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International Conference on Computer and Information Technology (ICCIT)

Quaternary quantum algorithm for determining properties of quaternary logic function

Mozammel H. A. Khan

Published Dec 1, 2011• 0 citations
Publisher

Abstract

A multiple-valued (d-dimensional with d >; 2) quantum system enables a much more compact and efficient information encoding than for binary (d = 2) quantum system. As multiple-valued quantum system is physically realizable, it is worth to explore multiple-valued quantum algorithm. In this paper, we propose quaternary (4-dimensional) quantum algorithm for testing whether a quaternary logic function is either constant or balanced. In classical computer, this test would require at least 4n-1+1 function evaluations, where n is the number of inputs of the function. But the proposed quantum algorithm requires only one function evaluation.

DOI
10.1109/iccitechn.2011.6164771

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Abstract Multi-valued quantum systems have more significant advantages in computational performance than binary quantum system, since the circuit design is more compact and efficient in multi-valued quantum systems. Hence, synthesizing multi-valued logic circuits is a better choice for future quantum computing technology. A subcircuit in the Grover’s algorithm is called an oracle, which inputs a set of quantum qudits and gives an output according to the search conditions. Greater-than/less-than/equality comparators are widely used in search conditions. In this paper, the ternary Toffoli gate proposed by Monfared and Haghparast is extended to multi-valued under certain conditions, and the existing optimal quaternary n-qudit comparator is optimized using the multi-valued Toffoli gate. In addition, we design universal multi-valued equality/greater-than/less-than comparators and complementary component, which constitute a comprehensive multi-valued n-qudit comparator. Compared to existing comparator circuits, these comparator circuits are more efficient in terms of quantum cost, garbage output, hardware complexity, and constant input.
We first propose herein a novel parallel computation, even though today's algorithm methodology for quantum computing, for all of the combinations of values in variables of a logical function. Our concern so far has been to obtain an attribute of some function. In fact such a task is only for one task problem solving. However, we could treat positively the plural evaluations of some logic function in parallel instead of testing the function for finding out its attribute. In fact, these evaluations of the function are naturally included and evaluated, in parallel, in normal quantum computing discussing a function in a Boolean algebra stemmed from atoms in it. As is naturally understandable with mathematics, quantum computing with qubit systems naturally is included and exemplified by a Boolean algebra, which treats only both 0 and 1. Namely, the theory in this paper is quite natural in logical sense even though physics domain. Therefore, quantum computing has an ability to solve some mathematical problems described in a Boolean algebra. The reason why we positively introduce a Boolean algebra here is because we have multiple evaluations of a function in quantum computing general.
Quaternary logic is very suitable for encoded realization of binary logic functions by grouping 2-bits together into quaternary digits. This sort of quaternary encoded reversible realization of binary logic function requires half times input/output lines than the original binary reversible realization. Quaternary decoder, multiplexer, and demultiplexer are very important building blocks of quaternary digital systems. In this paper, we show reversible realization of these circuits using quaternary reversible gates like quaternary shift gates (QSG), quaternary controlled shift gates (QCSG), and quaternary Toffoli gates (QTG). We also show the realization of multi-digit QCSG and QTG using QSG and QCSG, which are realizable using liquid ion-trap quantum technology and other reversible technologies.
We define and study a new type of quantum oracle, the quantum conditional oracle, which provides oracle access to the conditional probabilities associated with an underlying distribution. Amongst other properties, we (a) obtain highly efficient quantum algorithms for identity testing, equivalence testing and uniformity testing of probability distributions; (b) study the power of these oracles for testing properties of boolean functions, and obtain an algorithm for checking whether an n-input m-output boolean function is balanced or e-far from balanced; and (c) give an algorithm, requiring O˜(n/e) queries, for testing whether an n-dimensional quantum state is maximally mixed or not.