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.
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.
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.
The research on quantum convolutional codes is aimed at protecting a flow of information over long distance communications. The polynomial representation of a quantum state is defined. Based on the Calderbank-Shor-Steane(CSS)-type construction of quantum codes, a new method for encoding and decoding of CSS-type quantum convolutional codes is presented and corresponding networks are described. The basis state of the code is transformed into the product of an information polynomial by the generator polynomial. Then networks can be realized by operations of polynomial multiplication. Finally, inspired by classical convolutional decoding idea, a quantum Viterbi algorithm with linear complexity is put forward.
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.
This chapter deals with the subject of quantum error correction and the related codes (QECC), which can be applied to noisy quantum channels and quantum memories with the purpose of preserving or protecting the information integrity. I first describe the basics of quantum repetition codes , as applicable to bit-flip and phase-flip quantum channels. Then I consider the 9-qubit Shor code , which has the capability of diagnosing and correcting any combination of bit-flip and phase-flip errors, up to one error of each type. Furthermore, it is shown that the Shor code is, in fact, capable of fully restoring qubit integrity under a continuum of bit or phase errors, a property that has no counterpart in the classical world of error-correction codes. But the exploration of QECC does not stop here! We shall discover the elegant Calderbank–Shor–Steane (CSS) codes, which have the capability of correcting any number of errors t , both bit-flip and phase-flip. As an application of the CSS code, I then describe the 7-qubit Hadamard–Steane code , which can correct up to one error on single qubits. A corresponding quantum circuit, based on an original generator-matrix example, is presented. Quantum repetition code In Chapter 11, we saw that the simplest form of error-correction code (ECC) is the repetition code , based on the principle of majority logic . The background assumption is that in a given message sequence, or bit string, the probability of a bit error is sufficiently small for the majority of bits to be correctly transmitted through the channel.
Usual scenarios of fault-tolerant computation are concerned with the fault-tolerant realization of quantum algorithms that compute classical functions, such as Shor's algorithm for factoring. In particular, this means that input and output to the quantum algorithm are classical. In contrast to stand-alone single-core quantum computers, in many distributed scenarios, quantum information might have to be passed on from one quantum information processing system to another one, possibly via noisy quantum communication channels with noise levels above fault-tolerant thresholds. In such situations, quantum information processing devices will have quantum inputs, quantum outputs or even both, which pass qubits among each other. Working in the fault-tolerant framework of [Kitaev, 1997], we show that any quantum circuit with quantum input and output can be transformed into a fault-tolerant circuit that produces the ideal circuit with some controlled noise applied at the input and output. The framework allows the direct composition of the statements, enabling versatile future applications. We illustrate this with a concrete application, namely, communication over a noisy channel with faulty encoding and decoding operations [Christandl and M{ü}ller-Hermes, 2024]. For communication codes with linear minimum distance, we construct fault-tolerant encoders and decoders for general noise (including coherent errors). For the weaker, but standard, model of local stochastic noise, we obtain fault-tolerant encoders and decoders for any communication code that can correct a constant fraction random errors.