Universal quantum computation can be realised by measuring individual particles in a specially entangled state of many particles, called a universal resource state. This model of quantum computation, called measurement-based quantum computation (MBQC), provides a framework for studying the intrinsic computational power of physical systems. In this thesis I will investigate how universal resource states may arise naturally as ground states of interacting spin systems. In particular, I will describe new 'phases' of quantum matter, which are characterised by having universal resource states as ground states. This direction of research allows us to draw on techniques from both many-body quantum physics and quantum information theory.
Universal quantum computation can be realised by measuring individual particles in a specially entangled state of many particles, called a universal resource state. This model of quantum computation, called measurement-based quantum computation (MBQC), provides a framework for studying the intrinsic computational power of physical systems. In this thesis I will investigate how universal resource states may arise naturally as ground states of interacting spin systems. In particular, I will describe new 'phases' of quantum matter, which are characterised by having universal resource states as ground states. This direction of research allows us to draw on techniques from both many-body quantum physics and quantum information theory.
While quantum computers can achieve dramatic speedups over the classical computers familiar to us, identifying the origin of this quantum advantage in physical systems remains a major goal of quantum information science. A useful tool here is measurement-based quantum computation (MQC), a computational framework utilizing the quantum entanglement found in many-body resource states. Not all resource states are useful for quantum computation however, so an important question is what properties of many-body entanglement characterize universal resource states, which can implement any quantum computation. Many-body states are also studied in condensed matter physics, where the collective behavior of quantum many-body systems sometimes define topological phases of matter. These phases are defined by nonlocal many-body entanglement, making topologically-ordered states natural candidates for MQC. We might wonder if these topological phases could be organized as phases of quantum computation, so that every state within the phase is universal for MQC. While phases of symmetry-protected topological order (SPTO) have arisen as natural candidates, previous attempts to demonstrate an MQC-SPTO correspondence were mostly limited to nonuniversal 1D spin chains, leaving the important 2D setting wide open. In this dissertation, we explore the wide and varied connections between MQC and SPTO, and obtain new results for 1D and 2D systems. After identifying a new MQC-SPTO correspondence within 1D spin chains, we move up and explore the operational use of 2D states with two complementary forms of SPTO. We create a new Union Jack resource state, whose different form of SPTO than previous 2D resource states permits a hierarchical notion of MQC universality. This state leads us to consider an idealized model of 2D SPTO, where we show that an additional symmetry condition makes these model states form universal resources for MQC only when they have nontrivial SPTO. We finally study the intrinsic complexity of SPTO-inspired states for classically intractable sampling, and identify inherent advantages of MQC for this purpose. Our work highlights the rich complexity available in states of entangled quantum matter, providing new evidence which sharpens our understanding of the diverse connections between MQC and SPTO.
We present the first examples of topological phases of matter with uniform power for measurement-based quantum computation (MBQC). This is possible due to a new framework for analyzing the computational properties of phases of matter that is more general than previous constructions, which have been limited to short-range entangled phases in one dimension. We show that ground states of the toric code in an anisotropic magnetic field yield a natural, albeit noncomputationally universal, application of our framework. We then present a new model with topological order the ground states of which are universal resources for MBQC. Both topological models are enriched by subsystem symmetries and these symmetries protect their computational power. Our framework greatly expands the range of physical models that can be analyzed from the computational perspective.
Measurement-based quantum computation (MBQC) performs quantum algorithms by local, adaptive measurements on an initially entangled many-body "resource" state. This thesis investigates MBQC through two tightly coupled lenses: (i) which physical states and phases of matter constitute universal resources, and (ii) how the temporal order induced by measurement adaptivity constrains and, in turn, is constrained by those resources. Resources from valence-bond constructions. We develop hybrid AKLT-type (valence-bond) resources on two-dimensional lattices and show how a generalized POVM reduces higher-spin degrees of freedom to effective qubits while preserving computational viability. We analyze worst-case and frustrated geometries (e.g., kagome and decorated lattices), identify planarity-restoring local operations, and use percolation-style diagnostics on the induced graph states to certify universality beyond idealized cluster-state settings. A two-parameter phase with noisy POVMs. Extending ground states across a phase connected to the AKLT point, we introduce a measurement-noise–aware framework with probabilistic restoration and closed-form weight formulae derived via a graphical (g-)calculus. Large-scale numerics that account for correlations map a two-parameter phase diagram and delineate regimes of universal vs. non-universal post-measurement graphs. These results exhibit a robust universal region that persists away from ideal measurements and clarify how noise and Hamiltonian deformation jointly control computational power. Temporal order and determinism. We develop a stabilizer-and-influence-matrix formalism that places temporal dependencies on the same footing as resource structure. Four theorems establish a two-way determination between (a) the classical processing relations (including the influence matrix that generates the partial order of measurements) and (b) the resource stabilizer in a normal form, up to local equivalence. This links gauge freedoms, byproduct handling, and flow/gflow–style constraints into a unified picture of when adaptive measurement schedules yield deterministic computation. Together, these contributions clarify when physically motivated ground states serve as universal MBQC resources under realistic measurement noise, and how temporal constraints fundamentally shape the space of admissible computations in MBQC.
Quantum phases of matter are resources for notions of quantum computation. In this work, we establish a new link between concepts of quantum information theory and condensed matter physics by presenting a unified understanding of symmetry-protected topological (SPT) order protected by subsystem symmetries and its relation to measurement-based quantum computation (MBQC). The key unifying ingredient is the concept of quantum cellular automata (QCA) which we use to define subsystem symmetries acting on rigid lower-dimensional lines or fractals on a 2D lattice. Notably, both types of symmetries are treated equivalently in our framework. We show that states within a non-trivial SPT phase protected by these symmetries are indicated by the presence of the same QCA in a tensor network representation of the state, thereby characterizing the structure of entanglement that is uniformly present throughout these phases. By also formulating schemes of MBQC based on these QCA, we are able to prove that most of the phases we construct are computationally universal phases of matter, in which every state is a resource for universal MBQC. Interestingly, our approach allows us to construct computational phases which have practical advantages over previous examples, including a computational speedup. The significance of the approach stems from constructing novel computationally universal phases of matter and showcasing the power of tensor networks and quantum information theory in classifying subsystem SPT order.
Quantum phases of matter are resources for notions of quantum computation. In this work, we establish a new link between concepts of quantum information theory and condensed matter physics by presenting a unified understanding of symmetry-protected topological (SPT) order protected by subsystem symmetries and its relation to measurement-based quantum computation (MBQC). The key unifying ingredient is the concept of quantum cellular automata (QCA) which we use to define subsystem symmetries acting on rigid lower-dimensional lines or fractals on a 2D lattice. Notably, both types of symmetries are treated equivalently in our framework. We show that states within a non-trivial SPT phase protected by these symmetries are indicated by the presence of the same QCA in a tensor network representation of the state, thereby characterizing the structure of entanglement that is uniformly present throughout these phases. By also formulating schemes of MBQC based on these QCA, we are able to prove that most of the phases we construct are computationally universal phases of matter, in which every state is a resource for universal MBQC. Interestingly, our approach allows us to construct computational phases which have practical advantages over previous examples, including a computational speedup. The significance of the approach stems from constructing novel computationally universal phases of matter and showcasing the power of tensor networks and quantum information theory in classifying subsystem SPT order.
Any computational process is built from a sequence of components. In this context, there are at least two ways in which “component” can be interpreted: physically and logically. The former can be understood as referring to the hardware for the computation, i.e., the physical substrate where the computation “takes place”, while the latter can be understood as referring to the software, i.e., the mathematical mapping of inputs to outputs. These two interpretations are by no means distinct: each logical component is implemented as the evolution of a physical component in some state. Accordingly, the logical requirements of a computation also specify the minimal physical requirements of the computational device. In quantum computational processes, the physical components are constructed from materials whose states and evolutions are described by quantum mechanics. At the logical level, this permits the use of quantum information theory: states are represented by vectors in a Hilbert space and evolutions are specified via unitary dynamics and measurements. In measurement-based quantum computational processes, the latter form of evolution, i.e. that driven by measurements, is most prominent. In this context, certain quantum states taken on the quality of a computational resource. That is to say, it is the combination of state and measurement together that produce the desired evolution for a given logical component. This thesis treats several topics related to the preparation and use of such resource states. In particular, much of the thesis will be concerned with states that support universal quantum computation; that is, for which any quantum computation is possible within the framework of MBQC. The first contribution in this regard consists in establishing a connection between a different framework of universal computation, namely that of parity quantum computing, and MBQC. This connection relates to both preparation and use since the former was developed in the context of quantum optimisation (a use) and utilises a specific type of encoding of quantum states (a preparation). The next sections consider the requirements under which specific sets of logical components suffice for universal quantum computation. These sets commonly arise in quantum information theory, and as state-measurement combinations in MBQC and parity quantum computing in particular. In addition to establishing results on the sizes of such sets, this thesis investigates the consequences for parity quantum computing and MBQC, including, e.g., considerations for the design of physical devices and the compilation of computations. Finally, several limits to the preparation and use of states for MBQC are investigated. In one case, an information-theoretic quantity is considered which can be used to quantify uncertainty regarding some piece of information. In the context of MBQC, the piece of information can pertain to, e.g., the choice of computation to perform on a given state, such as in the context of blind quantum computing. Thereafter, a method is developed capable of witnessing the inability to prepare various states under certain constraints that are relevant in, e.g., the context of quantum networks.