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IEEE International Symposium on Multiple-Valued Logic (ISMVL)

New Two-Qubit Gate Library with Entanglement

Md. Belayet Ali, Takashi Hirayama, Katsuhisa Yamanaka, Yasuaki Nishitani

Published May 1, 2016• 1 citation
Publisher

Abstract

The NCV (NOT, CNOT and Controlled-V/V) gate set is most commonly used for the realization of classical reversible functions and implementation of quantum circuits. Logic operations in quantum circuits are performed by individual gates or can be expressed as matrix-vector multiplication. Thispaper presents a two-qubit quantum gate library which consists most commonly used elementary quantum gates NOT, CNOT, Controlled-V and Controlled-V. The new library contains all possible two-qubit quantum gates which do not produce entangled states in the final output state. Two approaches have been appliedto generate the new two-qubit quantum gate library because it hasbeen observed that some entangled gates act like non-entangledif those entangled gates are merged with other valid gates in thelibrary. The experimental result shows the difference betweenthe libraries with and without considering entangled gates in theprocess of generating new gates.

DOI
10.1109/ismvl.2016.26

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The NCV (NOT, CNOT and Controlled-V/V) gate set is most commonly used for the realization of classical reversible functions and implementation of quantum circuits. Logic operations in quantum circuits are performed by individual gates or can be expressed as matrix-vector multiplication. Thispaper presents a two-qubit quantum gate library which consists most commonly used elementary quantum gates NOT, CNOT, Controlled-V and Controlled-V. The new library contains all possible two-qubit quantum gates which do not produce entangled states in the final output state. Two approaches have been appliedto generate the new two-qubit quantum gate library because it hasbeen observed that some entangled gates act like non-entangledif those entangled gates are merged with other valid gates in thelibrary. The experimental result shows the difference betweenthe libraries with and without considering entangled gates in theprocess of generating new gates.
Owing to the exponential nature of the memory or run-time complexity,many existing methods can only synthesize 3-qubit logic circuits using quantum logic gate library,however,a few can optimally synthesize 3-qubit logic circuits for quantum NOT,CNOT,Controlled-V and Controlled-V+(NCV)gates,the key approach reduces the NCV quantum circuit synthesis problem to four-valued logic synthesis.This paper proposes using NCV gates to create a new quantum logic gate library,which is exactly the same as NCV gate library in the synthesis of all optimal 3-qubit circuits,thus it also reduces the four-valued logic synthesis to easily solved two-valued logic synthesis.We present a 3-qubit efficient synthesis algorithms based on perfect hash function,which can quickly construct all optimal 3-qubit circuits——the average speed that synthesizes circuits with minimum cost is nearly 127 times faster than that of the best result of Maslov 2007.
In previous work and motivated by a theoretical discussion on physical realizations, a new quantum gate library (the NCV-v1 library) for electronic design automation of quantum circuits has been proposed. Here, qudits instead of qubits are assumed, i.e. a basic building block which does not rely on a two level quantum system but a (multiple-valued) d-level quantum system is assumed. However, the descriptions on the foundation of this new library remained brief. This technical report provides an extended description of the applied ideas and concepts.
Multiple Control Toffoli (MCT) gates are the main constituent of reversible circuits. For quantum circuits, MCT gates are realized using quantum gate libraries such as NCV or NCVW which are composed of universal quantum gates like NOT, CNOT, V/V+, and W/W+. In order to improve the design of quantum circuits, the mapping of MCT gates to a cascade of these quantum gates has to be improved as well. In this work, we propose reconfigured structures of quantum gates realizing MCT gates. To this end, we rely on the established mapping structure, but append redundant gates which can be used afterwards for simplifications. Eventually, the proposed design is mapped to an NCVW quantum circuit. We have successfully tested our mapping technique. The obtained results have experimentally been compared to related work.
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.