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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Matrix-based algorithm for 4-Qubit reversible circuits synthesis

Dong Wang, Shengyao Sun, Hanwu Chen

Published Mar 13, 2013• 0 citations
Publisher

Abstract

The mathematical model of the quantum reversible circuits is unitary matrix. Matrix can better reflect the quantum state evolution and further reflect the physical properties of the quantum computation. Matrix transformation based algorithm for 4-qubit reversible logic circuits synthesis is proposed in this paper. The algorithm skillfully uses the matrix representation of the quantum circuit and the circuit of the adjacent matrix to construct 4-qubit circuit with lower cost.

DOI
10.1117/12.2013683

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The mathematical model of the quantum reversible circuits is unitary matrix. Matrix can better reflect the quantum state evolution and further reflect the physical properties of the quantum computation. Matrix transformation based algorithm for 4-qubit reversible logic circuits synthesis is proposed in this paper. The algorithm skillfully uses the matrix representation of the quantum circuit and the circuit of the adjacent matrix to construct 4-qubit circuit with lower cost.
Quantum reversible logic circuits synthesis is one of the key technologies to construct quantum computer. The algebraic model for quantum information processing is a unitary matrix operator. Matrix can better reflect the quantum state evolution and the properties of quantum computation. Bidirectional matrix-based algorithm for quantum reversible logic circuits synthesis is proposed in this paper. The matrix representation of quantum reversible circuit and the circuit transformation rules of adjacent matrix are employed to construct any quantum reversible circuit in this paper. Compared with, the computational complexity of our algorithm has been decreased exponentially and the speed has been increased by about 105times. In addition, the types of the quantum reversible circuits synthesized by our algorithm are extended from only even permutations in to even and odd ones. we have synthesized 13!=6227020800 quantum reversible circuits, which can't be done by other algorithms.
Reversible quantum logic synthesis is to study given quantum gates and quantum circuits of the constraints and limitations and find the smallest or smaller quantum cost to achieve the desired quantum logic circuits.The quantum logic gate functions of the matrix is indicated by the mathematical model. The synthesis and optimization are achieved simultaneously by the genetic algorithm as global search tool. Genetic algorithm is applied to quantum reversible logic synthesis.The fourth-order quantum circuit experiment has achieved good results,and further analysis of this method in high-level synthesis of quantum circuits and its application is completed.
Quantum Boolean circuit is an interdisciplinary subject between classical reversible computing and quantum computing.It is of great significance to find a synthesis method of quantum Boolean circuit. A novel hybrid synthesis method is proposed which is based on the elementary transformation of matrices. Toffoli gate set is chosen as the basis gate library.Each logic gate in the library is a unitary matrix that can be decomposed into a product of a series of elementary transformation which is referred to a transformation path.Combined with some heuristics,the circuit structure composing of Toffoli gates can be constructed from the transformation path.In the end,a 3-qubit Boolean circuit is synthesized through the new method.
The synthesis method of quantum reversible logic circuits based on the genetic algorithm was studied.The multi-objective optimal design of reversible logic circuit's function,the number of quantum gates,the number of garbage bits and quantum cost was achieved by new synthesis method.The mathematical model for quantum reversible logic circuit synthesis was established,the matrix encoding method of quantum reversible logic circuits was proposed,the evolution operations and the multi-objective evolutionary algorithm for quantum reversible logic circuit design were developed.The 8 bits reversible multiplier was designed as an example,the experimental results show that the proposed multi-objective evolutionary design method for quantum reversible logic circuit synthesis is correct and effective.
Synthesis of quantum reversible logic circuits means to automatically construct desired quantum reversible logic circuit with minimal quantum cost.We present a novel and efficient algorithm which can construct almost all optimal 4-qubit reversible logic circuits with various types of gates and minimum length cost based on constructing the shortest coding and the specific topological compression, whose lossless compression ratios of the space of n-qubit circuits is near2×n!.We firstly have created all 3120218828 optimal 4-qubit circuits whose length is less than 9 for the Controlled-Not gate,NOT gate and Toffoli gate library,and our method can achieve 16 steps through cascading created circuits.Our algorithm can not only synthesizes all the 4-qubit benchmark circuits,but also runs extremely fast.
In order to efficiently automatically construct quantum reversible logic circuits with low cost,a novel method for quantum circuit's synthesis is proposed.Through the line topology transformation and truth table permutation,it converts an n-qubit circuit synthesis problem into single quantum circuit synthesis using traditional recursive thought.Then directly generate relevant circuit.After optimization,the quantum reversible logic circuit is synthesized finally.All 3-qubit reversible logic circuits have been synthesized by this method.The average number of EGT gates is only 6.41,and the number is down to 5.22 after optimization.Experimental results show that the number of gates to construct reversible logic circuits is less than other methods.For any n-qubit binary logic function,the number of EGT gates is less than n2n-1.Meanwhile,it voids the exponential nature of the memory or run-time complexity,and is very simple to implement in classical computer.