Information-Scrambling-Enhanced Quantum Sensing Beyond the Standard Quantum Limit

TL;DR

This work tackles the quantum sensing challenge of decoherence and entanglement generation at scale by introducing butterfly metrology, a scrambling-enhanced protocol that converts local interactions into delocalized, metrologically useful correlations. Implemented on a 9-qubit cross-shaped superconducting processor, the protocol uses forward and backward evolution to create a butterfly state and employs Loschmidt echo and out-of-time-ordered correlators (OTOCs) to quantify information scrambling. The experiment demonstrates sensing beyond the standard quantum limit (SQL) with a maximal inverse sensitivity of $1/\eta = 3.78$ for $N=9$, approaching the bound $\eta \approx 2/N$ (i.e., $1/\eta \approx N/2$), and shows robustness to both coherent control errors and probe noise. These results establish a scalable, noise-resilient path toward practical quantum sensing advantages via scrambling across programmable quantum processors.

Abstract

Quantum sensing promises measurement precision beyond classical limits, but its practical realization is often hindered by decoherence and the challenges of generating and stabilizing entanglement in large-scale systems. Here, we experimentally demonstrate a scalable, scrambling-enhanced quantum sensing protocol, referred to as butterfly metrology, implemented on a cross-shaped superconducting quantum processor. By harnessing quantum information scrambling, the protocol converts local interactions into delocalized metrologically useful correlations, enabling robust signal amplification through interference of the scrambled and polarized quantum states. We validate the time-reversal ability via Loschmidt echo measurements and quantify the information scrambling through out-of-time-ordered correlators, establishing the essential quantum resources of our protocol. Our measurements reveal that the sensing sensitivity surpasses the standard quantum limit (SQL) with increasing qubit number, reaching 3.78 in a 9-qubit configuration, compared to the SQL of 3.0. The scheme further exhibits inherent robustness to coherent control errors and probed signal noise. This work demonstrates a readily scalable path toward practical quantum sensing advantages with prevalent experimental platforms.

Figures (5)

• Figure 1: Concept and protocol for quantum sensing enhanced by information scrambling.a, Schematic of the experimental system and the butterfly quantum state. The lower panel shows the architecture of the 9-transmon superconducting quantum processor. The upper panel illustrates the butterfly state, an entanglement of a polarized state $\lvert 0 \rangle^{\otimes 9}$ and a scrambled state $\lvert \psi_{\textrm{sc}} \rangle$, engineered to approach a sensitivity scaling of half the Heisenberg limit. b, Interferometric representation of the protocol. The sensing signal $\phi$ is imprinted as a global phase on each branch of the butterfly state $\lvert \psi_{B} \rangle$, analogous to a quantum-enhanced Mach--Zehnder interferometer. c, The full metrological protocol: (i) Butterfly state preparation via a localized perturbation $(I + i V)/\sqrt{2}$ and forward unitary evolution $U(t)$, (ii) Signal acquisition during a sensing time $t$, and (iii) Readout via a rotation operation and projective measurement of the central qubit.
• Figure 2: a, Quantum circuit for measuring the Loschmidt echo. b, Experimental decay of the Loschmidt echo overlap fidelity, $F_{fid}(t)$, for $N=9$ qubits. Data points (diamonds) show experimental measurements with error bars. The solid line shows numerical simulation with zero frequency detuning ($\Delta f = 0~\mathrm{MHz}$), which shows excellent agreement with the experimental data. The dashed line shows simulation with Gaussian-distributed detunings ($\sigma_f = 3~\mathrm{MHz}$), demonstrating how frequency disorder perturbs the time-reversal evolution. c, Quantum circuit for probing information scrambling via the OTOC. d, Experimental measurement of the OTOC, $C(t)$, showing its characteristic decay at a evolution time of $500~\mathrm{ns}$, signaling the onset of scrambling. Data points show experimental measurements for different operator combinations $C$, $C_p$, and $C_m$, while dashed lines show corresponding numerical simulations.
• Figure 3: Quantum metrology in the 9-qubit configuration.a,Random initial state: experimental interference pattern. Measured local expectation value $\langle \sigma_x \rangle$ as a function of signal phase $\phi$ and evolution time $t$, quantifying interference between the scrambled quantum state and the polarized sensing state. Data correspond to an ensemble of random initial states. b,Random initial state: numerical simulation. Theoretically predicted $\langle \sigma_x \rangle$ from exact many-body simulation of the experimental protocol. c,Random initial state: extracted sensitivity. Inverse sensitivity $1/\eta$ as a function of time, derived from the interference pattern in panel a. The shaded regions indicate metrological performance relative to the standard quantum limit (SQL): below SQL (gray) corresponds to classical sensing, while above SQL (red) demonstrates quantum-enhanced sensing. d--f,Specific initial state. Same sequence as panels a--c, but for a deterministically prepared initial state, highlighting enhanced sensitivity. g,Interference fringe. Cross-section of pannel at the optimal time (50 ns), showing the amplified fringe curvature that yields maximal inverse sensitivity $1/\eta = 3.78$ (fourier fit, dashed line), surpassing the SQL.
• Figure 4: Inverse sensitivity scaling of the butterfly metrology protocol. The figure illustrates the scaling behavior of the inverse sensitivity as a function of the qubit number $N$, with the background divided into four distinct performance regimes: the classical metrology region (below the SQL), the butterfly metrology region (between the SQL and half of the Heisenberg limit), the super-butterfly metrology regime (between half of the Heisenberg limit and the Heisenberg limit), and the super-Heisenberg region (above the Heisenberg limit). The butterfly metrology limit at $\eta \simeq 2/N$ represents the theoretical upper bound for the butterfly metrology protocol. Experimental scaling is shown for two scenarios: the downward-oriented green triangle corresponds to measurements using a fixed initial state, while the upward-oriented blue triangle represents the average over randomly sampled initial states. Both cases exhibit favorable scaling beyond the SQL, approaching the butterfly metrology bound and demonstrating the metrological advantage enabled by scrambling dynamics.
• Figure 5: Measured inverse sensitivity under Gaussian noise. The protocol's robustness is quantified by injecting noise during the time‑reversed state preparation and signal sensing stages. a,Qubit‑frequency noise: Inverse sensitivity $1/\eta$ versus $t$ for five values of $\sigma_\omega/2\pi$ (in MHz). b,Signal‑phase noise: Inverse sensitivity $1/\eta$ versus $t$ for five values of $\sigma_\phi$ (in rad). The protocol retains quantum enhancement (sensitivity above SQL) for noise strengths up to $\sigma_\omega/2\pi \approx 0.3\,\mathrm{MHz}$ and $\sigma_\phi \approx 0.2\,\mathrm{rad}$.