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Cambridge University Press eBooks

Quantum convolutional codes

Mark M. Wilde

Published Sep 5, 2013• 1 citation
Publisher

Abstract

Introduction Many of the quantum codes discussed in this book are quantum block codes. Quantum block codes are useful both in quantum computing and in quantum communication, but one of the drawbacks for quantum communication is that, in general, the sender must have her block of qubits ready before encoding takes place. This preparation may be a heavy demand on the sender when the size of the block code is large. Quantum convolutional coding theory [OT03, OT04, FGG07] offers a paradigm different from quantum block coding and has numerous benefits for quantum communication. The convolutional structure is useful when a sender possesses a stream of qubits to transmit to a receiver. It is possible to encode and decode quantum convolutional codes with quantum shift-register circuits [W09b], the natural “quantization” of a classical shift register circuit. These quantum circuits ensure a low complexity for encoding and decoding while also providing higher performance than a block code with equivalent encoding complexity [FGG07]. Quantum shift-register circuits have the property that the sender Alice and the receiver Bob can respectively send and receive qubits in an “online” fashion. Alice can encode an arbitrary number of information qubits without worrying beforehand how many she may want to send over the quantum communication channel.

DOI
10.1017/cbo9781139034807.011

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Introduction Many of the quantum codes discussed in this book are quantum block codes. Quantum block codes are useful both in quantum computing and in quantum communication, but one of the drawbacks for quantum communication is that, in general, the sender must have her block of qubits ready before encoding takes place. This preparation may be a heavy demand on the sender when the size of the block code is large. Quantum convolutional coding theory [OT03, OT04, FGG07] offers a paradigm different from quantum block coding and has numerous benefits for quantum communication. The convolutional structure is useful when a sender possesses a stream of qubits to transmit to a receiver. It is possible to encode and decode quantum convolutional codes with quantum shift-register circuits [W09b], the natural “quantization” of a classical shift register circuit. These quantum circuits ensure a low complexity for encoding and decoding while also providing higher performance than a block code with equivalent encoding complexity [FGG07]. Quantum shift-register circuits have the property that the sender Alice and the receiver Bob can respectively send and receive qubits in an “online” fashion. Alice can encode an arbitrary number of information qubits without worrying beforehand how many she may want to send over the quantum communication channel.
We present a general theory of entanglement-assisted quantum convolutional coding. The codes have a convolutional or memory structure, they assume that the sender and receiver share noiseless entanglement prior to quantum communication, and they are not restricted to possess the Calderbank-Shor-Steane structure as in previous work. We provide two significant advances for quantum convolutional coding theory. We first show how to "expand" a given set of quantum convolutional generators. This expansion step acts as a preprocessor for a polynomial symplectic Gram-Schmidt orthogonalization procedure that simplifies the commutation relations of the expanded generators to be the same as those of entangled Bell states (ebits) and ancilla qubits. The above two steps produce a set of generators with equivalent error-correcting properties to those of the original generators. We then demonstrate how to perform online encoding and decoding for a stream of information qubits, halves of ebits, and ancilla qubits. The upshot of our theory is that the quantum code designer can engineer quantum convolutional codes with desirable error-correcting properties without having to worry about the commutation relations of these generators. © 2010 Springer Science+Business Media, LLC.
A quantum-shift-register circuit acts on a set of input qubits and memory qubits, outputs a set of output qubits and updated memory qubits, and feeds the memory back into the device for the next cycle (similar to the operation of a classical shift register). Such a device finds application as an encoding and decoding circuit for a particular type of quantum error-correcting code called a quantum convolutional code. Building on the Ollivier-Tillich and Grassl-R\"otteler encoding algorithms for quantum convolutional codes, I present a method to determine a quantum-shift-register encoding circuit for a quantum convolutional code. I also determine a formula for the amount of memory that a Calderbank-Shor-Steane (CSS) quantum convolutional code requires. I then detail primitive quantum-shift-register circuits that realize all of the finite- and infinite-depth transformations in the shift-invariant Clifford group (the class of transformations important for encoding and decoding quantum convolutional codes). The memory formula for a CSS quantum convolutional code then immediately leads to a formula for the memory required by a CSS entanglement-assisted quantum convolutional code.
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 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.
We outline a quantum convolutional coding technique for protecting a stream of classical bits and qubits. Our goal is to provide a framework for designing codes that approach the "grandfather" capacity of an entanglement-assisted quantum channel for sending classical and quantum information simultaneously. Our method incorporates several resources for quantum redundancy: fresh ancilla qubits, entangled bits, and gauge qubits. The use of these diverse resources gives our technique the benefits of both active and passive quantum error correction. We can encode a classical-quantum bit stream with periodic quantum gates because our codes possess a convolutional structure. We end with an example of a "grandfather" quantum convolutional code that protects one qubit and one classical bit per frame by encoding them with one fresh ancilla qubit, one entangled bit, and one gauge qubit per frame. We explicitly provide the encoding and decoding circuits for this example and discuss its error-correcting capability.
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.