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IEEE Congress on Evolutionary Computation (CEC)

Synthesis of Hybrid and d-valued Quantum Logic Circuits

Faisal Shah Khan, Marek A. Perkowski

Published Sep 22, 2006• 0 citations
Publisher

Abstract

Recent research in generalizing quantum computation from 2-valued qudits to d-valued qudits has shown practical advantages for scaling up a quantum computer. A further generalization leads to quantum computing with hybrid qudits where two or more qudits have different finite dimensions. In both cases, a quantum computation is performed when a unitary evolution operator, acting as a quantum logic gate, transforms the state of qudits in a quantum system. Unitary operators can be represented by unitary matrices, and therefore for a quantum system consists of multiple qudits, a gate may be synthesized by matrix decomposition techniques such as QR factorization and the Cosine-sine Decomposition (CSD). In this article, we present a CSD based synthesis method for n qudit hybrid and d-valued quantum circuits.

DOI
10.1109/cec.2006.1688607

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Recent research in generalizing quantum computation from 2-valued qudits to d-valued qudits has shown practical advantages for scaling up a quantum computer. A further generalization leads to quantum computing with hybrid qudits where two or more qudits have different finite dimensions. In both cases, a quantum computation is performed when a unitary evolution operator, acting as a quantum logic gate, transforms the state of qudits in a quantum system. Unitary operators can be represented by unitary matrices, and therefore for a quantum system consists of multiple qudits, a gate may be synthesized by matrix decomposition techniques such as QR factorization and the Cosine-sine Decomposition (CSD). In this article, we present a CSD based synthesis method for n qudit hybrid and d-valued quantum circuits.
Recent research in multi-valued logic for quantum computing has shown practical advantages for scaling up a quantum computer. Multivalued quantum systems have also been used in the framework of quantum cryptography, and the concept of a qudit cluster state has been proposed by generalizing the qubit cluster state. An evolutionary algorithm based synthesizer for ternary quantum circuits has recently been presented, as well as a synthesis method based on matrix factorization.In this paper, a recursive synthesis method for ternary quantum circuits based on the Cosine-Sine unitary matrix decomposition is presented.
Because it has the ability to avoid energy losses and the possibility to solve some exponentially hard problems in polynomial time, quantum computation is getting more and more attention. Multi-valued logic for quantum computing has shown practical advantages for scaling up a quantum computer. In this paper, some reversible logic gates for the multi-valued quantum logic and some synthesis methods for reversible logic are introduced, including the Cosine-Sine decomposition and the evolutionary quantum logic synthesis.
In this paper, the problem of constructing an efficient quantum circuit for the imple-mentation of an arbitrary quantum computation is addressed. To this end, a basic blockbased on the cosine-sine decomposition method is suggested which contains l qubits.In addition, a previously proposed quantum-logic synthesis method based on quantumShannon decomposition is recursively applied to reach unitary gates over l qubits. Then,the basic block is used and some optimizations are applied to remove redundant gates.It is shown that the exact value of l affects the number of one-qubit and CNOT gates inthe proposed method. In comparison to the previous synthesis methods, the value of l isexamined consequently to improve either the number of CNOT gates or the total numberof gates. The proposed approach is further analyzed by considering the nearest neighborlimitation. According to our evaluation, the number of CNOT gates is increased by atmost a factor of 5/3 if the nearest neighbor interaction is applied.
Given an arbitrary ${2}^{w}\ifmmode\times\else\texttimes\fi{}{2}^{w}$ unitary matrix UUU, a powerful matrix decomposition can be applied, leading to four different syntheses of a www-qubit quantum circuit performing the unitary transformation. The demonstration is based on a recent theorem by H. F\"uhr and Z. Rzeszotnik [Linear Algebra Its Appl. 484, 86 (2015)] generalizing the scaling of single-bit unitary gates (w=1w=1w=1) to gates with arbitrary value of www. The synthesized circuit consists of controlled one-qubit gates, such as negator gates and phasor gates. Interestingly, the approach reduces to a known synthesis method for classical logic circuits consisting of controlled not gates in the case that UUU is a permutation matrix.
In this paper, the problem of constructing an efficient quantum circuit for the implementation of an arbitrary quantum computation is addressed. To this end, a basic block based on the cosine-sine decomposition method is suggested which contains lll qubits. In addition, a previously proposed quantum-logic synthesis method based on quantum Shannon decomposition is recursively applied to reach unitary gates over lll qubits. Then, the basic block is used and some optimizations are applied to remove redundant gates. It is shown that the exact value of lll affects the number of one-qubit and CNOT gates in the proposed method. In comparison to the previous synthesis methods, the value of lll is examined consequently to improve either the number of CNOT gates or the total number of gates. The proposed approach is further analyzed by considering the nearest neighbor limitation. According to our evaluation, the number of CNOT gates is increased by at most a factor of 53\frac{5}{3}35​ if the nearest neighbor interaction is applied.