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Frameworks for Quantum Algorithms

Stacey Jeffery

Published Aug 26, 2014• 13 citations
OpenAlex

Abstract

Due to the difficulty of constructing new quantum algorithms, frameworks that facilitate this construction are of great importance in quantum computing. These frameworks reduce the problem of coming up with a quantum algorithm to that of constructing some combinatorial object that is often much simpler to reason about. The implementation and analysis of an algorithm for the specified problem follow from the properties of this object. A number of such frameworks have been extremely successful in leading to the development of numerous quantum algorithms for a variety of problems. In this thesis, we build on two of these frameworks, the quantum walk search framework, and the span program framework, extending their algorithmic potential. \n \nThe quantum walk search framework gives a generic quantum analogue to a specific type of classical algorithm based on random walks. If one can construct a classical algorithm of this form, a corresponding quantum algorithm with better complexity immediately follows. In this framework, a generic algorithm is constructed from several subroutines for which implementations must be given for each application. One of these subroutines, a checking subroutine, is run many times throughout the algorithm. This subroutine may be implemented by any quantum algorithm that satisfies the required functionality, including another quantum walk algorithm. By making a slight modification to the quantum walk framework, we can show how to nest a quantum walk algorithm in the checking subroutine of another quantum walk algorithm in a way that gives better complexity than the naive nesting. This modification allows us to reproduce a number of upper bounds previously obtained in another framework, the learning graph framework, including upper bounds for triangle finding, and more generally, subgraph finding for constant-sized subgraphs. Porting these upper bounds over to the setting of quantum walks is desirable because the algorithms achieved in the quantum walk search framework are much more explicit than those of the learning graph framework, making them easier to work with, modify, and build on, as needed. Our efficient nested checking idea has already been used to come up with new quantum algorithms for finding sub-hypergraphs. \n \nAnother subroutine that is called repeatedly by the generic quantum walk search algorithm is the update subroutine. It was not clear how to use a quantum walk algorithm to perform this step, but by making another slight modification to the quantum walk search framework, we are able to show how to nest a quantum walk in the update step of another quantum walk in an efficient way. Our technique for doing this is a special case of a technique that allows the update to be implemented with garbage --- i.e., some unwanted data in an auxiliary register, entangled with the desired state. This technique may have other applications. \n \nUsing the nested update technique, we are able to improve the best known upper bounds on the time complexity of $k$-distinctness. Previously the best known upper bound on the time complexity was $n^{k/(k+1)}$, due to Ambainis. Belovs had recently improved the query complexity of $k$-distinctness to $o(n^{3/4})$ for all $k$, but since this upper bound was obtained in a framework called span programs, which only gives upper bounds on quantum query complexity, there was no known time efficient implementation of his $k$-distinctness algorithm. We use ideas from his construction and our nested update technique to get a quantum time upper bound for $3$-distinctness of $n^{5/7}$, matching his quantum query upper bound. We can generalize our algorithm get a time upper bound of $n^{(k-1)/k}$ for any $k>3$, slightly improving on the best previous upper bound. \n \nAnother framework, the span program framework, is known to be equivalent to quantum query complexity, in the sense that for any Boolean decision problem, there exists a span program construction that yields a tight upper bound on its quantum query complexity. We explore several variations of this framework. First, we slightly modify the definition of a span program so that we can show that for \\emph{any} (not necessarily Boolean) decision problem, there is a span program construction that yields a tight upper bound on its quantum query complexity. Previously this was only known up to logarithmic factors. We also explore several approximate versions of span programs, and show them to be equivalent. Finally, we explore the structure of span program witnesses, and use this structure to present an algorithm for evaluating span programs that is straightforward and intuitive. We also show how to evaluate approximate span programs, opening the possibility for the construction of new upper bounds using approximate span programs.

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Due to the difficulty of constructing new quantum algorithms, frameworks that facilitate this construction are of great importance in quantum computing. These frameworks reduce the problem of coming up with a quantum algorithm to that of constructing some combinatorial object that is often much simpler to reason about. The implementation and analysis of an algorithm for the specified problem follow from the properties of this object. A number of such frameworks have been extremely successful in leading to the development of numerous quantum algorithms for a variety of problems. In this thesis, we build on two of these frameworks, the quantum walk search framework, and the span program framework, extending their algorithmic potential. \n \nThe quantum walk search framework gives a generic quantum analogue to a specific type of classical algorithm based on random walks. If one can construct a classical algorithm of this form, a corresponding quantum algorithm with better complexity immediately follows. In this framework, a generic algorithm is constructed from several subroutines for which implementations must be given for each application. One of these subroutines, a checking subroutine, is run many times throughout the algorithm. This subroutine may be implemented by any quantum algorithm that satisfies the required functionality, including another quantum walk algorithm. By making a slight modification to the quantum walk framework, we can show how to nest a quantum walk algorithm in the checking subroutine of another quantum walk algorithm in a way that gives better complexity than the naive nesting. This modification allows us to reproduce a number of upper bounds previously obtained in another framework, the learning graph framework, including upper bounds for triangle finding, and more generally, subgraph finding for constant-sized subgraphs. Porting these upper bounds over to the setting of quantum walks is desirable because the algorithms achieved in the quantum walk search framework are much more explicit than those of the learning graph framework, making them easier to work with, modify, and build on, as needed. Our efficient nested checking idea has already been used to come up with new quantum algorithms for finding sub-hypergraphs. \n \nAnother subroutine that is called repeatedly by the generic quantum walk search algorithm is the update subroutine. It was not clear how to use a quantum walk algorithm to perform this step, but by making another slight modification to the quantum walk search framework, we are able to show how to nest a quantum walk in the update step of another quantum walk in an efficient way. Our technique for doing this is a special case of a technique that allows the update to be implemented with garbage --- i.e., some unwanted data in an auxiliary register, entangled with the desired state. This technique may have other applications. \n \nUsing the nested update technique, we are able to improve the best known upper bounds on the time complexity of kkk-distinctness. Previously the best known upper bound on the time complexity was nk/(k+1)n^{k/(k+1)}nk/(k+1), due to Ambainis. Belovs had recently improved the query complexity of kkk-distinctness to o(n3/4)o(n^{3/4})o(n3/4) for all kkk, but since this upper bound was obtained in a framework called span programs, which only gives upper bounds on quantum query complexity, there was no known time efficient implementation of his kkk-distinctness algorithm. We use ideas from his construction and our nested update technique to get a quantum time upper bound for 333-distinctness of n5/7n^{5/7}n5/7, matching his quantum query upper bound. We can generalize our algorithm get a time upper bound of n(k−1)/kn^{(k-1)/k}n(k−1)/k for any k>3k>3k>3, slightly improving on the best previous upper bound. \n \nAnother framework, the span program framework, is known to be equivalent to quantum query complexity, in the sense that for any Boolean decision problem, there exists a span program construction that yields a tight upper bound on its quantum query complexity. We explore several variations of this framework. First, we slightly modify the definition of a span program so that we can show that for \emph{any} (not necessarily Boolean) decision problem, there is a span program construction that yields a tight upper bound on its quantum query complexity. Previously this was only known up to logarithmic factors. We also explore several approximate versions of span programs, and show them to be equivalent. Finally, we explore the structure of span program witnesses, and use this structure to present an algorithm for evaluating span programs that is straightforward and intuitive. We also show how to evaluate approximate span programs, opening the possibility for the construction of new upper bounds using approximate span programs.
We develop a new framework that extends the quantum walk framework of Magniez, Nayak, Roland, and Santha, by utilizing the idea of quantum data structures to construct an efficient method of nesting quantum walks. Surprisingly, only classical data structures were considered before for searching via quantum walks. The recently proposed learning graph framework of Belovs has yielded improved upper bounds for several problems, including triangle finding and more general subgraph detection. We exhibit the power of our framework by giving a simple explicit constructions that reproduce both the O(n 35/27) and O(n 9/7) learning graph upper bounds (up to logarithmic factors) for triangle finding, and discuss how other known upper bounds in the original learning graph framework can be converted to algorithms in our framework. We hope that the ease of use of this framework will lead to the discovery of new upper bounds.
We develop a new framework that extends the quantum walk framework of Magniez, Nayak, Roland, and Santha, by utilizing the idea of quantum data structures to construct an efficient method of nesting quantum walks. Surprisingly, only classical data structures were considered before for searching via quantum walks. The recently proposed learning graph framework of Belovs has yielded improved upper bounds for several problems, including triangle finding and more general subgraph detection. We exhibit the power of our framework by giving a simple explicit constructions that reproduce both the O(n35/27) and O(n9/7) learning graph upper bounds (up to logarithmic factors) for triangle finding, and discuss how other known upper bounds in the original learning graph framework can be converted to algorithms in our framework. We hope that the ease of use of this framework will lead to the discovery of new upper bounds.
In the first part of the thesis we construct a new, simple framework which amplifies to a constant the success probability of any abstract search algorithm. The total query complexity is given by the quantum hitting time of the resulting operator, which we show that it is of the same order as the quantum hitting time of the original algorithm. As a major application of our framework, we show that for any reversible walk PPP and a single marked state, the quantum walk corresponding to PPP can find the solution using a number of queries that is quadratically smaller than the classical hitting time of PPP. Our algorithm is more general and simpler to implement than the solution known previously in the literature (Krovi, Magniez, Ozols, and Roland, 2015), which was developed specifically for quantum walks; we also prove that, for the particular case of quantum walks, we can embed their algorithm into our framework, thus simulating it exactly. Finally, we show that we can implement amplitude amplification using our tool. In the second part of the thesis we give a new lower bound in the query model which proves that Grover's algorithm for unordered searching is exactly optimal. Similar to existing methods for proving lower bounds, we bound the amount of information we can gain from a single oracle query, but we bound this information in terms of angles. This allows our proof to be simple, self-contained, based on only elementary mathematics, capturing our intuition, while obtaining at the same an exact bound. We then turn our attention to non-adaptive algorithms for the same problem of searching an unordered set. In this model, we obtain a lower bound and we give an algorithm which matches the lower bound, thus showing that the lower bound is exactly optimal.
This thesis is on quantum algorithms. It has three main themes: \n(1) quantum walk based search algorithms, \n(2) quantum rejection sampling, and \n(3) the Boolean function hidden shift problem. \nThe first two parts deal with generic techniques for constructing quantum algorithms, and the last part is on quantum algorithms for a specific algebraic problem. \n \nIn 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. \n \nIn 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. \n \nIn 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.
In the thesis, we use a recently developed tight characterisation of quantum query complexity, the adversary bound, to develop new quantum algorithms and lower bounds. Our results are as follows: * We develop a new technique for the construction of quantum algorithms: learning graphs. * We use learning graphs to improve quantum query complexity of the triangle detection and the kkk-distinctness problems. * We prove tight lower bounds for the kkk-sum and the triangle sum problems. * We construct quantum algorithms for some subgraph-finding problems that are optimal in terms of query, time and space complexities. * We develop a generalisation of quantum walks that connects electrical properties of a graph and its quantum hitting time. We use it to construct a time-efficient quantum algorithm for 3-distinctness.