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Cambridge University Press eBooks

The synthesis of a quantum circuit

Alexis De Vos, Stijn De Baerdemacker

Published Jan 1, 2014• 4 citations
OpenAlex

Abstract

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.

Related papers

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

Alexis De Vos, Stijn De Baerdemacker

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.

Journal of Physics Conference Series•Apr 13, 2015•9 citations

The Decomposition of U(n) into XU(n) and ZU(n)

Alexis De Vos, Stijn De Baerdemacker

Any matrix of the unitary group U(n) can (up to a global phase) be decomposed into 2n-1 matrices from two subgroups, denoted XU(n) and ZU(n). This leads to the decomposition of an arbitrary quantum circuit into NEGATOR circuits and PHASOR circuits. The NEGATOR circuits are closely related to classical reversible computation.

IEEE International Symposium on Multiple-Valued Logic (ISMVL)•May 1, 2014•3 citations

The decomposition of U(n) into XU(n) and ZU(n)

Alexis De Vos, Stijn De Baerdemacker

Any matrix of the unitary group U(n) can be decomposed into matrices from two subgroups, denoted XU(n) and ZU(n). This leads to decompositions of an arbitrary quantum circuit into NEGATOR circuits and PHASOR circuits. The NEGATOR circuits are closely related to classical reversible computation.

•Jan 1, 2016•0 citations

The NEGATOR as a Basic Building Block for Quantum Circuits

Alexis De Vos, Stijn De Baerdemacker

Between (classical) reversible computation and quantum computation there exists an intermediate computational world, represented by unitary matrices that have all line sums equal to 1. All of these quantum circuits can be synthesized with the help of merely two building blocks: the NEGATOR and the singly controlled square root of NOT.

Open Systems & Information Dynamics•Mar 1, 2013•17 citations

Block- ZXZ synthesis of an arbitrary quantum circuit

Alexis De Vos, Stijn De Baerdemacker

Given an arbitrary ${2}^{w}\ifmmode\times\else\texttimes\fi{}{2}^{w}$ unitary matrix $U$, a powerful matrix decomposition can be applied, leading to four different syntheses of a $w$-qubit quantum circuit performing the unitary transformation. The demonstration is based on a recent theorem by H. F\"uhr and Z. Rzeszotnik [Linear Algebra Its Appl. 484, 86 (2015)] generalizing the scaling of single-bit unitary gates ($w=1$) to gates with arbitrary value of $w$. The synthesized circuit consists of controlled one-qubit gates, such as negator gates and phasor gates. Interestingly, the approach reduces to a known synthesis method for classical logic circuits consisting of controlled not gates in the case that $U$ is a permutation matrix.

arXiv:1512.07240•Physical Review A•Nov 14, 2016•28 citations

Quantum Computing with the Subgroup YU(n) of the Unitary Group U(n)

Alexis De Vos

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.

Open Systems & Information Dynamics•Mar 1, 2025•0 citations
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.
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.
Given an arbitrary ${2}^{w}\ifmmode\times\else\texttimes\fi{}{2}^{w}$ unitary matrix UUU, a powerful matrix decomposition can be applied, leading to four different syntheses of a www-qubit quantum circuit performing the unitary transformation. The demonstration is based on a recent theorem by H. F\"uhr and Z. Rzeszotnik [Linear Algebra Its Appl. 484, 86 (2015)] generalizing the scaling of single-bit unitary gates (w=1w=1w=1) to gates with arbitrary value of www. The synthesized circuit consists of controlled one-qubit gates, such as negator gates and phasor gates. Interestingly, the approach reduces to a known synthesis method for classical logic circuits consisting of controlled not gates in the case that UUU is a permutation matrix.
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.