Circular CNOT Circuits: Definition, Analysis and Application to Fault-Tolerant Quantum Circuits

TL;DR

This paper introduces circular CNOT circuits, a novel quantum circuit formalism that enables fault-tolerant quantum computation by analyzing cyclic permutations and stabiliser transformations within a circular framework.

Contribution

It defines circular CNOT circuits, develops a Boolean model for stabiliser transformations, and demonstrates their application to scalable, fault-tolerant quantum computing.

Findings

Circular CNOT circuits can represent universal fault-tolerant quantum computation.

A Boolean model for stabiliser transformations is established.

The approach supports scalable quantum circuit design.

Abstract

The work proposes an extension of the quantum circuit formalism where qubits (wires) are circular instead of linear. The left-to-right interpretation of a quantum circuit is replaced by a circular representation which allows to select the starting point and the direction in which gates are executed. The representation supports all the circuits obtained after computing cyclic permutations of an initial quantum gate list. Two circuits, where one has a gate list which is a cyclic permutation of the other, will implement different functions. The main question appears in the context of scalable quantum computing, where multiple subcircuits are used for the construction of a larger fault-tolerant one: can the same circular representation be used by multiple subcircuits? The circular circuits are sufficient for constructing computationally universal, fault-tolerant circuits formed entirely of qubit initialisation, CNOT gates and qubit measurements. The main result of modelling circular CNOT circuits is that a derived Boolean representation allows to define a set of equations for $X$ and $Z$ stabiliser transformations. Through a well defined set of steps it is possible to reduce the initial equations to a set of stabiliser transformations given a series of cuts through the circular circuit.