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Effective Mode Approximation for Probabilistic Verification of Collective Hamiltonians in Large Continuous-Variable Quantum Systems

José R. Rosas-Bustos, Jesse Van Griensven Thé, Roydon Andrew Fraser, Nadeem Said, Sebastian Ratto Valderrama, Mark Pecen, Alexander Truskovsky, Andy Thanos

Published May 2, 2026• 0 citations
Publisher

Abstract

The Effective Mode Approximation (EMA) is a verification-oriented framework for characterizing collective Hamiltonian dynamics in large continuous-variable (CV) quantum systems from experimentally accessible collective measurements. Rather than reconstructing a full mode-resolved Hamiltonian, EMA maps the observed dynamics onto a canonically normalized collective mode and tests whether summed quadrature trajectories are consistent with an effective harmonic description. We validate EMA using time-resolved homodyne sampling in Gaussian simulations of ring-coupled multi-qu-mode optical systems with N=8,16,32, and 64 modes. One-tone and two-tone sinusoidal models, selected using the Akaike Information Criterion (AIC), recover a stable dominant collective frequency across system size and produce residuals that remain centred near zero. The results show that EMA can verify dominant collective behaviour with a fixed number of effective parameters even when full microscopic reconstruction is impractical. EMA is therefore best understood not as a full-state ansatz, but as a low-overhead tool for validating collective dynamics under realistic measurement constraints in scalable CV hardware.

DOI
10.3390/e28050514

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Non-Gaussian interactions are a key ingredient for achieving universality in continuous-variable quantum computation, yet their experimental characterization and the validation of their correct implementation remain challenging tasks. In this work, we focus on a three-mode non-Gaussian trilinear Hamiltonian that has recently been realized in superconducting microwave platforms, and present a comprehensive theoretical analysis of the computational resources it generates, together with experimentally accessible protocols to validate their presence. We systematically investigate its ability to generate two key resources for quantum computation: multipartite entanglement and Wigner negativity. In particular, using displaced-parity Bell tests, we demonstrate the generation of nonlocal states and thereby provide an operational certification of multipartite entanglement in the non-Gaussian states produced by the dynamics. We further quantify the Wigner logarithmic negativity and benchmark it against that of established non-Gaussian resource states. Building on this resource-based characterization, we introduce a measurement-efficient protocol for the experimental validation of the Hamiltonian implementation without requiring full reconstruction of the Wigner function. The protocol combines the measurement of zero-variance observables (nullifiers and stabilizers) with a limited number of targeted phase-space measurements, leading to a drastic reduction of the experimental overhead.
Quantum state reconstruction is an essential element in quantum information processing. However, efficient and reliable reconstruction of nontrivial quantum states in the presence of hardware imperfections can be challenging. This task is particularly demanding for high-dimensional states encoded in continuous-variable (CV) systems, where a large number of grid-based measurements are often used to adequately sample relevant regions of phase space. In this work, we introduce an efficient and robust technique of optimized reconstruction with excitation-number sampling (ORENS) based on the idea of the generalized Q function. We use a standard bosonic circuit quantum electrodynamics (cQED) setup to experimentally demonstrate effective state reconstruction using the theoretically minimum number of measurements. Our investigation highlights that ORENS is naturally free of parasitic system dynamics and resilient to decoherence effects in the hardware, enabling it to outperform the conventional reconstruction techniques in cQED such as Wigner tomography. Finally, ORENS relies only on the ability to accurately measure the excitation number of a given CV state, making it a versatile and accessible tool for a wide range of CV platforms and readily scalable to multimode systems. Thus, our work provides a crucial and valuable primitive for practical quantum information processing using bosonic modes.
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Abstract We aim to devise feasible, efficient verification schemes for bosonic channels. To this end, we construct an average-fidelity witness that yields a tight lower bound for average fidelity plus a general framework for verifying optimal quantum channels. For both multi-mode unitary Gaussian channels and single-mode amplification channels, we present experimentally feasible average-fidelity witnesses and reliable verification schemes, for which sample complexity scales polynomially with respect to all channel specification parameters. Our verification scheme provides an approach to benchmark the performance of bosonic channels on a set of Gaussian-distributed coherent states by employing only two-mode squeezed vacuum states and local homodyne detections. Our results demonstrate how to perform feasible tests of quantum components designed for continuous-variable quantum information processing.
Cross-platform verification compares output states of independently operated bosonic and quantum optical systems when classical predictions and joint measurements are unavailable. Continuous-variable distributed inner product estimation (CV-DIPE) provides a local measurement approach, but infinite-dimensional state spaces require rigorous control of truncation error. We develop three CV-DIPE protocols for the natural class of energy-constrained continuous-variable quantum states. One uses coordinated displaced parity, another coordinated Haar measurements, and the third independent homodyne shadows. For arbitrary multimode inputs in this class, including entangled and non-Gaussian states, we establish truncation error bounds and worst-case sample complexity guarantees for the exact inner product. The leading terms of these bounds favor coordinated Haar measurements, whose global implementation is experimentally challenging. These protocols provide certified strategies for cross-platform verification of continuous-variable systems using only local measurement data.