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arXiv:2501.07786

Decomposition of unitary matrix into quantum gates

Dmytro Fedoriaka

Published Jan 14, 2025• 1 citation
arXivHTMLPDF

Abstract

An algorithm is proposed to convert arbitrary unitary matrix to a sequence of $X$ gates and fully controlled $R_y, R_z$ and $R_1$ gates. This algorithm is used to generate Q# implementation for arbitrary unitary matrix. Some optimizations are considered and complexity of the result is analyzed.

DOI
10.48550/arxiv.2501.07786

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An algorithm is proposed to convert arbitrary unitary matrix to a sequence of XXX gates and fully controlled Ry,RzR_y, R_zRy​,Rz​ and R1R_1R1​ gates. This algorithm is used to generate Q# implementation for arbitrary unitary matrix. Some optimizations are considered and complexity of the result is analyzed.
Constructing appropriate unitary matrix operators for new quantum algorithms and finding the minimum cost gate sequences for the implementation of these unitary operators is of fundamental importance in the field of quantum information and quantum computation. Evolution of quantum circuits faces two major challenges: complex and huge search space and the high costs of simulating quantum circuits on classical computers. Here, we use the group leaders optimization algorithm to decompose a given unitary matrix into a proper-minimum cost quantum gate sequence. We test the method on the known decompositions of Toffoli gate, the amplification step of the Grover search algorithm, the quantum Fourier transform, and the sender part of the quantum teleportation. Using this procedure, we present the circuit designs for the simulation of the unitary propagators of the Hamiltonians for the hydrogen and the water molecules. The approach is general and can be applied to generate the sequence of quantum gates for larger molecular systems.
Unitary decomposition is a widely used method to map quantum algorithms to an arbitrary set of quantum gates. Efficient implementation of this decomposition allows for the translation of bigger unitary gates into elementary quantum operations, which is key to executing these algorithms on existing quantum computers. The decomposition can be used as an aggressive optimization method for the whole circuit, as well as to test part of an algorithm on a quantum accelerator. For the selection and implementation of the decomposition algorithm, perfect qubits are assumed. We base our decomposition technique on Quantum Shannon Decomposition, which generates O(344n) controlled-not gates for an n-qubit input gate. In addition, we implement optimizations to take advantage of the potential underlying structure in the input or intermediate matrices, as well as to minimize the execution time of the decomposition. Comparing our implementation to Qubiter and the UniversalQCompiler (UQC), we show that our implementation generates circuits that are much shorter than those of Qubiter and not much longer than the UQC. At the same time, it is also up to 10 times as fast as Qubiter and about 500 times as fast as the UQC.
Optimal implementation of quantum gates is crucial for designing a quantum computer. We consider the matrix representation of an arbitrary multiqubit gate. By ordering the basis vectors using the Gray code, we construct the quantum circuit which is optimal in the sense of fully controlled single-qubit gates and yet is equivalent with the multiqubit gate. In the second step of the optimization, superfluous control bits are eliminated, which eventually results in a smaller total number of the elementary gates. In our scheme the number of controlled NOT gates is O(4(n)) which coincides with the theoretical lower bound.
Constructing appropriate unitary matrix operators for new quantum algorithms and finding the minimum cost gate sequences for the implementation of these unitary operators is of fundamental importance in the field of quantum information and quantum computation. Evolution of quantum circuits faces two major challenges: complex and huge search space and the high costs of simulating quantum circuits on classical computers. Here, we use the group leaders optimization algorithm to decompose a given unitary matrix into a proper-minimum cost quantum gate sequence. We test the method on the known decompositions of Toffoli gate, the amplification step of the Grover search algorithm, the quantum Fourier transform, and the sender part of the quantum teleportation. Using this procedure, we present the circuit designs for the simulation of the unitary propagators of the Hamiltonians for the hydrogen and the water molecules. The approach is general and can be applied to generate the sequence of quantum gates for larger molecular systems.
Quantum algorithms may be described by sequences of unitary transformations called quantum gates and measurements applied to the quantum register of n quantum bits, qubits. A collection of quantum gates is called universal if it can be used to construct any n-qubit gate. In 1995, the universality of the set of one-qubit gates and controlled NOT gate was shown by Barenco et al. using QR decomposition of unitary matrices. Almost ten years later the decomposition was improved to include essentially fewer elementary gates. In addition, the cosine-sine matrix decomposition was applied to efficiently implement decompositions of general quantum gates. In this chapter, we review the different types of general gate decompositions and slightly improve the best known gate count for the controlled NOT gates to (23/48)4^n in the leading order. In physical realizations, the interaction strength between the qubits can decrease strongly as a function of their distance. Therefore, we also discuss decompositions with the restriction to nearest-neighbor interactions in a linear chain of qubits.