Periodic Reporting for period 3 - QAFA (Quantum Algorithms from Foundations to Applications)
Periodo di rendicontazione: 2022-05-01 al 2023-10-31
Quantum computers are designed to use quantum mechanics to go beyond the power of any standard computer based only on classical physics. Quantum computers could solve important computational problems with impact on major societal challenges. These include simulation of quantum-mechanical systems, with applications to designing novel materials that enable more efficient batteries and solar cells; breaking cryptographic codes; and solving hard optimisation problems more efficiently than standard methods, with applications to logistics and communication network design, among many other areas.
Following intensive experimental efforts, so-called “quantum computational supremacy” was demonstrated in 2019, where a quantum computer solved a problem more quickly than the fastest algorithm then known, running on the world's fastest supercomputer. However, many urgent questions remain regarding the usefulness of quantum computers for problems of real practical interest, and the timescale on which such usefulness will be achieved. The overall goal of this project is to address the most significant near-term and long-range theoretical challenges involved in bringing quantum algorithms to practical applications.
Particular objectives are to develop quantum algorithms that accelerate general classical algorithmic frameworks (for example, within the domains of optimisation and constraint satisfaction); to develop efficient quantum communication protocols; to develop a better understanding of the features of problems that allow a quantum speedup; to demonstrate larger quantum-classical separations than were previously known; and to find new quantum algorithms for key problems in quantum physics, such as simulating, learning and testing quantum systems.
Following intensive experimental efforts, so-called “quantum computational supremacy” was demonstrated in 2019, where a quantum computer solved a problem more quickly than the fastest algorithm then known, running on the world's fastest supercomputer. However, many urgent questions remain regarding the usefulness of quantum computers for problems of real practical interest, and the timescale on which such usefulness will be achieved. The overall goal of this project is to address the most significant near-term and long-range theoretical challenges involved in bringing quantum algorithms to practical applications.
Particular objectives are to develop quantum algorithms that accelerate general classical algorithmic frameworks (for example, within the domains of optimisation and constraint satisfaction); to develop efficient quantum communication protocols; to develop a better understanding of the features of problems that allow a quantum speedup; to demonstrate larger quantum-classical separations than were previously known; and to find new quantum algorithms for key problems in quantum physics, such as simulating, learning and testing quantum systems.
During the reporting period, 15 research works have been completed that span the range of work undertaken in this project. Three of the key achievements presented in these works are as follows.
We have developed quantum procedures for accelerating general-purpose classical optimisation algorithms. First, we addressed branch-and-bound algorithms. Branch-and-bound is a general classical technique which is used to solve optimisation problems, often achieving a significant speedup over naive search techniques by exploring only a subset of possible solutions. Our algorithm based on quantum walks reduces the complexity scaling of classical branch-and-bound methods to approximately the square root of what it was originally. Second, we considered numerical optimisation algorithms. In a joint project with 4 undergraduate students and Prof Noah Linden, we found quantum speedups for a variety of numerical optimisation algorithms used classically (global optimisation algorithms under a Lipschitz constraint, backtracking line search, and the Nelder-Mead algorithm).
We have developed quantum algorithms for solving prominent problems relating to partial differential equations (PDEs) more efficiently than classical computers. PDEs are used throughout science and engineering to model physical systems. In the first of two works relating to quantum algorithms for PDEs, we undertook a comprehensive study of the heat equation, a prototypical PDE which models the movement of heat within an object. We showed that a quantum speedup is possible compared with classical algorithms, but that this speedup is relatively modest, as compared with the exponential speedup that might be expected. We also designed a quantum speedup of a technique known as the multilevel Monte Carlo method, which is one of the leading ways of solving certain differential equations efficiently, and applied this to stochastic differential equations that are important in mathematical finance.
We have also made contributions to understanding the complexity of quantum systems. We have made significant steps in classifying which quantum systems are "universal" (in the sense of being able to encode any other quantum system) for systems of qudits (the generalisation of quantum bits (qubits) to higher dimensions). We have also started a programme of work developing efficient quantum algorithms which run on near-term quantum computers. This includes a project giving a detailed theoretical and numerical analysis of applying quantum computing to speed up density matrix embedding theory, which is a method of solving a large quantum system via reducing it to solving a smaller embedded system, and a project applying quantum computing to solve a quantum model known as the antiferromagnetic Heisenberg model on the kagome lattice, which has been a well-studied and significant challenge for standard numerical methods.
We have developed quantum procedures for accelerating general-purpose classical optimisation algorithms. First, we addressed branch-and-bound algorithms. Branch-and-bound is a general classical technique which is used to solve optimisation problems, often achieving a significant speedup over naive search techniques by exploring only a subset of possible solutions. Our algorithm based on quantum walks reduces the complexity scaling of classical branch-and-bound methods to approximately the square root of what it was originally. Second, we considered numerical optimisation algorithms. In a joint project with 4 undergraduate students and Prof Noah Linden, we found quantum speedups for a variety of numerical optimisation algorithms used classically (global optimisation algorithms under a Lipschitz constraint, backtracking line search, and the Nelder-Mead algorithm).
We have developed quantum algorithms for solving prominent problems relating to partial differential equations (PDEs) more efficiently than classical computers. PDEs are used throughout science and engineering to model physical systems. In the first of two works relating to quantum algorithms for PDEs, we undertook a comprehensive study of the heat equation, a prototypical PDE which models the movement of heat within an object. We showed that a quantum speedup is possible compared with classical algorithms, but that this speedup is relatively modest, as compared with the exponential speedup that might be expected. We also designed a quantum speedup of a technique known as the multilevel Monte Carlo method, which is one of the leading ways of solving certain differential equations efficiently, and applied this to stochastic differential equations that are important in mathematical finance.
We have also made contributions to understanding the complexity of quantum systems. We have made significant steps in classifying which quantum systems are "universal" (in the sense of being able to encode any other quantum system) for systems of qudits (the generalisation of quantum bits (qubits) to higher dimensions). We have also started a programme of work developing efficient quantum algorithms which run on near-term quantum computers. This includes a project giving a detailed theoretical and numerical analysis of applying quantum computing to speed up density matrix embedding theory, which is a method of solving a large quantum system via reducing it to solving a smaller embedded system, and a project applying quantum computing to solve a quantum model known as the antiferromagnetic Heisenberg model on the kagome lattice, which has been a well-studied and significant challenge for standard numerical methods.
During the project we have developed many new quantum algorithms that outperform the best algorithms known previously, for tasks such as optimisation, solving partial differential equations, and identifying an unknown graph. We have also carried out theoretical and numerical analyses to understand the performance of previously proposed algorithms more accurately than has been achieved previously. All of these developments aim to extend the reach of quantum computing to new applications which will ultimately be of practical importance.
By the end of the project, we expect to have developed more quantum algorithms, and in particular near-term ones for problems relating to simulation of quantum systems, which may be practically important for solving problems in materials science. We will also have tested some of the algorithms developed on leading quantum computing hardware. Additionally, we expect to have gained a deeper understanding of the power of quantum computation, and the origin of the speedup that is achieved by quantum computers over classical methods.
By the end of the project, we expect to have developed more quantum algorithms, and in particular near-term ones for problems relating to simulation of quantum systems, which may be practically important for solving problems in materials science. We will also have tested some of the algorithms developed on leading quantum computing hardware. Additionally, we expect to have gained a deeper understanding of the power of quantum computation, and the origin of the speedup that is achieved by quantum computers over classical methods.