TY - GEN

T1 - On the quantum complexity of closest pair and related problems

AU - Aaronson, Scott

AU - Chia, Nai Hui

AU - Lin, Han Hsuan

AU - Wang, Chunhao

AU - Zhang, Ruizhe

N1 - Funding Information:
Funding SA was supported by a Vannevar Bush Fellowship from the US Department of Defense, a Simons Investigator Award, and the Simons “It from Qubit” collaboration. NHC, HHL, and CW were supported by SA’s Vannevar Bush Faculty Fellowship. RZ received support from the National Science Foundation (grant CCF-1648712).

PY - 2020/7/1

Y1 - 2020/7/1

N2 - The closest pair problem is a fundamental problem of computational geometry: given a set of n points in a d-dimensional space, find a pair with the smallest distance. A classical algorithm taught in introductory courses solves this problem in O(n log n) time in constant dimensions (i.e., when d = O(1)). This paper asks and answers the question of the problem's quantum time complexity. Specifically, we give an Oe(n2/3) algorithm in constant dimensions, which is optimal up to a polylogarithmic factor by the lower bound on the quantum query complexity of element distinctness. The key to our algorithm is an efficient history-independent data structure that supports quantum interference. In polylog(n) dimensions, no known quantum algorithms perform better than brute force search, with a quadratic speedup provided by Grover's algorithm. To give evidence that the quadratic speedup is nearly optimal, we initiate the study of quantum fine-grained complexity and introduce the Quantum Strong Exponential Time Hypothesis (QSETH), which is based on the assumption that Grover's algorithm is optimal for CNF-SAT when the clause width is large. We show that the naïve Grover approach to closest pair in higher dimensions is optimal up to an no(1) factor unless QSETH is false. We also study the bichromatic closest pair problem and the orthogonal vectors problem, with broadly similar results.

AB - The closest pair problem is a fundamental problem of computational geometry: given a set of n points in a d-dimensional space, find a pair with the smallest distance. A classical algorithm taught in introductory courses solves this problem in O(n log n) time in constant dimensions (i.e., when d = O(1)). This paper asks and answers the question of the problem's quantum time complexity. Specifically, we give an Oe(n2/3) algorithm in constant dimensions, which is optimal up to a polylogarithmic factor by the lower bound on the quantum query complexity of element distinctness. The key to our algorithm is an efficient history-independent data structure that supports quantum interference. In polylog(n) dimensions, no known quantum algorithms perform better than brute force search, with a quadratic speedup provided by Grover's algorithm. To give evidence that the quadratic speedup is nearly optimal, we initiate the study of quantum fine-grained complexity and introduce the Quantum Strong Exponential Time Hypothesis (QSETH), which is based on the assumption that Grover's algorithm is optimal for CNF-SAT when the clause width is large. We show that the naïve Grover approach to closest pair in higher dimensions is optimal up to an no(1) factor unless QSETH is false. We also study the bichromatic closest pair problem and the orthogonal vectors problem, with broadly similar results.

UR - http://www.scopus.com/inward/record.url?scp=85089395303&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85089395303&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.CCC.2020.16

DO - 10.4230/LIPIcs.CCC.2020.16

M3 - Conference contribution

AN - SCOPUS:85089395303

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 35th Computational Complexity Conference, CCC 2020

A2 - Saraf, Shubhangi

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 35th Computational Complexity Conference, CCC 2020

Y2 - 28 July 2020 through 31 July 2020

ER -