We analyze the quantum limit of sensitivity in four-dimensional scanning transmission electron microscopy (4D-STEM), which has emerged as a favored technique for imaging the structure of a wide variety of materials, including biological and other radiation-sensitive materials. 4D-STEM is an indirect (computational) imaging technique, which uses a scanning beam and records the scattering distribution in momentum (diffraction) space for each beam position. We find that, in measuring a sample's electrostatic potential, the quantum Fisher information from 4D-STEM can match that from real-space phase-contrast imaging. Near-optimum quantum Fisher information is achieved using a delocalized speckled probe. However, owing to the detection in the diffraction plane, 4D-STEM ultimately enables only about half of the quantum limit, whereas Zernike phase-contrast imaging enables the quantum limit for all spatial frequencies admitted by the optical system. On the other hand, 4D-STEM can yield information on spatial frequencies well beyond those accessible by phase-contrast TEM. Our conclusions extend to analogous imaging modalities using coherent scalar visible light and x rays.
We analyze the quantum limit of sensitivity in four-dimensional scanning transmission electron microscopy (4D-STEM), which has emerged as a favored technique for imaging the structure of a wide variety of materials, including biological and other radiation-sensitive materials. 4D-STEM is an indirect (computational) imaging technique, which uses a scanning beam and records the scattering distribution in momentum (diffraction) space for each beam position. We find that, in measuring a sample's electrostatic potential, the quantum Fisher information from 4D-STEM can match that from real-space phase-contrast imaging. Near-optimum quantum Fisher information is achieved using a delocalized speckled probe. However, owing to the detection in the diffraction plane, 4D-STEM ultimately enables only about half of the quantum limit, whereas Zernike phase-contrast imaging enables the quantum limit for all spatial frequencies admitted by the optical system. On the other hand, 4D-STEM can yield information on spatial frequencies well beyond those accessible by phase-contrast TEM. Our conclusions extend to analogous imaging modalities using coherent scalar visible light and x rays.
What are the ultimate limits of accuracy for techniques in transmission electron microscopy (TEM)? For example, for a given electron dose, what is the ultimate accuracy that can be achieved for the atomic structure of a material? And what determines this accuracy? In this work, we show how quantum estimation theory can be used to provide some quantitative answers to such “big” questions. Quantum estimation theory (also known as quantum metrology theory) is concerned with measurements made at the quantum level. In recent years, the theory has been applied to an increasingly broad range of scientific techniques [1,2]. Here, we apply it to the problem of multiple electron scattering from an atomic structure in a TEM [3]. We focus on cases where the parameters of interest are those which characterize the sample, an extremely relevant and very general scenario. We report the following: We have obtained a quantitative prediction of the ultimate accuracy allowed by quantum mechanics for the precision of atom coordinates within a sample. As expected, this ultimate precision is a function of the electron dose (among other factors), and so this result has important consequences for the precision achievable for the atomic structure of radiation-sensitive materials. We demonstrate that the ultimate (quantum) limit for a simultaneous estimation of multiple sample parameters can be achieved under the condition of weak electron scattering, such as when the phase-object or weak-phase-object approximation is applicable. We demonstrate that this same quantum limit cannot be achieved under the condition of strong multiple scattering, due to the incompatibility of the projection measurements required. Hence strong multiple scattering precludes the quantum limit (with few exceptions) - a fascinating result. In the theory, a TEM experiment (which could be imaging, diffraction, holography, etc.) is divided into three stages: the incident electron beam |ψ0⟩, the beam scattered by the sample S|ψ0⟩, and the beam incident on the electron detector AS|ψ0⟩. Here, S and A are mathematical operators that account, respectively, for the scattering by the sample and the post-sample electron optics (e.g., lens aberrations). This description applies to many TEM modalities, which allows the theory to make general predictions. Fig. 1 shows the specific case of TEM imaging. As will be shown, the quantum limit for an arbitrary set of sample parameters can be achieved by phase-contrast imaging under the Zernike phase condition. Fig. 2 shows a quantitative prediction of the ultimate precision achievable for the lateral coordinate of an atom in a material, as measured in the TEM. Higher doses allow better precision, as expected. For a given dose, better precision is possible for heavier atoms, and for lower beam energies. The results show a strong and unexpected "threshold" effect which occurs at the onset of the asymptotic regime (where the dose is high enough for the statistics to become Gaussian). Finally, while our main focus here is TEM, the generality of the theory means that several of our important conclusions extend to other techniques which utilize the scattering of coherent radiation, such as coherent x-ray scattering in a synchrotron and neutron scattering. Optics for TEM imaging, showing the mathematical notation relevant to this work. Ultimate limits for the precision of a (lateral) atom position measured using TEM. Results are shown for two representative atomic species and beam energies 30-300 keV. Squares and diamonds show results from maximum likelihood estimation (MLE) applied to Monte Carlo image simulations.
New-generation transmission electron microscopes (TEMs) are equipped with detectors that approach the shot-noise limit. Hence it is pertinent to ask: What are the quantum limits of electron scattering experiments in the TEM? For example, for a given electron dose, what is the ultimate accuracy allowed by quantum mechanics for the atomic structure of a material? We provide quantitative answers based on quantum estimation theory. We also show that, for an arbitrary set of sample parameters, the quantum limit is achievable under conditions of weak scattering, but not strong multiple scattering (this conclusion extends to scattering of other types of radiation). Implications for structure determination of radiation-sensitive materials are discussed.
Coherent spin resonance techniques, such as nuclear and electron spin resonance spectroscopy, have revolutionized non-invasive imaging by providing spectrally resolved information about spin dynamics. Motivated by the recent emergence of electron microscopy methods capable of sensing microwave-excitations, we establish a theoretical framework for Spin Resonance Spectroscopy (SRS) in transmission electron microscopy (TEM). This technique combines microwave pump fields with focused electron probe beams to enable state-selective spin imaging at the atomic scale. Using scattering theory, we model the interaction between free-space electrons and electron spin systems, capturing both elastic and inelastic processes. The strongest effect of the spin system on the free electron is a magnetic phase shift. Our simulations demonstrate that phase shifts from individual electron spins are detectable in both image mode and diffraction mode. In principle, differential measurements under microwave control allow the extraction of local resonance frequencies that are influenced by the surrounding spin environment. By evaluating the Classical Fisher Information (CFI), we identify imaging conditions that maximize the signal-to-noise ratio (SNR), showing how defocus and beam width affect the measurement sensitivity. These findings establish a foundation for integrating SRS with high-resolution TEM, bridging spin spectroscopy and atomic-scale imaging, and enabling new capabilities in quantum spin research and nanoscale materials characterization.
Zernike phase-contrast imaging is unique among imaging techniques in that it enables the upper limit of Fisher information allowed by quantum mechanics. Here we show that, in a departure from an ideal setting, using an incident beam of finite width, and a π/2 phase plate having a finite cutoff, the technique can still deliver >95% of the quantum limit. We point out that the Zernike method is, in principle, applicable to any incident beam. As an example, we sketch an approximate implementation of the method for an incident speckle beam, and show that it too can deliver >95% of the quantum limit.
Abstract Recent advances in transmission electron microscopy (TEM) have opened the path toward spin resonance spectroscopy with single-spin sensitivity. To assess this potential, we investigate the quantum precision limits for sensing magnetic moments with free-electron probes. Using a scattering model where an electron wavepacket interacts with a localized spin, we study two metrological tasks: estimating the magnitude of the magnetic moment and discriminating the presence of a spin. The sensitivity for a given measurement setting is generally determined by the classical Fisher information, which we benchmark against the quantum bound optimized over all measurements. We find that conventional TEM imaging can saturate the quantum bound when backaction of the probe electron onto the spin state is negligible. We also find that, when backaction is relevant, one could do better by realizing a measurement of the electron's orbital angular momentum state. These results establish the quantum limits of spin sensing in TEM and guide the development of future experiments probing individual electron spins or nanoscale ensembles of nuclear spins.
Transmission electron microscopy (TEM) routinely achieves sub-Ångström resolution for imaging atomic structures in materials. Accurate simulations are essential for quantitative microscopy, supporting interpretation of high-resolution phase contrast and multidimensional S/TEM modalities. Classical multislice simulations [1] scale as O(N2log N) per slice on an N × N grid, becoming costly for large fields of view, thick specimens, and systematic parameter sweeps, motivating quantum approaches. We present a quantum algorithmic framework for CTEM image formation using fault-tolerant quantum circuits. Our approach amplitude-encodes the electron wavefield on an N × N grid into a 2log2N-qubit register [2] (Fig. 1a). Free-space propagation and lens aberrations use quantum Fourier transforms (QFTs) and diagonal phase operators, while specimen interaction is modeled under the weak phase object approximation (WPOA) as a diagonal unitary Uobj. The quantum circuit (Fig. 1b) maps each CTEM stage to unitary gates. We validate against classical simulations using abTEM package [3] for MoS2 on a 128 × 128 grid at 80 kV. Fig. 1c-f shows exact agreement at Δf = -80 nm and -10 nm with C3 = 1.3 mm, achieving ρ = 1.000000 and MSE ~10-24. Quantum-computed CTFs match analytical expressions across 80-300 kV, and projected potentials agree with abTEM references to within a few percent. A 256 × 256 circuit requires 58 logical qubits and ~6 × 105 T-gates [4] with polylogarithmic scaling for fixed structures. For the MoS2 benchmark (18 atoms), gate counts increase only 35% from 4 × 4 to 1024 × 1024 grids. However, full N × N image reconstruction requires O(N2/ε2) shots, reaching ~1010 evaluations for 1024 × 1024 images at 1% precision, precluding advantage over GPU-based codes. Quantum advantage lies in three complementary regimes that avoid exhaustive pixel readout: (i) Fourier-space observables - estimating specific structure factors, Bragg peak intensities, or diffraction pattern moments via partial measurement in the QFT basis, requiring O(1) to O(polylog N) samples rather than O(N2); (ii) phase-coherent discriminators - exploiting ancilla-assisted protocols to distinguish structures with classically degenerate intensity signatures by probing the full complex wavefunction ψ = Aeiφ rather than intensity |ψ|2 alone; (iii) extended physics beyond WPOA – simulating inelastic scattering, dynamical diffraction, and many-body correlation effects that scale exponentially for classical tensor-network methods but remain polynomial in system size for quantum circuits. As an additional benchmark, we implemented a 4 × 4 instance of the CTEM circuit on the ibm_torino superconducting quantum processor [5], achieving a classical fidelity of 0.9984 between hardware and ideal simulations after 4096 shots. Figure 2a shows the input projected potential, Fig. 2b the ideal statevector probability distribution, Fig. 2c the measured distribution on IBM hardware, and Fig. 2d the difference map showing sub-1% deviations, demonstrating that small-scale CTEM circuits are already within reach of current NISQ platforms. To demonstrate phase-coherent discrimination, Fig. 2e shows ancilla Y-basis expectation values ⟨Ya⟩ for conjugate ±V phase objects and independent random potentials, while Fig. 2f shows the phase-discrimination signal Δ⟨Ya⟩ = ⟨Ya⟩(+V) – ⟨Ya⟩(–V), achieving robust separation of sign-flipped weak phase objects that produce identical classical intensity images. This framework [7,8] establishes a complete, physics-grounded mapping from established CTEM theory to fault-tolerant quantum circuits with explicit arithmetic circuit constructions for specimen potentials, extending beyond prior oracle-based proposals [6] to gate-level synthesis validated against classical codes. While near-term quantum advantage for full CTEM image reconstruction remains unlikely due to the measurement bottleneck, the framework provides a rigorous baseline for extending toward computationally hard regimes where classical methods struggle, including thick specimens requiring many slices, real-time parameter optimization over high-dimensional microscope settings, and correlation effects in quantum materials [9]. (a) CTEM column and quantum circuit mapping. (b) 4-qubit circuit implementation. (c–f) Quantum vs. classical simulations for 128 × 128 MoS2 at Δf = −80 nm and −10 nm with C3 = 1.3 mm, 80 kV (ρ = 1.000000, MSE ~10−24). Phase discrimination protocol. (a–d) 4 × 4 grid: input potential, ideal and IBM Torino distributions, difference map. (e–f) Ancilla ⟨Ya⟩ values showing robust ±V separation.