A general consideration to the relation between the general SU(2) formulation for quantum searching with certainty and its corresponding Hamiltonian is undertaken in this work. We have derived the general Hamiltonian from the SU(2) formulation and reduced it to all known search Hamiltonians. A quantum mechanical algorithm is a complete and straightforward set of directions, based on the quntum physical process which possess the unique features of superpositions and entanglement, to accomplish a computation in a finite number of steps. Each step in the algorithm is designed by a sequence of unitary operators that transform the initial state to a desired one. We usually consider that these transformations are composed of several universal quantum logic gates. Therefore, this way to think about the construction of a quantum mechanical algorithm is called the quantum circuit model for quantum computing. Famous examples include the quantum factorizing algorithm discorvered by Shor[1] and the quantum searching algorithm well developed by Grover[2]. The latter is what we intend to discuss in this work.
A general consideration to the relation between the general SU(2) formulation for quantum searching with certainty and its corresponding Hamiltonian is undertaken in this work. We have derived the general Hamiltonian from the SU(2) formulation and reduced it to all known search Hamiltonians. A quantum mechanical algorithm is a complete and straightforward set of directions, based on the quntum physical process which possess the unique features of superpositions and entanglement, to accomplish a computation in a finite number of steps. Each step in the algorithm is designed by a sequence of unitary operators that transform the initial state to a desired one. We usually consider that these transformations are composed of several universal quantum logic gates. Therefore, this way to think about the construction of a quantum mechanical algorithm is called the quantum circuit model for quantum computing. Famous examples include the quantum factorizing algorithm discorvered by Shor[1] and the quantum searching algorithm well developed by Grover[2]. The latter is what we intend to discuss in this work.
A general quantum search algorithm with arbitrary unitary transformations and an arbitrary initial state is considered in this work. To search a marked state with certainty, we have derived, using an SU(2) representation: (1) the matching condition relating the phase rotations in the algorithm, (2) a concise formula for evaluating the required number of iterations for the search, and (3) the final state after the search, with a complex phase in its amplitude. Moreover, the optimal choices and modifications of the phase angles in the Grover kernel are also studied.
IntroductionFor a search problem associated with a unsorted database, the remarkable Grover's quantum algorithm (1) provides a quadratic speedup over its classical counterpart.The search problem can be described as follows: for a given function f , there exists one unknown element in the set {0, 1, ..., N -1} that satisfies f (x)=-1, say x = τ, whereas the other N -1o n e sg i v e f (x)=1.How many times of evaluations of f are required to determine the element τ for f (τ)=-1?Through a conventional algorithm, one needs O(N) trials to achieve this aim.How about the utility of quantum algorithm for the search?For the scenario in the quantum world, the search problem can be rephrased in the quantum mechanical language: for a given unitary operator I τ , that is sometimes called the oracle operator, and a set of state vectors (orthonormal basis): s = {|0 , |1 , ..., |N -1 }, I τ |x = |x for all states in the set except I τ |x = -|x for x = τ.H owma n yqueriesofI τ are required to determine |τ ?By Grover's algorithm, one needs only O( √ N) quantum mechanical steps to find the marked state |τ out.It has been shown that Grover's algorithm is optimal since it needs minimal oracle calls to perform a quantum search (2).The quantum searching process will be briefly reviewed as follows.The first step of Grover's algorithm is to prepare a superposition state of all elements with uniform probability amplitude:Then apply the Grover kernel G = -I η I τ to |s ,w h er eI η is a unitary operator and contains no bias against the marked state.For large N,a f t e ra b o u tm = π √ N/4 repetitions of G operations, the probability to observe |τ is close to one, i.e., G m |s ∼ |τ .( 2 ) Since every single G involves one query of I τ ,onlyO( √ N) searching steps are required for a quantum search task.In what follows, we will first investigate on the general SU(2) formulation for the kernel of Grover's searching operator G.The discussions of quantum searching certainty, robustness, and the analog analogue version of the Grover's algorithm will be given afterwards.
Models of Grover’s search algorithm is reviewed to build the foundation for the other algorithms. Thereafter, some preliminary modifications of the original algorithms by others are stated, that increases the applicability of the search procedure. A general quantum computation on an isolated system can be represented by a unitary matrix. In order to execute such a computation on a quantum computer, it is common to decompose the unitary into a quantum circuit, i.e., a sequence of quantum gates that can be physically implemented on a given architecture. There are different universal gate sets for quantum computation. Here we choose the universal gate set consisting of CNOT and single-qubit gates. We measure the cost of a circuit by the number of CNOT gates as they are usually more difficult to implement than single qubit gates and since the number of single-qubit gates is bounded by about twice the number of CNOT’s.
These notes begin in Chapter 1 with a review of linear algebra and the postulates of quantum mechanics, leading to an explanation of single- and multi-qubit gates. Chapter 2 explores the challenge of constructing arbitrary quantum states from given initial states, and introduces circuits for building oracles. Chapter 3 presents foundational algorithms such as entanglement creation, quantum teleportation, Deutsch-Jozsa, Bernstein-Vazirani, and Simon's algorithm. Chapters 4 and 5 cover algorithms based on the quantum Fourier transform, including phase estimation, period finding, factoring, and logarithm computation. These chapters also include complexity analysis and detailed quantum circuits suitable for implementation in code. Chapter 6 introduces Grover's algorithm for quantum search and amplitude amplification, including its realization via Hamiltonian simulation and a method for derandomization. Chapter 7 discusses basic techniques for Hamiltonian simulation, such as Lie-Trotter decomposition, sparse Hamiltonians, and the linear combination of unitaries. It also provides example circuits for simulating Hamiltonians expressed as linear combinations of Pauli operators. Chapter 8 introduces variational quantum algorithms, and Chapter 9 presents an algorithm for simulating fermionic many-particle systems, with an emphasis on molecular Hamiltonians. It also outlines the key transformations needed to map a molecular Hamiltonian to a form suitable for simulation on a quantum computer.
Summary form only given, as follows. Gives a system theoretic approach to the concept of quantum computing. A quantum computer is envisaged to be a system of quantum circuits, acting on a state space, which is a finite-say 2/sup n/-dimensional-complex Hilbert space. The circuitry is a sequence of unitary transforms U/sub t/ /spl epsiv/ SU(2/sup n/) followed by a measurement. These transforms, so-called quantum gates, are controlled by a classical computer, usually especially electromagnetic fields. The state space of a quantum computer has the structure of a Hermitian vector space. Thus it allows "simultaneous" superposition of orthogonal basis states (corresponding to classical states) with the possibility of constructive and destructive interference between different paths of computation. This principle allows the usage of so-called entangled states by preparing the superposition of special "bent" configurations of basis vectors in the product Hilbert space, e.g., as they are known from error control and cryptography. This latter "entanglement" not only promises to make a quantum computer much more powerful than a probabilistic one, but it also allows a method of parallelism. The reason for this lies not only in the fact that the 2/sup n/ dimensional Hilbert space is the n-fold tensor product of 2-dimensional spaces as it is given by spin-1/2-particles, like photons, representing a quantum bit. First of all, we shall address the fact that the power of quantum computing lies in the properties of entangled states as opposed to those of separable states which essentially would resemble classical circuits without giving the feature of exponential speed-up. We shall briefly describe possible physical realisations of quantum (optic, electronic, magnetic,...) devices allowing the engineering of Hamiltonians needed for quantum systems. Basic quantum gates, especially the quantum Horner-Toffoli gate will be discussed to display the principles and methodology for designing quantum circuits. An outlook towards the possible availability of nano- and mesoscopic technologies supporting this new architecture in future generations concludes based on a speculative worst/best case forecast of possible applications, e.g., in public key and conventional cryptography.
We recast Grover's generalised search algorithm in a geometric language even when the states are not approximately orthogonal. We provide a possible search algorithm based on an arbitrary unitary transformation which can speed up the steps still further. We discuss the lower and upper bounds on the transition matrix elements when the unitary operator changes with time, thereby implying that quantum search process can not be too fast or too slow. This is a remarkable feature of quantum computation unlike classical one. Quantum mechanical uncertainty relation puts bounds on search process. Also we mention the problems of perturbation and other issues in time-dependent search operation.