Periodic Reporting for period 4 - QAFA (Quantum Algorithms from Foundations to Applications)
Reporting period: 2023-11-01 to 2024-10-31
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 were 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.
These goals have all been achieved. The project has developed new quantum algorithms, especially for solving hard optimisation problems and for simulating quantum systems; has significantly reduced the complexity of solving key problems relating to materials science and other applications of quantum simulation; and has carried out the most complex simulations performed to date of certain quantum systems on quantum computing hardware. Overall, the project has made substantial progress in bringing quantum computing closer to real-world applications, and in understanding the power of quantum computers.
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 were 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.
These goals have all been achieved. The project has developed new quantum algorithms, especially for solving hard optimisation problems and for simulating quantum systems; has significantly reduced the complexity of solving key problems relating to materials science and other applications of quantum simulation; and has carried out the most complex simulations performed to date of certain quantum systems on quantum computing hardware. Overall, the project has made substantial progress in bringing quantum computing closer to real-world applications, and in understanding the power of quantum computers.
The project’s initial perspective was to develop the long-term theory of quantum algorithms and quantum computational complexity. Following the start of the project, to capitalise on exciting developments in the performance of quantum computing hardware and in the quantum industry, the quantum software startup Phasecraft (co-founded by the PI) was added as a partner, and the project’s focus shifted to include a greater emphasis on near-term quantum algorithms.
- New quantum algorithms and communication protocols
The new algorithmic ideas developed in this component of the project have allowed us, in some cases, to accelerate entire families of existing classical algorithms which are widely used in practical applications. For example, we gave a quantum algorithm for speeding up quadratically any classical algorithm based on the widely used technique called branch-and-bound. We developed techniques for speeding up some well-known numerical optimization algorithms, in a collaboration with 4 undergraduate students carried out as a summer project. We also developed techniques, based on previous work of the PI, for speeding up multilevel Monte Carlo methods, a fundamental technique in finance.
In other cases, we have been able to characterise the complexity of fundamental mathematical problems. We developed algorithms and complexity bounds on a variety of problems relating to graphs and matrices. The latter case gave an essentially complete characterisation of the complexity of the very natural problem of computing functions of sparse matrices, based on approximate polynomial degree. We carried out a comprehensive study on the performance of quantum algorithms for solving partial differential equations, evaluating many different methods and finding that the speedup over classical was less than previously expected.
- Stronger quantum-classical separations
Solving hard optimization problems has long been proposed as an important application of quantum computers, yet theoretical and numerical evidence that near-term quantum algorithms could outperform their classical counterparts has been lacking. We have shown for the first time that near-term quantum optimization algorithms can outperform classical ones and can even achieve exponential speedups over industry-standard classical algorithms. On the fundamental side, we have characterised the complexity of solving ground-state energy estimation problems for general quantum systems beyond qubits. This has identified an intricate mathematical structure behind many of these problems and has enabled us to determine large families of physical systems which are universal, in the sense that they can simulate any other quantum-physical system.
- Quantum algorithms for simulating quantum-mechanical systems
Simulating the static or dynamic properties of physical systems where quantum mechanics plays a key role is anticipated to be one of the most important applications of quantum computers. We have contributed to the development of new quantum algorithms for modelling materials which achieve speedups by factors of more than a million compared with the best previous quantum algorithms. Our approach brought materials modelling to within touching distance of the performance of near-term quantum hardware.
One of the most well-known benchmarks for the performance of quantum simulation algorithms is the notorious Fermi-Hubbard model, which may provide insights into high-temperature superconductivity. We implemented efficient quantum algorithms for producing low-energy states of instances of the Fermi-Hubbard model, which we had previously developed, on Google’s world-leading quantum hardware. Our efficient algorithms and error mitigation procedures enabled us to address systems that were 4x larger than the best previous result. We also gave a detailed analysis of the complexity of incorporating quantum algorithms for solving the Fermi-Hubbard model together with the Density Matrix Embedding Theory (DMET) method, which enables larger problem instances to be solved than the size of the quantum hardware allows to be addressed directly. More recently, we have designed new algorithms for simulating the time-dynamics of quantum systems more efficiently than the best previous algorithms.
- New quantum algorithms and communication protocols
The new algorithmic ideas developed in this component of the project have allowed us, in some cases, to accelerate entire families of existing classical algorithms which are widely used in practical applications. For example, we gave a quantum algorithm for speeding up quadratically any classical algorithm based on the widely used technique called branch-and-bound. We developed techniques for speeding up some well-known numerical optimization algorithms, in a collaboration with 4 undergraduate students carried out as a summer project. We also developed techniques, based on previous work of the PI, for speeding up multilevel Monte Carlo methods, a fundamental technique in finance.
In other cases, we have been able to characterise the complexity of fundamental mathematical problems. We developed algorithms and complexity bounds on a variety of problems relating to graphs and matrices. The latter case gave an essentially complete characterisation of the complexity of the very natural problem of computing functions of sparse matrices, based on approximate polynomial degree. We carried out a comprehensive study on the performance of quantum algorithms for solving partial differential equations, evaluating many different methods and finding that the speedup over classical was less than previously expected.
- Stronger quantum-classical separations
Solving hard optimization problems has long been proposed as an important application of quantum computers, yet theoretical and numerical evidence that near-term quantum algorithms could outperform their classical counterparts has been lacking. We have shown for the first time that near-term quantum optimization algorithms can outperform classical ones and can even achieve exponential speedups over industry-standard classical algorithms. On the fundamental side, we have characterised the complexity of solving ground-state energy estimation problems for general quantum systems beyond qubits. This has identified an intricate mathematical structure behind many of these problems and has enabled us to determine large families of physical systems which are universal, in the sense that they can simulate any other quantum-physical system.
- Quantum algorithms for simulating quantum-mechanical systems
Simulating the static or dynamic properties of physical systems where quantum mechanics plays a key role is anticipated to be one of the most important applications of quantum computers. We have contributed to the development of new quantum algorithms for modelling materials which achieve speedups by factors of more than a million compared with the best previous quantum algorithms. Our approach brought materials modelling to within touching distance of the performance of near-term quantum hardware.
One of the most well-known benchmarks for the performance of quantum simulation algorithms is the notorious Fermi-Hubbard model, which may provide insights into high-temperature superconductivity. We implemented efficient quantum algorithms for producing low-energy states of instances of the Fermi-Hubbard model, which we had previously developed, on Google’s world-leading quantum hardware. Our efficient algorithms and error mitigation procedures enabled us to address systems that were 4x larger than the best previous result. We also gave a detailed analysis of the complexity of incorporating quantum algorithms for solving the Fermi-Hubbard model together with the Density Matrix Embedding Theory (DMET) method, which enables larger problem instances to be solved than the size of the quantum hardware allows to be addressed directly. More recently, we have designed new algorithms for simulating the time-dynamics of quantum systems more efficiently than the best previous algorithms.
During the project we have developed many new quantum algorithms that outperform the best algorithms known previously, for tasks such as optimisation, modelling quantum-physical systems such as materials and chemicals, solving partial differential equations, and identifying an unknown graph. In some cases, these have reduced the cost of solving important problems for practical applications by factors of 1,000,000 or more. 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.