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Time Hypothesis on a Quantum Computer the Implications Of The implications of breaking the strong exponential time hypothesis on a quantum computer Jorg Van Renterghem Student number: 01410124 Supervisor: Prof. Andris Ambainis Master's dissertation submitted in order to obtain the academic degree of Master of Science in de informatica Academic year 2018-2019 i Samenvatting In recent onderzoek worden reducties van de sterke exponenti¨eletijd hypothese (SETH) gebruikt om ondergrenzen op de complexiteit van problemen te bewij- zen [51]. Dit zorgt voor een interessante onderzoeksopportuniteit omdat SETH kan worden weerlegd in het kwantum computationeel model door gebruik te maken van Grover's zoekalgoritme [7]. In de klassieke context is SETH wel nog geldig. We hebben dus een groep van problemen waarvoor er een klassieke onder- grens bekend is, maar waarvoor geen ondergrens bestaat in het kwantum com- putationeel model. Dit cre¨eerthet potentieel om kwantum algoritmes te vinden die sneller zijn dan het best mogelijke klassieke algoritme. In deze thesis beschrijven we dergelijke algoritmen. Hierbij maken we gebruik van Grover's zoekalgoritme om een voordeel te halen ten opzichte van klassieke algoritmen. Grover's zoekalgoritme lost het volgende probleem op in O(pN) queries: gegeven een input x ; :::; x 0; 1 gespecifieerd door een zwarte doos 1 N 2 f g die queries beantwoordt, zoek een i zodat xi = 1 [32]. We beschrijven een kwantum algoritme voor k-Orthogonale vectoren, Graaf diameter, Dichtste paar in een d-Hamming ruimte, Alle paren maximale stroom, Enkele bron bereikbaarheid telling, 2 sterke componenten, Geconnecteerde deel- graaf en S; T -bereikbaarheid. We geven ook nieuwe ondergrenzen door gebruik te maken van reducties en de sensitiviteit methode. Voor Dichtste paar in een d-Hamming ruimte, Enkele bron bereikbaarheid telling, 2 sterke componenten en Geconnecteerde deelgraaf geven we een ondergrens aan hun kwantum query complexiteit. Voor Dynamische problemen is het veel moeilijker om een goed kwantum algoritme te vinden. De oorzaak hiervan is dat kwantum algoritmes in het algemeen hun voordeel halen uit het feit dat tussenresultaten niet noodzakelijk moeten geweten zijn. Deze tussenresultaten zijn echter vaak handig om sneller de oplossing voor een licht aangepast probleem te vinden. Als laatste bekijken we ook een volledig ander model voor het beschrijven van kwantum algoritmen: span programma's [37]. Dit model is interessant omdat het toelaat om grafen problemen zoals st-connectiviteit op een nieuwe manier op te lossen [12]. Pogingen om dit algoritme aan te passen zodat het ook st-pad lengte kan oplossen mislukken echter. De oorzaak hiervan is het feit dat span programma's het mogelijk maken om lineaire combinaties te maken met negative factoren. Dit gaf de aanzet om kegel programma's te beschouwen om kwantum al- goritmes voor te stellen. We botsten echter op het probleem dat kegel pro- gramma's, ondanks het feit dat ze heel gelijkaardig zijn aan span programma's, moeilijker te vertalen zijn naar kwantum algoritmes. Dit wordt veroorzaakt door het gebrek aan orthogonaliteit tussen vectoren in een kegel en die er buiten. Een vector w kan zich binnen een kegel bevinden terwijl w er buiten ligt. Maar het isj onmogelijki om w en w te onderscheiden− gebruikmakend j i van een kwantum meeting. j i − j i ii Summary In recent research the strong exponential time hypothesis (SETH) in combina- tion with fine grained reductions have been used to prove lower bounds on the complexity of problems [51]. This provides an interesting research opportunity, because SETH is valid in a classical context, but can be broken using Grover search in the quantum query model [7]. We thus have a set of problems for which a classical lower bound is known, but no such lower bound exist in the quantum query model. This creates the potential for finding quantum algorithms which are faster than any classical algorithm. In this thesis we provide such algorithms, making use of Grover search to get our advantage over classical algorithms. Grover search solves the following problem in O(pN) queries: Given an input x ; :::; x 0; 1 specified by a 1 N 2 f g black box that answers queries, find an i such that xi = 1 [32]. We provide new quantum algorithms for k-Orthogonal vectors, Graph di- ameter, Closest pair in d-Hamming space, All pairs max flow, Single source reachability count, 2 Strong components, Connected subgraph and S; T - reach- ability. We also provide new quantum lower bounds for these problems using re- ductions and the sensitivity method. For 2-Orthogonal vectors, Closest pair in d-Hamming space,Single source reachability count, 2 Strong components and Connected subgraph we give a new lower bound on their quantum query com- plexity. Dynamic problems are much harder to improve upon using quantum algo- rithms, because quantum algorithms in general get their advantage by the fact that not all intermediate results are necessary. Yet these intermediate results are often useful to find faster results for a slightly modified problem. Finally we look at a completely different model for describing quantum algo- rithms: span programs [37]. This program paradigm provides a new way to look at graph problems such as st-connectivity [12]. We try to modify the algorithm for st-connectivity to provide an algorithm for st-distance. This approach is unsuccessful. After a close examination the cause is found to be the fact that span programs allow linear combinations using negative factors. This was a reason to look into Cone programs, which are similar to span programs, to describe quantum algorithms. But Cone programs, while closely related to span programs, are much harder to translate into a quantum algo- rithm. This is caused by the lack of orthogonality in a cone program. A vector w can be inside a cone while a vector w is outside the cone. Yet it is im- possiblej i to differentiate between w and− j wi using a quantum measurement. j i − j i iii The implications of breaking the strong exponential time hypothesis on a quantum computer. Jorg Van Renterghem Supervisor: Andris Ambainis Abstract In this study we provide quantum algorithms for problems which have a proven classical lower bound under the the strong exponential time hypothesis. We provide new quantum algorithms for k-Orthogonal vectors, Graph diameter, Closest pair in d-Hamming space, All pairs max flow, Single source reachability count, 2 Strong components, Connected subgraph and S, T -reachability. All of these algorithms are faster than the best known classical algorithm and most of them break the classical lower bound. For Closest pair in d-Hamming space, Single source reachability count, 2 Strong components and Connected subgraph we also give a new lower bound on their quantum query complexity. In a second part of this study we also look at cone programs as an alternative to span programs for defining quantum algorithms. 1 Introduction with clause size at most k (the so called k-SAT prob- (1 )n lem) and n variables cannot be solved in O(2 − ) In recent research the strong exponential time hy- time even by a randomized algorithm. pothesis (SETH) in combination with fine grained re- ductions have been used to prove lower bounds on the As the clause size k grows, the lower bound given complexity of problems [24]. This provides an inter- by SETH converges to 2n. SETH also implies that esting research opportunity, because SETH is valid in general CNF-SAT on formulas with n variables and n o(n) a classical context, but can be broken using Grover m clauses requires 2 − poly(m) time. search in a quantum query model [5]. SETH is motivated by the lack of fast algorithms We thus have a set of problems for which a classi- for k-SAT as k grows. It is a much stronger assump- cal lower bound is known, but no such lower bound tion than P = NP which assumes that SAT requires exist in the quantum query model. This creates the superpolynomial6 time. A weaker version, the Expo- potential for finding quantum algorithms which are nential Time Hypothesis (ETH) asserts that there is faster than any classical algorithm. some constant δ > 0 such that CNF-SAT requires Ω(2δn). 1.1 SETH In recent years there have been a considerable num- Impagliazzo, Paturi and Zane [14] introduced SETH ber of problems whose hardness has been proven un- to address the complexity of conjunctive normal form der SETH. Arguably the first such problem was the satisfiability problem (CNF-SAT). At the time they Orthogonal vector problem(OV) which was shown by only considered deterministic algorithms, but nowa- Williams [23] to require quadratic time under SETH. days it is common to extend SETH to allow random- Many conditional hardness results based on OV ization. and SETH have been discovered. For example for Graph diameter [20], All pairs max flow [16], dynamic Hypothesis 1 (SETH) For every > 0 there ex- graph problems [17],... A more extensive listing can ists an integer k 3 such that CNF-SAT on formulas be found in [24]. ≥ iv 1.2 Grover This measure is useful because it provides a lower bound on the complexity of f. In this thesis we provide such algorithms, making use of Grover search to get our advantage over classical Theorem 1 Given a function f :[q]n [l] with → algorithms. The problem Grover search solves can be q, n, l N. Ω( s(f)) is a lower bound on the quan- described as follows: tum query∈ complexity of f p Definition 1 (Search) Given an input x1, ..., xN 0, 1 specified by a black box that answers queries.∈ 2 Quantum algorithms In{ a} query, we input i to the black box and it outputs xi.
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