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SCIENTIA SINICA Informationis, Volume 47, Issue 10: 1277-1299(2017) https://doi.org/10.1360/N112017-00178

Quantum Computing

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  • ReceivedSep 11, 2017
  • AcceptedSep 18, 2017
  • PublishedOct 16, 2017

Abstract

Quantum computing exploits quantum mechanical properties to perform computations. It enables quantum parallelism and provides much more powerful data processing capabilities than classical computers. With a quantum computer, one can exponentially accelerate quantum system simulation and accelerate some important classical algorithms. Traditional quantum algorithms use unitary evolution to process information. A quantum computing process is the product of a series of unitary operators. In 1994, Shor invented a quantum prime factorization algorithm that exponentially accelerates the factorization of an integer. In 1996, Grover proposed a quantum search algorithm that accelerates the search of an unsorted database using square-root steps. Afterwards, the development of quantum algorithms slowed down, leading to a subsequent query by Shor in 2003 regarding why no more significant quantum algorithms have been found. Since 2009, many important new quantum algorithms have been proposed, such as a quantum algorithm for solving linear equations with the capability of exponential acceleration, a quantum algorithm for sparse Hamiltonian simulation using linear combinations of unitary operators, and a novel algorithm for Hamiltonian simulation, which provides exponential improvements in precision. In this paper, we first describe the basic principles of quantum computation. We then describe the Shor algorithm and Grover/Long search algorithm. These quantum algorithms are the general quantum algorithms that use unitary operations in their computational processes. Next, we introduce the basic principles of duality quantum computing, which was proposed in 2002. In contrast to traditional quantum computing, duality quantum computing allows the use of linear combinations of unitary operators in the computation process, meaning the multiplication, division, addition, and subtraction of unitary operators are all possible in duality quantum computing. Thus, duality quantum computing provides more flexibility for constructing quantum algorithms and the techniques used in classical algorithm design can be directly used to construct quantum algorithms. We then review recent work regarding newer quantum algorithms that enable the linear combination of unitary operators, which are actually duality quantum algorithms. Additionally, a duality quantum simulation algorithm for open quantum systems is introduced. This algorithm not only reduces computational complexity, but also improves accuracy exponentially. Finally, a summary and the future prospects of quantum algorithms are provided.


Funded by

国家重点基础研究发展计划(973)(2011CB9216002)

国家重点研发计划(2017YFA0303700)

国家自然科学基金(91221205,11175094)


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  • Figure 1

    Geometrical illustration of Grover algorithm, modified from [36]

  • Figure 2

    (Color online) An illustration for a three-slits duality quantum computer. The input is entered from the second slits marked as 1, andthe input is divided into three sub waves by three slits of the middle screen. After the middle screen, the sub waves are performed individual operations in different slits.The output of the duality quantum computation is obtained from three slits on the right screen, and the outputs at different slits correspond to different quantum calculating results

  • Figure 3

    (Color online) The multi-output duality quantum computing circuit. $|\Psi\rangle$ denotes the initial state of work qubit, and $|0\rangle$ is the initial state of the controlling auxiliary qudit

  • Figure 4

    (Color online) Quantum circuit for the BCCKS algorithm in duality quantum computing. Part A is the quantum circuit of duality computing. $|\Psi\rangle$ is the initial state of duality quantum computer and there are $ K $ auxiliary controlling qubits $|0\rangle$ and $ K $ auxiliary controlling qudits $ |0\rangle_{L}$with $ L$ energy level system . Part B is to illustrate that each unitary operation $U_{0}$ is composed of $ H_{1}, H_{2}, \ldots, H_{L-1}, H_{L}$. The unitary operations $U_{0}$ are activated only when the $ L$ level $ |0\rangle_{L}$ auxiliary controlling qudits hold the values indicated in respective circles

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