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arXiv:2506.11909

Measurement-based quantum computation with variable-range interacting systems

Ghosh, Debkanta, Keshav Das Agarwal, Pritam Halder, Aditi Sen De

Published Jun 13, 2025• 0 citations
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Abstract

We demonstrate that weighted graph states (WGS) generated via variable-range interacting Ising spin systems where the interaction strength decays with distance as a power law, characterized by the fall-off rate, can successfully implement single- and two-qubit gates with fidelity exceeding classical limits by performing suitable measurements. In the regime of truly long-range interactions (small fall-off rate), optimizing over local unitary operations, while retaining the local measurement scheme in the original measurement-based quantum computation (MBQC) set-up, enables the scheme to achieve nonclassical average fidelities. Specifically, we identify a threshold fall-off rate of the interaction above which the fidelity of both universal single- and two-qubit gates consistently exceeds $90\%$ accuracy. Moreover, we exhibit that the gate-implementation protocol remains robust under two realistic imperfections -- noise in the measurement process, modeled via unsharp measurements, and disorder in the interaction strengths. These findings confirm WGS produced through long-range systems as a resilient and effective resource for MBQC.

DOI
10.48550/arxiv.2506.11909

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We demonstrate that weighted graph states (WGS) generated via variable-range interacting Ising spin systems where the interaction strength decays with distance as a power law, characterized by the fall-off rate, can successfully implement single- and two-qubit gates with fidelity exceeding classical limits by performing suitable measurements. In the regime of truly long-range interactions (small fall-off rate), optimizing over local unitary operations, while retaining the local measurement scheme in the original measurement-based quantum computation (MBQC) set-up, enables the scheme to achieve nonclassical average fidelities. Specifically, we identify a threshold fall-off rate of the interaction above which the fidelity of both universal single- and two-qubit gates consistently exceeds 90%90\%90% accuracy. Moreover, we exhibit that the gate-implementation protocol remains robust under two realistic imperfections -- noise in the measurement process, modeled via unsharp measurements, and disorder in the interaction strengths. These findings confirm WGS produced through long-range systems as a resilient and effective resource for MBQC.
The cluster state acquired by evolving the nearest-neighbor (NN) Ising model from a completely separable state is the resource for measurement-based quantum computation. Instead of an NN system, a variable-range power-law interacting Ising model can generate a weighted graph state (WGS) whose genuine multipartite entanglement may reveal intrinsic characteristics of the evolving Hamiltonian. We establish that the generalized geometric measure (GGM) in the time evolved state with an arbitrary number of qubits is sensitive to falloff rates and the range of interactions of the evolving Hamiltonian. We report that the time-derivative and time-averaged GGM at a particular time can detect the transition points present in the falloff rates of the interaction strength, separating different regions, namely, long-range, quasilocal, and local ones in one- and two-dimensional lattices with deformation. Moreover, we illustrate that in the quasilocal and local regimes there exists a minimum coordination number in the evolving Ising model for a fixed total number of qubits which can mimic the GGM of the long-range model. In order to achieve a finite-size subsystem from the entire system, we design a local measurement strategy that allows a WGS of an arbitrary number of qubits to be reduced to a local unitarily equivalent WGS having fewer qubits with modified weights.
Weighted graph states are a natural generalization of graph states, which are generated by applying controlled-phase gates, instead of controlled-Z gates, to a separable state. In this paper, we show that uniformly weighted graph states on a suitable planar graph constitute universal resources for measurement-based quantum computation for an arbitrary nonzero constant weight. To our knowledge, this is the first example of universal resources prepared with only non-maximally entangling gates and has potential applications to weakly interacting systems, such as photonic systems.
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 computers offer impressive computational speed-ups over their present-day (classical) counterparts. In the measurement-based model, quantum computation is driven by single-site measurements on a large entangled quantum state known as a cluster state. This thesis explores extensions of the measurement-based model for quantum computation in qubit and continuous-variable systems. Within the qubit setting, we consider the task of characterizing how well a small-scale measurement-based quantum device can perform logic gates. We adapt a pre-existing scheme known as randomized benchmarking into the setting of measurement-based quantum computation on a one-dimensional cluster state. A key feature of randomized benchmarking is that it uses random sequences of gates. We show how the intrinsic randomness of measurement-based quantum computation can be harnessed when implementing them. Within the continuous-variable setting, we consider optical cluster states that can be generated with current technology. We propose a compact method for generating universal cluster states based on optical-parametric-oscillator technology. We consider how finite squeezing effects manifest in computation and show that pre-existing measurement-based protocols are suboptimal. We propose new measurement-based protocols that have better noise properties, compactness, and circuit flexibility. As an application, we introduce a measurement-based method for implementing interferometry. In this model, the finite squeezing noise can be dealt with as a photon-loss process. Building further on this work, we investigate the resource requirements of a measurement-based boson-sampling device, proving simultaneous efficiency in time, space, and squeezing (energy) resources. These results offer new insights into how to build, use, and characterize a measurement-based quantum computer.
Weighted graph states naturally arise when spin systems interact via an Ising-type interaction. First, we abstractly define the class of weighted graph states and demonstrate its computational accessibility. We show how reduced density matrices of a small number of spins (≈10) can be computed from arbitrarily large systems using weighted graph techniques and projected entangled pair techniques, and we discuss various entanglement measures accessible from these reduced density matrices. Second, we apply these findings to spin chains and lattices with long-range interactions and analytically derive area laws for the scaling of block-wise entanglement. Then, we turn to disordered spin systems, spin gases, which are connected to random weighted graph states and which share their entanglement properties. Finally, we use a spin gas as a bath that introduces decoherence in single as well as multipartite spin systems. The microscopic, exact decoherence model we obtain can operate in different regimes and exhibit non-Markovian features as well as spatially correlated noise effects.
Recently, it was shown that the non-local correlations needed for measurement-based quantum computation (MBQC) can be revealed in the ground state of the Affleck–Kennedy–Lieb–Tasaki (AKLT) model involving nearest-neighbour spin-3/2 interactions on a honeycomb lattice. This state is not singular but resides in the disordered phase of the ground states of a large family of Hamiltonians characterized by short-range-correlated valence bond solid states. By applying local filtering and adaptive single-particle measurements, we show that most states in the disordered phase can be reduced to a graph of correlated qubits that is a scalable resource for MBQC. At the transition between the disordered and Néel ordered phases, we find a transition from universal to non-universal states as witnessed by the scaling of percolation in the reduced graph state.