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arXiv:quant-ph/0410066 · Physical Review A

Quantum circuits with uniformly controlled one-qubit gates

Ville Bergholm, Juha J. Vartiainen, Mikko Möttönen, Martti M. Salomaa

Published May 27, 2005• 159 citations
arXivHTMLPDF

Abstract

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 $n$-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 $n$-qubit gate. To illustrate the versatility of these gates we then apply them to the decomposition of a general $n$-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.

DOI
10.1103/physreva.71.052330

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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 2n\ensuremath−1\ensuremath−1{2}^{n\ensuremath{-}1}\ensuremath{-}12n\ensuremath−1\ensuremath−1 controlled-NOT gates, 2n\ensuremath−1{2}^{n\ensuremath{-}1}2n\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.
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.
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.
In quantum computation every unitary operation can be decomposed into quantum circuits---a series of single-qubit rotations and a single type entangling two-qubit gates, such as controlled-not(cnot) gates. Two measures are important when judging the complexity of the circuit: the total number of cnot gates needed to implement it and the depth of the circuit, measured by the minimal number of computation steps needed to perform it. Here we give an explicit and simple quantum circuit scheme for preparation of arbitrary quantum states, which can directly utilize any decomposition scheme for arbitrary full quantum gates, thus connecting the two problems. Our circuit reduces the depth of the best currently known circuit by a factor of 222. It also reduces the total number of cnot gates from 2n{2}^{n}2n to 23242n\frac{23}{24}{2}^{n}2423​2n in the leading order for even number of qubits. Specifically, the scheme allows us to decrease the upper bound from 111111 cnot gates to 999 and the depth from 111111 to 555 steps for four qubits. Our results are expected to help in designing and building small-scale quantum circuits using present technologies.
Multicontrolled unitary gates have been a subject of interest in quantum computing since their conception and are widely used in quantum algorithms. The current state-of-the-art approach to implementingnnn-qubit multicontrolled gates with a single target without relying on auxiliary qubits or approximate results 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 special unitary, SU(2). The decomposition of annnn-qubit multicontrolled SU(2) gate requires a circuit with a number of CNOT gates proportional to28n28n28n. In this work, we present a new decomposition ofnnn-qubit multicontrolled SU(2) gates that require a circuit with a number of CNOT gates proportional to20n20n20nand proportional to16n16n16nif the SU(2) gate has at least one real-valued diagonal. The proposed algorithms produce the most efficient known circuits and improve 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.
Circuit synthesis of multi-controlled gates is crucial for qudit (ddd-level) quantum computing. This paper presents efficient synthesis schemes that reduce the elementary gate count for multi-controlled single-qudit gates. For synthesizing general (n−1)(n-1)(n−1)-controlled unitaries on nnn qudits, we reduce the controlled-increment (CINC) and generalized controlled-XXX (GCX) gate counts to O(n2)O(n^2)O(n2), improving upon existing O(n2+log⁡2d)O(n^{2+\log_2 d})O(n2+log2​d) CINC and O(n3)O(n^3)O(n3) GCX bounds. For (n−1)(n-1)(n−1)-controlled special unitaries, this complexity is further reduced to O(n)O(n)O(n). Furthermore, we present a method for approximately synthesizing multi-controlled unitary gates with a linear complexity in the number of control qudits. By utilizing the proposed circuit, we present qudit-based circuit constructions for isometries and quantum channels from nnn to mmm qudits. Moreover, for the first time, we present a circuit synthesis scheme for single-controlled gates using SUM gates and single-qudit gates when ddd is prime. This enables all CINC-based circuits for various quantum operations to be converted into SUM-gate circuits while preserving the same asymptotic complexity. Finally, we establish a theoretical lower bound on the number of SUM and CINC gates required to synthesize general nnn-qudit unitaries.
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.