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arXiv:quant-ph/0504100

Decompositions of general quantum gates

Mikko Möttönen, Juha J. Vartiainen

Published Apr 14, 2005• 86 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.quant-ph/0504100

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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.
Uniformly controlled one-qubit gates are quantum gates which can be represented as direct sums of two-dimensional unitary operators acting on a single qubit. We present a quantum gate array which implements any nnn-qubit gate of this type using at most ${2}^{n\ensuremath{-}1}\ensuremath{-}1$ controlled-NOT gates, ${2}^{n\ensuremath{-}1}$ one-qubit gates, and a single diagonal nnn-qubit gate. To illustrate the versatility of these gates we then apply them to the decomposition of a general nnn-qubit gate and a state preparation procedure. Moreover, we study their implementation using only nearest-neighbor gates. We give upper bounds for the one-qubit and controlled-NOT gate counts for all the aforementioned applications. In all four cases, the proposed circuit topologies either improve on or achieve the previously reported upper bounds for the gate counts. Thus, they provide the most efficient method for general gate decompositions currently known.
A general scheme is presented to decompose a d-by-d unitary matrix as the product of two-level unitary matrices with additional structure and prescribed determinants. In particular, the decomposition can be done by using two-level matrices in d - 1 classes, where each class is isomorphic to the group of 2 × 2 unitary matrices. The proposed scheme is easy to apply, and useful in treating problems with the additional structural restrictions. A Matlab program is written to implement the scheme, and the result is used to deduce the fact that every quantum gate acting on n-qubit registers can be expressed as no more than 2n-1(2n-1) fully controlled single-qubit gates chosen from 2n-1 classes, where the quantum gates in each class share the same n - 1 control qubits. Moreover, it is shown that one can easily adjust the proposed decomposition scheme to take advantage of additional structure evolving in the process.
Any unitary transformation of quantum computational networks is explicitly decomposed, in an exact and unified form, into a sequence of a limited number of one-qubit quantum gates and the two-qubit diagonal gates that have diagonal unitary representation in usual computational basis. This decomposition may be simplified greatly with the help of the properties of the finite-dimensional multiple-quantum operator algebra spaces of a quantum system and the specific properties of a given quantum algorithm. As elementary building blocks of quantum computation, the two-qubit diagonal gates and one-qubit gates may be constructed physically with one- and two-body interactions in a two-state quantum system and hence could be conveniently realized experimentally. The present work will be helpful for implementing generally any N-qubit quantum computation in those feasible two-state quantum systems and determining conveniently the time evolution of these systems in course of quantum computation.
In this article we study n-qubit quantum gates which may be represented as direct sums of two-dimensional unitary operators. We call this class of gates uniformly controlled one-qubit gates. We present a quantum circuit which implements any gate of this type using 2^{n-1} - 1 controlled-NOT gates, 2^{n-1} one-qubit gates and a single n-qubit diagonal gate. The construction is based on a modified version of the so-called quantum multiplexor circuit. We illustrate the versatility of the uniformly controlled gates by applying them to the decomposition of a general n-qubit gate and an n-qubit state-preparation procedure. Moreover, we study the implementation of these circuits using only nearest-neighbor gates and give the worst-case one-qubit and CNOT gate counts for all of the aforementioned applications. In all four cases, the proposed method either improves or achieves the previously reported upper bounds for the gate counts.
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.