Journal of Northwest University

Generalized commutative gates and applications

Qin Liu

Published Jan 1, 2010• 0 citations

Abstract

Aim To establish the relationship between generalized controlled-NOT gates U(k,l)CN and U(l,k)CN.Methods Using some methods of operator theory.Results Generalized commutative gates C(k,l)n are introduced so as to exchange the kth quantum bit and lth quantum bit,a calculation formula is given.Conclusion A relationship between generalized controlled-NOT gates U(k,l)CN and U(l,k)CN is established,and the matrix representations of related gates are obtained.

Related papers

ANALYTIC ONE-BIT AND CNOT GATE CONSTRUCTIONS OF GENERAL n-QUBIT CONTROLLED GATES

Yang Liu, GUI LU LONG, Yang Sun

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.

International Journal of Quantum InformationJun 1, 200870 citations

Construction of General Unitary Transformation for Conditional Quantum Dynamics and Realization of Quantum Controlled Gates

Xiaoguang Wang, YU Rong-jin, Jian-zheng Fu

By factorizing a general one-bit transformation matrix for one-bit gates in quantum computer, a general unitary transformation is constructed, which can serve as (controlled) m gate for the conditional quantum dynamics. When the quantized single-mode electromagnetic field containing n or no photons acts as the controlled bit, the quantum controlled-NOT gates and square root of the controlled-NOT gates in cavity-quantum electrodynamics are realized.

Chinese Physics LettersJun 1, 19980 citations

Realization of a new permutative gate library using controlled-kth-root-of-NOT quantum gates for exact minimization of quantum circuits

Zhiqiang Li, Xiaoyu Song B, Marek A. Perkowski +2 more

Since non-permutative quantum gates have more complex rules than permutative quantum gates, it is very hard to synthesize quantum logic circuits using non-permutative quantum gates, such as controlled-square-root-of-NOT gates (CV∕CV+ gates). In the efficient synthesis algorithm, direct use of quantum non-permutative gates should be avoided. Rather, the key method is to use quantum gates to create new permutative quantum gates that then replace non-permutative quantum gates. This method assumes the library of quantum gate primitives are constructed so as to have the lowest possible quantum cost. In this paper, we first propose some new CV∕CV+-like gates, i.e. controlled-kth-root-of-NOT gates where k = 2,4,8,…, and give all corresponding matrixes. Further, we also present a novel generic method to quickly and directly construct this new optimal quantum logic gate library using CNOT and these non-permutative quantum gates. Our method introduces new means to find permutative quantum gates with lower quantum cost.

International Journal of Quantum InformationAug 1, 20142 citations

Geometric Control Methods for Quantum Computations

Zakaria Giunashvili

The applications of geometric control theory methods on Lie groups and homogeneous spaces to the theory of quantum computations are investigated. These methods are shown to be very useful for the problem of constructing an universal set of gates for quantum computations: the well-known result that the set of all one-bit gates together with almost any one two-bit gate is universal is considered from the control theory viewpoint.

arXiv:quant-ph/0404050Apr 8, 20040 citations

Entangled states of trapped atoms for quantum logic and spectroscopy

Christopher R. Monroe, D. M. Meekhof, B. E. King +3 more

Summary form only given. A quantum computer is composed of quantum bits that can support superposition states and hence allow parallel processing for certain problems such as factorization. All quantum computations can, in principle, be reduced to a series of two-bit quantum logic gates accompanied by simple single-bit rotations. An example of a two-bit quantum logic gate is the controlled-NOT (CN) gate, which rotates one quantum bit by 180 dependent on the state of a second quantum bit. We demonstrate operation of a two bit CN quantum logic gate on a trapped ion.

Quantum Electronics and Laser Science ConferenceJun 7, 19960 citations

Remarks on generalized quantum gates

Zhaoqi Wu, Shifang Zhang, Chuanxi Zhu

In this paper, we give a characterization of generalized quantum gates.We also show that many important operators are generalized quantumgates, moreover, some of these operators can be represented as theconvex combination of only two unitary operators. Our results answerwhat kinds of operations a duality quantum computing admits. Wepoint out that the set of all generalized quantum gates coincides withthe set of all restricted allowable generalized quantum gates. Thus,our results are also valid for restricted allowable generalized quantumgates

Hacettepe Journal of Mathematics and StatisticsJun 1, 20145 citations