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arXiv:quant-ph/9904049 · Physical Review A

Generalized quantum search with parallelism

Robert M. Gingrich, Colin P. Williams, Nicolas J. Cerf

Published Apr 19, 2000• 41 citations
arXivHTMLPDF

Abstract

We generalize Grover's unstructured quantum search algorithm to enable it to work with arbitrary starting superpositions and arbitrary unitary operators. We show that the generalized quantum search algorithm, when cast in a special orthonormal basis, can be understood as performing an exact rotation of a starting superposition into a target superposition. We derive a formula for the success probability of the generalized quantum search algorithm after n rounds of amplitude amplification. We then use this formula to determine the optimal strategy for a punctuated quantum search algorithm, i.e., one in which the amplitude amplified state is observed before the point of maximum success probability. On average, the optimal strategy is about 12% better than the naive use of Grover's algorithm. The speedup obtained is not dramatic but it illustrates that a hybrid use of quantum computing and classical computing techniques can yield a performance that is better than either alone. In addition, we show that a punctuated quantum algorithm that takes the same average computation time as Grover's standard algorithm only requires half the coherence time. We then extend the analysis to the case of a society of k quantum searches acting in parallel. We derive an analytic formula that connects the degree of parallelism with the expected computation time for k-parallel quantum search. The resulting parallel speedup scales as $O(\sqrt{k}),$ while the minimum number of agents needed to ensure success, k, decreases as the inverse of the square of the achievable coherence time. This result has practical significance for the design of rudimentary quantum computers that are likely to have a limited coherence time.

DOI
10.1103/physreva.61.052313

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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.

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In chapter 2 various parameterizations for the orbits under local unitary transformations of three-qubit pure states are analyzed. It is shown that the entanglement monotones of any multipartite pure state uniquely determine the orbit of that state. It follows that there must be an entanglement monotone for three-qubit pure states which depends on the Kempe invariant defined in [1]. A form for such an entanglement monotone is proposed. A theorem is proved that significantly reduces the number of entanglement monotones that must be looked at to find the maximal probability of transforming one multipartite state to another. In chapter 3 Grover's unstructured quantum search algorithm is generalized to use an arbitrary starting superposition and an arbitrary unitary matrix. A formula for the probability of the generalized Grover's algorithm succeeding after n iterations is derived. This formula is used to determine the optimal strategy for using the unstructured quantum search algorithm. The speedup obtained illustrates that a hybrid use of quantum computing and classical computing techniques can yield a performance that is better than either alone. The analysis is extended to the case of a society of k quantum searches acting in parallel. In chapter 4 the positive map Г : p → (Trρ) - ρ is introduced as a separability criterion. Any separable state is mapped by the tensor product of Г and the identity in to a non-negative operator, which provides a necessary condition for separability. If Г acts on a two-dimensional subsystem, then it is equivalent to partial transposition and therefore also sufficient for 2 x 2 and 2 x 3 systems. Finally, a connection between this map for two qubits and complex conjugation in the magic basis [2] is displayed.

•Sep 1, 2002•0 citations
We generalize Grover's unstructured quantum search algorithm to enable it to work with arbitrary starting superpositions and arbitrary unitary operators. We show that the generalized quantum search algorithm, when cast in a special orthonormal basis, can be understood as performing an exact rotation of a starting superposition into a target superposition. We derive a formula for the success probability of the generalized quantum search algorithm after n rounds of amplitude amplification. We then use this formula to determine the optimal strategy for a punctuated quantum search algorithm, i.e., one in which the amplitude amplified state is observed before the point of maximum success probability. On average, the optimal strategy is about 12% better than the naive use of Grover's algorithm. The speedup obtained is not dramatic but it illustrates that a hybrid use of quantum computing and classical computing techniques can yield a performance that is better than either alone. In addition, we show that a punctuated quantum algorithm that takes the same average computation time as Grover's standard algorithm only requires half the coherence time. We then extend the analysis to the case of a society of k quantum searches acting in parallel. We derive an analytic formula that connects the degree of parallelism with the expected computation time for k-parallel quantum search. The resulting parallel speedup scales as O(k),O(\sqrt{k}),O(k​), while the minimum number of agents needed to ensure success, k, decreases as the inverse of the square of the achievable coherence time. This result has practical significance for the design of rudimentary quantum computers that are likely to have a limited coherence time.
Grover's algorithm provides a quadratic speedup over classical algorithms for searching an unstructured database and is known to be strictly optimal in oracle query complexity, with tight bounds on its success probability. Although the standard Grover search cannot be further accelerated in the full-search setting, a trade-off between accuracy and query complexity gives rise to the partial search problem. The Grover-Radhakrishnan-Korepin (GRK) algorithm is the standard and most extensively studied protocol for this task. In this work, we provide systematic numerical evidence that the GRK operator sequence gives the highest success probability in all examined cases, supporting it as the optimal ansatz among admissible compositions of global and local Grover operators. Guided by this numerically supported GRK ansatz, we derive an asymptotically tight upper bound on the maximal success probability within the GRK family and establish the corresponding lower bound on the minimal expected number of oracle queries. Furthermore, we investigate parallel quantum search within the partial-search framework. While a direct GRK-based parallelization does not outperform established parallel Grover schemes, we demonstrate that a hybrid strategy combining partial and full search protocols yields a strict, though subleading, improvement over the outer parallel Grover scheme. Our results clarify the fundamental limits of quantum partial search and its role in optimizing parallel quantum search algorithms.
Search-base algorithms have widespread applications in different scenarios. Grover's quantum search algorithms and its generalization, amplitude amplification, provide a quadratic speedup over classical search algorithms for unstructured search. We consider the problem of searching with prior knowledge. More preciously, search for the solution among N items with a prior probability distribution. This letter proposes a new generalization of Grover's search algorithm which performs better than the standard Grover algorithm in average under this setting. We prove that our new algorithm achieves the optimal expected success probability of finding the solution if the number of queries is fixed.
Background: The future computations need parallelism, and a way to achieve the same is through quantum-based specific properties. Quantum computation supports exponential operation to be performed in parallel over a single execution step and hence achieves computational speedup. Objective: One of the quantum-based algorithms allows to search the key element over an index based unstructured database and succeeds to obtain the speedup. In the whole context of article writing, our approach reviews the significance of quantum computing, the basics required for quantum computation and their quantum logic-based circuit operations. Method: The article focuses on Grover’s search algorithm and its variations, and then illustrates the relevant amplitude amplification & phase matching techniques in accordance with the advantages and limitation to the specific perspective. Results: We made the comparative analysis between the amplitude amplification-based static & dynamic Grover’s searching and phase estimation technique. Along with this, we justified the possibilities of the empirical results and observed facts over Grover’s variant approaches. Conclusion: Finally, we conclude the analytical comparison between quantum-based searching techniques along with the applicability of Grover’s algorithm in practical computational science.
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.
We present an optimized adiabatic quantum schedule for unstructured search building on the original approach of Roland and Cerf [Phys. Rev. A 65, 042308 (2002)]. Our schedule adiabatically varies the Hamiltonian even more rapidly at the endpoints of its evolution, preserving Grover's well-known quadratic quantum speedup. In the errorless adiabatic limit, the probability of successfully obtaining the marked state from a measurement increases directly proportional to time, suggesting efficient parallelization. Numerical simulations of an appropriate reduced two-dimensional Schrödinger system confirm adiabaticity while demonstrating superior performance in terms of probability compared to existing adiabatic algorithms and Grover's algorithm, benefiting applications with possible premature termination. We introduce a protocol that ensures a marked-state probability at least ppp in time of order N(1+p/ε)\sqrt{N}(1+p/\varepsilon)N​(1+p/ε), and analyze its implications for realistic bounded-resource scenarios. Our findings suggest that quantum advantage may still be achievable under constrained coherence times (where other algorithms fail), provided the hardware allows for them to be sufficiently long.
In chapter 2 various parameterizations for the orbits under local unitary transformations of three-qubit pure states are analyzed. It is shown that the entanglement monotones of any multipartite pure state uniquely determine the orbit of that state. It follows that there must be an entanglement monotone for three-qubit pure states which depends on the Kempe invariant defined in [1]. A form for such an entanglement monotone is proposed. A theorem is proved that significantly reduces the number of entanglement monotones that must be looked at to find the maximal probability of transforming one multipartite state to another. In chapter 3 Grover's unstructured quantum search algorithm is generalized to use an arbitrary starting superposition and an arbitrary unitary matrix. A formula for the probability of the generalized Grover's algorithm succeeding after n iterations is derived. This formula is used to determine the optimal strategy for using the unstructured quantum search algorithm. The speedup obtained illustrates that a hybrid use of quantum computing and classical computing techniques can yield a performance that is better than either alone. The analysis is extended to the case of a society of k quantum searches acting in parallel. In chapter 4 the positive map Г : p → (Trρ) - ρ is introduced as a separability criterion. Any separable state is mapped by the tensor product of Г and the identity in to a non-negative operator, which provides a necessary condition for separability. If Г acts on a two-dimensional subsystem, then it is equivalent to partial transposition and therefore also sufficient for 2 x 2 and 2 x 3 systems. Finally, a connection between this map for two qubits and complex conjugation in the magic basis [2] is displayed.