From Quantum Matter to Quantum Computers

The quest for quantum computers is motivated by their potential for solving problems that defy existing, classical, computers. The theory of computational complexity provides a rigorous framework for classifying the hardness of problems according to the computational resources, most notably time, needed to solve them. Its extension to quantum computers allows the relative power of quantum computers to be analyzed. This framework identifies families of problems which are likely hard for classical computers (“NP-complete”) and those which are likely hard for quantum computers (“QMA-complete”) by indirect methods. That is, they identify problems of comparable worst-case difficulty without directly determining the individual hardness of any given instance. Statistical mechanical methods can be used to complement this classification by directly extracting information about particular families of instances by studying random ensembles of them. These pose interesting (quantum) statistical mechanical questions and the results shed light on the difficulty of problems for large classes of algorithms as well as providing a window on the contrast between typical and worst case complexity. In this lecture, I will provide a brief overview of the basic ideas of how to quantify algorithmic complexity with reference to the circuit model of quantum computing. I will then discuss basic classical and quantum complexity classes and related canonical hard problems, such as quantum satisfiability. I will also introduce some of the related notions of 'hard' free energy landscapes in spin glasses and how ensembles of (quantum) optimization problems may exhibit intricate phase diagrams, with transitions between classically easy, classically hard and entangled phases. Time permitting, we will close with an snaphsot of analog quantum approaches to optimization problems, as implemented in the D-Wave and 2D Rydberg platforms.

Quantum Error Correction (QEC) is a key component of any large scale quantum computation. In stabilizer codes QEC  is implemented by periodically measuring a set of commuting operators, the outcome of which is called a syndrome. The problem of decoding is then to infer from the syndrome a recovery operation that corrects errors that might have occurred between syndrome measurements. It is well known that the decoding of so called "zero rate" stabilizer codes —  that are codes where the number of encoded qubits does not scale with the number of physical qubits — can be mapped onto statistical mechanics models with quenched disorder. I will introduce this mapping and then present some work on generalizing it to codes with finite rate — that is codes where the number of encoded qubits does scale with the number of physical qubits. As an application of this generalization, I will discuss the decoding of hyperbolic surface codes.

Practical quantum computing will require error rates that are well below what is achievable with physical qubits. Quantum error correction offers a path to algorithmically-relevant error rates by encoding logical qubits within many physical qubits, where increasing the number of physical qubits enhances protection against physical errors. However, introducing more qubits also increases the number of error sources, so the density of errors must be sufficiently low in order for logical performance to improve with increasing code size. Here, we report the measurement of logical qubit performance scaling across multiple code sizes, and demonstrate that our system of superconducting qubits has sufficient performance to overcome the additional errors from increasing qubit number. We find our distance-5 surface code logical qubit modestly outperforms an ensemble of distance-3 logical qubits on average, both in terms of logical error probability over 25 cycles and logical error per cycle (2.914%±0.016% compared to 3.028%±0.023%). To investigate damaging, low-probability error sources, we run a distance-25 repetition code and observe a $1.7 \times 10^{-6}$ logical error per round floor set by a single high-energy event ($1.6 \times 10^{-7}$ when excluding this event). We are able to accurately model our experiment, and from this model we can extract error budgets that highlight the biggest challenges for future systems. These results mark the first experimental demonstration where quantum error correction begins to improve performance with increasing qubit number, illuminating the path to reaching the logical error rates required for computation.

Spin Glasses, with their intrinsic disorder and frustration, have become paradigms in various disciplines such as statistical physics, optimization tasks and the theory of neural networks. Classical spin glasses can be promoted to quantum models by the introduction of a transverse field. The resulting quantum spin glass models not only form an interesting playground to study the interplay of disorder and frustration with quantum effects, but their quantumness may be exploited to shortcut optimisation tasks, for instance through quantum annealing. The quantum Sherrington-Kirkpatrick (SK) model is the textbook example of a quantum spin glass model. Whilst the famous Parisi Solution provides a full solution to the classical SK model, many open questions remain for the quantum SK model. In the talk I will give a brief introduction to the quantum SK model and present a variational ansatz for the ground states of the quantum SK model. I will show that the ansatz very well captures the fundamental aspects of the model. The ansatz further enables us to study some previously unexplored features, such as the entanglement structure of the ground states.

The field of thermodynamics is one of the crown jewels of classical physics. Thanks to the advent of experiments in cold atomic systems with long coherence times, our understanding of the connection of thermodynamics to quantum statistical mechanics has seen remarkable progress. Extending these ideas and concepts to the non-equilibrium setting is a challenging topic, in itself of perennial interest. Here, we present perhaps the simplest non-equilibrium class of quantum problems, namely Floquet systems, i.e. systems whose Hamiltonians depend on time periodically, H(t + T) = H(t). For these, there is no energy conservation, and hence not even a natural concept of temperature. We find that certain structures from equilibrium thermodynamics are lost, while entirely new non-equilibrium phenomena can arise, including a spectacular spatiotemporal `time-crystalline' form of order, recently observed experimentally on google AI's sycamore NISQ platform. References: for an introductory overview, see Nature Physics 13, 424–428 (2017). For an in-depth review, see arxiv:1910.10745 . NISQ experiment: Nature 601, 531 (2022).

The generic noisiness of the environment interacting with a spin system motivates us to study the dissipative aspects of its dynamics. Typically, dissipation causes energy relaxation and wave function decoherence, and thus appears to be destructive. However, it can also be harnessed as useful resources, especially, in stabilizing interesting dynamical steady states. In this talk, I will focus on one example, where a cooperative dissipation can induce a steady entanglement between two spin qubits. I will briefly introduce some recent approaches in studying magnetic systems from an open quantum system point of view, and if time allows, comment on the connection to the semiclassical description.

While quantum computers try to exploit the exponential growth of the Hilbert space as a resource, it renders simulations of generic quantum many-body systems on classical computers challenging. However, it turns out that many states of physical interest have only a moderate entanglement, and can still be represented efficiently and almost exactly with a tensor network ansatz. We will discuss one such ansatz class, namely matrix product states (MPS) for one-dimensional systems. We will look into a small code implementing the density matrix renormalization group (DMRG) algorithm, and employ it to find MPS representations of ground states.

In this talk, I would like to convey some of the ways in which condensed matter physics research progresses in real life, which often differs from the impression that can be given in text books. Starting from some general remarks on serendipity, I will then tell a story about research on my favourite superconductor, Sr2RuO4, starting as far back as 1988. I will try to use that story to illustrate a number of points, including the importance of achieving high levels of material perfection, the need to be constantly on the lookout for systematic errors, and the way in which scientific questions can drive unexpected experimental advances (another variation on the theme of serendipity). Finally, I will stress the need to ‘reality check’ any exciting new observation or theory by checking for its consistency with a remarkable but sometimes under-appreciated theory dating back to the 19th century – thermodynamics. Andrew Mackenzie

In recent years, it has become experimentally possible to microscopically implement Quantum Electrodynamics (QED) within materials. This is achieved in a novel regime, which is not relativistic, not in vacuum, and not perturbative (the effective fine-structure constant being of order one). It carries the promise of technological applications in quantum materials’ control, as well as quantum nonlinear optics and information processing. At the same time, the underlying many-body physics poses serious challenges to theoreticians. After introducing the basics of the implementations of QED with quantum matter both in solid state and ultracold atomic gases, I will discuss the main classes of quantum many-body models that can be realised, part of which are already experimentally available. Finally, I will briefly discuss some of our current research projects.

Quantum materials are a collection of atoms with interacting electrons and nuclei displaying emergent behaviour and topological properties—properties robust to local defects. The past two decades has witnessed an explosion in the field of topological materials: from weak interacting electrons to strongly correlated ones, topological materials represent one of the most exciting application for future technologies. High performance electronics, quantum information or ultrafast spintronics are just a few of the possible technologies that can be developed based on these materials. In this talk I will discuss the route to go from pure mathematical prediction of topological properties, through high through-put materials search to device fabrication. I will discuss both topological insulators, in non magnetic and magnetic phases as well as topological (chiral) semimetals using the the modern theory of topological band structure—Topological Quantum Chemistry — built upon symmetry-based considerations and complemented with chemical theories of bonding, ionization, and covalence. Consequently, it describes the universal global properties of all possible band structures and materials.

Spins associated to optically-active defects in diamond provide a platform to explore quantum simulations, computations and networks. In this lecture, I will introduce how we can control such spin systems and use them for quantum simulations of many-body physics. I will explain how we can use a single electron spin to detect and control up to 50 nuclear spin qubits in its environment, forming a flexible quantum simulator. Additionally, I will discuss how the capability to connect these systems into networks through optical entanglement links might provide future scalability to large systems for distributed quantum computations and simulations.

Three dimensional magnetic systems promise significant opportunities for both fundamental physics, and technological applications, for example providing higher density devices and new functionalities associated with complex topology and greater degrees of freedom [1,2]. With recent advances in both characterization and nanofabrication techniques, the experimental investigation of three dimensional magnetic systems is now possible [3,4,5], opening the door to the elucidation of new physical properties, and representing the first steps towards higher dimensional magnetic devices. Harnessing these recent advances in X-ray magnetic tomographic imaging, we have been able to map both the static configuration [3,5], and dynamical behaviour [4,6], of topological magnetic structures [3-8]. Understanding these complex configurations is challenging: recent advances in analytical techniques [7] have provided new capabilities to locate and identify 3D magnetic solitons, leading to the first observation of nanoscale magnetic vortex rings [7,9]. As well as existing within bulk and thin film systems, 3D spin textures can be introduced via the patterning of complex 3D magnetic nanostructures [10]. Such textures are not limited to the magnetization: we have recently observed highly coupled curvilinear nanosystems leading to the formation of topological textures within the magnetic stray field [8]. These new experimental capabilities for 3D magnetic systems open the door to complex three-dimensional magnetic structures, and their dynamic behaviour. In this talk I will discuss these new experimental capabilities of X-ray magnetic tomography and nanofabrication, and how they open the door to the elucidation of complex three-dimensional magnetic structures, and their dynamic behaviour. References [1] Fernández-Pacheco et al., Nature Communications 8, 15756 (2017). [2] C. Donnelly and V. Scagnoli, J. Phys. D: Cond. Matt. 32, 213001 (2020). [3] C. Donnelly et al., Nature 547, 328 (2017). [4] C. Donnelly et al., Nature Nanotechnology 15, 356 (2020). [5] K. Witte, et al., Nano Letters 20, 1305 (2020). [6] S. Finizio et al., Nano Letters (2022) [7] C. Donnelly et al., Nat. Phys. 17, 316 (2020) [8] C. Donnelly et al., Nature Nanotechnology 17, 136 (2022) [9] N. Cooper, PRL. 82, 1554 (1999). [10] L. Skoric et al., Nano Letters 20, 184 (2020).

Reinforcement learning (RL) is, alongside supervised and unsupervised learning, one of the three main pillars of modern machine learning. In RL, an artificial intelligence agent interacts with its environment in order to solve a task, by maximizing a reward signal. I will give an intuitive introduction to the basics of reinforcement learning and its mathematical framework (environment, states, actions, rewards, Markov decision processes, etc.). We will then discuss simple RL algorithms, such as policy gradient and Q-learning, recently used to teach AI agents to play video and board games. Finally, we will cover recent applications of RL to quantum control.

Quantum control is essential for most modern quantum technologies such as quantum simulation, computation, and metrology. These technologies are based on harnessing a large number of interacting quantum particles. However, the exponential growth of the Hilbert space dimension with the number of qubits makes it challenging to classically simulate quantum many-body systems and consequently, to devise reliable and robust optimal control protocols. In this talk, I will present a novel framework for efficiently controlling quantum many-body systems based on tensor networks and machine learning. I will introduce the concepts of matrix product states as efficient representations of quantum many-body states and reinforcement learning as a tool for performing optimal control. I will show you how these can be combined to tackle the problem of quantum many-body control. Finally, I will demonstrate how our technique performs on different state preparation tasks and showcase some useful features of the approach.

Symmetry-protected topological phases of matter have long been classified based on mappings from the Brillouin zone to the space of projectors, or flat band Hamiltonians. This is the basis of the ten fold way classification scheme. Here, we show effectively non-interacting topological phases are associated with mappings from the Brillouin zone to the space of other observables. We specifically consider topological phases of the spin degree of freedom of the ground state, protected by a generalized particle-hole symmetry, which are independent of the topological phases of the full set of degrees of freedom of the ground state. We show these phases realize distinctive momentum-space skyrmionic spin textures, and exhibit two types of bulk-boundary correspondence. We present recipes for constructing toy models, and also explore consequences of this physics in models for transition metal oxide superconductors as the generalized particle-hole symmetry occurs in centrosymmetric superconductors. When spin is not conserved, we find two kinds of topological phase transitions are possible. The second kind occurs without the closing of an energy gap while respecting the symmetry protecting the topological phase, due to the minimum spin magnitude becoming zero. This contradicts the flat band limit assumption used to construct the ten fold way classification scheme.