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.
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.
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.
Quantum metrology exploits quantum mechanical effects to increase the precision of measurements of physical quantities. A wide variety of applications are currently being developed for scientific and technological purposes, however, most research relies on the use of highly entangled resource states that are challenging to generate and control in a given physical system. Here, we study the use of weighted graph states as more accessible resources for quantum metrology, which yield a favorable precision beyond the classical limit, approaching the Heisenberg limit. We find a notable robustness to variation in weights and less challenging weight requirements compared to standard graph states, which require a maximal weight at all edges. Both of these aspects reduce the practical demands in a physical setup, with the latter implying significantly less entanglement is required to gain a quantum advantage in metrology. We study the quantum Fisher information and optimized estimator variance of two identified sub classes of weighted graph states for an arbitrary number of N qubits, providing analytical forms and investigating their scaling. Our work opens up opportunities for using weakly entangled states in quantum-enhanced metrology.
Quantum metrology seeks to push the boundaries of measurement precision by harnessing quantum phenomena. Conventional methods often rely on maximally entangled resources, with states that are usually challenging to produce and sustain in practical setups. Here, we show that the maximally entangled constraint can be lifted by experimentally realizing a photonic two-qubit weighted graph state with an arbitrarily tunable graph weight. We use the generated state as a resource for quantum-enhanced phase sensing. We experimentally characterize the state and study its minimum estimator variance for two distinct local measurement bases as the graph weight varies from the maximally entangled to weakly entangled limit. We find excellent quantitative agreement with theoretical predictions, and observe a gain in precision beyond the classically attainable precision limit for graph weights substantially below the maximally entangled limit. This confirms that considerably less entanglement is required to achieve a quantum advantage. Albeit non-scalable in our test setup, this work represents the first experimental realization of weighted graph states with a tunable graph weight using linear optics. We expect more scalable versions of the model to be possible in an on-chip photonic platform.
We introduce a new family of models for measurement-based quantum computation which are deterministic and approximately universal. The resource states which play the role of graph states are prepared via 2-qubit gates of the form exp(−iπ2nZ⊗Z) . When n=2 , these are equivalent, up to local Clifford unitaries, to graph states. However, when n>2 , their behaviour diverges in two important ways. First, multiple applications of the entangling gate to a single pair of qubits produces non-trivial entanglement, and hence multiple parallel edges between nodes play an important role in these generalised graph states. Second, such a state can be used to realise deterministic, approximately universal computation using only Pauli Z and X measurements and feed-forward. Even though, for n>2 , the relevant resource states are no longer stabiliser states, they admit a straightforward, graphical representation using the ZX-calculus. Using this representation, we are able to provide a simple, graphical proof of universality. We furthermore show that for every n>2 this family is capable of producing all Clifford gates and all diagonal gates in the n -th level of the Clifford hierarchy.
The direct compilation of algorithm-specific graph states in measurement-based quantum computation (MBQC) can lead to resource reductions in terms of circuit depth, entangling gates, and even the number of physical qubits. In this work, we extend previous studies on algorithm-tailored graph states to periodic sequences of generalized Pauli rotations, which commonly appear in, e.g., Trotterized Hamiltonian simulation. We first implement an enhanced simulated-annealing-based algorithm to find optimal periodic graph states within local-Clifford equivalent MBQC resources. In addition, we derive a novel scheme for the preparation of resource states based on a graph state and a ladder of CNOT gates, which we term anticommutation-based MBQC, since it uncovers a direct relationship between the graph state and the anticommutation matrix for the set of Hamiltonians generating the computation. We also deploy our two approaches to derive universal resource states from minimal universal sets of generating Hamiltonians. Finally, we demonstrate and compare both of our methods based on various examples from condensed matter physics and universal quantum computation.
Weighted graph states extend standard graph states by associating phases with entangling edges, and may serve as resources for measurement-based quantum computation. We analyze how the two main fusion operations, Type-I and type-II, act on weighted graph states. Type-I fusion operates identically to the unweighted case, merging two one-dimensional weighted graphs, while preserving edge weights and success probabilities. In addition, the pool of two-qubit weighted graph states can be generated easily from Greenberger-Horne-Zeilinger states or Bell pairs. In contrast, type-II fusion requires a logical qubit, which can be formed only for specific weight configurations, and with a success probability below one-half. When successful, it fuses the states correctly, but its failure outcomes destroy the structure of the graphs, removing the good-failure feature known from ordinary graph states. These two obstacles can be overcome by using extra qubits and π weights in the one-dimensional weighted graph states that are being fused, which enables scalability. We compute the change in the entanglement entropy of the resulting link due to the fused states being weighted graph states (for generalized fusion), and classify the resulting states of a general non-Bell projection. These results define the practical limits of the fusion-based construction of weighted graph states. To our knowledge, this is the first time that fusion processes have been used to construct weighted graph states.