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arXiv:2503.12870 · Physical Review A

Low-overhead tailoring and learning of noise in graph states

Guedong Park, Jinzhao Sun, Hyunseok Jeong

Published Sep 2, 2026• 0 citations
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Abstract

Graph and hypergraph states are important resource states for realizing universal quantum computation and diverse non-local physical phenomena. However, noise learning in such states is challenging due to their large entanglement and magic. This work establishes a low Clifford hierarchy circuit-based scheme for tailoring and learning noise in (hyper)graph states generated by multiple controlled-Z gates. The key is to convert the noisy input state into diagonal form and derive the convolution equation for the diagonal noise rates, which has a lower resource overhead. Such a diagonal form can be used in real quantum simulation by using the same conversion technique on the graph state input. We demonstrate that a single-depth Bell measurement is sufficient in our scheme for arbitrary graph states, while noise tailoring can be done via Pauli operations. After that, we suggest the Walsh-Hadamard transform and some approximation method for decoding the convolution equation to estimate the tailored noise. We prove that constant sampling and polynomial time complexities are sufficient to guarantee bounded noise estimation error with respect to the $l_2$-norm, which usually requires exponential complexity with existing noise estimation methods. The polynomial complexities are maintained even for the more stringent $l_1$ approximation under the sparse noise assumption. Conclusively, we propose novel methods for characterizing the noise properties of an entangled state at a lower cost than that of state generation. Moreover, compared with state verification methods, we can attain richer information on noise rates and enable noise mitigation, thereby providing guarantees for the preparation of graph states.

DOI
10.1103/rr8y-yz5m

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Graph states are an important class of entangled states that serve as a key resource for distributed information processing and communication in quantum networks. In this work, we propose a protocol that utilizes a Bell sampling subroutine to characterize the diagonal elements in the graph basis of noisy graph states distributed across a network. Our approach offers significant advantages over direct diagonal estimation using unentangled single-qubit measurements in terms of scalability. Specifically, we prove that estimating the full vector of diagonal elements requires a sample complexity that scales linearly with the number of qubits (O(n)\mathcal{O}(n)O(n)), providing an exponential reduction in resource overhead compared to the best known O(2n)\mathcal{O}(2^n)O(2n) scaling of direct estimation. Furthermore, we demonstrate that global properties, such as state fidelity, can be estimated with a sample complexity independent of the network size. Finally, we present numerical results indicating that the estimation in practice is more efficient than the derived theoretical bounds. Our work thus establishes a promising technique for efficiently estimating noisy graph states in large networks under realistic experimental conditions.
Efficiently characterizing large quantum states and processes is a central yet notoriously challenging task in quantum information science, as conventional tomography methods typically require resources that grow exponentially with system size. Here, we introduce a structure-agnostic learning framework for noisy nnn-qubit quantum circuits under~i.i.d.~single-qubit noise. We first prove that quantum states with unital noise channels admit an efficient learnable representation in the logarithmic-depth regime. We then extend this framework to quantum process tomography under constant noise, deriving a unified protocol that applies to both unital and non-unital noisy channels and retains efficient guarantees for logarithmic-depth circuits. This process-learning formulation is input-agnostic and imposes no distributional assumptions on the input quantum states. We further study a more general regime with arbitrary noise strength. In this setting, low-weight Pauli propagation induces a terminal truncation whose threshold depends logarithmically on the inverse accuracy, leading to quasi-polynomial complexity and near-unit success probability in the average case. In contrast to the preceding two results, this arbitrary-noise guarantee does not impose any restriction on the circuit depth, and therefore covers arbitrary-depth circuits, including both the noiseless limit (γ=0γ= 0γ=0) and the strong-decoherence regime (γ=Θ(1)γ= Θ(1)γ=Θ(1)). Numerical simulations of two-dimensional Hamiltonian dynamics further demonstrate the accuracy and robustness of the approach, including for structured circuits beyond the random-circuit setting assumed in the theoretical analysis. These results provide a scalable and practically relevant route toward characterizing large-scale noisy quantum devices, addressing a key bottleneck in the development of quantum technologies.
Hypergraph states are generalizations of graph states where controlled-ZZZ gates on edges are replaced with generalized controlled-ZZZ gates on hyperedges. Hypergraph states have several advantages over graph states. For example, certain hypergraph states, such as the Union Jack states, are universal resource states for measurement-based quantum computing with only Pauli measurements, while graph state measurement-based quantum computing needs non-Clifford basis measurements. Furthermore, it is impossible to classically efficiently sample measurement results on hypergraph states unless the polynomial hierarchy collapses to the third level. Although several protocols have been proposed to verify graph states with only sequential single-qubit Pauli measurements, there was no verification method for hypergraph states. In this paper, we propose a method for verifying a certain class of hypergraph states with only sequential single-qubit Pauli measurements. Importantly, no i.i.d. property of samples is assumed in our protocol: any artificial entanglement among samples cannot fool the verifier. As applications of our protocol, we consider verified blind quantum computing with hypergraph states, and quantum computational supremacy demonstrations with hypergraph states.
Learning unknown noise processes in quantum systems reveals their physical origin and informs error suppression, mitigation, and correction. Characterizing a general quantum process requires exponentially many parameters inferred from noncommuting measurements. Because these measurements cannot be performed simultaneously, the sample complexity grows exponentially. For Pauli channels, quantum memory and entangling operations can transform this task into measurements of commuting observables, reducing complexity exponentially. However, noise in these resources increases the overhead, leaving open whether any advantage remains in realistic devices. Here, we introduce error-mitigated entanglement-enhanced learning, analyze it theoretically, and demonstrate it experimentally. We quantify the noise-induced overhead, perform hypothesis testing with up to 64 qubits, and learn intrinsic noise in parallel-gate layers using up to 16 qubits of a superconducting processor. We show that noisy quantum memory provides a learning advantage, with a current experimental overhead of 1.33 ± 0.05 per qubit, below the no-entanglement lower bound of 2. The authors show that entanglement with noisy quantum memory can significantly speed up learning of quantum processes. With error mitigation, their method characterizes quantum noise at scale and outperforms entanglement-free approaches.
Magic states are essential for universal quantum computation and are widely viewed as a key source of quantum advantage, yet in realistic devices they are inevitably noisy. In this work, we characterize how noise on injected magic resources changes the classical simulability of quantum circuits and when it induces a transition from classically intractable behavior to efficient classical simulation. We adopt a resource-centric noise model in which only the injected magic components are noisy, while the baseline states, operations, and measurements belong to an efficiently simulable family. Within this setting, we develop an approximate classical sampling algorithm with controlled error and prove explicit noise-dependent conditions under which the algorithm runs in polynomial time. Our framework applies to both qubit circuits with Clifford baselines and fermionic circuits with matchgate baselines, covering representative noise channels such as dephasing and particle loss. We complement the analysis with numerical estimates of the simulation cost, providing concrete thresholds and runtime scaling across practically relevant parameter regimes.
Quantum circuits are considered more powerful than classical circuits and require exponential resources to simulate classically. Clifford circuits are a special class of quantum circuits that can be simulated in polynomial time but still show important quantum effects such as entanglement. In this work, we present an algorithm that simulates Clifford circuits by performing Gaussian elimination on a modified adjacency matrix derived from the circuit structure. Our work builds on a ZX-calculus tensor network representation of Clifford circuits that reduces to quantum graph states. We give a concise formula of amplitudes of graph states based on the LDL decomposition of matrices over GF(2), and use it to get efficient algorithms for strong and weak simulation of Clifford circuits using a tree-decomposition-based sparse LDL algorithm. The complexity of our algorithm matches the state of the art for weak graph state simulation and improves the state of the art for strong graph state simulation by taking advantage of Strassen-like fast matrix multiplication. Our algorithm is also efficient when computing many amplitudes or samples of a Clifford circuit. Further, our amplitudes formula provides a new characterization of locally Clifford equivalent graph states as well as an efficient protocol to learn graph states with low-rank adjacency matrices.