX observing period