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arXiv:1702.06272

General structures of reversible and quantum gates

Kishore Thapliyal, Anirban Pathak

Published Feb 21, 2017• 0 citations
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Abstract

The most general structure (in matrix form) of a single-qubit gate is presented. Subsequently, used that to obtain a set of conditions for testing (a) whether a given 2-qubit gate is genuinely a 2-qubit gate, i.e., not decomposable into two single qubit gates and (b) whether a given single qubit gate is self-inverse? Relevance of the results reported here is discussed in the context of optimization of reversible and quantum circuits, especially for the optimization of quantum cost. A systematic procedure is developed for the identification of the non-decomposable 2-qubit gates. Such a non-decomposable 2-qubit gate along with all possible single qubit gates form a universal quantum gate library. Further, some possible applications of the present work are also discussed.

DOI
10.48550/arxiv.1702.06272

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The most general structure (in matrix form) of a single-qubit gate is presented. Subsequently, used that to obtain a set of conditions for testing (a) whether a given 2-qubit gate is genuinely a 2-qubit gate, i.e., not decomposable into two single qubit gates and (b) whether a given single qubit gate is self-inverse? Relevance of the results reported here is discussed in the context of optimization of reversible and quantum circuits, especially for the optimization of quantum cost. A systematic procedure is developed for the identification of the non-decomposable 2-qubit gates. Such a non-decomposable 2-qubit gate along with all possible single qubit gates form a universal quantum gate library. Further, some possible applications of the present work are also discussed.
It is well known that most of the frequently used reversible logic gates (e.g., NOT, CNOT, SWAP, Toffoli, Fredkin) are self-inverse and are represented by square matrices that are unitary and Hermitian. However, with a simple minded argument, it is established that the most of the allowed reversible gates are non-self-inverse (unitary but non-Hermitian) in nature. It is also shown that the % of non-Hermitian gates increases with the dimension. For example, 58.33% of the 2-bit gates, 98.10% of the 3-bit gates and 99.99% of the 4-bit gates are non-Hermitian. As classical reversible gates are essentially permutation gates, above statistics is strictly valid for classical reversible gates, but the argument can be easily extended to include quantum gates and to establish that the majority of the quantum gates are also non-self-inverse. Further, the % of genuinely 2-bit reversible gates (i.e., 2-bit gates that cannot be decomposed as a product of two single bit gates) among all possible gates has been computed as 83.3%, and the applicability of this analysis in the optimization of circuit cost is discussed.
In order to demonstrate non-trivial quantum computations experimentally, such as the synthesis of arbitrary entangled states, it will be useful to nderstand how to decompose a desired quantum computation into the shortest possible sequence of one-qubit and two-qubit gates. We contribute to this effort by providing a method to construct an optimal quantum circuit for a general two-qubit gate that requires at most 3 CNOT gates and 15 elementary one qubit gates. We then prove that these constructions are optimal with respect to the family of CNOT, y-rotation, z-rotation, and phase gates.
In order to demonstrate nontrivial quantum computations experimentally, such as the synthesis of arbitrary entangled states, it will be useful to understand how to decompose a desired quantum computation into the shortest possible sequence of one-qubit and two-qubit gates. We contribute to this effort by providing a method to construct an optimal quantum circuit for a general two-qubit gate that requires at most 3 controlled-NOT (CNOT) gates and 15 elementary one-qubit gates. Moreover, if the desired two-qubit gate corresponds to a purely real unitary transformation, we provide a construction that requries at most 2 CNOT and 12 one-qubit gates. We then prove that these constructions are optimal with respect to the family of CNOT, yyy-rotation, zzz-rotation, and phase gates.
A critical task in utilizing quantum physics in many application fields is circuit design using reversible quantum gates. Using decomposition techniques enables transformation of unitary matrices into fundamental quantum gates. Any 3x3 reversible quantum gate can be decomposed into single-qubit rotation gates and two qubit CNOT gates. In this chapter, quantum implementations of FRSG1, URG, JTF1 and R gates into CNOT gates and single qubit U3 gates with different optimization levels on a platform provided by IBM have been discussed. FRSG1 and JTF1 gates are important in applications like Stochastic computing, fingerprint authentication system, and parity generation circuits. URG gate is better in terms of number of complex functions and can be utilized to design quantum comparator circuits. R gate plays an important role in inverting and duplicating a signal. In FRSG1, URG, JTF1 and R gates, the implementation count of single qubit gates decreases by 56%, 11%, 71%, and 62%, respectively and the count of two qubit gates reduces by 15%, 26%, 41%, and 5% respectively after optimization.
Most of the frequently used quantum gates (e.g., NOT, Hadamard, CNOT, SWAP, Toffoli, Fredkin and Pauli gates) are self-inverse (Hermitian). However, with a simple minded argument it is established that most of the allowed quantum gates are non-Hermitian (non-self-inverse). It is also shown that the % of non-Hermitian gates increases with the dimension. For example, 58.33% of the 2-qubit gates, 98.10% of the 3-qubit gates and 99.99% of the 4-qubit gates are non-Hermitian. As classical reversible gates are essentially permutation gates so the above statistics is strictly valid for classical reversible gates. Further, since Hermiticity is not of much interest in context of the classical reversible gate, hence the result implies that most of the allowed classical reversible gates are non-self-inverse.