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IEEE International Symposium on Multiple-Valued Logic (ISMVL)

Synthesis of Ternary Grover's Algorithm

Sudhindu Bikash Mandal, Amlan Chakrabarti, Susmita Sur‐Kolay

Published May 1, 2014• 11 citations
Publisher

Abstract

Grover's search algorithm is one of the well-studied quantum computing algorithms, which plays a key role in many applications such as graph coloring, triangle finding, Boolean satisfiability. Although there are many works on circuit synthesis for Grover's algorithm in the binary quantum domain, only a few exist for the ternary version of the algorithm. In this paper, we first propose a new ternary superposition operator and utilize it to synthesize a ternary quantum logic circuit for Grover's algorithm. Our proposed circuits for a ternary oracle and the ternary Grover's diffusion operator require 4 and 6 gate levels respectively per iteration, and for the ternary oracle 1 ancilla qutrit. To the best of our knowledge, this is the first circuit for the Grover's diffusion operator using ternary gates. Finally, we use these two circuits to present a ternary quantum circuit for the vertex coloring problem. The oracle circuit for this problem has smaller gate count compared to existing solutions.

DOI
10.1109/ismvl.2014.40

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Grover's search algorithm is one of the well-studied quantum computing algorithms, which plays a key role in many applications such as graph coloring, triangle finding, Boolean satisfiability. Although there are many works on circuit synthesis for Grover's algorithm in the binary quantum domain, only a few exist for the ternary version of the algorithm. In this paper, we first propose a new ternary superposition operator and utilize it to synthesize a ternary quantum logic circuit for Grover's algorithm. Our proposed circuits for a ternary oracle and the ternary Grover's diffusion operator require 4 and 6 gate levels respectively per iteration, and for the ternary oracle 1 ancilla qutrit. To the best of our knowledge, this is the first circuit for the Grover's diffusion operator using ternary gates. Finally, we use these two circuits to present a ternary quantum circuit for the vertex coloring problem. The oracle circuit for this problem has smaller gate count compared to existing solutions.
Many applications such as Graph coloring, Triangle finding, Boolean satisfiability, Traveling salesman problem can be solved by Grover's search algorithm, which is one of the remarkable quantum computing algorithm. To design a quantum circuit for a given quantum algorithm, which involves Grover's search, we need to define an oracle circuit specific to the given algorithm and the diffusion operator for amplification of the desired quantum state. In this paper, we propose a quantum circuit implementation for the oracle of the vertex coloring problem based on Grover's algorithmic approach. To the best of our knowledge, this is a first of its kind approach in regards to the quantum circuit synthesis of the vertex coloring oracle in binary quantum domain. We have performed the synthesis of the proposed oracle circuit for the six commonly available Physical Machine Description (PMD) gate libraries using the Fault Tolerant Quantum Logic Synthesis (FTQLS) tool. The synthesis results have been presented to understand the cost estimation of the oracle circuit for the various PMDs in terms of number of quantum operations and number of cycles.
Basic logic gates and their operations in ternary quantum domain are involved in the synthesis of ternary quantum circuits. Only a few works define ternary algebra for ternary quantum logic realization. In this paper, a ternary logic function is expressed in terms of projection operations including a new one. A method to realize new multi-qutrit ternary gates in terms of generalized ternary gates and projection operations is also presented. We also introduced ten simplification rules for reducing ancilla qutrits and gate levels. Our method yields lower gate cost and fewer gate levels and ancilla qutrits than that obtained by earlier methods for the ternary benchmark circuits. The nnn qutrit ternary sum function is synthesized without any ancilla qutrit by our proposed methodology.
Synthesis of ternary quantum circuits involves basic ternary gates and logic operations in the ternary quantum domain. Works that define ternary algebra and their applications for ternary quantum logic realization, are very few. In this paper, we express a ternary logic function in terms of projection operations including a new one. We demonstrate how to realize a few new multi-qutrit ternary gates in terms of generalized ternary gates and projection operations. We then employ our synthesis method to design ternary adder circuits which have better cost than that obtained by earlier method.
Abstract—Multiple-valued quantum circuits are promising choices for future quantum computing technology, since the multiple-valued quantum system is more compact than the corresponding binary quantum system. Grover’s quantum search algorithm requires a sub-circuit called oracle, which takes a set of inputs and gives an output stating whether a given search condition on the inputs is satisfied or not. Equality, less-than, and greater-than comparisons are widely used as search conditions. In this paper, we show design of quantum ternary equality, less-than, and greater-than comparators on the top of ion-trap realizable 1-qutrit gates and 2-qutrit Muthukrishnan-Stroud gates. Index Terms—Multiple-valued logic, quantum logic, reversible logic, ternary comparators, ternary logic I.
Abstract — This paper investigates a universal approach of synthesizing arbitrary ternary logic circuits in quantum computation based on the truth table technology. It takes into account of the relationship of classical logic and quantum logic circuits. By adding inputs with constant value and garbage outputs, the classical non-reversible logic can be transformed into reversible logic. Combined with group theory, it provides an algorithm using the ternary Swap gate, ternary NOT gate and ternary Toffoli gate library. Simultaneously, the main result shows that the numbers of qutrits we use are minimal compared to other methods. We also illustrate with two examples to test our approach. Index Terms — ternary quantum computation, reversible logic, circuit synthesis, group theory I.