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arXiv:1405.6741 · ACM Journal on Emerging Technologies in Computing Systems

Decomposition of Diagonal Hermitian Quantum Gates Using Multiple-Controlled Pauli Z Gates

Mahboobeh Houshmand, Morteza Saheb Zamani, Mehdi Sedighi, Mona Arabzadeh

Published Dec 30, 2014• 13 citations
arXivHTMLPDF

Abstract

Quantum logic decomposition refers to decomposing a given quantum gate to a set of physically implementable gates. An approach has been presented to decompose arbitrary diagonal quantum gates to a set of multiplexed-rotation gates around z axis. In this article, a special class of diagonal quantum gates, namely diagonal Hermitian quantum gates, is considered and a new perspective to the decomposition problem with respect to decomposing these gates is presented. It is first shown that these gates can be decomposed to a set that solely consists of multiple-controlled Z gates. Then a binary representation for the diagonal Hermitian gates is introduced. It is shown that the binary representations of multiple-controlled Z gates form a basis for the vector space that is produced by the binary representations of all diagonal Hermitian quantum gates. Moreover, the problem of decomposing a given diagonal Hermitian gate is mapped to the problem of writing its binary representation in the specific basis mentioned previously. Moreover, CZ gate is suggested to be the two-qubit gate in the decomposition library, instead of previously used CNOT gate. Experimental results show that the proposed approach can lead to circuits with lower costs in comparison with the previous ones.

DOI
10.1145/2629526

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Quantum logic decomposition refers to decomposing a given quantum gate to a set of physically implementable gates. An approach has been presented to decompose arbitrary diagonal quantum gates to a set of multiplexed-rotation gates around z axis. In this article, a special class of diagonal quantum gates, namely diagonal Hermitian quantum gates, is considered and a new perspective to the decomposition problem with respect to decomposing these gates is presented. It is first shown that these gates can be decomposed to a set that solely consists of multiple-controlled Z gates. Then a binary representation for the diagonal Hermitian gates is introduced. It is shown that the binary representations of multiple-controlled Z gates form a basis for the vector space that is produced by the binary representations of all diagonal Hermitian quantum gates. Moreover, the problem of decomposing a given diagonal Hermitian gate is mapped to the problem of writing its binary representation in the specific basis mentioned previously. Moreover, CZ gate is suggested to be the two-qubit gate in the decomposition library, instead of previously used CNOT gate. Experimental results show that the proposed approach can lead to circuits with lower costs in comparison with the previous ones.
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.
Efficient decomposition of multiqubit operators is critical in the execution of quantum algorithms. In this paper, we introduce decompositions of the multiple controlled-ZZZ (ccz and cccz) gates tailored to various qubit-coupling graphs. Specifically, we demonstrate that the ccz gate is realized with cz-depth 4 on a square-shaped qubit-coupling graph utilizing one auxiliary qubit. As for the cccz gate, previous research indicated that the decomposition requires 14 cz gates in a fully connected topology. However, our findings reveal that only four specific qubit couplings are needed to achieve a decomposition using the same number of cz gates, 14. Our research employs an optimization method to improve the alignment of parametrized quantum circuits with their intended quantum gates, which facilitates these efficient decompositions. This methodology is versatile and can be applied to decompose any quantum gates, not just the ccz and cccz gates. These advancements in decomposing multiqubit gates, coupled with our ccz and cccz decompositions, are poised to reduce quantum circuit execution times and enhance the efficiency of complex quantum algorithms in imminent quantum computing applications.
Quantum gates typically represent either rotations of controlled gates. Rotation gates usually involve one qubit, although Ising coupling gates involve two qubits. On the other hand, controlled gates can have any number of control qubits with at least a target qubit. In this paper, the decomposition of controlled gates in terms of rotation gates around a given axis has been carried out. The focus herein has been set on theX\mathbf{X}Xaxis, such that AND-CmX gates and OR-CmX gates have been decomposed and proved form={1,2,3,4}\mathbf{m}=\{\mathbf{1, 2, 3, 4}\}m={1,2,3,4}and different polarities.
Multi-controlled unitary gates have been a subject of interest in quantum computing since its inception, and are widely used in quantum algorithms. The current state-of-the-art approach to implementing n-qubit multi-controlled gates involves the use of a quadratic number of single-qubit and CNOT gates. However, linear solutions are possible for the case where the controlled gate is a special unitary SU(2). The most widely-used decomposition of an n-qubit multi-controlled SU(2) gate requires a circuit with a number of CNOT gates proportional to 28n. In this work, we present a new decomposition of n-qubit multi-controlled SU(2) gates that requires a circuit with a number of CNOT gates proportional to 20n, and proportional to 16n if the SU(2) gate has at least one real-valued diagonal. This new approach significantly improves the existing algorithm by reducing the number of CNOT gates and the overall circuit depth. As an application, we show the use of this decomposition for sparse quantum state preparation. Our results are further validated by demonstrating a proof of principle on a quantum device accessed through quantum cloud services.
Universal gate sets for quantum computation, when single and two qubit operations are accessible, include both Hermitian and non-Hermitian gates. Here we utilize the fact that any single-qubit operator may be implemented as two Hermitian gates, and thus a purely Hermitian universal set is possible. This implementation can be used to prepare high fidelity single-qubit states in the presence of amplitude errors, and helps to achieve a high fidelity single-qubit gate decomposition using four Hermitian gates. An implementational convenience can be that non-identity single-qubit Hermitian gates are equivalent to π rotations up to a global phase. We show that a gate set comprised of π rotations about two fixed axes, along with the CNOT gate, is universal for quantum computation. Moreover, we show that two π rotations can transform the axis of any multi-controlled unitary, a special case being a single CNOT sufficing for any controlled π rotation. These gates simplify the process of circuit compilation in view of their Hermitian nature. We exemplify by designing efficient circuits for a variety of controlled gates, and achieving a CNOT count reduction for the four-controlled Toffoli gate in LNN-restricted qubit connectivity.
Efficient decompositions of multi-qubit gates are essential in NISQ applications, where the number of gates or the circuit depth is limited. This paper presents efficient decompositions of CCZ and CCCZ gates, typical multi-qubit gates, under several qubit connectivities. We can construct the CCZ gate with only four CZ-depth when the qubit is square-shaped, including one auxiliary qubit. In T-shaped qubit connectivity, which has no closed loop, we can decompose the CCCZ gate with 17 CZ gates. While previous studies have shown a CCCZ gate decomposition with 14 CZ gates for the fully connected case, we found only four connections are sufficient for 14 CZ gates' implementation. The search for constraint-sufficient decompositions is aided by an optimization method we devised to bring the parameterized quantum circuit closer to the target quantum gate. We can apply this scheme to decompose any quantum gates, not only CCZ and CCCZ. Such decompositions of multi-qubit gates, together with the newly found CCZ and CCCZ decompositions, shorten the execution time of quantum circuits and improve the accuracy of complex quantum algorithms on near future QPUs.