S imilarly to classical systems, quantum algorithms are a set of step-by-step instructions or rules designed to perform a specific task or solve a particular problem. They process information to solve a problem, by encoding information in the state of a single qubit, or the joint state of many qubits, and performing operations over them. Some algorithms are simple instructions and enable other algorithms, while others consist of many instructions to address more complex problems. To properly understand how algorithms work, it is important to take some factors into account, shown in Figure 4.1 , and which we review in this section.
S imilarly to classical systems, quantum algorithms are a set of step-by-step instructions or rules designed to perform a specific task or solve a particular problem. They process information to solve a problem, by encoding information in the state of a single qubit, or the joint state of many qubits, and performing operations over them. Some algorithms are simple instructions and enable other algorithms, while others consist of many instructions to address more complex problems. To properly understand how algorithms work, it is important to take some factors into account, shown in Figure 4.1 , and which we review in this section.
Measurement is a fundamental operation in quantum computing and has many important use cases in quantum algorithms. This article provides a comprehensive overview of the basic measurement operations in quantum computing and represents a selected set of their applications in quantum algorithms. Our goal is to provide one of the first algorithmic overviews of measurement processes in quantum computing. From the quantum information-theoretical perspective, measurements are either a method to access the result of a quantum computation or a technique to modify a quantum state. We also identify measurement-based methods to solve quantum computational challenges, such as error mitigation and circuit cutting. We discuss three main categories of measurements: performing measurements in static quantum circuits, modifying the quantum state in dynamic quantum circuits via measurements, and addressing challenges in quantum computing with measurements. Based on the reviewed topics, the measurement operations are frequently not at the center of the quantum algorithm design. However, the most novel and error-prone quantum algorithms will likely require sophisticated measurement schemes. Thus, the central message of this article is to broaden the view of measurement operations and highlight their importance at every level of quantum algorithm design.
In the Name of the Pasta, and of the Sauce, and of the Holy Meatballs … – Bobby Henderson, The Gospel of the Flying Spaghetti Monster, 2006 After the conceptual comes the practical. While the study of quantum foundations is as old as quantum theory itself, the field of quantum computing is relatively new. So new in fact that large-scale, practical quantum computing is still not a reality. A typical ‘quantum computer’ takes many months to set up before performing such astounding tasks as factoring 6 into 3 × 2. Nonetheless, if those machines would exist, we know that we would gain amazing speed-ups in solving some hard (classical) computational problems, such as those involved in breaking a huge portion of cryptographic systems in use today. Before we get into ‘quantum computing’, we should say a couple of things about ‘computing’. What is computing? Our answer by now probably won't come as such a shock: it's a process theory! Computation is indeed all about wiring the inputs and outputs of small processes together to make bigger processes. More specifically, a computation consists of a (finite) set of basic processes, which are wired together according to some (also finite) instructions, which we refer to as an algorithm or simply a program . The only essential difference between classical and quantum computation is the contents of the basic processes. For classical computation, these operations consist of things like logical operations (e.g. XOR) or reading/writing locations in memory. For quantum computation, we can extend this with quantum processes and classical-quantum interactions such as measurements. So quantum computing is all about figuring out how to write new kinds of programs that exploit these new building blocks to build faster algorithms or accomplish new kinds of tasks that aren't possible classically. The first quantum algorithms were ‘proofs of concept’, in the sense that they solved some problem much faster than a classical computer, but the kinds of problems they solved were not particularly interesting in their own right. However, this changed drastically with the advent of Grover's quantum search and Shor's factoring algorithms.
Abstract Quantum computers are designed to outperform standard computers by running quantum algorithms. Areas in which quantum algorithms can be applied include cryptography, search and optimisation, simulation of quantum systems and solving large systems of linear equations. Here we briefly survey some known quantum algorithms, with an emphasis on a broad overview of their applications rather than their technical details. We include a discussion of recent developments and near-term applications of quantum algorithms.
A quantum algorithm is a set of instructions for a quantum computer, however, unlike algorithms in classical computer science their results cannot be guaranteed. A quantum system can undergo two types of operation, measurement and quantum state transformation, operations themselves must be unitary (reversible). Most quantum algorithms involve a series of quantum state transformations followed by a measurement. Currently very few quantum algorithms are known and no general design methodology exists for their construction.
All the data stored and processed by our computers is encoded as sequences of zeros and ones, called bits. Quantum computers offer an alternative to this traditional way of encoding and manipulating information. The information in a collection of qubits can be a combination, also called a superposition, over many possible sequences of zeros and ones. Where classical computation consists of flipping zeros and ones, quantum computation consists of manipulating high-dimensional vectors called quantum states. Researchers have found ways to use this to engineer quantum algorithms that solve some tasks much faster than any classical computer would. These are called quantum advantages, and are proven for specific tasks, such as breaking some encryption schemes and solving certain large linear systems. The goal of this thesis was to find new tasks for which quantum computers provably have an advantage over classical computers. In this thesis, I present two such advantages and associated results. The first one revolves around a physically relevant learning task. In other words, I explore how quantum computers can be used to help solve specialized machine learning tasks. The second one addresses the simulation of classical systems. One of the first high-potential and natural uses of quantum computers was the simulation of quantum systems. A lot of real-world problems, however, are ruled by classical physics. It turns out that quantum computers are well suited for the simulation of some exponentially large classical systems. In this thesis, I add a new problem to this list.
Rapid progress in quantum technology in this decade has dramatically increased the coherence time of qubits (quantum bits) and the fidelity of operations on qubits.Although currently available quantum computers are noisy and intermediate in scale, it is believed that they will be used in some specific applications such as quantum chemistry, machine learning, optimization, and sampling in the near future.Along with progress in quantum computing hardware, development of the software stack has also become more active.A quantum compiler, which translates a quantum program into a sequence of control instructions, is one of the most essential software components to provide a human-friendly programming interface for quantum algorithms.In quantum compilers, optimizing the output instructions is very important: For noisy quantum computers, it increases the probability of completing all the operations before any qubit decoheres, yielding computational results with higher fidelity.Even for fault-tolerant quantum computers, optimization increases the throughput.In this thesis, we address two important optimization tasks in quantum compilers: quantum circuit mapping and quantum circuit scheduling.We provide their formulations as classical optimization problems considering commutation between quantum gates, which are not fully considered in previous studies.We also provide exact and heuristic algorithms for solving them.We examine how considering gate commutation in our algorithms improves the quality of solutions by computational experiments in practical settings.The thesis is organized as follows.After reviewing studies on quantum compilers, we consider the problem of mapping quantum circuits to quantum computers, whose operations are limited by their coupling architecture, with as few additional gates as possible.We propose exact and heuristic algorithms for quantum circuit mapping, which use gate commutation rules and a gate transformation rule of Bridge gate.We compare these algorithms with the state-of-the-art algorithms by computational experiments with circuits of a commonly used benchmark dataset.Next, we consider the problem of scheduling operations in a given circuit with the shortest total execution time.We demonstrate that quantum circuit scheduling can be seen as an optimization problem only when we consider gate commutation and can be interpreted as a special type of job-shop problem.We provide Constraint Programming and Mixed Integer Programming formulations and show how solving circuit scheduling independently improved the schedule length through experiments with real circuits and a compiler.We believe that our algorithms are useful in practical quantum compilers and the developed techniques provide new insight into optimization in quantum compilers.We hope that they will be helpful to advance software stack development and improve the utility of quantum computers.