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arXiv:1512.07240 · Physical Review A

Block- ZXZ synthesis of an arbitrary quantum circuit

Alexis De Vos, Stijn De Baerdemacker

Published Nov 14, 2016• 28 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.94.052317

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Given an arbitrary 2w\ifmmode×\else\texttimes\fi2w{2}^{w}\ifmmode\times\else\texttimes\fi{}{2}^{w}2w\ifmmode×\else\texttimes\fi2w 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.
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.
This paper proposes a new optimized quantum block-ZXZ decomposition method [7,8,10] that results in more optimal quantum circuits than the quantum Shannon decomposition (QSD)[27], which was introduced in 2006 by Shende et al. The decomposition is applied recursively to generic quantum gates, and can take advantage of existing and future small-circuit optimizations. Because our method uses only one-qubit gates and uniformly controlled rotation-Z gates, it can easily be adapted to use other types of multi-qubit gates. With the proposed decomposition, a general 3-qubit gate can be decomposed using 19 CNOT gates (rather than 20). For general nnn-qubit gates, the proposed decomposition generates circuits that have 22484n−322n+53\frac{22}{48}4^n - \frac{3}{2}2^n +\frac{5}{3}4822​4n−23​2n+35​ CNOT gates, which is less that the best known exact decomposition algorithm by (4n−2−1)/3(4^{n-2} -1)/3(4n−2−1)/3 CNOT gates.
One of the challenges in quantum computing is the synthesis of unitary operators into quantum circuits with polylogarithmic gate complexity. Exact synthesis of generic unitaries requires an exponential number of gates in general. We propose a novel approximate quantum circuit synthesis technique by relaxing the unitary constraints and interchanging them for ancilla qubits via block encodings. This approach combines smaller block encodings, which are easier to synthesize, into quantum circuits for larger operators. Due to the use of block encodings, our technique is not limited to unitary operators and can be applied for the synthesis of arbitrary operators. We show that operators which can be approximated by a canonical polyadic expression with a polylogarithmic number of terms can be synthesized with polylogarithmic gate complexity with respect to the matrix dimension.
In this article, the problem of synthesizing a general Hermitian quantum gate into a set of primary quantum gates is addressed. To this end, an extended version of the Jacobi approach for calculating the eigenvalues of Hermitian matrices in linear algebra is considered as the basis of the proposed synthesis method. The quantum circuit synthesis method derived from the Jacobi approach and its optimization challenges are described. It is shown that the proposed method results in multiple-control rotation gates around the y axis, multiple-control phase shift gates, multiple-control NOT gates, and a middle diagonal Hermitian matrix, which can be synthesized to multiple-control Pauli Z gates. Using the proposed approach, it is shown how multiple-control U gates, where U is a single-qubit Hermitian quantum gate, can be implemented using a linear number of elementary gates in terms of circuit lines with the aid of one auxiliary qubit in an arbitrary state.
In this paper, the problem of constructing an efficient quantum circuit for the implementation of an arbitrary quantum computation is addressed. To this end, a basic block based on the cosine-sine decomposition method is suggested which contains lll qubits. In addition, a previously proposed quantum-logic synthesis method based on quantum Shannon decomposition is recursively applied to reach unitary gates over lll qubits. Then, the basic block is used and some optimizations are applied to remove redundant gates. It is shown that the exact value of lll affects the number of one-qubit and CNOT gates in the proposed method. In comparison to the previous synthesis methods, the value of lll is examined consequently to improve either the number of CNOT gates or the total number of gates. The proposed approach is further analyzed by considering the nearest neighbor limitation. According to our evaluation, the number of CNOT gates is increased by at most a factor of 53\frac{5}{3}35​ if the nearest neighbor interaction is applied.