Phase matching in Grover's algorithm

Li, Panchi; Li, Shiyong

doi:10.1016/j.physleta.2007.02.029

Cited by 28 publications

(44 citation statements)

References 8 publications

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“…Several results [10,11] improve the performance of fixed-point algorithms on wide ranges of M=N; however, these algorithms are numerical, and as such, their time scaling cannot be assessed. Indeed, the π=3 algorithm was shown to be optimal in time [12], ostensibly proving it impossible to find a search algorithm that both avoids the soufflé problem and provides a quantum advantage.…”

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confidence: 99%

Fixed-Point Quantum Search with an Optimal Number of Queries

2014

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Grover's quantum search and its generalization, quantum amplitude amplification, provide a quadratic advantage over classical algorithms for a diverse set of tasks but are tricky to use without knowing beforehand what fraction λ of the initial state is comprised of the target states. In contrast, fixed-point search algorithms need only a reliable lower bound on this fraction but, as a consequence, lose the very quadratic advantage that makes Grover's algorithm so appealing. Here we provide the first version of amplitude amplification that achieves fixed-point behavior without sacrificing the quantum speedup. Our result incorporates an adjustable bound on the failure probability and, for a given number of oracle queries, guarantees that this bound is satisfied over the broadest possible range of λ. Grover's quantum search algorithm [1] provides a quadratic speedup over classical algorithms for solving a broad class of problems. Included are the many important, yet computationally prohibitive nondeterministic polynomial time (NP) problems [2], which can always be solved, albeit inefficiently, by searching the space of possible solutions. Because the problem Grover's algorithm solves is so simple to understand-given an oracle function that recognizes marked items, locate one of M such marked items among N unsorted items-its classical time complexity OðN=MÞ is obvious, making the quantum speedup that much more conclusive.Conceptually also, Grover's algorithm is compellingthe iterative application of the oracle and initial state preparation rotates from a superposition of mostly unmarked states to a superposition of mostly marked states in just Oð ffiffiffiffiffiffiffiffiffiffi ffi N=M p Þ steps [3]. This interpretation of Grover's algorithm as a rotation is very natural because the Grover iterate is a unitary operator. However, this same unitarity is also a weakness. Without knowing exactly how many marked items there are, there is no knowing when to stop the iteration. This leads to the soufflé problem [4], in which iterating too little "undercooks" the state, leaving mostly unmarked states, and iterating too much "overcooks" the state, passing by the marked states and leaving us again with mostly unmarked states.The most direct solution of the soufflé problem is to estimate M by either using full-blown quantum counting [5,6] or a trial-and-error scheme where iterates are applied an exponentially increasing number of times [5,7]. Although scaling quantumly, these strategies are unappealing for search as they work best not by monotonically amplifying marked states, but rather by getting "close enough" before resorting to classical random sampling.An alternative approach, in line with what we advocate here, is to construct, either recursively or dissipatively, operators that avoid overcooking by always amplifying marked states. Such algorithms are known as fixed-point searches. For example, running Grover's π=3 algorithm [8] or the comparable ancilla algorithm [9] longer can only ever improve its success probability....

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confidence: 99%

Fixed-Point Quantum Search with an Optimal Number of Queries

2014

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“…Step 4: Prepare the initial state |ψ = H |0 , and perform on |ψ the Yoder's sequence l j=1 G (φ j , ϕ j ), where l = c k−1 is the number of iterations and the phases {φ j , ϕ j } satisfy the multiphase matching condition of Eq. (7). Step 5: Measure the system.…”

Section: Hybrid Fixed-point and Trail-and-error Quantum Searchmentioning

confidence: 99%

Quantum search for unknown number of target items by hybridizing fixed-point method with trail-and-error method*

Li¹,

Zhang²,

Fu³

et al. 2019

Chinese Phys. B

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For the unsorted database quantum search with the unknown fraction λ of target items, there are mainly two kinds of methods, i.e., fixed-point or trail-and-error. (i) In terms of the fixed-point method, Yoder et al. [Phys. Rev. Lett. 113, 210501 (2014)] claimed that the quadratic speedup over classical algorithms has been achieved. However, in this paper, we point out that this is not the case, because the query complexity of Yoder's algorithm iswhere λ0 is a known lower bound of λ.(ii) In terms of the trail-and-error method, currently the algorithm without randomness has to take more than 1 times queries or iterations than the algorithm with randomly selected parameters. For the above problems, we provide the first hybrid quantum search algorithm based on the fixed-point and trail-and-error methods, where the matched multiphase Grover operations are trialed multiple times and the number of iterations increases exponentially along with the number of trials. The upper bound of expected queries as well as the optimal parameters are derived. Compared with Yoder's algorithm, the query complexity of our algorithm indeed achieves the optimal scaling in λ for quantum search, which reconfirms the practicality of the fixed-point method. In addition, our algorithm also does not contain randomness, and compared with the existing deterministic algorithm, the query complexity can be reduced by about 1/3. Our work provides an new idea for the research on fixed-point and trial-and-error quantum search.

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“…From this matrix it is also possible to see that if the target state is changed, then the angles found through the variational algorithm will give the same probabilities. An arbitrary phase applied by the oracle was first proposed in [17] although only remarks regarding the use of an arbitrary phase to get higher probabilities for the searched item were done, afterwards several studies regarding the validity of replacing Grover's oracle and diffusion operator with an arbitrary phase version were made [18][19][20][21]. The main conclusion is that a phase matching condition is required.…”

Section: Problem I4 (Variational Oracle and Diffusion)mentioning

confidence: 99%

Variational learning of Grover's quantum search algorithm

2018

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Given a parameterized quantum circuit such that a certain setting of these real-valued parameters corresponds to Grover's celebrated search algorithm, can a variational algorithm recover these settings and hence learn Grover's algorithm? We studied several constrained variations of this problem and answered this question in the affirmative, with some caveats. Grover's quantum search algorithm is optimal up to a constant. The success probability of Grover's algorithm goes from unity for two-qubits, decreases for three-and four-qubits and returns near unity for five-qubits then oscillates ever-so-close to unity, reaching unity in the infinite qubit limit. The variationally approach employed here found an experimentally discernible improvement of 5.77% and 3.95% for three-and four-qubits respectively. Our findings are interesting as an extreme example of variational search, and illustrate the promise of using hybrid quantum classical approaches to improve quantum algorithms. This paper further demonstrates that to find optimal parameters one doesn't need to vary over a family of quantum circuits to find an optimal solution. This result looks promising and points out that there is a set of variational quantum problems with parameters that can be efficiently found on a classical computer for arbitrary number of qubits.Grover's algorithm [2] is one of the most celebrated quantum algorithms, enabling quantum computers to quadratically outperform classical computers at database search provided database access is restricted to a 'black box' -called the oracle model. In addition to the wide application scope of database search, Grover's algorithm has further applications as a subroutine used in a variety of other quantum algorithms.Variational hybrid quantum/classical algorithms have recently become an area of significant interest [3][4][5][6][7][8][9][10]. These algorithms have shown several advantages such as robustness to quantum errors and low coherence time requirements [11], which make them ideal for implementations in current quantum computer architectures. Here we take inspiration from algorithms such as the variational quantum eigensolver (VQE) [3] and the quantum approximate optimization algorithm (QAOA) [4]. The general procedure of these variational hybrid quantum/classical algorithms is the following:

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Phase matching in Grover's algorithm

Cited by 28 publications

References 8 publications

Fixed-Point Quantum Search with an Optimal Number of Queries

Fixed-Point Quantum Search with an Optimal Number of Queries

Quantum search for unknown number of target items by hybridizing fixed-point method with trail-and-error method*

Variational learning of Grover's quantum search algorithm

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