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arXiv:quant-ph/0703231

Quantum Search in an Ordered List via Adaptive Learning

Michael Ben-Or, Avinatan Hassidim

Published Mar 24, 2007• 12 citations
arXivHTMLPDF

Abstract

We use a Bayesian approach to optimally solve problems in noisy binary search. We deal with two variants: 1. Each comparison can be erroneous with some probability $1 - p$. 2. At each stage $k$ comparisons can be performed in parallel and a noisy answer is returned We present a (classic) algorithm which optimally solves both variants together, up to an additive term of O(\log \log(n)), and prove matching information theoretic lower bounds. We use the algorithm to improve the results of Farhi et al \cite{FGGS99} presenting a quantum (error free) search algorithm in an ordered list of expected complexity less than (\log_2n) / 3.

DOI
10.48550/arxiv.quant-ph/0703231

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We use a Bayesian approach to optimally solve problems in noisy binary search. We deal with two variants: 1. Each comparison can be erroneous with some probability 1−p1 - p1−p. 2. At each stage kkk comparisons can be performed in parallel and a noisy answer is returned We present a (classic) algorithm which optimally solves both variants together, up to an additive term of O(\log \log(n)), and prove matching information theoretic lower bounds. We use the algorithm to improve the results of Farhi et al \cite{FGGS99} presenting a quantum (error free) search algorithm in an ordered list of expected complexity less than (\log_2n) / 3.
Ordered search is the problem of locating a target element in a sorted list of size nnn using comparison queries. Classically, binary search requires ⌈log⁡2n⌉\lceil \log_2 n\rceil⌈log2​n⌉ queries, which is optimal. Quantum algorithms offer a constant-factor speedup, but the precise constant has been a longstanding open question. We close this gap by exhibiting two new quantum algorithms for ordered search, each using the optimal 1πln⁡n+o(log⁡n)\frac{1}π\ln n+o(\log n)π1​lnn+o(logn) queries. The first, discovered by Claude Fable 5, is a simple zero-error algorithm derived from a continuum relaxation of the problem. The second, discovered by GPT-5.6-Sol (informed by Claude's zero-error algorithm), is an exact algorithm based on an analytic solution of the polynomial program of Farhi, Goldstone, Gutmann, and Sipser.
We use a Bayesian approach to optimally solve problems in noisy binary search. We deal with two variants:1. Each comparison is erroneous with independent probability 1-p. 2. At each stage k comparisons can be performed in parallel and a noisy answer is returned. We present a (classical) algorithm which solves both variants optimally (with respect to p and k), up to an additive term of O(loglog n), and prove matching information-theoretic lower bounds. We use the algorithm to improve the results of Farhi et al., presenting an exact quantum search algorithm in an ordered list of expected complexity less than (log2n)/3.
We consider the quantum complexities of searching an ordered list and sorting an un-ordered list. For searching an ordered list of N elements, we prove a lower bound of \\frac{1}{\\pi}(\\ln(N)-1) on the number of oracle queries that access the list elements. This improves the previously best lower bound of ({1/12}\\log_2(N) - O(1)) due to Ambainis. For sorting N numbers, we prove a lower bound of \\frac{N}{2\\pi}(\\ln(N)-1) on the number of binary comparisons. The previously best lower bound is \\Omega(N). Our proofs are based on a weighted all-pairs inner product argument, and our results generalize to bounded error quantum algorithms. Both results are proven in the so-called quantum black box model, a quantum analogue of classical decision trees. In addition to our lower bound results, we give an exact quantum algorithm for ordered searching using (\\log_3(N) + O(1)) queries, which is roughly 0.631 \\log_2(N). Although our algorithm is worse than that of Farhi, Goldstone, Gutmann and Sipser, which makes 0.526 \\log_2(N) queries, its philosophy is completely different. Our algorithm is a quantum version of the classical binary search algorithm, and it uses a quantum routine for traversing through a binary search tree faster than classically.
Unstructured database search is a fundamental problem that requires finding one or more elements satisfying a specific property within a set of candidates. While Grover's algorithm provides a quadratic speedup for finding a marked element when the number of solutions is known, finding all solutions when their number is unknown presents additional challenges. In this paper, we thoroughly analyze this problem by considering all the different regimes of combinations of classical, Grover and generalized fixed-point quantum search instances. We unify these into an adaptive Bayesian approach that maximizes the expected success probability per unit execution time at each searching instance. For a problem with a known number of solutions, much lower than the database size, we prove that our proposed approach is asymptotically optimal within the considered search model. Furthermore, we prove that for any probability distribution characterizing the likely number of solutions, in regimes where the classical verification time is at most 25% of the quantum amplitude amplification time, hybrid classical and Grover approaches are preferred to generalized fixed-point. Outside of this regime, we derive an upper bound on the maximal improvement achievable by fixed-point search compared to the best Grover-based score. Additionally, we determine the optimal transition point between classical sampling and quantum searching using an integral approximation of the probability distribution of the number of solutions. The analytical results are also supported by numerical simulations of the proposed algorithm and by comparisons between the generalized fixed-point and Grover-based search policies.
We prove that any exact quantum algorithm searching an ordered list of N elements requires more than \frac{1}π(\ln(N)-1) queries to the list. This improves upon the previously best known lower bound of {1/12}\log_2(N) - O(1). Our proof is based on a weighted all-pairs inner product argument, and it generalizes to bounded-error quantum algorithms. The currently best known upper bound for exact searching is roughly 0.526 \log_2(N). We give an exact quantum algorithm that uses \log_3(N) + O(1) queries, which is roughly 0.631 \log_2(N). The main principles in our algorithm are an quantum parallel use of the classical binary search algorithm and a method that allows basis states in superpositions to communicate.
Ordered search is the task of finding an item in an ordered list using comparison queries. The best exact classical algorithm for this fundamental problem uses ⌈log⁡2n⌉\lceil \log _{2}{n}\rceil⌈log2​n⌉ queries for a list of length n . Quantum computers can achieve a constant-factor speedup, but the best possible coefficient of log⁡2n\log _{2}{n}log2​n for exact quantum algorithms is only known to lie between (ln⁡2)/π≈0.221(\ln {2})/\pi \approx 0.221(ln2)/π≈0.221 and 4/log⁡2605≈0.4334/\log _{2}{605} \approx 0.4334/log2​605≈0.433 . We consider a special class of translation-invariant algorithms with no workspace, introduced by Farhi, Goldstone, Gutmann, and Sipser, that has been used to find the best known upper bounds. First, we show that any bounded-error, k -query quantum algorithm for ordered search can be implemented by a k -query algorithm in this special class. Second, we use linear programming to show that the best exact 5-query quantum algorithm can search a list of length 7265, giving an ordered search algorithm that asymptotically uses 5log⁡7265n≈0.390log⁡2n5 \log _{7265}{n} \approx 0.390 \log _{2}{n}5log7265​n≈0.390log2​n quantum queries.