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IEEE Conference on Nanotechnology (IEEE NANO)

Quantum boolean circuits construction using tabulation method

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

Published Apr 1, 2005• 2 citations
Publisher

Abstract

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.

DOI
10.1109/nano.2004.1392431

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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.
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.
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.
An important property of reversible circuits is that they can reduce the energy consumption which is now a big problem for the advanced circuit design. If a circuit is reversible, it can reduce the energy consumption caused by information loss. The tabulation method is more efficient than other simplification methods for combination logic circuits. But the classical tabulation method is not directly applicable to reversible circuits because the basic logic gates, except the NOT gate, are not reversible gates. In this paper, we propose a method to solve the problem so that the tabulation method can be applied to the reversible circuit synthesis. Our algorithm provides a systematic method to simplify the reversible circuit. This can produce the resulting expression in exclusive-sum form and transform it into a final reversible circuit with lower quantum cost. Moreover, we can realize permutations to be reversible circuits with lower quantum cost and without unnecessary garbage bits. We can also convert irreversible circuits by adding qubits to make the circuits reversible. The experimental results show that the average saving in quantum cost is 15.82% compared with previous approaches.
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.
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.
Functional verification is an important design method for verifying functional equivalence between a simplified quantum Boolean circuit and original one. During the design process, checking the equivalence of two quantum Boolean circuits is necessary. In this paper, we present an algorithm that can efficiently and easily verify two quantum Boolean circuits by using the back propagation method. For a set of input vectors, the idea of the algorithm is to find the checking vectors that output vectors are different from input vectors by a backward tracking process.