Experimental critical quantum metrology with the Heisenberg scaling

The study addresses whether adiabatic critical quantum metrology can achieve Heisenberg-limited precision in a feasible experimental setting. Conventional quantum metrology enhances precision beyond the shot-noise limit using entanglement, but practical noise and the need for highly entangled states or long coherent evolutions limit performance. Critical quantum metrology leverages enhanced sensitivity near quantum critical points and robustness of adiabatic evolution, yet prior schemes often rely on continuous QPTs only present at the thermodynamic limit with fixed minimal gap, making experiments difficult. The authors propose and test an approach based on a first-order QPT in a minimal two-spin system, enabling tunable gaps and practical adiabatic passages to encode and estimate a magnetic field with Heisenberg scaling in time.

Prior work established quantum metrology limits (shot-noise and Heisenberg) and challenges under noise. Adiabatic critical metrology was suggested to exploit ground-state sensitivity near criticality, but typically for continuous QPTs requiring large systems and suffering from critical slowing due to fixed minimal gaps. Studies on fidelity susceptibility and ground-state overlap linked criticality to enhanced estimation, and local adiabatic evolution was proposed to speed quantum searches by adapting to local gaps. Robustness of adiabatic quantum computation and noisy metrology bounds further motivated adiabatic schemes. Recent works also explored finite-component transitions, dynamic critical sensing, and methods like shortcuts to adiabaticity. This paper builds on these by using a first-order QPT in a two-spin Ising model with a tunable transverse field to lift degeneracies and control the gap, making experimental realization feasible.

Theory and model: The probe is a two-spin-1/2 Ising model used for estimating a longitudinal magnetic field Bz. A small transverse field Bx is added to lift level crossings at the first-order critical points (Bz = ±1) and open a tunable gap. In the symmetric triplet subspace, the effective two-level Hamiltonian near the critical point is Heff = Bz I2 + (1 − Bz^2) σz + √2 Bx σx, with ground state parameterized by θ via tan θ = √2 Bx / Bz. The quantum Fisher information (QFI) of the ground state near Bz = 1 behaves as FQ ≈ 2/Bx^2 for fixed state dependence, but achieving it requires an adiabatic passage whose time depends on the minimal gap Δmin ∝ Bx. Adiabatic control: The system evolves under Had[A(s)] = [1 − A(s)] HIsing(Bz0) + A(s) HIsing(Bzc), with normalized time s = t/T, interpolating from a large initial field Bz0 to a final field Bzc near the critical point. A control field Bc(t) is used to implement a time-dependent effective longitudinal field while preserving symmetry; experimentally Bx and Bc(t) are combined. Adiabatic paths considered include linear (A(s) = s) with T ∝ 1/Δmin^2 and local adiabatic paths with ds/dt ∝ Δ^2(s) yielding shorter times. Numerical optimization: A path is numerically constructed to maintain fidelity above a threshold Pc = 0.9999 at each step by adaptively choosing A(s) increments, directly controlling the evolved ground-state fidelity and minimizing time. This optimized path achieves T scaling on the order of 1/√F0, enabling Heisenberg scaling. Experimental platform: Nuclear magnetic resonance (NMR) on 13C-labeled chloroform (spins 13C and 1H) at room temperature using a Bruker Avance III 400 MHz spectrometer. The natural scalar coupling J ≈ 214.5 Hz sets the time unit; control uses on-resonance RF along x for transverse field and frequency offset for z-field. State preparation: Initialize to a pseudopure state and prepare the ground state of the initial Hamiltonian. Adiabatic evolution implementation: Use trotterized evolution with M+1 = 100 discrete segments of duration Δt each, approximating the optimized path by linear interpolation between A(i). For Bx = 0.1, Δt ≈ 0.36 (in units of 2π/ω) ensures >99.8% fidelity between exact and trotterized segment evolution; total time T = M Δt = 36, with constant c = T Bx ≈ 3.6. For Bx = 0.2 and 0.3, keep c = 3.6 and M fixed, giving T = 18 and 12, respectively. Decoherence (T1, T2 for 13C and 1H) is included in optimizing Δt and M. Measurement: To saturate the quantum Cramér-Rao bound, perform the optimal projective measurement corresponding to the symmetric logarithmic derivative. In practice, implement a parameter-dependent unitary UO(Bz) transforming the optimal basis to local observables measurable in NMR (e.g., Oloc = σx ⊗ I). The eigenbasis of the optimal observable is constructed analytically as functions of θ (hence Bz); UO(Bz) is decomposed into single-qubit rotations and CNOTs for implementation. Classical Fisher information equals the QFI under this optimal measurement. Estimation procedure: For various stopping points (final Bz from 0.1 to 2.7) and different transverse fields Bx = 0.1, 0.2, 0.3 controlling T, measure outcome probabilities and approximate derivatives via finite differences at Bz ± δ (δ ≈ 0.03) to compute Fisher information. Additional experiments tune Bx to vary T and measure scaling of QFI with time at the critical point (Bz ≈ 1). Numerical simulations with the full Hamiltonian benchmark the experimental data.

- Enhanced sensitivity at criticality: For all tested transverse fields (Bx = 0.1, 0.2, 0.3), the experimentally obtained QFI is significantly higher near the critical point (Bz ≈ 1) than away from it. The QFI per unit time (FQ/T) also peaks near criticality, demonstrating practical metrological advantage. The total relative deviation between experiment and simulation in these scans is about 8.8%. - Heisenberg scaling in time: By tuning Bx to control the adiabatic evolution time T and stopping at the critical point, the measured QFI scales as FQ ∝ T^2, i.e., Heisenberg scaling. Plotting √FQ versus T shows linear behavior with coefficient of determination R^2 ≈ 98.6% and fitted slope ≈ 0.31 ± 0.0032. The total relative deviation from simulations in this scaling test is about 5.1%. - Tunable energy gap and bandwidth–precision trade-off: Introducing a small transverse field Bx lifts the first-order level crossing and opens a gap Δmin ∝ Bx at Bz = ±1. This enables control over adiabatic time and the width/rate of the ground-state transition, allowing practical tuning between precision and bandwidth. - Robustness to decay: The adiabatic protocol maintains the system in its ground state, conferring inherent robustness against decoherence; simulations indicate the scheme can outperform standard (non-adiabatic) metrology protocols under noise.

The findings show that exploiting a first-order QPT in a minimal two-spin system, aided by a small transverse field, enables practical adiabatic passages to criticality with a tunable energy gap, thereby avoiding critical slowing and realizing Heisenberg scaling of precision with total probe time. The numerically optimized adiabatic path, constrained by high ground-state fidelity, outperforms linear and locally gapped-based paths in time efficiency. The enhanced QFI and QFI per unit time near the critical point validate criticality as a metrological resource in small systems. Because the scheme remains in the ground state, it exhibits robustness to decay, enabling advantages over standard dynamical metrology under realistic noise. For broader unknown parameter ranges, adaptive strategies can shift the effective field near the critical point to retain the advantage. This approach bridges precision limits with adiabatic speed, suggesting that metrological bounds can inform fundamental speed limits for adiabatic evolution under noise.

This work proposes and experimentally demonstrates an adiabatic critical quantum metrology scheme using a two-spin Ising model undergoing a first-order QPT with a tunable transverse field. The method encodes the magnetic field parameter in the ground state, controls the energy gap and adiabatic time via Bx, and uses a numerically optimized adiabatic path to maintain high fidelity. NMR experiments confirm a substantial QFI enhancement near the critical point and Heisenberg scaling (FQ ∝ T^2) of precision with probe time. The approach is experimentally accessible, robust to decoherence, and adaptable through gap tuning and adaptive estimation. Future directions include applying the scheme to larger systems and other platforms (e.g., NV centers, cold atoms, superconducting circuits), exploring adiabatic speed limits under noise, and investigating beyond-adiabatic strategies such as shortcuts to adiabaticity or quench dynamics to further improve performance.

- Local estimation regime: The demonstrated protocol assumes prior knowledge to operate near the critical point; broader unknown ranges require adaptive two-step procedures. - Small system size: Results are shown for a two-spin system; scaling to larger systems may introduce control complexity and decoherence challenges. - Discretized adiabatic evolution: Practical limitations in implementing many segments necessitated approximations (trotterization and interpolation), which may introduce errors and require careful optimization against decoherence. - Noise and decoherence: Although the adiabatic ground-state evolution is robust, finite T1/T2 impose limits on total evolution time and path discretization in experiments.