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On two subgroups of U(n), important for quantum computing

Alexis De Vos, Stijn De Baerdemacker

Published Jan 1, 2014• 0 citations
OpenAlex

Abstract

ZU(n) and XU(n) are subgroups of the unitary group U(n): the group XU(n) consists of all n X n unitary matrices with all 2n line sums (i.e. all n row sums and all n column sums) equal to 1 and the group ZU(n) consists of all n X n unitary diagonal matrices with first entry equal to 1.

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As two basic building blocks for any quantum circuit, we consider the 1-qubit PHASOR circuit ϕ( θ ) and the 1-qubit NEGATOR circuit N ( θ ). Both are roots of the IDENTITY circuit. Indeed: both ϕ(0) and N (0) equal the 2 × 2 unit matrix. Additionally, the NEGATOR is a root of the classical NOT gate. Quantum circuits (acting on w qubits) consisting of controlled PHASORs are represented by matrices from ZU(2 w ); quantum circuits consisting of controlled NEGATORs are represented by matrices from XU(2 w ). Here, ZU( n ) and XU( n ) are subgroups of the unitary group U( n ): the group XU( n ) consists of all n × n unitary matrices with all 2 n line sums (i.e. all n row sums and all n column sums) equal to 1 and the group ZU( n ) consists of all n × n unitary diagonal matrices with first entry equal to 1. Any U( n ) matrix can be decomposed into four parts: U = exp( iα ) Z 1 XZ 2 , where both Z 1 and Z 2 are ZU( n ) matrices and X is an XU( n ) matrix. We give an algorithm to find the decomposition. For n = 2 w it leads to a four-block synthesis of an arbitrary quantum computer.
Starting from the three Pauli matrices, it is possible to construct a synthesis procedure for an arbitrary quantum circuit. A prominent role is played by X-circuits and Y-circuits. They constitute the [Formula: see text]-dimensional subgroups XU([Formula: see text]) and YU([Formula: see text]) of the [Formula: see text]-dimensional group U([Formula: see text]). The former are described by unitary matrices with all line sums equal to 1. The latter are described by unitary matrices with all weighted line sums equal to 1. Whereas the group XU([Formula: see text]) has been studied before, investigation of the group YU([Formula: see text]) is new.
Quantum computation on w qubits is represented by the infinite unitary group U(2 w ); classical reversible computation on w bits is represented by the finite symmetric group S 2 w . In order to establish the relationship between classical reversible computing and quantum computing, we introduce two Lie subgroups XU( n ) and ZU( n ) of the unitary group U( n ). The former consists of all unitary n × n matrices with all line sums equal to 1; the latter consists of all unitary diagonal n × n matrices with first entry equal to 1. Such a group structure also reveals the relationship between matrix calculus and diagrammatic zx-calculus of quantum circuits.
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.