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arXiv:2608.20787

Quantum phase estimation for nondestructive monitoring and Wigner tomography of bosonic fields

Lucas R. S. Santos, Ciro M. Diniz, Daniel Z. Rossatto, Celso J. Villas-Boas

quant-phSubmitted Aug 21, 2026• Announced Aug 24, 2026• 0 citations
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Abstract

Quantum phase estimation is usually introduced as an algorithmic primitive for extracting eigenphases of unitary operators. Here we show that, when implemented through a dispersive light-matter interaction, it can also be used as a nondestructive measurement tool for bosonic fields. We consider a bosonic mode coupled to a multi-qubit register and calibrate the photon-number dependent phase shifts so that the register performs a number-resolved quantum phase estimation readout. Repeating this readout during dissipative evolution enables nondestructive monitoring of photon-number dynamics. We then show that the same readout can be converted into a Wigner tomography reconstruction by applying phase-space displacements before the quantum phase estimation block. Numerical reconstructions for Fock, coherent, and even/odd Schrödinger cat states show the expected nonclassical phase-space structures and near-unity Wigner overlap fidelities. The protocol provides a unified route to nondestructive monitoring and state tomography of bosonic fields, with direct relevance for bosonic-state characterization, calibration, and control in superconducting quantum architectures.

Comments
14 pages, 3 figures

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Quantum phase estimation is usually introduced as an algorithmic primitive for extracting eigenphases of unitary operators. Here we show that, when implemented through a dispersive light-matter interaction, it can also be used as a nondestructive measurement tool for bosonic fields. We consider a bosonic mode coupled to a multi-qubit register and calibrate the photon-number dependent phase shifts so that the register performs a number-resolved quantum phase estimation readout. Repeating this readout during dissipative evolution enables nondestructive monitoring of photon-number dynamics. We then show that the same readout can be converted into a Wigner tomography reconstruction by applying phase-space displacements before the quantum phase estimation block. Numerical reconstructions for Fock, coherent, and even/odd Schrödinger cat states show the expected nonclassical phase-space structures and near-unity Wigner overlap fidelities. The protocol provides a unified route to nondestructive monitoring and state tomography of bosonic fields, with direct relevance for bosonic-state characterization, calibration, and control in superconducting quantum architectures.
Quantum phase estimation is a cornerstone algorithm for determining eigenvalues of unitary operators with Heisenberg-limited precision. Conventional implementations rely on digital controlled-unitary operations together with phase-extraction circuits, which generally results in substantial circuit depth and hardware overhead. Here, we propose a hardware-native alternative that replaces digital controlled-unitary operations by analog bosonic interactions, naturally available in circuit quantum electrodynamics. The protocol extracts the phase through a sequence of binary threshold tests. A bosonic mode serves as an efficient quantum memory where the binary digits of the phase are encoded into the direction of phase-space rotations. These digits are then read out sequentially via high-fidelity homodyne measurements. We show that the protocol preserves Heisenberg scaling in estimation precision while simultaneously providing exponentially suppressed failure probability. By exploiting bosonic degrees of freedom and analog dispersive interactions, the scheme provides a hardware-efficient realization of quantum phase estimation and establishes a natural route toward implementing high-precision phase estimation on circuit quantum electrodynamics platforms.
Bosonic quantum systems provide a hardware-efficient platform for quantum information processing but remain challenging to characterise due to their large Hilbert space and the high measurement cost of state tomography. Existing approaches estimate the fidelity with respect to a single target state, making them unsuitable for applications in which physically equivalent states differ by phase space translations, rotations, or other transformations. Here, we introduce an adaptive reconstruction technique that estimates the fidelity with respect to a family of bosonic states while reconstructing the underlying Wigner function from a small number of measurements. The method combines a physics-informed parametric model with Bayesian inference, bootstrap, and active learning to iteratively select the most informative phase space sampling points. We implement the approach on a circuit quantum electrodynamics platform and benchmark it on Schrödinger cat states with amplitudes α∈[1,3]α\in[1,3]α∈[1,3]. The reconstruction yields reproducible fidelity estimates within a few minutes, remains robust to substantial displacements and rotations in phase space despite using a mismatched prior, and is sensitive to subtle state imperfections. We further compare the adaptive strategy with existing Wigner function sampling protocols experimentally, demonstrating the advantage of adaptive sampling for measurement-efficient fidelity estimation with respect to a family of cat states. Finally, we incorporate the reconstructed fidelity into the figure of merit used in a proof-of-principle closed-loop quantum optimal control experiment, demonstrating the applicability of the method to autonomous optimisation of bosonic quantum states.
Bosonic modes constitute a central resource in a wide range of quantum technologies, providing long-lived degrees of freedom for the storage, processing, and transduction of quantum information. Such modes naturally arise in platforms including circuit quantum electrodynamics, quantum acoustodynamics, and trapped-ion systems. In these architectures, coherent control and high-fidelity readout of the bosonic degrees of freedom are achieved via coupling to an auxiliary qubit. When operated in the strong dispersive regime, this interaction enables parity measurements of the mode which, in combination with phase-space displacements, constitute a standard experimental tool for full Wigner-function tomography. Here, we propose a protocol based on displaced parity measurements that allows for the preparation of a variety of bosonic quantum states. As a first example, we demonstrate the generation of squeezed states, achieving up to ~9 dB of squeezing after only three parity measurements, and show that the protocol is robust against experimental imperfections. Finally, we generalize our approach to the preparation of other paradigmatic bosonic states, including cat and Gottesman-Kitaev-Preskill states.
The quantum phase of a multiparticle system has remained a long-standing mystery in quantum physics. The absence of a phase operator that faithfully represents this physical quantity has prevented the formulation of a meaningful uncertainty relation with particle number. Here, we resolve this problem by measuring, for the first time, the quantum phase of multiparticle systems containing up to fifteen particles. We achieve this through a new quantum state tomography scheme that combines homodyne interferometry with photon-number-resolving detection. This approach enables us to reconstruct complementary photon-number and phase representations and establish the corresponding number-phase uncertainty relation. Our measurements further provide access to the mutually unbiased basis of a multiparticle system, a fundamental resource for multiparticle quantum information processing. Furthermore, we report the spatially resolved characterization of the quantum phase. These results provide an operational framework for measuring quantum phase in multiparticle systems, with implications for fundamental quantum physics and scalable multiparticle quantum technologies.
We propose and analyze an all-mechanical route to coherent control and quantum-state reconstruction of the fundamental flexural mode of a suspended carbon nanotube (CNT) operated in the anharmonic (Duffing/Kerr). A nearby atomic force microscope (AFM) provides a single, localized actuator that applies calibrated, time-dependent forces to the CNT. In the presence of mechanical anharmonicity this enables spectrally selective control of the lowest vibrational transition and thus supports effective two-level protocols such as Rabi oscillations and Ramsey interferometry. The same actuator also implements phase-space displacements required for Wigner function tomography via displaced-parity sampling, thereby unifying control and tomography without optical heating and without dedicated on-chip microwave drive lines at the CNT resonator. We develop explicit pulse sequences and a master equation framework that connect experimentally accessible signals to energy relaxation and phase coherence times and to parity-based quantum signatures, including negative regions of the Wigner function. The approach is compatible with multiple readout modalities, including direct AFM-based detection and dispersive coupling to superconducting circuitry such as Cooper-pair box, and/or a microwave cavity. Together, these techniques provide complete access to populations, coherence, and parity within a single device architecture. This minimal scheme provides a practical route to all-mechanical quantum control and state-resolved characterization of decoherence in mesoscopic mechanical systems.