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arXiv:1904.11250

Fundamental building blocks for unitary matrices and quantum logic gates

Ted Hurley

Published Apr 25, 2019• 0 citations
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Abstract

A unitary matrix is shown to be a product of certain {\em basic} unitary matrices and the product is unique up to order. A basic unitary matrix itself is a product of {\em minimal basic} matrices. A unitary $n\times n$ matrix can be expressed as a product of at most $n$ basic matrices. A basic matrix is defined in terms of an idempotent matrix; the idempotent matrix used in the definition of a minimal basic unitary matrix is a {\em density} matrix. This gives builders for unitary matrices. Quantum logic gates are represented by unitary matrices thus giving unique building material for quantum logic gates.

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A unitary matrix is shown to be a product of certain {\em basic} unitary
matrices and the product is unique up to order. A basic unitary matrix itself
is a product of {\em minimal basic} matrices. A unitary n×nn\times nn×n matrix can
be expressed as a product of at most nnn basic matrices. A basic matrix is
defined in terms of an idempotent matrix; the idempotent matrix used in the
definition of a minimal basic unitary matrix is a {\em density} matrix. This gives builders for unitary matrices. Quantum logic gates are represented
by unitary matrices thus giving unique building material for quantum logic
gates.
Quantum computing fundamentals are based on Hilbert spaces and linear algebra, which allows to describe quantum gates as matrix representations. In this paper, a matrix approach is taken in order to analyze basic quantum circuits, composed of basic quantum gates of 1, 2, and 3 qubits. The most common cases of all those gates are analyzed. One-qubit gates are expressed as 2x2 matrices, accounting for a rotation around a determined axis. Two-qubit gates are described as 4x4 matrices, standing for controlled gates. Three-quibit gates are defined as 8x8 matrices, extending the concept of controlled gates and rotations.
As two basic building blocks for any quantum circuit, we consider the 1-qubit NEGATOR(θ) circuit and the 1-qubit PHASOR(θ) circuit, extensions of the NOT gate and PHASE gate, respectively: NEGATOR(π) = NOT and PHASOR(π) = PHASE.Quantum circuits (acting on w qubits) consisting of controlled NEGATORs are represented by matrices from XU(2 w ); quantum circuits (acting on w qubits) consisting of controlled PHASORs are represented by matrices from ZU(2 w ).Here, XU(n) and ZU(n) are subgroups of the unitary group U(n): the group XU(n) consists of all n × n unitary matrices with all line sums equal to 1 and the group ZU(n) consists of all n × n unitary diagonal matrices with first entry equal to 1.We conjecture that any U(n) matrix can be decomposed into four parts: U = e iα Z1XZ2, where both Z1 and Z2 are ZU(n) matrices and X is an XU(n) matrix.For n = 2 w , this leads to a decomposition of a quantum computer into simpler blocks.
This chapter briefly describes the basic concepts and principles of quantum computing. Firstly, the concepts of qubit, quantum coherence, quantum decoherence, quantum entanglement, quantum density operators, linear operators, inner products, outer products, tensor products, Hermite operators, and unitary operators are described. Then, the four basic assumptions of quantum mechanics are introduced, focusing on the measurement assumptions of quantum mechanics. Finally, the definition of commonly used quantum logic gates is given including single qubit gates, double qubit gates, and multiple qubit gates. These contents provide the necessary theoretical basis for subsequent chapters.