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Quantum Information Processing

Simple exact quantum search

Raj Alexandru Guţoiu, Andrei Tănăsescu, Pantelimon George Popescu

Published Oct 18, 2024• 2 citations
Publisher

Abstract

Abstract While Grover’s search algorithm is asymptotically optimal, it does not always result in a real solution. If the search fails, the algorithm must be ran again from the beginning, conditionally doubling the effective number of oracle calls. Previous research attempted to fix this issue by modifying the oracle or alternating between numerically optimized reflectors. In this paper, we present an optimal initial state and reflector that produce an exact search with Grover’s algorithm at the cost of at most one additional oracle call beyond the optimum, a cost which can be nullified if we know a non-solution. We do this without modifying the oracle, without changing the diffuser at each step and even without any numerical optimization procedure required.

DOI
10.1007/s11128-024-04548-x

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Abstract While Grover’s search algorithm is asymptotically optimal, it does not always result in a real solution. If the search fails, the algorithm must be ran again from the beginning, conditionally doubling the effective number of oracle calls. Previous research attempted to fix this issue by modifying the oracle or alternating between numerically optimized reflectors. In this paper, we present an optimal initial state and reflector that produce an exact search with Grover’s algorithm at the cost of at most one additional oracle call beyond the optimum, a cost which can be nullified if we know a non-solution. We do this without modifying the oracle, without changing the diffuser at each step and even without any numerical optimization procedure required.
Grover's search algorithm drives a quantum system from an initial state $|s\ensuremath{\rangle}$ to a desired final state $|t\ensuremath{\rangle}$ by using selective phase inversions of these two states. Earlier, we studied a generalization of Grover's algorithm that relaxes the assumption of the efficient implementation of Is{I}_{s}Is​, the selective phase inversion of the initial state, also known as a diffusion operator. This assumption is known to become a serious handicap in cases of physical interest. Our general search algorithm works with almost any diffusion operator Ds{D}_{s}Ds​ with the only restriction of having $|s\ensuremath{\rangle}$ as one of its eigenstates. The price that we pay for using any operator is an increase in the number of oracle queries by a factor of O(B)O(B)O(B), where BBB is a characteristic of the eigenspectrum of Ds{D}_{s}Ds​ and can be large in some situations. Here we show that by using a quantum Fourier transform, we can regain the optimal query complexity of Grover's algorithm without losing the freedom of using any diffusion operator for quantum searching. However, the total number of operators required by the algorithm is still O(B)O(B)O(B) times more than that of Grover's algorithm. So our algorithm offers an advantage only if the oracle operator is computationally more expensive than the diffusion operator, which is true in most search problems.
The success probability of a search of MMM targets from a database of size NNN, using Grover's search algorithm depends critically on the number of iterations of the composite operation of the oracle followed by Grover's diffusion operation. Although the required number of iterations scales as O(N)\mathcal{O}(\sqrt{N})O(N​) for large NNN, the asymptote is not a good indicator of the optimal number of iterations when M/N\sqrt{M/N}M/N​ is not small. A scheme for the determination of the exact number of iterations, subject to a threshold set for the success probability of the search (probability of detecting the target state(s)), is crucial for the efficacy of the algorithm. In this work, a general scheme for the construction of nnn-qubit Grover's search algorithm with 1≤M≤N1 \leq M \leq N1≤M≤N target states is presented, along with the procedure to find the optimal number of iterations for a successful search. It is also shown that for given NNN and MMM, there is an upper-bound on the success probability of the algorithm.
Quantum search is among the most important algorithms in quantum computing. At its core is quantum amplitude amplification, a technique that achieves a quadratic speedup over classical search by combining two global reflections: the oracle, which marks the target, and the diffusion operator, which reflects about the initial state. We show that this speedup can be preserved when the oracle is the only global operator, with all other operations acting locally on non-overlapping partitions of the search register. We present a recursive construction that, when the initial and target states both decompose as tensor products over these chosen partitions, admits an exact closed-form solution for the algorithm's dynamics. This is enabled by an intriguing degeneracy in the principal angles between successive reflections, which collapse to just two distinct values governed by a single recursively defined angle. Applied to unstructured search, a problem that naturally satisfies the tensor decomposition, the approach retains the O(N)O(\sqrt{N})O(N​) oracle complexity of Grover search when each partition contains at least log⁡2(log⁡2N)\log_2(\log_2 N)log2​(log2​N) qubits. On an 18-qubit search problem, partitioning into two stages reduces the non-oracle circuit depth by as much as 51%-96% relative to Grover, requiring up to 9% additional oracle calls. For larger problem sizes this oracle overhead rapidly diminishes, and valuable depth reductions persist when the oracle circuit is substantially deeper than the diffusion operator. More broadly, these results show that a global diffusion operator is not necessary to achieve the quadratic speedup in quantum search, offering a new perspective on this foundational algorithm. Moreover, the scalar reduction at the heart of our analysis inspires and motivates new directions and innovations in quantum algorithm design and evaluation.
A quantum computer encodes information in quantum states and runs quantum algorithms to surpass the classical counterparts by exploiting quantum superposition and quantum correlation. Grover's quantum search algorithm is a typical quantum algorithm that proves the superiority of quantum computing over classical computing. It has a quadratic reduction in the query complexity of a database search and is known to be optimal when no a priori information about the elements of the database is provided. In this work we consider a probabilistic Grover search algorithm allowing nonzero probability of failure for a database with a general a priori probability distribution of the elements and minimize the number of oracle calls by optimizing the initial state of the quantum system and the reflection axis of the diffusion operator. The initial state and the reflection axis are allowed to not coincide and thus the quantum search algorithm rotates the quantum system in a three-dimensional subspace spanned by the initial state, the reflection axis, and the search target state in general. The number of oracle calls is minimized by a variational method, and formal results are obtained with the assumption of low failure probability. The results show that for a nonuniform a priori distribution of the database elements, the number of oracle calls can be significantly reduced given a small decrease in the success probability of the quantum search algorithm, leading to a lower average query complexity to find the solution of the search problem. The results are applied to a simple but nontrivial database model with two-value a priori probabilities to show the power of the optimized quantum search algorithm. The paper concludes with a discussion about the generalization to higher-order results that allows for a larger failure probability for the quantum search algorithm.
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.
While powerful tools have been developed to analyze quantum query complexity, there are still many natural problems that do not fit neatly into the black box model of oracles. We create a new model that allows multiple oracles with differing costs. This model captures more of the difficulty of certain natural problems. We test this model on a simple problem, Search with Two Oracles, for which we create a quantum algorithm that we prove is asymptotically optimal. We further give some evidence, using a geometric picture of Grover's algorithm, that our algorithm is exactly optimal.