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arXiv:1108.5774

Measurement-based quantum computation--a quantum-mechanical toy model for spacetime?

Robert Raussendorf, Pradeep Kiran Sarvepalli, Tzu-Chieh Wei, Poya Haghnegahdar

Published Aug 29, 2011• 2 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.1108.5774

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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.
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.
Measurement-based quantum computation (MBQC) is a paradigm for quantum computation where computation is driven by local measurements on a suitably entangled resource state. In this work we show that MBQC is related to a model of quantum computation based on Clifford quantum cellular automata (CQCAs). Specifically, we show that certain MBQCs can be directly constructed from CQCAs which yields a simple and intuitive circuit model representation of MBQC in terms of quantum computation based on CQCA. We apply this description to construct various MBQC-based Ans\"atze for parametrized quantum circuits, demonstrating that the different Ans\"atze may lead to significantly different performances on different learning tasks. In this way, MBQC yields a family of hardware-efficient Ans\"atze that may be adapted to specific problem settings and are particularly well suited for architectures with translationally invariant gates such as neutral atoms.
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.
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.
Circuit-based quantum computing (CBQC), also known as the gate model, is currently regarded as the leading paradigm for performing quantum computation and has garnered widespread interest due to its relatively better-established implementation and close intuitive resemblance to classical computation. However, an alternative model that solely utilizes single-qubit measurements on entangled states to perform computation, namely the measurement-based quantum computing (MBQC), has started to attract further research interest as it is deemed more naturally suited for certain quantum hardware architectures, though its actual implementation remains largely underexplored. Recently, there have been preliminary efforts to characterize the specific regimes in which MBQC may potentially be more advantageous compared to CBQC. Nevertheless, a more systematic assessment is necessary to assess the suitability of each model across a broader range of more practical cases. In this work, we present a detailed comparative analysis of CBQC and MBQC, beginning with reviewing the theoretical foundations of each model before highlighting its distinguishing features that set the two paradigms apart. Furthermore, building on previous work, we propose a generalized regime analysis that more accurately captures the conditions under which a given quantum algorithm is more favorably executed in MBQC or CBQC. Subsequently, we identify critical parameters that likely determine whether a quantum algorithm is better suited to CBQC or MBQC, categorizing detailed key factors and performance metrics into layers of quantum computation. Next, we propose a decision flow that, given specific hardware constraints and algorithm requirements, guides the selection to the most appropriate model, discussing suitability in two representative worked examples of quantum algorithm applications: Quantum Fourier Transform (QFT) and Variational Quantum Algorithm (VQE). Last, we discuss the potential to combine both models as a hybrid approach to leverage the best of both worlds and explore its benefits for practical quantum computing applications.