Reversible Circuit Synthesis and Multiple-Valued Quantum Circuit Verification
Date Issued
2010
Date
2010
Author(s)
Lu, Chin-Yung
Abstract
Reversible circuits have applications in many areas such as digital signal processing, low power design, and quantum computing. If a circuit is reversible, it can reduce the energy consumption caused by information loss. The Karnaugh map method is faster and easier to apply than other simplification methods for combination logic circuits. But the classical Karnaugh map is not directly applicable to reversible circuits because the basic logic gates, except the NOT gate, are not reversible gates. In this dissertation, we propose a method to solve the problem so that the Karnaugh map can be applied to the reversible circuit synthesis. Our algorithm provides a systematic method for simplifying the reversible circuit. This can generate 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.
So far there are no synthesis algorithms that can find all the optimal reversible circuits except an exhaustive algorithm. We propose a method based on the divide and conquer approach which can significantly improve the performance of the existing synthesis algorithms to synthesize reversible circuits. A reversible circuit is first divided into two subcircuits. The smaller subcircuit will input all possible combinations of m gates, except for those combinations with the same functionality only the one with fewest gates is selected for input. The other subcircuit can be synthesized with an existing algorithm. The two subcircuits are then combined to form all the possible results and from these results we can choose the most simplified one. According to the experimental results on all the 3-variable reversible functions, we can see that the performance of the existing algorithms can be significantly improved by using our method. Therefore the synthesized reversible circuits are much more simplified than previous results.
Moreover, in order to efficiently represent a quantum operation, we propose X-decomposition Quantum Decision Diagram (XQDD) which can easily perform matrix operations. XQDD can be used to verify quantum and reversible circuits even if the reversible circuits have different number of garbage qubits. It is more efficient in terms of space and time. We extend the binary-valued XQDD to multiple-valued quantum logic. The extended XQDD can represent a multiple-valued quantum operation and perform matrix operations. It can be applied to verify the equivalence of two multiple-valued quantum or reversible circuits which are synthesized by different approaches. According to the simulation results, it is much better than multiple-valued QuIDD and very close to QMDD in terms of time. Besides, we show that the space in multiple-valued XQDD is less than other representations.
Finally, previous quantum secure direct communication (QSDC) protocols usually consume one Einstein-Podolsky-Rosen (EPR) pair to transmit a single qubit. If Alice wants to transmit an n-bit message, she needs at least n/2 EPR pairs when dense coding scheme is used. We propose a new QSDC protocol based on EPR pairs. If both Alice and Bob pre-share 2c+1 EPR pairs with the trusted server where c is a constant, Alice can transmit arbitrary number of qubits to Bob. The 2c EPR pairs are used by Alice and Bob to authenticate each other and the remaining EPR pair is used to encode and decode the message qubit. Thus the total number of EPR pairs used for one communication is a constant no matter how many bits will be transmitted. It does not need to transmit EPR pairs before transmitting the secret message except the pre-shared constant number of EPR pairs. It reduces both the utilization of the quantum channel and the risk. In addition, after the authentication, the server is not involved in the message transmission. Thus we can prevent the server from knowing the message.
Subjects
Reversible Circuits
Logic Synthesis
Quantum Computing
Multiple Valued Logic
Verification
Authentication
SDGs
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