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International Conference on Computational Science and Computational Intelligence (CSCI)

Quantum Cost Reduction of Reversible Circuits Using New Toffoli Decomposition Techniques

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

Published Dec 1, 2015• 18 citations
Publisher

Abstract

Quantum cost is the most important criteria to evaluate reversible and quantum circuits. Also the fundamental building blocks of reversible and quantum circuits are Multiple-Control Toffoli (MCT) gates. The synthesis of MCT based reversible circuits are usually conducted into two steps. First, MCT circuits are decomposed into quantum circuits and then they are optimized using various techniques such as template matching, moving rules to reduce the quantum cost of reversible circuits. In this paper, we propose new techniques to decompose the Toffoli gates, in which MCT based circuits are mapped into a corresponding quantum realization. The main improvement is that the resulting quantum realization of MCT based circuits makes significantly better realization than those achieved in the earlier approaches and further reduction is possible using some other optimization techniques. Experimental results show that our new techniques enable to get sub-optimal realization of the MCT based reversible circuits in decomposition stage and quantum cost reduction of the reversible circuits is achieved by using that sub-optimal realization.

DOI
10.1109/csci.2015.41

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Quantum cost is the most important criteria to evaluate reversible and quantum circuits. Also the fundamental building blocks of reversible and quantum circuits are Multiple-Control Toffoli (MCT) gates. The synthesis of MCT based reversible circuits are usually conducted into two steps. First, MCT circuits are decomposed into quantum circuits and then they are optimized using various techniques such as template matching, moving rules to reduce the quantum cost of reversible circuits. In this paper, we propose new techniques to decompose the Toffoli gates, in which MCT based circuits are mapped into a corresponding quantum realization. The main improvement is that the resulting quantum realization of MCT based circuits makes significantly better realization than those achieved in the earlier approaches and further reduction is possible using some other optimization techniques. Experimental results show that our new techniques enable to get sub-optimal realization of the MCT based reversible circuits in decomposition stage and quantum cost reduction of the reversible circuits is achieved by using that sub-optimal realization.
State-of-the-art studies on quantum layout synthesis have proposed various approaches based on the assumption that the input circuit is only composed of single-qubit and two-qubit gates. This assumption greatly simplifies the layout synthesis problem, and thus they only require ensuring that all controlled-NOT (CNOT) gates satisfy the hardware constraints imposed by a given coupling graph. However, during the design of quantum circuits, multi-controlled Toffoli (MCT) gates are usually used to better characterize the function of the circuits. Directly decomposing them into single and two-qubit gates with a fixed routine ignores the flexibility and optimization opportunity provided by various decomposition (i.e., logic synthesis) possibilities and thus suffers from sub-optimal results. This article proposes a co-optimization approach for quantum logic and layout synthesis. The MCT gates are first decomposed into Toffoli gates, and an efficient qubit mapping checking process is proposed to optimally solve the SWAP-free layout synthesis problem by automatically determining the decomposition result of each Toffoli gate. If a SWAP-free result cannot be found, the proposed algorithm flow is then used to obtain a solution that minimizes the total cost that simultaneously counts the cost caused by the different decomposition methods for Toffoli gates and the cost induced by SWAP gates. Compared with a state-of-the-art method, the proposed approach reduces the number of additional CNOT gates by 16% with around 25X runtime speedup for the cases with SWAP-free solutions, which cannot be obtained without the co-optimization approach.
The paper presents an approach for optimizing a reversible netlist consisting of multiple-control Toffoli (MCT) gates with the objective of reducing quantum cost. The MCT gates are first decomposed into smaller gates with up to three control lines, and then a template matching method is applied on the netlist for optimization. The templates are constructed by considering all possible 3-gate sequences with up to four lines. This work constitutes an extension of a previous work by Dueck et al., where 2-gate sequences were considered in the templates. The input gate netlists are generated from the reversible benchmarks available in RevLib, followed by Barenco decomposition to produce Toffoli gates with at most three control connections. The proposed approach is applied on these netlists, and the final quantum cost calculated. Experimental results on standard benchmarks show that the method can lead to reductions in quantum cost.
It is considered to be very important to reduce so-called the T-depth of a target quantum circuit for realizing future fault-tolerant quantum computation. Thus, there have been many researches to consider how to decompose Multiple Controlled Toffoli (MCT) gates with few T-depth; these existing methods do not consider the T-depth of each qubit when they decompose one MCT gate. In contrast to existing methods, this paper proposes to consider the T-depth of each qubit when we decompose an MCT gate; we show that we can decrease the T-depth by considering the T-depth of each qubit and appropriately selecting ancilla qubits and decomposition methods. In addition, to consider the impact of the decomposition of one MCT gate on the decomposition of subsequent gates, our method utilizes beam search to select a possibly best decomposition for each MCT gate. We confirmed that the proposed method can reduce the T-depth by an average of 17.1% compared to existing methods.
Abstract The construction of quantum computers is based on the synthesis of low-cost quantum circuits. The quantum circuit of any Boolean function expressed in a Positive Polarity Reed-Muller (PPRM) expansion can be synthesized using Multiple-Control Toffoli (MCT) gates. This paper proposes two algorithms to construct a quantum circuit for any Boolean function expressed in a PPRM expansion. The Boolean function can be expressed with various algebraic forms, so there are different quantum circuits can be synthesized for the Boolean function based on its algebraic form. The proposed algorithms aim to map the MCT gates into the NCV gates for any quantum circuit by generating a simple algebraic form for the Boolean function. The first algorithm generates a special algebraic form for any Boolean function by rearrangement of terms of the Boolean function according to a predefined degree of term d term , then synthesizes the corresponding quantum circuit. The second algorithm applies the decomposition methods to decompose MCT circuit into its elementary gates followed by applying a set of simplification rules to simplify and optimize the synthesized quantum circuit. The proposed algorithms achieve a reduction in the quantum cost of synthesized quantum circuits when compared with relevant work in the literature. The proposed algorithms synthesize quantum circuits that can applied on IBM quantum computer.
The synthesis and optimization of quantum circuits are essential for the construction of quantum computers. This paper proposes two methods to reduce the quantum cost of 3-bit reversible circuits. The first method utilizes basic building blocks of gate pairs using different Toffoli decompositions. These gate pairs are used to reconstruct the quantum circuits where further optimization rules will be applied to synthesize the optimized circuit. The second method suggests using a new universal library, which provides better quantum cost when compared with previous work in both cost015 and cost115 metrics; this proposed new universal library “Negative NCT” uses gates that operate on the target qubit only when the control qubit’s state is zero. A combination of the proposed basic building blocks of pairs of gates and the proposed Negative NCT library is used in this work for synthesis and optimization, where the Negative NCT library showed better quantum cost after optimization compared with the NCT library despite having the same circuit size. The reversible circuits over three bits form a permutation group of size 40,320 (23!), which is a subset of the symmetric group, where the NCT library is considered as the generators of the permutation group.
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.