Showing papers by "Marek Perkowski published in 2011"

Improved Complexity of Quantum Oracles for Ternary Grover Algorithm for Graph Coloring

Abstract: The paper presents a generalization of the well-known Grover Algorithm to operate on ternary quantum circuits. We compare complexity of oracles and some of their commonly used components for binary and ternary cases and various sizes and densities of colored graphs. We show that ternary encoding leads to quantum circuits that have significantly less qud its and lower quantum costs. In case of serial realization of quantum computers, our ternary algorithms and circuits are also faster.

Synthesis of Quantum Circuits in Linear Nearest Neighbor Model Using Positive Davio Lattices

Abstract: We present a logic synthesis method based on lattices that realize quan- tum arrays in One-Dimensional Ion Trap technology. This means that all gates are built from 2x2 quantum primitives that are located only on neighbor qubits in a one- dimensional space (called also vector of qubits or Linear Nearest Neighbor (LNN) architecture). The Logic circuits designed by the proposed method are realized only with 3*3 Toffoli, Feynman and NOT quantum gates and the usage of the commonly used multi-input Toffoli gates is avoided. This realizatio n method of quantum cir- cuits is different from most of reversible circuits synthes is methods from the literature that use only high level quantum cost based on the number of quantum gates. Our synthesis approach applies to both standard and LNN quantum cost models. It leads to entirely new CAD algorithms for circuit synthesis and substantially decreases the quantum cost for LNN quantum circuits. The drawback of synthesizing circuits in the presented LNN architecture is the addition of ancilla qubits.

Synthesis of Reversible Synchronous Counters

Abstract: Reversible logic is very important in low-power circuit design and quantum computing. Though a significant number of works has been done on reversible combinational logic synthesis, only few papers have been published on reversible sequential logic synthesis and per mutative quantum automata. The reported works on reversible sequential logic discuss designs of reversible flip-flops and suggest synthesizing reversible sequential circuits by replacing the flip-flops and combinational parts of traditional sequential circuit designs by their reversible counterparts. In this paper, we discuss direct design of reversible synchronous counters based on positive polarity Reed-Muller expressions. Design results show that the direct design method is more efficient than the replacement method. The method can be also applied to per mutative quantum automata that have quantum memories external to the circuit.

Quantum Phase Estimation Using Multivalued Logic

Abstract: Quantum phase estimation (QPE) is one of the most important quantum algorithms which is used as a subroutine for other important quantum algorithms like Shor's factoring algorithm, simulation of quantum systems, quantum counting and QFT on arbitrary Zp. In this paper we develop the theoretical framework for the multivalued quantum logic version of the QPE algorithm using d valued qudits and show a quantum circuit to implement QPE with a complexity of O(nlogn) single qudit operations. The multivalued QPE algorithm, when compared to the binary quantum logic version, turns out to be more robust and leads to a significant decrease in the number of qudits required along with drastic improvement in the precision and success probability. We derive the requirements to amplify the probability of success to a value very close to 1 (for a given precision), thereby generalizing the previously obtained result in the binary case. Also, we note that the failure probability of QPE algorithm decreases exponentially as d increases.

Evolutionary Quantum Logic Synthesis of Boolean Reversible Logic Circuits Embedded in Ternary Quantum Space using Heuristics

Abstract: It has been experimentally proven that realizing universal quantum gates using higher-radices logic is practically and technologically possible. We developed a Parallel Genetic Algorithm that synthesizes Boolean reversible circuits realized with a variety of quantum gates on qudits with various radices. In order to allow synthesizing circuits of medium sizes in the higher radix quantum space we performed the experiments using a GPU accelerated Genetic Algorithm. Using the accelerated GA we compare heuristic improvements to the mutation process based on cost minimization, on the adaptive cost of the primitives and improvements due to Baldwinian vs. Lamarckian GA. We also describe various fitness function formulations that allowed for various realizations of well known universal Boolean reversible or quantum-probabilistic circuits.

Reversible Function Synthesis of Large Reversible Functions with No Ancillary Bits Using Covering Set Partitions

Abstract: This paper presents a synthesis algorithm, Covering Set Partitions (CSP), for reversible binary functions with no ancillary (garbage) bits. Existing algorithms are constrained to functions of small number of variables because they store the entire truth table of 2n terms in memory or require a huge amount of time to yield results because they must calculate all possible permutations of an input vector. In contrast, the CSP algorithm harnesses the natural mathematical properties of binary numbers, partially ordered sets and covering graph theory, to construct input vectors which are guaranteed to produce valid results. A randomly selected subset of all valid input vectors are processed where the best input vector sequence wins. The CSP algorithm is capable of synthesizing functions of large number of variables (30 bits) in a reasonable amount of time.

Decomposition of Reversible Logic Function Based on Cube-Reordering

Abstract: We present a novel approach to the synthesis of incompletely specified reversible logic functions. The method is based on cube grouping; the first step of the synthesis method analyzes the logic function and generates groupings of same cubes in such a manner that multiple sub-functions are realized by a single Toffoli gate. This process also reorders the function in such a manner that not only groups of similarly defined cubes are joined together but also don't care cubes. T he proposed method is verified on standard benchmarks for both reversible and irre versible logic functions. The obtained results show that for functions with a significa nt portion of don't cares the proposed method outperforms previously proposed synthesis methods.

Application of Genetic Algorithm for Synthesis of Large Reversible Circuits using Covered Set Partitions

Abstract: We present the results of application of Evolutionary Algorithms to the problem of synthesizing quantum circuits which belong to the class of reversible circuits, represented as an input/output mapping vectors. The paper specifically focuses on large quantum circuits where many valid solutions exist in an exponentially inflating search space. Valid solutions represent the set of all input vector permutations (arrangements) which satisfy the circuit specification. The search space for circuits with large number of variables grows exponentially making it impossible to discover the set of optimal solutions. The paper compares three methods for selecting valid solutions of input vector sequences: 1) randomly, 2) genetic algorithm, 3) Tabu search. The objective function calculates the number of elementary quantum gates needed to represent the solution such that lower number of gates represents better solutions. In addition to the choice of selection algorithm, we illustrate the impact of using different partition depths for the Covered Set Partitions algorithm used to construct valid input vector sequences.

Image processing and machine learning for the diagnosis of melanoma cancer

Abstract: Melanoma cancer is one of the most dangerous and potentially deadly types of skin cancer; however, if diagnosed early, it is nearly one-hundred percent curable (UnderstMel09). Here we propose an efficient system which helps with the early diagnosis of melanoma cancer. Different image processing techniques and machine learning algorithms are evaluated to distinguish between cancerous and non-cancerous moles. Two image feature databases were created: one compiled from a dermatologist-training tool for melanoma from Hosei University and the other created by extracting features from digital pictures of lesions using a software called Skinseg. We then applied various machine learning techniques on the image feature database using a Python-based tool called Orange. The experiments suggest that among the methods tested, the combination of Bayes machine learning with Hosei image feature extraction is the best method for detecting cancerous moles. Then, using this method, a computer tool was developed to return the probability that an image is cancerous. This is a very practical application as it allows for at-home findings of the probability that a mole is cancerous. This does not replace visits to a doctor, but provides early information that allows people to be proactive in the diagnosis of melanoma cancer.

Comparison of Influence of Two Data-Encoding Methods for Grover Algorithm on Quantum Costs

Abstract: It is important to be able to calculate realistic estimates of quantum costs for real oracles used in quantum algorithms. In this paper, we compare Perkowski's[1] oracle data encoding method with Hogg's[2] encoding method for Grover algorithm[3], to examine the decrease in Oracle gate cost, if any, for four common constraint satisfaction problems: Graph coloring, Satisfiability, Send-More-Money and Max Clique.