We show that, in contrast to ones of distinguishable particles, random bosonic states of any purity typically achieve the Heisenberg scaling in lossy phase sensing. A photon-counting measurement then does the job, while the desired states can be simulated with help of short random optical circuits.
Quantum sensors have been shown to enable more precise measurements than classical sensors. By using entanglement and coherent evolution, the estimation error can be made to scale better in interrogation time, photon count, and other resources, achieving Heisenberg-limited scaling. However, environmental noise and hardware errors typically destroy the coherence of the sensitive quantum states. This motivates the use of fault-tolerant techniques to create more robust quantum sensing protocols. Such fault-tolerant quantum sensing approaches have been shown to enhance sensitivity for qubit systems, but many practical sensors require bosonic probes. In this paper, we propose a fault-tolerant protocol using bosons to estimate displacements in the presence of dephasing noise. We construct an optimal error-correcting code under a mean photon-number constraint. Controlled displacements couple the probe to a repetition-encoded qubit register, while repeated boson-parity measurements and resets remove probe noise without erasing the stored logical phase. Under circuit-level bit-flip noise, a distance-d code suppresses logical failures to order p(d+1)/2. We show that, for a decay rate γA, a total sensing time T and a mean photon number budget N, and for sufficiently small residual signals, the Fisher information scales as NT2 over interrogation times of order γA−(d+1)/2. This is a significant improvement over the non-fault-tolerant approach that achieves Heisenberg scaling only for a total time O(γA−1).
We study how useful random states are for quantum metrology, i.e., surpass the classical limits imposed on precision in the canonical phase estimation scenario. First, we prove that random pure states drawn from the Hilbert space of distinguishable particles typically do not lead to super-classical scaling of precision even when allowing for local unitary optimization. Conversely, we show that random states from the symmetric subspace typically achieve the optimal Heisenberg scaling without the need for local unitary optimization. Surprisingly, the Heisenberg scaling is observed for states of arbitrarily low purity and preserved under finite particle losses. Moreover, we prove that for such states a standard photon-counting interferometric measurement suffices to typically achieve the Heisenberg scaling of precision for all possible values of the phase at the same time. Finally, we demonstrate that metrologically useful states can be prepared with short random optical circuits generated from three types of beam-splitters and a non-linear (Kerr-like) transformation.
Quantum metrology typically demands the preparation of exotic quantum probe states, such as entangled or squeezed states, to surpass classical limits. However, the need for carefully calibrated system parameters and finely optimized quantum controls imposes limitations on scalability and robustness. Here, we circumvent these limitations by introducing an echoed random process that achieves sensitivity approaching the Heisenberg limit while remaining blind to the random probe state. We demonstrate that by simply driving a Kerr nonlinear mode with random pulses, the emergence of sub-Planck phase-space structures grants high sensitivity, eliminating the need for complex quantum control. The protocol is statistically robust, yielding high performance across broad driving parameter ranges while exhibiting resilience to control fluctuations and photon loss. Broadly applicable to both bosonic and qubit platforms, our work reveals a practical, hardware-efficient, scalable, and optimization-free route to quantum-enhanced metrology in high-dimensional Hilbert spaces.
Optical phase measurement through its application in quantum metrology has pushed the precision limit with which some physical quantities can be measured accurately. At the very fundamental level, the laws of quantum mechanics dictate that the uncertainty in phase estimations scales as 1/N, where N is the number of quantum resources employed in the protocol [1]. This is the well known Heisenberg limit (HL) which is quadratically better than the traditional precision limit known as the standard quantum limit (SQL) with uncertainty asymptotically scaling as 1/sqrt{N} [1]. Several experiments have demonstrated that the SQL can be beaten by using an entangled state as the probe and a specific measurement scheme for ab initio estimation of unknown phases [2,3]. It has also been shown experimentally that even in the absence of the entanglement one can measure an unknown phase with imprecision scaling at the HL [4]. In this work we first present a new protocol able to estimate an optical phase at the Heisenberg limit, and then experimentally explore fundamental and practical issues in generating high-quality novel entangled states, for use in this protocol and beyond. Our aim in this study is to measure an unknown phase in the interval [0,2pi) with uncertainty attaining the exact HL. There is a condition that should be met to address this objective: preparation of an optimal state [5]. This would cover part of the presentation through which we explain how to experimentally realise such an optimal state with the current technological limitations and the feasibility of the scheme. In particular, we generate an entangled 3-photon (2-photon) state of specific superposition of GHZ (Bell) states. Our numerical simulation of the phase measurement gate together with the experimental outcomes show that the created state should have a high fidelity and purity to be able to have the phase uncertainty achieving the exact HL. Therefore, we briefly explain the modelling for experimental imperfections and finally present the results of experimental phase measurements.
Quantum metrology offers the potential to surpass its classical counterpart, pushing the boundaries of measurement precision toward the ultimate Heisenberg limit. This enhanced precision is normally attained by utilizing large squeezed states or multiparticle entangled quantum states, both of which are often challenging to implement and prone to decoherence in real quantum devices. In this work, we present a versatile and on-demand protocol for deterministic parameter estimation that leverages two efficient state-transfer operations on a single bosonic mode. Specifically, we demonstrate this protocol in the context of phase estimation using the superposition of coherent states in the bosonic circuit quantum electrodynamics (cQED) platform. With low average photon numbers of only up to 1.76, we achieve quantum enhanced precision approaching the Heisenberg scaling, reaching a metrological gain of 7.5 ( 6 ) dB. Importantly, we show that the gain or sensitivity range can be further enhanced on the fly by tailoring the input states, with different superposition weights, based on specific system constraints. The realization of this versatile and efficient scheme affords a promising path toward practical quantum enhanced sensing, not only for bosonic cQED hardware but also readily extensible to other continuous-variable platforms.
Abstract Two-mode interferometers lay the foundations for quantum metrology. Instead of exploring quantum entanglement in the two-mode interferometers, a single bosonic mode also promises a measurement precision beyond the shot-noise limit (SNL) by taking advantage of the infinite-dimensional Hilbert space of Fock states. Here, we demonstrate a single-mode phase estimation that approaches the Heisenberg limit (HL) unconditionally. Due to the strong dispersive nonlinearity and long coherence time of a microwave cavity, quantum states of the form (∣0⟩+∣N⟩)/2 0 + N ∕ 2 can be generated, manipulated and detected with high fidelities, leading to an experimental phase estimation precision scaling as ∼ N −0.94 . A 9.1 dB enhancement of the precision over the SNL at N = 12 is achieved, which is only 1.7 dB away from the HL. Our experimental architecture is hardware efficient and can be combined with quantum error correction techniques to fight against decoherence, and thus promises quantum-enhanced sensing in practical applications.