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Journal of Applied Mathematics and Physics

The Mapping and Optimization Method of Quantum Circuits for Clifford + T Gate

Xianya He, Zhijin Guan, Fei Ding

Published Jan 1, 2019• 5 citations
Publisher

Abstract

In order to solve the fault tolerance and reliability problems of quantum circuit, a series of structural equivalence rules and optimization operation strategies of quantum circuit are proposed to minimize the number of T gates, increase T gate depth, minimize circuit level, reduce fault tolerance implementation costs and increase circuit reliability. In order to satisfy the nearest neighbor constraints of some quantum systems, a LNN (linear nearest neighbor) arrangement algorithm based on Clifford + T gate quantum circuit is presented. Experiments are done on some benchmarks of RevLib, the results show that the optimization rate of most functions and the running time of the algorithm are better than those of the existing literature.

DOI
10.4236/jamp.2019.711192

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In order to solve the fault tolerance and reliability problems of quantum circuit, a series of structural equivalence rules and optimization operation strategies of quantum circuit are proposed to minimize the number of T gates, increase T gate depth, minimize circuit level, reduce fault tolerance implementation costs and increase circuit reliability. In order to satisfy the nearest neighbor constraints of some quantum systems, a LNN (linear nearest neighbor) arrangement algorithm based on Clifford + T gate quantum circuit is presented. Experiments are done on some benchmarks of RevLib, the results show that the optimization rate of most functions and the running time of the algorithm are better than those of the existing literature.
Optimization based on QC (quantum cost) for quantum circuits and the LNN (Linear Nearest Neighbor) Architecture are the bases for mapping quantum logic circuits into physical systems. The correlation between the underlying optimization and the upper layer of quantum logic design is an important issue that needs to be emphasized during the mapping process. In order to improve the efficiency and robustness for the above problems, a fast optimization algorithm for LNN architecture based on quantum weight is proposed. Based on quantum priority, a wire sequence initialization method is designed. Then the neighboring arrangement rule of the NCV gate library circuit and the optimization algorithm are given. The experimental results show that the proposed strategies can effectively reduce the QC and the number of inserted SWAP gates. The average optimization rate reaches 12.3% compared to the corresponding results.
In order to realize the linear nearest neighbor{(LNN)} of the quantum circuits and reduce the quantum cost of linear reversible quantum circuits, a method for synthesizing and optimizing linear reversible quantum circuits based on matrix multiplication of the structure of the quantum circuit is proposed. This method shows the matrix representation of linear quantum circuits by multiplying matrices of different parts of the whole circuit. The LNN realization by adding the SWAP gates is proposed and the equivalence of two ways of adding the SWAP gates is proved. The elimination rules of the SWAP gates between two overlapped adjacent quantum gates in different cases are proposed, which reduce the quantum cost of quantum circuits after realizing the LNN architecture. We propose an algorithm based on parallel processing in order to effectively reduce the time consumption for large-scale quantum circuits. Experiments show that the quantum cost can be improved by 34.31\% on average and the speed-up ratio of the GPU-based algorithm can reach 4 times compared with the CPU-based algorithm. The average time optimization ratio of the benchmark large-scale circuits in RevLib processed by the parallel algorithm is {95.57\%} comparing with the serial algorithm.
In order to realize the linear nearest neighbor(LNN) of quantum circuits and reduce the quantum cost of the linear reversible quantum circuits, a method of synthesizing and optimizing linear reversible quantum circuits based on matrix multiplication is proposed. This method shows the matrix representation of linear quantum circuits by proposing the matrix representation of the CNOT gate and the SWAP gate. The rule of judging whether a two-qubit quantum circuit is in LNN state is proposed. The LNN realization by adding SWAP gates is proposed and the equivalence of two ways of adding SWAP gates is proved. The elimination rules of SWAP gates between two overlapped adjacent quantum gates in different cases are proposed, which reduce the quantum cost of quantum circuits after realizing the LNN architecture. We propose a particular algorithm to realize the LNN, which can effectively reduce the quantum cost and improve calculation efficiency by inserting the SWAP gates on parallel computing devices. Experiments show that the quantum cost can be improved by 34.31% on average and the speed-up ratio of the GPU-based algorithm can reach 4 times compared with the CPU-based algorithm.
In order to solve the problem of mapping a quantum circuit to IBM QX architecture and satisfy connectivity constraints. This paper presents a mapping method, which is divided into two parts: Qubit initial mapping and Nearest-neighbor optimization. To solve the problem of initial mapping of qubits, according to the execution order and interaction of quantum gates in quantum circuit, a qubit mapping order method based on the weight of qubits is proposed. Then, according to the qubit interaction degree in the coupling graph, the initial qubit mapping is completed. In order to solve the nearest-neighbor problem in the mapping process, the quantum cost of non-neighbor two-qubit gate is determined by introducing the lookahead quantum cost method. Our algorithm is evaluated on IBM Q 20. Experimental results show that the proposed algorithm can effectively complete the mapping task even in large-scale benchmark tests. The proposed method is verified by experiments and compared with the existing methods, the optimized results were obtained in 74% of the test circuits.
In some practical quantum physical architectures, the qubits need to be distributed on 2-dimensional (2-D) grid structure to implement quantum computation. In order to map an 1-dimensional (1-D) quantum circuit into a 2-D grid structure and satisfy the nearest neighbor constraint of qubit interaction in the grid structure, a mapping method from 1-D quantum circuit to 2-D grid structure is proposed in this paper. This method firstly determines the order of placing qubits, and then presents the layout strategy of qubits in 2-D grid. We also proposed an algorithm for establishing interaction paths between non-adjacent qubits in 2-D grid structure, which can satisfy the physical constraints of the interaction of quantum bits in the grid in the process of mapping an 1-D quantum circuit to a 2-D grid structure. For some benchmark circuits, after using the method of this paper to place qubits, it is possible to make every 2-qubit gate in the circuit have a nearest neighbor, so that there is no need to use SWAP gate to establish channel routing. Compared with the latest available methods, the average optimization rate is 82.38%.
The linear nearest neighbor (LNN) restriction, present in several current implementations of 1D and 2D quantum circuits, limits the interaction of qubits to those which are adjacent to each other. While there have been several proposals to optimize the process of achieving LNN compliance in 1D circuits, there are few proposals for 2D circuits. Here, we present a new perspective on this problem for 2D quantum circuits. We propose to see this as a multiobjective optimization problem with two objectives: minimizing the size of the 2D grid in the circuit, and minimizing the number of SWAP gates required to achieve LNN compliance. We present some preliminary results which show that these are two contradictory objectives. Since this is common in multiobjective problems, these results indicate that a multiobjective algorithm might be a suitable way to address this problem, since it would make considerations which currently available methods do not make.