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arXiv:quant-ph/9902053 · Annual Symposium on Foundations of Computer Science

A better lower bound for quantum algorithms searching an ordered list

Andris Ambainis

Published Jan 20, 2003• 41 citations
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Abstract

We show that any quantum algorithm searching an ordered list of n elements needs to examine at least (log,n)/12-O(1) of them. Classically, log/sub 2/ n queries are both necessary and sufficient. This shows that quantum algorithms can achieve only a constant speedup for this problem.
DOI
10.1109/sffcs.1999.814606

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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 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.
Withdrawn by the author due to irreparable errors. We present a quantum algorithm that in the black-box model performs a search in an ordered list of N elements. Using 3/4 log N + O(1) queries, it achieves a success probability of at least 1/2, whereas classically, log N - O(1) queries are needed to obtain constant success probability. Moreover, our algorithm employs the Haar transform and thus differs substantially from Grover's search algorithm and from algorithms relying on the quantum Fourier transform.
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.
We prove lower bounds on the error probability of a quantum algorithm for searching through an unordered list of N items, as a function of the number T of queries it makes. In particular, if T=O(sqrt{N}) then the error is lower bounded by a constant. If we want error <1/2^N then we need T=Omega(N) queries. We apply this to show that a quantum computer cannot do much better than a classical computer when amplifying the success probability of an RP-machine. A classical computer can achieve error <=1/2^k using k applications of the RP-machine, a quantum computer still needs at least ck applications for this (when treating the machine as a black-box), where c>0 is a constant independent of k. Furthermore, we prove a lower bound of Omega(sqrt{log N}/loglog N) queries for quantum bounded-error search of an ordered list of N items.
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.