Quantum Stabilizer Research Articles

Grassmannian Packings From Operator Reed–Muller Codes

This paper introduces multidimensional generalizations of binary Reed-Muller codes where the codewords are projection operators, and the corresponding subspaces are widely separated with respect to the chordal distance on Grassmannian space. Parameters of these Grassmannian packings are derived and a low complexity decoding algorithm is developed by modifying standard decoding algorithms for binary Reed-Muller codes. The subspaces are associated with projection operators determined by Pauli matrices appearing in the theory of quantum error correction and this connection with quantum stabilizer codes may be of independent interest. The Grassmannian packings constructed here find application in noncoherent wireless communication with multiple antennas, where separation with respect to the chordal distance on Grassmannian space guarantees closeness to the channel capacity. It is shown that the capacity of the noncoherent multiple-input-multiple-output (MIMO) channel at both low and moderate signal-to-noise ratio (SNR) (under the constraint that only isotropically distributed unitary matrices are used for information transmission) is closely approximated by these packings.

Toric residue codes: I

In this paper, we begin exploring the construction of algebraic codes from toric varieties using toric residues. Though algebraic codes have been constructed from toric varieties, they have all been evaluation codes, where one evaluates the sections of a line bundle at a collection of rational points. In the present paper, instead of evaluating sections of a line bundle at rational points, we compute the residues of differential forms at these points. We show that this method produces codes that are close to the dual of those produced by the first technique. We conclude by studying several examples, and also discussing applications of this technique to the construction of quantum stabilizer codes and also to decryption of toric evaluation codes.

Quantum stabilizer codes for correlated and asymmetric depolarizing errors

We study the performance of common quantum stabilizer codes in the presence of asymmetric and correlated errors. Specifically, we consider the depolarizing noisy quantum memory channel and perform quantum error correction via the five and seven-qubit stabilizer codes. We characterize these codes by means of the entanglement fidelity as function of the error probability and the degree of memory. We show that their performances are lowered by the presence of correlations and we compute the error probability threshold values for codes effectiveness. Furthermore, we uncover that the asymmetry in the error probabilities does not affect the performance of the five-qubit code while it does affect the performance of the seven-qubit code which results less effective when considering correlated and symmetric depolarizing errors but more effective for correlated and asymmetric errors.

Family of fast quantum stabilizer codes constructions

In this letter, we proposed a new family of graphical quantum stabilizer codes (QSCs) based on block Jacket transform and its fast construction approach. The approach provides great flexibility in designing a stabilizer with a large block length and a high efficiency in terms of recursive relations. According to the graph theory, the present quantum code construction method has a low coding complexity and the codes are asymptotically good.

Efficient Quantum Stabilizer Codes: LDPC and LDPC-Convolutional Constructions

Existing quantum stabilizer codes constructed from the classic binary codes exclusively belong to the special subclass of Calderbank-Shor-Steane (CSS) codes. This paper fills in the gap by proposing five systematic constructions for non-CSS stabilizer codes, the first four of which are based on classic binary quasi-cyclic low-density parity-check (LDPC) codes and last on classic binary LDPC-convolutional codes. These new constructions exploit structured sparse graphs, make essential use of simple and powerful coding techniques including concatenation, rotation and scrambling, and generate rich classes of codes with a wide range of lengths and rates. We derive the sufficient, and in some cases also the necessary, conditions for each construction to satisfy the general symplectic inner product (SIP) condition, and develop practical decoder algorithms for these codes. The resulting codes are the first classes of non-CSS quantum LDPC codes and non-CSS quantum convolutional codes rooted from classic binary codes (rather than codes in GF(4 <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">)</i> ), and some of them perform as well as or better than the existing quantum codes.

Classical spin systems and the quantum stabilizer formalism: General mappings and applications

We present general mappings between classical spin systems and quantum physics. More precisely, we show how to express partition functions and correlation functions of arbitrary classical spin models as inner products between quantum stabilizer states and product states, thereby generalizing mappings for some specific models established in our previous work [M. Van den Nest et al., Phys. Rev. Lett. 98, 117207 (2007)]. For Ising- and Potts-type models with and without external magnetic field, we show how the entanglement features of the corresponding stabilizer states are related to the interaction pattern of the classical model, while the choice of product states encodes the details of the interaction. These mappings establish a link between the fields of classical statistical mechanics and quantum information theory, which we utilize to transfer techniques and methods developed in one field to gain insight into the other. For example, we use quantum information techniques to recover well known duality relations and local symmetries of classical models in a simple way and provide new classical simulation methods to simulate certain types of classical spin models. We show that in this way all inhomogeneous models of q-dimensional spins with pairwise interaction pattern specified by a graph of bounded tree width can be simulated efficiently. Finally, we show relations between classical spin models and measurement-based quantum computation.

COMBINATORIAL CONSTRUCTION OF LINEAR QUANTUM CODES FROM QUATERNARY BCH CODES

Classical BCH codes that contain their (Euclidean or Hermitian) dual codes can be used to construct quantum stabilizer codes. But for given restricted length n, good quantum BCH codes are very sparse. In this paper, by puncturing and pasting check matrices of Hermitian dual containing BCH codes over the quaternary field, we construct many linear quantum codes with good parameters, and some of them have parameters exceeding the finite Gilbert-Varshamov bound for stabilizer quantum codes.

A Family of Asymptotically Good Quantum Codes Based on Code Concatenation

We explicitly construct an infinite family of asymptotically good concatenated quantum stabilizer codes where the outer codes use quantum Reed-Solomon codes and the inner codes use a set of special quantum codes. In the field of quantum error-correcting codes, this is the first time that a family of asymptotically good quantum codes is derived from bad codes. Its occurrence supplies a gap in quantum coding theory.

Generalized concatenated quantum codes

We discuss the concept of generalized concatenated quantum codes. This generalized concatenation method provides a systematical way for constructing good quantum codes, both stabilizer codes and nonadditive codes. Using this method, we construct families of single-error-correcting nonadditive quantum codes, in both binary and nonbinary cases, which not only outperform any stabilizer codes for finite block length but also asymptotically meet the quantum Hamming bound for large block length.

Upper bound on quantum stabilizer codes

By studying sets of operators having constant weight, we present an analytical upper bound on the pure quantum stabilizer codes whose underlying quantum system can be of arbitrary dimension, which outperforms the well-known quantum Hamming bound, the optimal analytical upper bound so far for small code length.

Asymmetric quantum codes: constructions, bounds and performance

Recently, quantum error-correcting codes have been proposed that capitalize on the fact that many physical error models lead to a significant asymmetry between the probabilities for bit- and phase-flip errors. An example for a channel that exhibits such asymmetry is the combined amplitude damping and dephasing channel, where the probabilities of bit and phase flips can be related to relaxation and dephasing time, respectively. We study asymmetric quantum codes that are obtained from the Calderbank–Shor–Steane (CSS) construction. For such codes, we derive upper bounds on the code parameters using linear programming. A central result of this paper is the explicit construction of some new families of asymmetric quantum stabilizer codes from pairs of nested classical codes. For instance, we derive asymmetric codes using a combination of Bose–Chaudhuri–Hocquenghem (BCH) and finite geometry low-density parity-check (LDPC) codes. We show that the asymmetric quantum codes offer two advantages, namely to allow a higher rate without sacrificing performance when compared with symmetric codes and vice versa to allow a higher performance when compared with symmetric codes of comparable rates. Our approach is based on a CSS construction that combines BCH and finite geometry LDPC codes.

Graph-based classification of self-dual additive codes over finite fields

Quantum stabilizer states over Fm can be represented as self-dual additive codes over Fm2. These codes can be represented as weighted graphs, and orbits of graphs under the generalized local complementation operationcorrespond to equivalence classes of codes. We have previously used this fact to classify self-dual additive codes over F4. In this paper we classify selfdual additive codes over F9, F16, and F25. Assuming that the classical MDSconjecture holds, we are able to classify all self-dual additive MDS codes over F9 by using an extension technique. We prove that the minimum distanceof a self-dual additive code is related to the minimum vertex degree in theassociated graph orbit. Circulant graph codes are introduced, and a computersearch reveals that this set contains many strong codes. We show that some ofthese codes have highly regular graph representations.

The geometry of quantum codes

We give a geometric interpretation of additive quantum stabilizer codes in terms of sets of lines in binary symplectic space. It is used to obtain synthetic geometric constructions and non-existence results. In particular several open problems are removed from Grassl’s database [13].

A Construction of Quantum Stabilizer Codes Based on Syndrome Assignment by Classical Parity-Check Matrices

In quantum coding theory, stabilizer codes are probably the most important class of quantum codes. They are regarded as the quantum analogue of the classical linear codes and the properties of stabilizer codes have been carefully studied in the literature. In this paper, a new but simple construction of stabilizer codes is proposed based on syndrome assignment by classical parity-check matrices. This method reduces the construction of quantum stabilizer codes to the construction of classical parity-check matrices that satisfy a specific commutative condition. The quantum stabilizer codes from this construction have a larger set of correctable error operators than expected. Its (asymptotic) coding efficiency is comparable to that of CSS codes. A class of quantum Reed-Muller codes is constructed, which have a larger set of correctable error operators than that of the quantum Reed-Muller codes developed previously in the literature. Quantum stabilizer codes inspired by classical quadratic residue codes are also constructed and some of which are optimal in terms of their coding parameters.

Nonbinary quantum error-correcting codes from algebraic curves

We give a new exposition and proof of a generalized CSS construction for nonbinary quantum error-correcting codes. Using this we construct nonbinary quantum stabilizer codes with various lengths, dimensions, and minimum distances from algebraic curves. We also give asymptotically good nonbinary quantum codes from a Garcia–Stichtenoth tower of function fields which are constructible in polynomial time.

On Quantum and Classical BCH Codes

Classical Bose-Chaudhuri-Hocquenghem (BCH) codes that contain their (Euclidean or Hermitian) dual codes can be used to construct quantum stabilizer codes; this correspondence studies the properties of such codes. It is shown that a BCH code of length n can contain its dual code only if its designed distance delta=O(radicn), and the converse is proved in the case of narrow-sense codes. Furthermore, the dimension of narrow-sense BCH codes with small design distance is completely determined, and - consequently - the bounds on their minimum distance are improved. These results make it possible to determine the parameters of quantum BCH codes in terms of their design parameters

Trellises for stabilizer codes: Definition and uses

Trellises play an important theoretical and practical role for classical codes. Their main utility is to devise complexity-efficient error estimation algorithms. Here, we describe trellis representations for quantum stabilizer codes. We show that they share the same properties as their classical analogs. In particular, for any stabilizer code it is possible to find a minimal trellis representation. Our construction is illustrated by two fundamental error estimation algorithms.

Conversion of a general quantum stabilizer code to an entanglement distillation protocol

We show how to convert a quantum stabilizer code to a one-way or two-way entanglement distillation protocol. The proposed conversion method is a generalization of those of Shor-Preskill and Nielsen-Chuang. The recurrence protocol and the quantum privacy amplification protocol are equivalent to the protocols converted from [[2,1]] stabilizer codes. We also give an example of a two-way protocol converted from a stabilizer better than the recurrence protocol and the quantum privacy amplification protocol. The distillable entanglement by the class of one-way protocols converted from stabilizer codes for a certain class of states is equal to or greater than the achievable rate of stabilizer codes over the channel corresponding to the distilled state, and they can distill asymptotically more entanglement from a very noisy Werner state than the hashing protocol.

Quantum codes [[6, 2, 3]]/sub p/ and [[7, 3, 3]]/sub p/ (p ≥ 3) exist

We prove that nonbinary quantum stabilizer codes with parameters [[n, k, d]]/sub p/ = [[6, 2, 3]]/sub p/ and [[7, 3, 3]]/sub p/ exist for all odd primes p by using graph machinery, given by Schlingemann and Werner(see Phys. Rev. A, vol.65, no.012308, 2001), with a little number theory and combinatorics.

Achievable rates for the Gaussian quantum channel

We study the properties of quantum stabilizer codes that embed a finite-dimensional protected code space in an infinite-dimensional Hilbert space. The stabilizer group of such a code is associated with a symplectically integral lattice in the phase space of 2N canonical variables. From the existence of symplectically integral lattices with suitable properties, we infer a lower bound on the quantum capacity of the Gaussian quantum channel that matches the one-shot coherent information optimized over Gaussian input states.