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IEEE Conference on Nanotechnology (IEEE NANO)

Modified Karnaugh map for quantum Boolean circuits construction

Shiou-An Wang, Chin-Yung Lu, I-Ming Tsai, Sy‐Yen Kuo

Published Jan 23, 2004• 12 citations
Publisher

Abstract

Karnaugh map is an efficient method of minimization for conventional logic design. Unfortunately, it is usually used for 3 or 4 variables, at most 6 variables. In this paper, we modify the Karnaugh map and propose a set of reduction rules for quantum Boolean circuit optimization. By applying these rules, we can efficiently simplify a quantum Boolean circuit that has an arbitrary number of input variables. In terms of the space consumption, we use only one auxiliary qubit as the output qubit, and keep all the input qubits unchanged.

DOI
10.1109/nano.2003.1230996

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Karnaugh map is an efficient method of minimization for conventional logic design. Unfortunately, it is usually used for 3 or 4 variables, at most 6 variables. In this paper, we modify the Karnaugh map and propose a set of reduction rules for quantum Boolean circuit optimization. By applying these rules, we can efficiently simplify a quantum Boolean circuit that has an arbitrary number of input variables. In terms of the space consumption, we use only one auxiliary qubit as the output qubit, and keep all the input qubits unchanged.
The tabulation method can be used with a computer to simplify Boolean logic functions with up to 6 or more variables, especially with a large number of variables. For the various applications, their circuits usually are complex and we must simplify the circuit design to the best of our ability. In this paper, we present an algorithm that can efficiently simplify a quantum Boolean circuit with an arbitrary number of input variables by using the tabulation method. In terms of the space consumption, we use only one auxiliary qubit as the output qubit, and keep all the input qubits unchanged.
Today, quantum computers are still in development and are an area actively studied by researchers.Although quantum computers have high potential, there are significant technical difficulties and obstacles to achieving practical applications.Many research labs and companies are working on increasing the number of qubits, solving stability problems, and improving error correction techniques, and scalability.These efforts aim at advances such as the development of more stable and reliable qubits, the discovery of more precise control and reading techniques, and the design of more advanced algorithms.This study aims to establish a connection between quantum computers and classical computers.To achieve this, research and work have employed a Karnaugh-based approach.Karnaugh maps serve as highly useful tools in the analysis of complex logic circuits.Depending on one's needs and design requirements, different logic functions can be analyzed and simplified using Karnaugh maps.In this study, the Karnaugh map-based method is utilized to obtain the Boolean function from the state table of the Quantum circuit generated from the RCViewer+ synthesis tool.Subsequently, the obtained function is transformed into a combinational logic circuit.As a result, an optimal combinational circuit corresponding to the quantum circuit can be obtained.
Implementing a quantum computer at the circuit level has emerged as an important field of research recently. An important topic of building a general-purpose quantum computer is to implement classical Boolean logic using quantum gates and devices. Since the Toffoli gate is universal in classical Boolean logic, any classical combinational circuit can be implemented by the straightforward replacement algorithm with auxiliary qubits as intermediate storage. However, this inefficient implementation causes a large number of auxiliary qubits to be used. In this paper, a systematic procedure is proposed to derive a minimum space quantum circuit for a given classical combinational logic. We first formulate the problem of transforming an m-to-n bit classical Boolean logic into a t-bit unitary quantum operation. The eligible solution set is then constructed such that a solution can be found simply by selecting any member from this set. Finally, we show that the algorithm is optimal in terms of the space consumption.
To build a general-purpose quantum computer, it is crucial for the quantum devices to implement classical boolean logic. A straightforward realization of quantum boolean logic is to use auxiliary qubits as intermediate storage. This inefficient implementation causes a large number of auxiliary qubits to be used. In this paper, we have derived a systematic way of realizing any general m-to-n bit combinational boolean logic using elementary quantum gates. Our approach transforms the m-to-n bit classical mapping into a t-bit unitary quantum operation with minimum number of auxiliary qubits, then a variation of Toffoli gate is used as the basic building block to construct the unitary operation. Finally, each of these building blocks can be decomposed into one-bit rotation and two-bit control-U gates. The efficiency of the network is taken into consideration by formulating it as a constrained set partitioning problem.
Every quantum algorithm is represented by set of quantum circuits. Any optimization scheme for a quantum algorithm and quantum computation is very important especially in the arena of quantum computation with limited number of qubit resources. Major obstacle to this goal is the large number of elemental quantum gates to build even small quantum circuits. Here, we propose and demonstrate a general technique that significantly reduces the number of elemental gates to build quantum circuits. This is impactful for the design of quantum circuits, and we show below this could reduce the number of gates by 60% and 46% for the four- and five-qubit Toffoli gates, two key quantum circuits, respectively, as compared with simplest known decomposition. Reduced circuit complexity often goes hand-in-hand with higher efficiency and bandwidth. The quantum circuit optimization technique proposed in this work would provide a significant step forward in the optimization of quantum circuits and quantum algorithms, and has the potential for wider application in quantum computation.
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.