Nature, at it deepest level, is notoriously difficult to model, as quantum mechanical effects cause the size of the problems to grow exponentially. This poses major challenges in the accurate simulation of molecules and crystals, thus limiting the power of computers to drive major advances in the development of new materials (from batteries and solar cells to superconductors), new chemical processes (designing better catalysts) and new drugs (engineering molecules for desired biological and pharmacological effects). Each of these challenges can be addressed through tools we will help establish in this Prosperity Partnership. As Feyman observed, the best (and indeed only) way to accurately compute the behaviour of such quantum mechanical systems is to build a computer whose inner workings are fundamentally based on those same quantum mechanical principles. Such so-called 'quantum computers' require radically new hardware able to represent and maintain information in exotic quantum states involving superposition (where quantum bits can be 0 and 1) and entanglement. Major advances are being made world-wide using a variety of hardware platforms, and amongst the leading quantum processors today are those being developed at Google, based on superconducting circuits. In 2018, Google expects to announce a processor with 49 high-quality quantum bits - although this number may seem small compared to the billions of transistors in conventional processor chips, this 49-qubit processor will, we expect, demonstrate the ability to solve a computational problem beyond the capabilities of our most capable supercomputers. This first demonstration of quantum 'advantage' using a quantum processor chip, opens the door for a new research approach looking to characterise and harness the the capabilities of this new hardware and develop applications in the simulation and modeling of materials and molecules. The Partnership brings together the University of Bristol and UCL and their research groups with long-standing expertise in the theory of quantum computing and simulation, and Google, a world leader in the design and development of advanced quantum computing hardware based on superconducting qubits. Our goal is to develop new and improved algorithms, verification techniques and benchmarks for simulation of quantum systems on near-term quantum computers, which we will implement and demonstrate on Google's hardware. Such an industrial-academic collaboration would have been impossible a few years ago; now working together in this way is essential to efficiently address the main challenges in this area, as our ability to able to run and test problems on real quantum hardware will have a dramatic effect on the pace of quantum application development. In addition, the Partnership includes two UK startups developing quantum software and "quantum-inspired" software sphere, playing a strong role in the development of commercial applications of the results of this project. Through the Partnership, we will therefore build the foundation of a quantum software industry in the UK, with a specific focus on quantum simulation. Our programme is organised around a set of four main Challenges: - How can we optimise quantum simulation algorithms for imperfect quantum computers? - How do we test the behaviour of a quantum machine if it is classically un-simulatable? - What are the potential applications of quantum simulations in the medium term? - Can we quantify the computational complexity of problems and use this to improve algorithms? Each of these raises issues that are both fundamental and practical: the former involving the development of tools that can reframe these questions in a quantifiable way and the latter in in the formulation of explicit practical tests that can be implemented on current devices. In addressing these questions, we aim to develop a firm basis for the development of quantum software well-adapted to current architectures

CCP-QC is a network linking computational scientists with quantum computing scientists and engineers, to develop some of the first useful applications of quantum computers. Quantum computing is promising fundamentally faster computation as part of broader quantum technology development that includes more secure communications, and more sensitive measurements and imaging. Our conventional computers, including those in mobile phones, modern cars, and powering the internet, are based on silicon semicoductor technology. After half a century of growth, silicon semiconductor computer chips have been at the limit of what they can do for the past decade. Faster computing requires more computers, which use more electricity and this growth is thus limited. Quantum computing uses a different logic, enabling much faster computing for some types of problems. The engineering challenges are formidable, and we are still at the stage equivalent to the first semiconductor chips in the early 1960s. Early quantum computers are already available: developing applications to suit the capabilities of this hardware is the next step, to enable us to take advantage of the opportunities they offer to speed up our computations. An important set of computational tasks in materials, chemistry, physics, biology, and engineering is developed by communities supported by collaborative computational projects (CCPs). CCP-QC will network across these CCPs and the quantum computing community, to enable the CCP communities to enhance their computations by using quantum computers. It will do this by organising joint meetings, holding training days to teach computational scientists about quantum computing, supporting small projects to develop proof-of-principle code and demonstrations on early quantum computing hardware, and providing an online information resource on early quantum computing applications. CCP-QC will interface with the new National Quantum Computing Centre, to be launched in April 2020 and based on the STFC Harwell campus in Oxfordshire. CCP-QC will enable quantum computing hardware providers to have their hardware tested with real problems of importance to the computational science communities. The outcomes of such tests can thus influence the development of quantum computing hardware, leading to faster development of useful applications that are adapted to extract the best advantage from the early quantum hardware. The simulations carried out by the CCP communities cover a wide range of important applications, from smart materials (e.g., better solar cells and batteries) to drug design (bio-molecular simulation). CCP-QC will thus contribute to the development of faster computational methods in many important applications with wide-ranging scientific, social and economic benefits.

The primary objective of the QC2 CDT is to train the upcoming generation of pioneering researchers, entrepreneurs, and business leaders who will contribute to positioning the UK as a global leader in the quantum-enabled economy by 2033. The UK government and industry have demonstrated their commitment by investing £1 billion in the National Quantum Technologies Programme (NQTP) since 2014. In its March 2023 National Quantum Strategy document, the UK government reaffirmed its dedication to quantum technologies, pledging £2.5 billion in funding over the next decade. This commitment includes the establishment of the UKRI National Quantum Computing Centre (NQCC). The fields of quantum computation and quantum communications are at a pivotal juncture, as the next decade will determine whether the long-anticipated technological advancements can be realized in practical, commercially-viable applications. With a wide-ranging spectrum of research group activities at UCL, the QC2 CDT is uniquely situated to offer comprehensive training across all levels of the quantum computation and quantum communications system stacks. This encompasses advanced algorithms and quantum error-correcting codes, the full range of qubit hardware platforms, quantum communications, quantum network architectures, and quantum simulation. The QC2 CDT has been co-developed through a partnership between UCL and a network of UK and international partners. This network encompasses major global technology giants such as IBM, Amazon Web Services and Toshiba, as well as leading suppliers of quantum engineering systems like Keysight, Bluefors, Oxford Instruments and Zurich Instruments. We also have end-users of quantum technologies, including BT, Thales, NPL, and NQCC, in addition to a diverse group of UK and international SMEs operating in both quantum hardware (IQM, NuQuantum, Quantum Motion, SeeQC, Pasqal, Oxford Ionics, Universal Quantum, Oxford Quantum Circuits and Quandela) and quantum software (Quantinuum, Phase Craft and River Lane). Our partners will deliver key components of the training programme. Notably, BT will deliver training in quantum comms theory and experiments, IBM will teach quantum programming, and Quantum Motion will lead a training experiment on semiconductor qubits. Furthermore, 17 of our partners will co-sponsor and co-supervise PhD projects in collaboration with UCL academics, ensuring a strong alignment between the research outcomes of the CDT and the critical research objectives of the UK quantum economy. In total the cash and in-kind contributions from our partners exceed £9.1 million, including £2.944 million cash contribution to support 46 co-sponsored PhD studentships. QC2 will provide an extensive cohort-based training programme. Our students will specialize in advanced research topics while maintaining awareness of the overarching system requirements for these technologies. Central to this programme is its commitment to interdisciplinary collaboration, which is evident in the composition of the leadership and supervisory team. This team draws expertise from various UCL departments, including Chemistry, Electronics and Electrical Engineering, Computer Science, and Physics, as well as the London Centre for Nanotechnology (LCN). QC2 will deliver transferable skills training to its students, including written and oral presentation skills, fostering an entrepreneurial mindset, and imparting techniques to maximize the impact of research outcomes. Additionally, the programme is committed to taking into consideration the broader societal implications of the research. This is achieved by promoting best practices in responsible innovation, diversity and inclusion, and environmental impact.

Quantum information science and technologies (QIST) are uniquely placed to disrupt and transform sectors across the board. Quantum technologies, by exploiting the distinctive phenomena of quantum physics, can perform functions fundamentally unachievable by technologies based solely upon classical physics. For example, when applied to computing, calculations and operations that would take the best supercomputers hundreds of years to complete could be resolved within seconds using quantum computers; as another example, QIST can also be used in sensing and imaging to obtain enhanced precision in a variety of measurements ranging from gas concentrations to gravitational waves, supporting established industries in sectors like manufacturing, energy and healthcare. Furthermore, the application of quantum technologies will have significant implications within communications and security given their ability to break traditional encryption methods used to protect data within financial transactions or military communications while at the same time offering a range of novel, secure solutions largely compatible with the existing infrastructures. The potential of quantum technologies is well demonstrated through its significant financial and strategic backing globally. Restricted to academic environments up until the start of the last decade, the worldwide investment into quantum initiatives has now reached $33 billion, with significant contributions made across China, the US, and Europe. In the UK, the strategic importance of quantum technologies is clear: with a strategic commitment of £2.5 billion over the next decade, EPSRC has listed Quantum Technologies a mission-inspired research priority and the Department for Science Innovation and Technology have named quantum technologies as one of their seven technology families within the UK's Innovation Strategy. It is clear that, around the world, quantum technologies are flourishing. While the technological potential and national importance of QIST to the UK is undeniable, a key challenge to realising our ambitions in this area is the ability to develop a quantum workforce of capable physicists, engineers, computer scientists, and mathematicians with both the requisite expertise in quantum information science and expertise in the technologies that will realise it. In addition, the leaders of the UK's quantum future must possess critical professional skills: they must be excellent communicators, leaders, entrepreneurs, and project managers. To meet this key ambition and its resultant needs, the programme offered by the Quantum Information Science and Technologies Centre for Doctoral Training (QIST CDT) is uniquely positioned to deliver the diversity of skills and experience needed to supply the UK with internationally renowned QIST leaders across policy, innovation, research, entrepreneurship, and science communication. QIST CDT students will receive academic training delivered by world-recognised top educators and researchers; undertake industrially-relevant training modules co-delivered with industry partners; gain hands-on experience within world-leading quantum research laboratories; receive one-to-one entrepreneurial mentorship; undergo intellectual property and science policy training; undertake on-site industry placements; and complete multi-faceted cohort projects designed to develop multidisciplinary teamwork. This combination of world-class academic research training, which can be undertaken in a vast array of quantum-technology-relevant sectors, with bespoke instruction in professional skills driven by the needs of current and future quantum industry, will produce graduates with a drive to make a difference in Quantum Technologies and the skills to make that happen.

This CDT will create a cohesive, internationally-leading, cross-domain training and research community fusing algebraic, geometric and quantum methods across Algebra, Geometry and Topology, Mathematical Physics, and their Interfaces. The scientific aim of our CDT is no less than to develop new foundations unifying all three disciplines, and in the process to bolster and future-proof UK capability in mathematics. The breadth of mathematical mastery necessary to achieve these aims, on which our training programme is based, is of the highest international standard, and training students in this area requires both the deep focus and the wide scope which only the resources of a CDT can enable. Our three scientific areas Algebra, Geometry and Quantum Fields are established, flagship, internationally-leading areas of UK mathematical strength. Algebra: quite simply *the* language, and controlling structure, of symbolic computation and symmetry. Geometry: the mathematically rigorous foundations of our human spatial and visual intuition. Quantum Fields: the mathematical incarnation of our quantum physical reality. A hallmark feature of 21st century mathematics is the dramatically increased synergy and inter-dependence between these three fundamental disciplines. Whereas in centuries past mathematics and physics interacted primarily through analysis and calculus, the advent of quantum mechanics posits a fundamentally different, fundamentally algebraic, set of laws for the universe. Geometry enters irrevocably when we pose quantum mechanical laws in the presence of fields, such as the electro-magnetic and gravitational fields, which permeate throughout time and space. A surprising and thrilling discovery of 21st century mathematics has been that the mathematically rigorous study of quantum fields yields some of the most powerful predictive theories within algebra and geometry, even to questions with no a priori physical formulation. These fundamental scientific developments have had a vast and direct impact on our modern world, and on a remarkably short timescale. Algebra, geometry and quantum fields are the driving force behind key developments such as internet search, quantum computation, machine learning, and both classical and quantum cryptography. Society and industry need the students we will train. Our graduates' skills are both fundamentally transferable and widely applicable across many external partnerships and stakeholders. The Deloitte report, commissioned by EPSRC, attributed 2.8M jobs and £200BN of the UK economy to mathematical sciences research, highlighting R&D, computing/tech, public administration, defence, aerospace and pharmaceuticals as economic sectors requiring graduates with advanced mathematical training. Sustainable energy consulting has since emerged as a further industry requiring ever-advanced mathematical sophistication. Crucially a physical and mathematical powerhouse needs to be a diverse powerhouse, yet the traditional structure of training in these areas has inhibited diversity of entrants, both to career academia and to industry. Building on our track record, and equipped with the resources and flexibility only a CDT can provide, we will create a diverse and confident cohort, equipped with the mathematical skillsets needed for our tech-led future to flourish, and able to influence a wide range of people, sectors and institutions.