We discuss the interdependence of resource state, measurement setting and temporal order in measurement-based quantum computation. The possible temporal orders of measurement events are constrained by the principle that the randomness inherent in quantum measurement should not affect the outcome of the computation. We provide a classification for all temporal relations among measurement events compatible with a given initial stabilizer state and measurement setting, in terms of a matroid. Conversely, we show that classical processing relations necessary for turning the local measurement outcomes into computational output determine the resource state and measurement setting up to local equivalence. Further, we find a symmetry transformation related to local complementation that leaves the temporal relations invariant.
We discuss the interdependence of resource state, measurement setting and temporal order in measurement-based quantum computation. The possible temporal orders of measurement events are constrained by the principle that the randomness inherent in quantum measurement should not affect the outcome of the computation. We provide a classification for all temporal relations among measurement events compatible with a given initial stabilizer state and measurement setting, in terms of a matroid. Conversely, we show that classical processing relations necessary for turning the local measurement outcomes into computational output determine the resource state and measurement setting up to local equivalence. Further, we find a symmetry transformation related to local complementation that leaves the temporal relations invariant.
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.
We propose measurement-based quantum computation (MBQC) as a quantum mechanical toy model for spacetime. Within this framework, we discuss the constraints on possible temporal orders enforced by certain symmetries present in every MBQC. We provide a classification for all MBQC temporal relations compatible with a given initial quantum state and measurement setting, in terms of a matroid. Further, we find a symmetry transformation related to local complementation that leaves the temporal relations invariant. After light cones and closed time-like curves have previously been found to have MBQC counterparts, we identify event horizons as a third piece of the phenomenology of General Relativity that has an analogue in MBQC.
Measurement-Based Quantum Computation (MBQC) is a model of quantum computation, which uses local measurements instead of unitary gates. Here we explain that the MBQC procedure has a fundamental basis in an underlying gauge theory. This perspective provides a theoretical foundation for global aspects of MBQC. The gauge transformations reflect the freedom of formulating the same MBQC computation in different local reference frames. The main identifications between MBQC and gauge theory concepts are: (i) the computational output of MBQC is a holonomy of the gauge field, (ii) the adaptation of measurement basis that remedies the inherent randomness of quantum measurements is effected by gauge transformations. The gauge theory of MBQC also plays a role in characterizing the entanglement structure of symmetry-protected topologically (SPT) ordered states, which are resources for MBQC. Our framework situates MBQC in a broader context of condensed matter and high energy theory.
The Heisenberg uncertainty principle is conventionally interpreted as a constraint on the precision of future measurement outcomes. We argue that this temporal asymmetry is not justified by the underlying mathematics of quantum mechanics, which is fundamentally time-symmetric. We propose that uncertainty applies equally in both temporal directions: the past has no greater ontological determinacy than the future. Between collapse events, prior states were no more determined than future states are now. Only the present moment of collapse constitutes anything approaching a definite physical state, and that moment is immediately receding into a past that was never fully determined before collapse occurred. We formalize this interpretation through a timesymmetric reading of the existing uncertainty relations and discuss its implications for the ontological status of the past, the measurement problem, and the foundations of physical determinism.
This thesis considers various paradigms of quantum computation in an attempt to understand the nature of the underlying physics. A standard approach is to consider unitary computation on pure input states, such that the outcome of the computation is determined by single computational basis measurement on the output state. It has been shown that there exists equivalent models of computation, such as measurement based quantum computing (MBQC), which provide insight into the role of entanglement and measurement. Furthermore, constraining or relaxing available resources can directly impacts the power of the computation, allowing one to gauge their role in the process. Here, we first extend known constructions such as Matrix Product States, MBQC and the oneclean qubit model to a mixed state formalism, in an attempt to develop computational models where noise acting on the physical resources, as might be experienced in laboratory settings, may be mapped to logical noise on the computation. Next, we introduce Measurement-Based Classical Computing, an essentially classical model of computation, wherein the complexity hard wired into probability distributions generated via quantum means yields surprising non classical results. Finally, we consider postselection the ability to discard displeasing measurement outcomes and argue that it may be used in a tame way, which does not provide a dramatic increase in computational power. From here, we develop a new Hamiltonian, based on a circuit to Hamiltonian construction, presenting evidence of QMA-hardness.
In measurement-based quantum computing, there is a natural ``causal cone'' among qubits of the resource state, since the measurement angle on a qubit has to depend on previous measurement results in order to correct the effect of by-product operators. If we respect the no-signaling principle, by-product operators cannot be avoided. Here we study the possibility of acausal measurement-based quantum computing by using the process matrix framework [Oreshkov, Costa, and Brukner, Nat. Commun. 3, 1092 (2012)]. We construct a resource process matrix for acausal measurement-based quantum computing restricting local operations to projective measurements. The resource process matrix is an analog of the resource state of the standard causal measurement-based quantum computing. We find that if we restrict local operations to projective measurements the resource process matrix is (up to a normalization factor and trivial ancilla qubits) equivalent to the decorated graph state created from the graph state of the corresponding causal measurement-based quantum computing. We also show that it is possible to consider a causal game whose causal inequality is violated by acausal measurement-based quantum computing.