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ACM International Conference on Computing Frontiers (CF)

Scalable Experimental Bounds for Dicke and GHZ States Fidelities

Shamminuj Aktar, Abdel‐Hameed A. Badawy, Andreas Bärtschi, Stephan Eidenbenz

Published May 9, 2023• 3 citations
Publisher

Abstract

Estimating the state preparation fidelity of highly entangled states on noisy intermediate-scale quantum (NISQ) devices is an important task for benchmarking and application considerations. Unfortunately, exact fidelity measurements quickly become prohibitively expensive, as they scale exponentially as O(3N) for N-qubit states, using full state tomography with measurements in all Pauli bases combinations. However, Somma et al. [20] established that the complexity could be drastically reduced when looking at fidelity lower bounds for states that exhibit symmetries, such as Dicke States and GHZ States. For larger states, these bounds still need to be tight enough to provide reasonable estimations on NISQ devices.

DOI
10.1145/3587135.3592197

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Estimating the state preparation fidelity of highly entangled states on noisy intermediate-scale quantum (NISQ) devices is an important task for benchmarking and application considerations. Unfortunately, exact fidelity measurements quickly become prohibitively expensive, as they scale exponentially as O(3N) for N-qubit states, using full state tomography with measurements in all Pauli bases combinations. However, Somma et al. [20] established that the complexity could be drastically reduced when looking at fidelity lower bounds for states that exhibit symmetries, such as Dicke States and GHZ States. For larger states, these bounds still need to be tight enough to provide reasonable estimations on NISQ devices.
Estimating the state preparation fidelity of highly entangled states on noisy intermediate-scale quantum (NISQ) devices is important for benchmarking and application considerations. Unfortunately, exact fidelity measurements quickly become prohibitively expensive, as they scale exponentially as O (3 N for N -qubit states, using full state tomography with measurements in all Pauli bases combinations. However, Somma et al.established that the complexity could be drastically reduced when looking at fidelity lower bounds for states that exhibit symmetries, such as Dicke states and GHZ states. These bounds must still be tight enough for larger states to provide reasonable estimations on NISQ devices. For the first time and more than 15 years after the theoretical introduction, we report meaningful lower bounds for the state preparation fidelity of all Dicke states up to N =10 and all GHZ states up to N =20 on Quantinuum H1 ion-trap systems using efficient implementations of recently proposed scalable circuits for these states. Our achieved lower bounds match or exceed previously reported exact fidelities on superconducting systems for much smaller states. Furthermore, we provide evidence that for large Dicke states ∣DN/2N⟩\left|\smash{D_{N/2}^{N}} \right\rangle∣DN/2N​⟩ , we may resort to a GHZ-based approximate state preparation to achieve better fidelity. This work provides a path forward to benchmarking entanglement as NISQ devices improve in size and quality.
Characterizing quantum states is essential for validating quantum devices, yet conventional quantum state tomography becomes prohibitively expensive as system size grows. Direct tomography offers a distinct route by enabling selective access to individual complex density-matrix elements, with a particular advantage for sparse target states and some verification tasks. Here we introduce a direct quantum state tomography scheme combining strong-measurement estimation with a fan-out coupling architecture. It enables mutually commuting interactions between system qubits and a single meter qubit, thereby achieving constant circuit depth, independent of system size. Notably, the involutory fan-out coupling reduces to the identity under repetition, enabling straightforward noise scaling for quantum error mitigation. We experimentally validate the scheme on a superconducting quantum processor via the IBM Quantum Platform, demonstrating four-qubit state reconstruction and single-circuit GHZ-state fidelity estimation up to 20 qubits with error mitigation. Consistent results with standard tomography and improved efficiency establish our scheme as a promising approach to reconstructing full quantum states and scalable verification tasks.
Reliable characterization of quantum states and quantum processes is central to the development of scalable quantum computing. Standard characterization techniques such as full quantum state tomography become rapidly impractical as system size increases. In this work we present two software-oriented fidelity estimation protocols that avoid full tomographic reconstruction while remaining experimentally relevant for noisy intermediate-scale quantum devices. The first protocol is direct fidelity estimation (DFE), where the fidelity between a target state and an experimentally prepared state is estimated from importance-sampled Pauli measurements. The second protocol is a survival-probability-based method that reconstructs fidelity from experimentally accessible moments of a noisy evolution and its approximate inverse, and is intrinsically robust against state-preparation-and-measurement (SPAM) errors. We discuss the theoretical basis of both approaches, analyze their noise sensitivity, and present numerical evidence on representative multiqubit states under depolarizing, dephasing, and amplitude-damping channels. For DFE we show that attenuation-aware correction improves robustness against measurement noise. For the survival-probability approach we show that an L2L_2L2​-optimized expansion substantially outperforms naive Taylor truncation in high-noise regimes.
Introduction: The scalability of superconducting quantum processors is fundamentally limited by environmental noise and the exponential measurement overhead required for multi-qubit entanglement verification. Methods: In this work, we introduce a hybrid machine learning (ML) framework utilizing convolutional neural networks (CNNs) to simultaneously characterize decoherence channels and verify multipartite entanglement. The model is trained on synthetic noisy syndrome measurement data augmented with empirical SPAM models and applied to a 5-qubit superconducting device, where the neural network acts as a non-linear entanglement witness. Results: sweeps. Furthermore, it successfully verifies genuine multipartite entanglement with a fidelity prediction of 99.1%, while requiring only 12% of the measurement bases demanded by full quantum state tomography (QST). Discussion: Crucially, our methodology bypasses two distinct challenges: it drastically reduces the measurement complexity inherent in multi-qubit entanglement verification, and it inherently filters out state preparation and measurement (SPAM) errors without requiring noise-amplifying matrix inversions. These numerical results demonstrate that ML-assisted protocols provide a highly resource-efficient pathway for the characterization of noisy intermediate-scale quantum (NISQ) devices.