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IEEE Information Theory Workshop (ITW)

On encoders for quantum convolutional codes

Markus Grassl, Martin Rötteler

Published Aug 1, 2010• 1 citation
Publisher

Abstract

We consider the problem of computing an encoding circuit for a quantum convolutional code given by a polynomial stabilizer matrix S(D) = (X(D) | Z(D)). We present an algorithm that is very similar to a polynomial-time algorithm for computing the Smith form of a polynomial matrix. This is a step towards the conjecture that any quantum convolutional code has an encoder with polynomially bounded depth.

DOI
10.1109/cig.2010.5592857

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We consider the problem of computing an encoding circuit for a quantum convolutional code given by a polynomial stabilizer matrix S(D) = (X(D) | Z(D)). We present an algorithm that is very similar to a polynomial-time algorithm for computing the Smith form of a polynomial matrix. This is a step towards the conjecture that any quantum convolutional code has an encoder with polynomially bounded depth.
We address the problems of constructing quantum convolutional codes (QCCs) and of encoding them. The first construction is a CSS-type construction which allows us to find QCCs of rate 2/4. The second construction yields a quantum convolutional code by applying a product code construction to an arbitrary classical convolutional code and an arbitrary quantum block code. We show that the resulting codes have highly structured and efficient encoders. Furthermore, we show that the resulting quantum circuits have finite depth, independent of the lengths of the input stream, and show that this depth is polynomial in the degree and frame size of the code.
Quantum convolutional codes can be used to protect streams of quantum bits against errors when sending them along a quantum channel.We review the underlying error model, conditions for quantum convolutional codes, and present a product code construction that yields families of quantum convolutional codes.Concerning the structure of encoding circuits for general quantum convolutional codes, we review the notion of catastrophic errors and show that each quantum convolutional code has encoders which are non-catastrophic, i. e., which have good error propagation properties.We present an algorithm to compute a quantum circuit which can be used to perform non-catastrophic encoding and inverse encoding in constant depth and give several examples for illustration.
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.
Quantum communication system is influenced by environment noise in the whole process. Quantum error-correcting codes can solve this problem efficiently. The polynomial representation of a quantum state is defined. Based on the structure of the classical convolutional codes, the basis state of the quantum convolutional codes is transformed into the multiplication of an information polynomial by the generator polynomial. A new method is proposed to encode and decode these quantum codes and the networks can be realized with the polynomial multiplication circuits. Finally, we provide a maximum likelihood error estimation algorithm with complexity growing linearly.
Quantum convolutional coding is a technique for encoding a stream of quantum information before transmitting it over a noisy quantum channel. Two important goals in the design of quantum convolutional encoders are to minimize the memory required by them and to avoid the catastrophic propagation of errors. In a previous paper, we determined minimal-memory, noncatastrophic, polynomial-depth encoders for a few exemplary quantum convolutional codes. In this paper, we elucidate a general technique for finding an encoder of an arbitrary quantum convolutional code such that the encoder possesses these desirable properties. We also provide an elementary proof that these encoders are nonrecursive. Finally, we apply our technique to many quantum convolutional codes from the literature. © 1963-2012 IEEE.
Quantum convolutional coding is a technique for encoding a stream of quantum information before transmitting it over a noisy quantum channel. Two important goals in the design of quantum convolutional encoders are to minimize the memory required by them and to avoid the catastrophic propagation of errors. In a previous paper, we determined minimal-memory, noncatastrophic, polynomial-depth encoders for a few exemplary quantum convolutional codes. In this paper, we elucidate a general technique for finding an encoder of an arbitrary quantum convolutional code such that the encoder possesses these desirable properties. We also provide an elementary proof that these encoders are nonrecursive. Finally, we apply our technique to many quantum convolutional codes from the literature.