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International Conference on Multimedia Technology (ICMT)

An efficient method for improving the performance of quantum boolean circuit synthesis algorithms

Shiou-An Wang, Chin-Yung Lu

Published Jul 1, 2011• 0 citations
Publisher

Abstract

So far there are no synthesis algorithms that can find all the optimal quantum boolean circuits except an exhaustive algorithm. In this paper, we propose a method based on the divide and conquer approach which can significantly improve the performance of the existing synthesis algorithms to synthesize quantum boolean circuits. A quantum boolean circuit is first divided into two subcircuits. The subcircuit with fewer gates will input all possible combinations of m gates excluding those with the same function specification. The other subcircuit can be synthesized by using the existing algorithm. The two subcircuits are combined and then we can choose the most simplified quantum boolean circuit. According to the experimental results of all the 3-variable functions, we can see that the performance of the existing algorithms can be significantly improved by using our method. Therefore the synthesized quantum boolean circuits are much more simplified than previous results.

DOI
10.1109/icmt.2011.6002505

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•May 1, 2015•7 citations
So far there are no synthesis algorithms that can find all the optimal quantum boolean circuits except an exhaustive algorithm. In this paper, we propose a method based on the divide and conquer approach which can significantly improve the performance of the existing synthesis algorithms to synthesize quantum boolean circuits. A quantum boolean circuit is first divided into two subcircuits. The subcircuit with fewer gates will input all possible combinations of m gates excluding those with the same function specification. The other subcircuit can be synthesized by using the existing algorithm. The two subcircuits are combined and then we can choose the most simplified quantum boolean circuit. According to the experimental results of all the 3-variable functions, we can see that the performance of the existing algorithms can be significantly improved by using our method. Therefore the synthesized quantum boolean circuits are much more simplified than previous results.
At present, only the exhaustive synthesis algorithm can find all the optimal quantum Boolean circuits. It is more difficult to synthesize a more complicated quantum Boolean circuit. In this paper, we use the method of circuit bipartition to partition a more complicated quantum Boolean circuit into simpler circuits in order to reduce the difficulty of the more complicated quantum Boolean circuit synthesis. And find these partitioned quantum Boolean circuits by searched the database of quantum Boolean circuits according the method of the shortest path in the circuit. Finally, combine these partitioned circuits to become the more complicated quantum Boolean circuit. This approach can significantly reduce the synthesis complexity of a more complicated quantum Boolean circuit. We can see that the results are very close to the optimal circuits according to the experimental results of all the 3-variable reversible functions.
Quantum computers, more specifically quantum circuits, take on the eyes with their computational promises such as reversibility. In this paper, we perform synthesis and optimization of quantum circuits. In the first part, we propose a fast synthesis algorithm that implements any given reversible Boolean function with quantum gates. Instead of an exhaustive search on every given function, our algorithm creates a library of essential functions and performs sorting. As an example, to implement 4 bit circuits we only use 120 essential functions out of all 20922789888000 functions. In the second part, we optimize our circuits by using new templates. The proposed templates mainly consist of Toffoli gates with negative and positive controlling lines. These templates also show us that optimum area solutions proposed in the literature are not actually optimum; they can be improved.
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 efficient synthesis of quantum circuits is an active research area. Since many of the known quantum algorithms include a large Boolean component (e.g. the database in the Grover search algorithm), quantum circuits are commonly synthesized in a two-stage approach. First, the desired function is realized as a reversible circuit making use of existing synthesis methods for this domain. Afterwards, each reversible gate is mapped to a functionally equivalent quantum gate cascade. In this paper, we propose an improved mapping of reversible circuits to quantum circuits which exploits a certain structure of many reversible circuits. In fact, it can be observed that reversible circuits are often composed of similar gates which only differ in the position of their target lines. We introduce an extension of reversible gates which allow multiple target lines in a single gate. This enables a significantly cheaper mapping to quantum circuits. Experiments show that considering multiple target lines leads to improvements of up to 85% in the resulting quantum cost.
This paper presents how to synthesize efficient Boolean quantum circuits (BQC) from Reed-Muller (RM) expansions of Boolean functions. Firstly, the method of Younes et al [A. Younes and J. F. Miller, International Journal of Electronics 91, 431 (2004)] is generalized to synthesize BQC for multiple outputs function. The transformation is straightforward. A technique rearranging the modified Toffoli gates is also presented to eliminate additive NOT gates as much as possible. Secondly, the approach synthesizing BQC without work qubits is proposed which is simple and high-efficient. The necessary conditions for the RM expansions which can be synthesized by the proposed method are derived too.
Quantum circuit synthesis is one of the major areas of current research in the field of quantum computing.Analogous to its Boolean counterpart, the task involves constructing arbitrary quantum gates using only those available within a small set of universal gates that can be realized physically.However, unlike the latter, there are an infinite number of single qubit quantum gates, all of which constitute the special unitary group SU(2).Realizing any given single qubit gate using a given universal gate family is a complex task.Although gates can be synthesized to arbitrary degree of precision as long as the set of finite strings of the gate family is a dense subset of SU(2), it is desirable to accomplish the highest level of precision using only the minimum number of universal gates within the string approximation.Almost all algorithms that have been proposed for this purpose are based on the Solovay-Kitaev algorithm.The crux of the Solovay-Kitaev algorithm is the use of a procedure to decompose a given quantum gate into a pair of group commutators with the pair being synthesized separately.The Solovay-Kitaev algorithm involves group commutator decomposition in a recursive manner, with a direct approximation of a gate into a string of universal gates being performed only at the last level, i.e. in the leaf nodes of the search tree representing the execution of the Solovay-Kitaev algorithm.The main contribution of this research is in integrating conventional optimization procedures within the Solovay-Kitaev algorithm.Two specific directions of research have been studied.Firstly, optimization is incorporated within the group commutator decomposition, so that a more optimal pair of group commutators are obtained.As the degree of precision of the synthesized gate is explicitly minimized by means of this optimization procedure, the enhanced algorithm allows for more accurate quantum gates to be synthesized than what the original Solovay-Kitaev algorithm achieves.Simulation results with random gates indicate that the obtained accuracy is an order of magnitude better than before.Two versions of the new algorithm are examined, with the optimization in the first version being invoked only at the bottom level of Solovay-Kitaev algorithm and when carried out across all levels of the search tree in the next.Extensive simulations show that the second version yields better results despite equivalent computation times.Theoretical analysis of the proposed algorithm is able to provide a more formal, quantitative explanation underlying the experimentally observed phenomena.The other direction of investigation of this research involves formulating the group commutator decomposition in the form of bi-criteria optimization.This phase of research relaxed the equality constraint in the previous approach and with relaxation, a bi-criteria optimization is proposed.This optimization algorithm is new and has been devised primarily when the objective needs to be relaxed in different stages.This bi-criteria approach is able to provide comparably accurate synthesis as the previous approach.