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Conference on Lasers and Electro-Optics Pacific Rim (CLEO-PR)

Analyzing and Implementing N-qubit Controlled Gates

Yang Liu, Yang Sun, Gui Lu Long

Published Aug 1, 2007• 1 citation
Publisher

Abstract

We present two recursive schemes which generally simulate N-qubit controlled gates with standard one-qubit rotation and CNOT gates respectively using exponential and polynomial complexity simulations. Explicit circuits and decomposition expressions are deduced. We compare exact numbers of basic operations required in these two approaches using a gate counting technique.

DOI
10.1109/cleopr.2007.4391567

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We present two recursive schemes which generally simulate N-qubit controlled gates with standard one-qubit rotation and CNOT gates respectively using exponential and polynomial complexity simulations. Explicit circuits and decomposition expressions are deduced. We compare exact numbers of basic operations required in these two approaches using a gate counting technique.
In this Letter, we present two analytic expressions that most generally simulate nnn-qubit controlled-UUU gates with standard one-qubit gates and CNOT gates using exponential and polynomial complexity respectively. Explicit circuits and general expressions of decomposition are derived. The exact numbers of basic operations in these two schemes are given using gate counting technique.
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.
General n-qubit controlled unitary gates are frequently used in quantum information processing tasks. Barenco, Bennett, Cleve, Di Vincenzo, Margolus and Shor [Phys. Rev. A52 (1995) 3457] have given the general construction methods, and explicit results for up-to-four-qubits controlled unitary gates. We extended their calculation and gave two analytic expressions for the construction of general n-qubit controlled unitary gates in terms of one-qubit and two-qubit CNOT gates. There are two expressions – one is exponential in the qubit number which is efficient for up to ten qubits, and the other is polynomial in the qubit number, which is efficient for more than ten qubits.
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.
This paper proposes an optimized quantum block-ZXZZXZZXZ decomposition method that results in more optimal quantum circuits than the quantum Shannon decomposition, which was presented in 2005 by M. M\"ott\"onen, and J. J. Vartiainen [in Trends in quantum computing research, edited by S. Shannon (Nova Science Publishers, 2006) Chap. 7, p. 149, arXiv:quant-ph/0504100]. 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 single-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 three-qubit gate can be decomposed using 19 cnot gates (rather than 20). For general nnn-qubit gates, the proposed decomposition generates circuits that have $\frac{22}{48}{4}^{n}\ensuremath{-}\frac{3}{2}{2}^{n}+\frac{5}{3}$ cnot gates, which is less than the best-known exact decomposition algorithm by $({4}^{n\ensuremath{-}2}\ensuremath{-}1)/3$ cnot gates.
We study the implementation of quantum channels with quantum computers while minimizing the experimental cost, measured in terms of the number of controlled-not (cnot) gates required (single-qubit gates are free). We consider three different models. In the first, the quantum circuit model (QCM), we consider sequences of single-qubit and cnot gates and allow qubits to be traced out at the end of the gate sequence. In the second (RandomQCM), we also allow external classical randomness. In the third (MeasuredQCM) we also allow measurements followed by operations that are classically controlled on the outcomes. We prove lower bounds on the number of cnot gates required and give near-optimal decompositions in almost all cases. Our main result is a MeasuredQCM circuit for any channel from mmm qubits to nnn qubits that uses at most one ancilla and has a low cnot count. We give explicit examples for small numbers of qubits that provide the lowest known cnot counts.