The quantum adversary method and classical formula size lower bounds

We introduce two new complexity measures for Boolean functions, or more generally for functions of the form f:S->T. We call these measures sumPI and maxPI. The quantity sumPI has been emerging through a line of research on quantum query complexity lower bounds via the so-called quantum adversary method [Amb02, Amb03, BSS03, Zha04, LM04], culminating in [SS04] with the realization that these many different formulations are in fact equivalent. Given that sumPI turns out to be such a robust invariant of a function, we begin to investigate this quantity in its own right and see that it also has applications to classical complexity theory. As a surprising application we show that sumPI^2(f) is a lower bound on the formula size, and even, up to a constant multiplicative factor, the probabilistic formula size of f. We show that several formula size lower bounds in the literature, specifically Khrapchenko and its extensions [Khr71, Kou93], including a key lemma of [Has98], are in fact special cases of our method. The second quantity we introduce, maxPI(f), is always at least as large as sumPI(f), and is derived from sumPI in such a way that maxPI^2(f) remains a lower bound on formula size. While sumPI(f) is always a lower bound on the quantum query complexity of f, this is not the case in general for maxPI(f). A strong advantage of sumPI(f) is that it has both primal and dual characterizations, and thus it is relatively easy to give both upper and lower bounds on the sumPI complexity of functions. To demonstrate this, we look at a few concrete examples, for three functions: recursive majority of three, a function defined by Ambainis, and the collision problem.Comment: Appears in Conference on Computational Complexity 200

Quantum Simulation Logic, Oracles, and the Quantum Advantage

Query complexity is a common tool for comparing quantum and classical computation, and it has produced many examples of how quantum algorithms differ from classical ones. Here we investigate in detail the role that oracles play for the advantage of quantum algorithms. We do so by using a simulation framework, Quantum Simulation Logic (QSL), to construct oracles and algorithms that solve some problems with the same success probability and number of queries as the quantum algorithms. The framework can be simulated using only classical resources at a constant overhead as compared to the quantum resources used in quantum computation. Our results clarify the assumptions made and the conditions needed when using quantum oracles. Using the same assumptions on oracles within the simulation framework we show that for some specific algorithms, like the Deutsch-Jozsa and Simon's algorithms, there simply is no advantage in terms of query complexity. This does not detract from the fact that quantum query complexity provides examples of how a quantum computer can be expected to behave, which in turn has proved useful for finding new quantum algorithms outside of the oracle paradigm, where the most prominent example is Shor's algorithm for integer factorization.Comment: 48 pages, 46 figure

Quantum algorithms for searching, resampling, and hidden shift problems

This thesis is on quantum algorithms. It has three main themes: (1) quantum walk based search algorithms, (2) quantum rejection sampling, and (3) the Boolean function hidden shift problem. The first two parts deal with generic techniques for constructing quantum algorithms, and the last part is on quantum algorithms for a specific algebraic problem. In the first part of this thesis we show how certain types of random walk search algorithms can be transformed into quantum algorithms that search quadratically faster. More formally, given a random walk on a graph with an unknown set of marked vertices, we construct a quantum walk that finds a marked vertex in a number of steps that is quadratically smaller than the hitting time of the random walk. The main idea of our approach is to interpolate the random walk from one that does not stop when a marked vertex is found to one that stops. The quantum equivalent of this procedure drives the initial superposition over all vertices to a superposition over marked vertices. We present an adiabatic as well as a circuit version of our algorithm, and apply it to the spatial search problem on the 2D grid. In the second part we study a quantum version of the problem of resampling one probability distribution to another. More formally, given query access to a black box that produces a coherent superposition of unknown quantum states with given amplitudes, the problem is to prepare a coherent superposition of the same states with different specified amplitudes. Our main result is a tight characterization of the number of queries needed for this transformation. By utilizing the symmetries of the problem, we prove a lower bound using a hybrid argument and semidefinite programming. For the matching upper bound we construct a quantum algorithm that generalizes the rejection sampling method first formalized by von~Neumann in~1951. We describe quantum algorithms for the linear equations problem and quantum Metropolis sampling as applications of quantum rejection sampling. In the third part we consider a hidden shift problem for Boolean functions: given oracle access to f(x+s), where f(x) is a known Boolean function, determine the hidden shift s. We construct quantum algorithms for this problem using the "pretty good measurement" and quantum rejection sampling. Both algorithms use the Fourier transform and their complexity can be expressed in terms of the Fourier spectrum of f (in particular, in the second case it relates to "water-filling" of the spectrum). We also construct algorithms for variations of this problem where the task is to verify a given shift or extract only a single bit of information about it.1 yea

Flexible molecular alignment: an industrial case study on quantum algorithmic techniques

Dissertação de mestrado em Engenharia FísicaFlexible molecular alignment is a complex and challenging problem in the area of Medic inal Chemistry. The current approach to this problem does not test all possible alignments, but makes a previous analysis of all the variables and chooses the ones with potentially greater impact in the posterior alignment. This procedure can lead to wrong ”best align ments” since not every data is considered. Quantum computation, due to its natural parallelism, may improve algorithmic solutions for this kind of problems because it may test and/or simulate all possible solutions in an execution cycle. As a case study proposed by BIAL and in collaboration with IBM, the main goal of this dissertation was to study and create quantum algorithms able to refactor the problem of molecular alignment in the new setting of quantum computation. Additionally, the comparison between both classical and quantum solutions was defined as a subsequent goal. During this dissertation and due to its complexity, in order to produce a practical solu tion to this problem, we resorted to a manageable number of conformations per molecule, revisited the classical solution and elaborated a corresponding quantum algorithm. Such algorithm was then tested in both a quantum simulator and a real device. Despite the privileged collaboration with IBM, the quantum simulations were not pro duced in viable time, making them impractical for industry applications. Nonetheless, tak ing in consideration the current point of development of quantum hardware, the suggested solutions still has potential for the future.O alinhamento de moléculas flexíveis é um problema complexo na área de Química Medicinal, onde, mesmo hoje em dia, é um desafio encontrar uma solução. A atual abordagem para este problema não testa todos os possíveis alinhamentos. Em vez disso, realiza uma análise prévia de todas as variáveis e escolhe aquelas com maior potencial de impacto no posterior alinhamento. Este procedimento pode levar a falsos “melhores alinhamentos” visto que nem todos os dados são considerados. A computação quântica, devido ao seu natural paralelismo, pode melhorar as soluções algorítmicas deste tipo de problemas visto que poderá testar e/ou simular todas as possíveis soluções num ciclo de execução. Partindo de um caso de estudo proposto pela BIAL, e em colaboração com a IBM, o objetivo principal desta dissertação foi estudar e criar algoritmos quânticos capazes reformular no contexto de computação quântica o problema de alinhamento de moléculas. Adicionalmente, e como objetivo subsequente, foi prevista a comparação entre os algoritmos clássicos e quânticos. Durante esta dissertação e devido à sua complexidade, de modo a produzir uma solução prática para este problema, foi utilizado um número tratável de conformações por molécula, revisitada a solução clássica e desenvolvido um algoritmo quântico correspondente. Tal algoritmo foi depois testado tanto num simulador quântico como num dispositivo real. Apesar da colaboração privilegiada com a IBM, as simulações quânticas não foram produzidas em tempo viável, tornando-as impraticáveis para aplicações industriais. Não obstante, tendo em consideração o ponto atual de desenvolvimento dos dispositivos quânticos, as soluções propostas terão potencial para o futuro