Rate-(n-2)/n unrestricted and CSS-type quantum convolutional codes with up to 4096 states and minimum distances up to 10 are constructed as stabilizer codes from classical self-orthogonal rate-1/n F4-linear and binary linear convolutional codes, respectively. These codes generally have higher rate and less decoding complexity than comparable quantum block codes or previous quantum convolutional codes. Rate-(n-2)/n block stabilizer codes with the same rate and error-correction capability and essentially the same decoding complexity are derived from these convolutional codes via tail-biting
Rate-(n-2)/n unrestricted and CSS-type quantum convolutional codes with up to 4096 states and minimum distances up to 10 are constructed as stabilizer codes from classical self-orthogonal rate-1/n F4-linear and binary linear convolutional codes, respectively. These codes generally have higher rate and less decoding complexity than comparable quantum block codes or previous quantum convolutional codes. Rate-(n-2)/n block stabilizer codes with the same rate and error-correction capability and essentially the same decoding complexity are derived from these convolutional codes via tail-biting
Simple rate-1/3 single-error-correcting unrestricted and CSS-type quantum convolutional codes are constructed from classical self-orthogonal F4-linear and F2-linear convolutional codes, respectively. These quantum convolutional codes have higher rate than comparable quantum block codes or previous quantum convolutional codes, and are simple to decode. A block single-error-correcting [9,3,3] tail-biting code is derived from the unrestricted convolutional code, and similarly a [15,5,3] CSS-type block code from the CSS-type convolutional code
We present a construction of self-orthogonal codes using product codes. From the resulting codes, one can construct both block quantum error-correcting codes and quantum convolutional codes. We show that from the examples of convolutional codes found, we can derive ordinary quantum error-correcting codes using tail-biting with parameters [42N, 24N, 3]2. While it is known that the product construction cannot improve the rate in the classical case, we show that this can happen for quantum codes: we show that a code [15, 7, 3]2is obtained by the product of a code [5,1,3]2with a suitable code
A new class of quantum stabilizer codes is presented.The problem of finding stabilizer quantum-error-correcting codes can be transformed into the problem of finding self-orthogonal codes over the Galois field GF(4) under a trace inner product.Based on this connection,the necessary and sufficient condition is proved under which classical convolutional codes over GF(4) are self-orthogonal with respect to the trace inner product,and then corresponding codes satisfying the condition are found.Tail-biting codes with a simple block structure are obtained by tail-biting transformation,which are self-orthogonal with respect to the trace inner product.Finally,relative quantum stabilizer codes are constructed.The code construction is simple,and the minimum distance of the code approaches the upper bound.
Stabilizer codes are an important branch of quantum codes, but the existing stabilizer codes constructed from the classical binary codes almost exclusively belong to the special subclass of CSS codes. This paper develops simple and systematic constructions for two rich classes of non-CSS quantum stabilizer codes: quantum low density parity check (LDPC) codes based on classical quasi-cyclic LDPC codes and quantum convolutional codes based on classical LDPC-convolutional codes. Both classes enjoy a wide range of lengths and rates, and offer stronger error correction capability than the existing codes.
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)), and some of them perform as well as or better than the existing quantum codes.
Quantum error correction codes were introduced as a means to protect quantum information from decoherance and operational errors. Based on their approach to error control, error correcting codes can be divided into two different classes: block codes and convolutional codes. There has been significant development towards finding quantum block codes, since they were first discovered in 1995. In contrast, quantum convolutional codes remained mainly uninvestigated. In this thesis, we develop the stabilizer formalism for quantum convolutional codes. We define distance properties of these codes and give a general method for constructing encoding circuits, given a set of generators of the stabilizer of a quantum convolutional stabilizer code, is shown. The resulting encoding circuit enables online encoding of the qubits, i.e., the encoder does not have to wait for the input transmission to end before starting the encoding process. We develop the quantum analogue of the Viterbi algorithm. The quantum Viterbi algorithm (QVA) is a maximum likehood error estimation algorithm, the complexity of which grows linearly with the number of encoded qubits. A variation of the quantum Viterbi algorithm, the Windowed QVA, is also discussed. Using Windowed QVA, we can estimate the most likely error without waiting for the entire received sequence.