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.
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.
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.
Atomic resolution transmission electron microscopy (TEM) is an indispensable tool for interdisciplinary investigations of radiation soft or hard materials that evolves rapidly because emerging technologies enable remarkable new experiments [1, 2]. Nowadays, many of them embrace the time domain [2-4]. Beyond materials research, electron microscopy is uniquely suited to study fundamental quantum-mechanical aspects. In fact, it remains concerning that aspects addressing the wave-particle duality, time-dependent measurements, or the entanglement of wave functions and their collapse at a detector still require detailed investigations to guide progress. Certainly, this is no academic matter because such aspects become the bottleneck for the deployment of microscopes that enable studying dynamic processes at high spatiotemporal resolution. It is established practice to describe the scattering of electron ensembles by either a particle picture (Figure 1a) or a wave picture (Figure 1b). Sometimes both pictures are mixed. In spite of calls for solving Schrödinger’s time dependent equation [5], the static Schrödinger equation of wave functions is almost exclusively used to reliably simulate the image formation process using multislice calculations for coherent-elastically scattered electron ensembles [6, 7]. Moreover, the Copenhagen convention of quantum mechanics [8] is adopted, which emphasizes the importance of an observation for the collapse of wave functions (Figure 1b). In order to foster further insight into fundamental quantum-mechanical aspects we pursue a series of experiments with low electron dose rates that guarantee the presence of single electrons in the microscope column at any given time [9]. Consequently, we address the scattering of single electrons (Figure 1c) and build up intensities from electron ensembles over time. In such conditions energy filtered TEM recently revealed the presence of image intensity at sample/vacuum interfaces that cannot be simulated using the established (static) calculation tools even if frozen phonon excitations are considered [10]. In contrast, these intensities can be explained by the time-dependent Schrödinger equation assuming that an energy exchange ΔE occurs during any interaction of the electron with the sample in the Heisenberg limit ΔE = ℏ/2 Δt−1 (coherent-inelastic scattering). In this process a wave package is formed by the superposition of partial waves and a decoherence phase of 0.5-1 radian [10]. The time-dependent Schrödinger equation yields two parameters that characterize the static and the dynamic behavior of any wave function ψ(r, t) = ψ(r) ϕ(t), namely xstatic = ℏ/(2mΔE)1/2 and xdynamic = ℏc/ΔE [11]. The former is the de Broglie wavelength if ΔE is a kinetic energy and the latter becomes an expression for self-coherence if the scattering process is coherent-inelastic in the Heisenberg limit. In these circumstances, the product of Planck’s constant and the speed of light hc is given by the product of the expression for temporal coherence λ2/Δλ and the energy loss ΔE (hc = (λ2/Δλ) ΔE). The validity of the relation is shown in Figure 2 [11]. Thus, the temporal part of Schrödinger’s equation adds contributions of a finite self-coherence length (or time if the electron velocity is considered) to any scattering process, which is neglected if only the static expression is exploited. Further, the formation of wave packages rationalizes the wave-particle duality. The width of wave packages is described by a self-coherence length ls that can vary by over 40 orders of magnitude depending on the strength of interactions [11]. For Coulomb interactions, ls explains the vanishing phase contrast from EF-HRTEM images when the energy loss exceeds values of ΔE > 200 eV [11]. A procedure to experimentally determine the self-coherence length was established earlier by measuring the energy dependent localization of scattered matter waves in surface proximity [10]. Therefore, electron microscopy can be used as a general tool to measure the self-coherence length in real space images on a scale of nanometers and relate this these values to a self-coherence time that reaches attoseconds [12]. Particle and wave descriptions of electron scattering. a) Scattering of electron ensembles in a particle picture. The process is often described by Feynman diagrams, which involves the emission of a virtual photon of lifetime given by Heisenbergs Uncertainty Principle. b) Scattering of electron ensembles in a wave picture that underlies the multi-slice calculations. Infinite wave functions entangle source, sample and detector. The Copenhagen convention requires a wave function collapse upon observation. c) Proposed scattering process of a single electron wave function. The energy exchange occurs in the Heisenberg limit ΔEloss = E2-E1= ℏ/2 (t2-t1)−1= ℏ/2 Δt−1 when a wave package is formed during Δt because of existing energy (wavelength) differences ΔEloss of partial waves [11]. Measurements of self-coherence length and the related self-coherence time at the calculated Heisenberg limit. The colored background marks the quantum realm. The solid lines represent the Heisenberg limit in space and time, which includes the constant ℏc [11].
Feynman once asked physicists to build better electron microscopes to be able to watch biology at work. While electron microscopes can now provide atomic resolution, electron beam induced specimen damage precludes high resolution imaging of sensitive materials, such as single proteins or polymers. Here, we use simulations to show that an electron microscope based on a multi-pass measurement protocol enables imaging of single proteins, without averaging structures over multiple images. While we demonstrate the method for particular imaging targets, the approach is broadly applicable and is expected to improve resolution and sensitivity for a range of electron microscopy imaging modalities, including, for example, scanning and spectroscopic techniques. The approach implements a quantum mechanically optimal strategy which under idealized conditions can be considered interaction-free.
We present a quantum algorithmic framework for simulating phase-contrast transmission electron microscopy (CTEM) image formation using a fault-tolerant, gate-based quantum circuit model. The electron wavefield on an N×N grid is amplitude-encoded into a 2log2N-qubit register. Free-space propagation and objective-lens aberrations are implemented via two-dimensional quantum Fourier transforms (QFTs) and diagonal phase operators in reciprocal space, while specimen interaction is modeled under the weak phase object approximation (WPOA) as a position-dependent phase grating. We validate projected potentials, contrast transfer function (CTF) behavior, and image contrast trends against classical multislice simulations for MoS2 over experimentally relevant parameters, and provide resource estimates and key assumptions that determine end-to-end runtime. While extracting complete N×N intensity images requires O(N2/ε2) measurements that preclude advantage for full-image reconstruction, the framework enables quantum advantage for tasks requiring Fourier-space queries, global image statistics, or phase-coherent observables inaccessible to classical intensity-only detection. This framework provides a physics-grounded mapping from CTEM theory to quantum circuits and establishes a baseline for extending toward full multislice and inelastic scattering models.
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.