Zhi-qiang LI, Sai CHEN*, Wei ZHU, Han-wu CHEN A Common Algorithm of Construction a New Quantum Logic Gate for Exact Minimization of Quantum Circuits Abstract: Sincenon-permutative quantum gates have more complex rules than permutative quantum gates, direct use of non-permutative quantum gates should be avoided in the efficient synthesis algorithm because it is very hard to synthesize. The key method is using quantum gates to create new permutative quantum gates to replace non-permutative quantum gates. In this paper, we propose an algorithm using CNOT and non-permutative quantum gates to construct new optimal quantum logic gates library automatically. Our method based on the idea of exhaustion finds the all combinations of quantum logic gates with lower quantum cost no matter how many the quantum lines are. Keywords: automatically; non-permutative quantum gates; exhaustive 1 Introduction Cascading and combining the quantum logical gates are the basic elements of reversible quantum logic circuits, and then, a quantum computer is constructed by quantum reversible logic circuits. According to the characteristics of the input and the output, quantum gates can be divided into non-permutative quantum gates and permutative quantum gates. In a quantum logic circuit, if the input is logical, the output must be logical and vice versa. But when the input and output are all logical, the internal circuit allows the superposition of quantum information and quantum entanglement that is non-permutative values. If only using quantum logic gates in a quantum logic circuit, the synthesis algorithms of the circuits are like the classic reversible logic synthesis algorithm. In addition, non-permutative quantum gates are also used, the superposition of information will make the process of synthesis more complicated and low-performance. Non-permutative quantum gates, such as NCV quantum gates library (including NOT gates [1], controll-NOT gates and controlled- square-root-of-NOT gates [2]), are used to construct new quantum permutative gates and gate libraries. *Corresponding author: Sai CHEN, College of Information Engineering, Yangzhou University Yangzhou, China, E-mail: [email protected] Zhi-qiang LI, Wei ZHU, College of Information Engineering, Yangzhou University, Yangzhou, China Han-wu CHEN, School of Computer Science and Engineering, Southeast University,Nanjing, China 372 A Common Algorithm of Construction a New Quantum Logic Gate To reduce the cost of the circuits, an excellent combination and optimization techniques is the key. The essence of constructing less cost quantum logic gates is the reversible logic synthesis [3]. With the further research of reversible circuit, many synthesis methods of reversible circuits also appeared [4-8]. The ultimate goal of quantum reversible logic synthesis algorithms is to efficiently construct optimum quantum logic circuits and automatically design reversible quantum logic circuits with less cost. However, these methods generally were designed for the entire logic circuits optimization. And the research of the foundational quantum logic gates construction is less. The optimization of quantum logic gates will directly affect the entire quantum logic circuits optimization. If the quantum logic gates can be optimized automatically, the synthesize algorithm which using this gates will have better performance and minimum cost. This will play an important role in the optimization of the entire circuits. People have done a lot of research and put forward many quantum circuits synthesis algorithms in which most of 3-qubit synthesis algorithms based on quantum logic gates have been presented [9-13]. However, the algorithms based on non-permutative quantum gates are few. There are several synthesis algorithm based on NCV gates right now [14-17]. Although a variety of 4-qubit algorithms have been proposed, these algorithms still based on quantum logic gates. In [18], new quantum logic gates can be constructed by using NVC gate library to compose four types Peres- like gates. But this method is not universal, and can only be used when the number of lines is fewer. If the quantum lines are increased, such as the number of lines greater than five, this method cannot be realized. Therefore, in this paper we proposed a universal algorithm to generate any lines of optimal new quantum logic gates automatically. Here the optimum means that in the new quantum logic gates can be no longer broken down into several cascading quantum logic gates with equivalent quantum cost. 2 Backgroung A quantum gate is the basic unit of quantum information processing and its cascade constitutes a quantum circuit. A quantum circuit is reversible. In the quantum computation, a quantum gate is corresponding to a unitary transformation. It is well known that the operation of each gate in an n-line reversible or quantum circuit can be represented by a square matrix of dimension 2n. The matrix of the NOT 0 1 1 0 = I = gate is N false, and 0 1 false represents the identity circuit. 1 0 As shown in Figure 1 is the NOT gates, control-NOT gates and Toffoli gates [19]. They are all typical permutative quantum gates. A Common Algorithm of Construction a New Quantum Logic Gate 373 x3 x3 x2 x2 x2 x2 ⊕ x x⊕ xx x1 x1 x1 xx121 1 23 ()a ()b ()c Figure 1. The permutative quantum gates. The Controlled-square-root-of-NOT gates contain a controlled-V (CV) gate and a controlled-V+ (CV+) gate as shown in Figure 2. They are all typical non-permutative quantum gates. Figure 2. Basic quantum algebra rules for CV/CV† gates. If the input and the output are logical, the gate must be a permutative quantum gate. If the input is not logical and the output is logic, the gate must not be a permutative quantum gate andvice versa. If the input and output are all not logical, the gate cannot be sure a permutative quantum gate. For example, control-NOT gate in Figure 1 (b), 1111+−ii 1 1 + i 01 11 1+−ii 1 = xx12⊕= × = set x2 to 1 and x1to v0 which v0 is 2211−+ii 0 1 − i false, then 10 22 1−+ii 1 false. We can clearly see the result is not logic. If the gate is replaced by a controlled-V gate, 01+ 1 11+−ii01+ 2 × set x2 to 1 and x1to false, the result is 2211−+ii false. It is clearly also not logical. 3 The common algorithm ofr ealizing an ewquantum logic gate In [18], new quantum logic gates were constructed by using NVC gate library to compose four types Peres-like gates. The new quantum logic gates along with the NOT gates and control-NOT gates together constituted the new quantum logic gate library (NCP4). The NCP4 gate library can construct optimal 3-qubit quantum logic circuit which the NCV gate library can also construct the equivalent. That means the function of the two methods are the same, but the construction methods are different. We can also say that the two gate libraries are equivalent when they synthesize 3-qubit logic circuits. In Figure 4, the new quantum logic gate was constructed by our hands when the quantum lines were five. This method totally spent 11 CNOT gates. The Table 374 A Common Algorithm of Construction a New Quantum Logic Gate 1 is the all kinds of inputs. We can clearly see that each line only eight U gates are to be used. So we can sure the U gates are controlled-Kth-root-of-NOT when K=8. Table 1. All kinds of inputs in fig. 4 X3X2X1X0 U1 U2 U3 U4 U5 U6 U7 U8 U9 U10 U11 U12 U13 U14 U15 0000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0001 1 0 0 0 0 0 1 0 0 1 1 1 1 1 1 0010 0 1 0 0 0 1 0 1 1 0 0 1 1 1 1 0011 1 1 0 0 0 1 1 1 1 1 1 0 0 0 0 0100 0 0 1 0 1 0 0 1 1 1 1 0 0 1 1 0101 1 0 1 0 1 0 1 1 1 0 0 1 1 0 0 0110 0 1 1 0 1 1 0 0 0 1 1 1 1 0 0 0111 1 1 1 0 1 1 1 0 0 0 0 0 0 1 1 1000 0 0 0 1 1 1 1 0 1 0 1 0 1 0 1 1001 1 0 0 1 1 1 0 0 1 1 0 1 0 1 0 1010 0 1 0 1 1 0 1 1 0 0 1 1 0 1 0 1011 1 1 0 1 1 0 0 1 0 1 0 0 1 0 1 1100 0 0 1 1 0 1 1 1 0 1 0 0 1 0 1 1101 1 0 1 1 0 1 0 1 0 0 1 1 0 0 1 1110 0 1 1 1 0 0 1 0 1 1 0 1 0 0 1 1111 1 1 1 1 0 0 0 0 1 0 1 0 1 0 1 x3 x3 x2 x2 x1 x1 x0 x0 z U1 U2 U3 U4 U5 U6 U7 U8 U9 U10 U11 U12 U13 U14 U15 z' Figure 4. The new quantum logic gate with four control lines was constructed by hands. Because the lines of the circuits are few and there are certain rules in the circuits, we can construct these circuits by our hands. But when the lines of the new quantum logic gates are increased, this method is difficult to implement. We must find a common algorithm. In [20], non-permutative quantum gates constructed new permutative quantum gates by using controlled-Kth -root-of-NOT gates.