The quantum circuit model essentially treats every quantum algorithm as a straight-line program. While this view is universal, recent work has shown that it is inconvenient for using different-length quantum subroutines in superposition. Using the quantum walk formalism of quantum algorithms, it is possible to model such branching behaviour, and get better composition in this setting.
We apply the above branching composition to Grover's algorithm, which gives a variable-time quantum search algorithm that is worse than previous work. The reason it is worse is because branching composition does not take into account another deviation from straight-line programs: looping. We show that by modifying branching composition to also include looping, we can get a complexity that matches previous work. This highlights the importance of properly modeling the program control flow when designing quantum algorithms.
The quantum circuit model essentially treats every quantum algorithm as a straight-line program. While this view is universal, recent work has shown that it is inconvenient for using different-length quantum subroutines in superposition. Using the quantum walk formalism of quantum algorithms, it is possible to model such branching behaviour, and get better composition in this setting.
We apply the above branching composition to Grover's algorithm, which gives a variable-time quantum search algorithm that is worse than previous work. The reason it is worse is because branching composition does not take into account another deviation from straight-line programs: looping. We show that by modifying branching composition to also include looping, we can get a complexity that matches previous work. This highlights the importance of properly modeling the program control flow when designing quantum algorithms.
An important tool in algorithm design is the ability to build algorithms from other algorithms that run as subroutines. In the case of quantum algorithms, a subroutine may be called on a superposition of different inputs, which complicates things. For example, a classical algorithm that calls a subroutine Q times, where the average probability of querying the subroutine on input i is pi, and the cost of the subroutine on input i is Ti, incurs expected cost Q∑ipiE[Ti] from all subroutine queries. While this statement is obvious for classical algorithms, for quantum algorithms, it is much less so, since naively, if we run a quantum subroutine on a superposition of inputs, we need to wait for all branches of the superposition to terminate before we can apply the next operation. We nonetheless show an analogous quantum statement (*): If qi is the average query weight on i over all queries, the cost from all quantum subroutine queries is Q∑iqiE[Ti]. Here the query weight on i for a particular query is the probability of measuring i in the input register if we were to measure right before the query. We prove this result using the technique of multidimensional quantum walks, recently introduced in arXiv:2208.13492. We present a more general version of their quantum walk edge composition result, which yields variable-time quantum walks, generalizing variable-time quantum search, by, for example, replacing the update cost with ∑u,vπuPu,vE[Tu,v2], where Tu,v is the cost to move from vertex u to vertex v. The same technique that allows us to compose quantum subroutines in quantum walks can also be used to compose in any quantum algorithm, which is how we prove (*).
Quantum algorithms for tasks such as factorization, search, and simulation rely on control flow such as branching and iteration that depends on the value of data in superposition. High-level programming abstractions for control flow, such as switches, loops, higher-order functions, and continuations, are ubiquitous in classical languages. By contrast, many quantum languages do not provide high-level abstractions for control flow in superposition, and instead require the use of hardware-level logic gates to implement such control flow. The reason for this gap is that whereas a classical computer supports control flow abstractions using a program counter that can depend on data, the typical architecture of a quantum computer does not analogously provide a program counter that can depend on data in superposition. As a result, the complete set of control flow abstractions that can be correctly realized on a quantum computer has not yet been established. In this work, we provide a complete characterization of the properties of control flow abstractions that are correctly realizable on a quantum computer. First, we prove that even on a quantum computer whose program counter exists in superposition, one cannot correctly realize control flow in quantum algorithms by lifting the classical conditional jump instruction to work in superposition. This theorem denies the ability to directly lift general abstractions for control flow such as the λ-calculus from classical to quantum programming. In response, we present the necessary and sufficient conditions for control flow to be correctly realizable on a quantum computer. We introduce the quantum control machine, an instruction set architecture featuring a conditional jump that is restricted to satisfy these conditions. We show how this design enables a developer to correctly express control flow in quantum algorithms using a program counter in place of logic gates.
Quantum algorithms for tasks such as factorization, search, and simulation rely on control flow such as branching and iteration that depends on the value of data in superposition. High-level programming abstractions for control flow, such as switches, loops, and higher-order functions, are ubiquitous in classical languages. By contrast, many quantum languages do not provide high-level abstractions for control flow in superposition, and instead require the use of hardware-level logic gates to implement such control flow. The reason for this gap is that whereas a classical computer supports control flow using a program counter that can depend on data, the typical architecture of a quantum computer does not provide a program counter that can depend on data in superposition. As a result, the complete set of control flow abstractions that can be correctly realized on a quantum computer has not yet been established. In this work, we provide a complete characterization of the properties of control flow abstractions that are correctly realizable on a quantum computer. First, we prove that even on a quantum computer whose program counter exists in superposition, one cannot correctly realize control flow in quantum algorithms by lifting the classical conditional jump instruction to work in superposition. This theorem denies the ability to directly lift general abstractions for control flow such as the λ-calculus from classical to quantum programming. In response, we present the necessary and sufficient conditions for control flow to be correctly realizable on a quantum computer. We introduce the quantum control machine, an instruction set architecture featuring a conditional jump that is restricted to satisfy these conditions. We show how this design enables a developer to correctly express control flow in quantum algorithms using a program counter in place of logic gates.
Quantum query complexity has several nice properties with respect to composition. First, bounded-error quantum query algorithms can be composed without incurring log factors through error reduction e x a c t n e s s . Second, through careful accounting t h r i f t i n e s s , the total query complexity is smaller if subroutines are mostly run on cheaper inputs -- a property that is much less obvious in quantum algorithms than in their classical counterparts. While these properties were previously seen through the model of span programs (alternatively, the dual adversary bound), a recent work by two of the authors (Belovs, Yolcu 2023) showed how to achieve these benefits without converting to span programs, by defining quantum Las Vegas query complexity . Independently, recent works, including by one of the authors (Jeffery 2022), have worked towards bringing thriftiness to the more practically significant setting of quantum t i m e complexity.In this work, we show how to achieve both exactness and thriftiness in the setting of time complexity. We generalize the quantum subroutine composition results of Jeffery 2022 so that, in particular, no error reduction is needed. We give a time complexity version of the well-known result in quantum query complexity, Q ( f ∘ g ) = O ( Q ( f ) ⋅ Q ( g ) ) , without log factors.We achieve this by employing a novel approach to the design of quantum algorithms based on what we call t r a n s d u c e r s , and which we think is of large independent interest. While a span program is a completely different computational model, a transducer is a direct generalisation of a quantum algorithm, which allows for much greater transparency and control. Transducers naturally characterize general state conversion, rather than only decision problems; provide a very simple treatment of other quantum primitives such as quantum walks; and lend themselves well to time complexity analysis.
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 nk/(k+1), due to Ambainis. Belovs had recently improved the query complexity of k-distinctness to o(n3/4) 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 n5/7, 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.
It is widely believed that quantum computers (if realized) will be more powerful than today’s computers from the viewpoint of computational complexity . However, it is not obvious how to utilize quantum computers in practical situations. For practical purposes, a Grover search may be one of the most promising quantum algorithms known so far. Thus, in this paper, we propose an efficient framework where we can use Grover search for general programming. Our framework has the following steps. (1) A programmer writes a program by using standard C++ programming language. (2) Some if expressions within for loops in the C++ source code are chosen as candidates to be performed as a Grover search on a quantum computer. (3) The framework automatically generates a corresponding quantum circuit for each Grover search chosen in (2). Unlike the existing quantum circuit design methods, we can treat large problems in our quantum circuit design. (4) By evaluating the number of primitive quantum gates in the quantum circuit generated in (3), the framework determines whether the processing time of the quantum circuit is faster than the processing time of the corresponding if expression on a classical computer. If the framework determines that the quantum circuit is faster, it generates some interface source codes for a classical computer. Thus, in our framework, a programmer can use a Grover search with almost no effort.