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Chapman & Hall/CRC eBooks

A Realization Scheme for Quantum Multi-Object Search

Zijian Diao, Goong Chen, Peter J.-S. Shiue

Published Sep 19, 2007• 0 citations
Publisher

Abstract

Abstract We study the quantum circuit design using 1-bit and 2-bit unitary gates for the iterations of the multi-object quantum search algorithm. The oracle block is designed in order to efficiently implement any sign-flipping operations. A chief ingredient in the design is the permutation operator which maps a set of search targets to another set on which the sign-flipping operation can be easily done. Such a proposed algorithmic approach implicates a minimal symmetric group generation problem: how to generate elements of a symmetric group using the smallest number of concatenations with a set of given generators. For the general case, this is an open problem. We indicate how the complexity issues depend on the solution of this problem through simple examples.

DOI
10.1201/9781584889007-8

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Abstract We study the quantum circuit design using 1-bit and 2-bit unitary gates for the iterations of the multi-object quantum search algorithm. The oracle block is designed in order to efficiently implement any sign-flipping operations. A chief ingredient in the design is the permutation operator which maps a set of search targets to another set on which the sign-flipping operation can be easily done. Such a proposed algorithmic approach implicates a minimal symmetric group generation problem: how to generate elements of a symmetric group using the smallest number of concatenations with a set of given generators. For the general case, this is an open problem. We indicate how the complexity issues depend on the solution of this problem through simple examples.
We present a circuit design realizing Grover’s algorithm based on 1-bit unitary gates and 2-bit quantum phase gates implementable with cavity QED techniques. In the first step, we express the circuit block which performs a key unitary transformation that flips only the sign of the state |11 · · · 11〉 using 1-bit and 2-bit gates. The Grover’s iteration operator can then be constructed using this key unitary transformation twice, plus other operations involving only 1-bit unitary gates on each qubit. Mathematical proofs are given to justify that the cricuiting satisfies the desired operator properties.
We observe an enormous increase in the computational power of digital computers. This was due to the revolution in manufacturing processes and controlling semiconductor structures on submicron scale, ultimately leading to the control of individual atoms. Eventually, the classical electric circuits encountered the barrier of quantum mechanics and its effects. However, the laws of quantum mechanics can be also used to produce computational devices that lead to extraordinary speed increases over classical computers. Thus quantum computing becomes a very promising and attractive research area. The Computer Aided Design for Quantum circuits becomes an essential ingredient for such emerging research which may lead to these powerful computers to be realized—an era of Quantum computing. This thesis presents an integrated theoretical study of software algorithms to design circuits of quantum oracles as well as methods for designing quantum oracles for Grover algorithm to solve combinatorial problems. An implementation of quantum algorithm involves the initialization of the input state and its manipulation with quantum gates followed by the measurements. In Grover algorithm the problem to be solved is specified by a permutative logic oracle – the fundamental problem is then how to build this oracle from quantum logic circuits and how to optimize these circuits. These problems are NP-hard and require search algorithms. In future, the search will be also done in quantum and this thesis leads to quantum algorithms to design quantum circuits more efficiently.
Several prominent quantum computing algorithms—including Grover’s search algorithm and Shor’s algorithm for finding the prime factorization of an integer—employ subcircuits termed ‘oracles’ that embed a specific instance of a mathematical function into a corresponding bijective function that is then realized as a quantum circuit representation. Designing oracles, and particularly, designing them to be optimized for a particular use case, can be a non-trivial task. For example, the challenge of implementing quantum circuits in the current era of NISQ-based quantum computers generally dictates that they should be designed with a minimal number of qubits, as larger qubit counts increase the likelihood that computations will fail due to one or more of the qubits decohering. However, some quantum circuits require that function domain values be preserved, which can preclude using the minimal number of qubits in the oracle circuit. Thus, quantum oracles must be designed with a particular application in mind. In this work, we present two methods for automatic quantum oracle synthesis. One of these methods uses a minimal number of qubits, while the other preserves the function domain values while also minimizing the overall required number of qubits. For each method, we describe known quantum circuit use cases, and illustrate implementation using an automated quantum compilation and optimization tool to synthesize oracles for a set of benchmark functions; we can then compare the methods with metrics including required qubit count and quantum circuit complexity.
The search for information in a system has been a continuous problem for a computer. This has resulted in the construction of a set of classical algorithms that can search for a set of data. This is why search systems can be divided into the type of information being searched, the number of solutions to find, and even the terms used for searching. With the emergence of quantum computing, new algorithms have been generated for this type of process. An example is the Grover algorithm, which performs theoretically better than traditional algorithms. This is why there has been research on optimizing it, applying it to new fields, and making it more accessible to industry users. Even if the algorithm is a promising alternative, one of the disadvantages of Grover’s algorithm is the use of an oracle function that must be generated for every set of search data. This review describes three sets of methodologies for generating quantum circuits that can be applied to constructing this oracle quantum circuit.
In a quantum computation, some algorithms use oracles (black boxes) for abstract computational objects. This paper presents an example for organizing Grover's quantum oracle by synthesizing several unitary gates such as CNOT gates, Toffoli gates, and Hadamard gates. As an example, we show a concrete quantum circuit for the knapsack problem, which belongs to the class of NP-complete problems. The time complexity of an oracle for the knapsack problem is estimated to be O(n2), where n is the number of variables. And the same order is obtained for space complexity.