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arXiv:quant-ph/9811046

Lower Bounds for Quantum Search and Derandomization

Harry Buhrman, Ronald de Wolf

Published Nov 18, 1998• 12 citations
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Abstract

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.

DOI
10.48550/arxiv.quant-ph/9811046

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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.
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.
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.
The problem of finding a local minimum of a black-box function is central for understanding local search as well as quantum adiabatic algorithms. For functions on the Boolean hypercube {0,1}n\left\{0,1\right\}^n{0,1}n, we show a lower bound of Ω(2n/4/n)\Omega\left(2^{n/4}/n\right)Ω(2n/4/n) on the number of queries needed by a quantum computer to solve this problem. More surprisingly, our approach, based on Ambainis's quantum adversary method, also yields a lower bound of Ω(2n/2/n2)\Omega\left(2^{n/2}/n^2\right)Ω(2n/2/n2) on the problem's classical randomized query complexity. This improves and simplifies a 1983 result of Aldous. Finally, in both the randomized and quantum cases, we give the first nontrivial lower bounds for finding local minima on grids of constant dimension d≥3d\geq3d≥3.
We examine the number of queries to input variables that a quantum algorithm requires to compute Boolean functions on {0,1} N in the black-box model. We show that the exponential quantum speed-up obtained for partial functions (i.e., problems involving a promise on the input) by Deutsch and Jozsa, Simon, and Shor cannot be obtained for any total function: if a quantum algorithm computes some total Boolean function f with small error probability using T black-box queries, then there is a classical deterministic algorithm that computes f exactly with O ( Ts 6 ) queries. We also give asymptotically tight characterizations of T for all symmetric f in the exact, zero-error, and bounded-error settings. Finally, we give new precise bounds for AND, OR, and PARITY. Our results are a quantum extension of the so-called polynomial method, which has been successfully applied in classical complexity theory, and also a quantum extension of results by Nisan about a polynomial relationship between randomized and deterministic decision tree complexity.