Quantum Shift Register Circuits

TL;DR

This paper introduces a method to construct quantum shift register circuits for quantum convolutional codes, providing formulas for memory requirements and realizing key transformations in the Clifford group.

Contribution

It presents a novel method to determine quantum shift register encoding circuits and formulas for memory needs in CSS quantum convolutional codes.

Findings

Derived a formula for memory requirements of CSS quantum convolutional codes.

Constructed primitive circuits for all shift-invariant Clifford transformations.

Extended memory formulas to entanglement-assisted quantum convolutional codes.

Abstract

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-Roetteler 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 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.