We show that quantum computational imaging is advantageous in the setting of low-dose electron microscopy of beam-sensitive specimens. Two qudits placed near the electron beam enable full transfer of quantum information between the electron microscope and a quantum computer in the proposed scheme, providing the specimen is a phase object. We present a quantum algorithm that identifies the correct image among n candidate images, where n is larger than the effective dimension of the Hilbert space of the imaging electron.
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4 pages, 1 figure. Supersedes the unpublished preliminary version in arXiv:2209.04819
We show that quantum computational imaging is advantageous in the setting of low-dose electron microscopy of beam-sensitive specimens. Two qudits placed near the electron beam enable full transfer of quantum information between the electron microscope and a quantum computer in the proposed scheme, providing the specimen is a phase object. We present a quantum algorithm that identifies the correct image among n candidate images, where n is larger than the effective dimension of the Hilbert space of the imaging electron.
We propose a simple design of a quantum electron microscope that ``queries'' a beam-sensitive phase object, such as a biological specimen, as part of quantum computation. Lower quantum query complexity, not the time complexity, of a quantum algorithm means less specimen damage, which translates to more data extracted from the specimen. Hence small-scale quantum computing offers provable quantum advantage in this context. A possible application of the proposed microscope is the Grover search for a true structure, out of a set of candidate structures.
One of the astounding consequences of quantum mechanics is that it allows the detection of a target using an incident probe, with only a low probability of interaction of the probe and the target. This 'quantum weirdness' could be applied in the field of electron microscopy to generate images of beam-sensitive specimens with substantially reduced damage to the specimen. A reduction of beam-induced damage to specimens is especially of great importance if it can enable imaging of biological specimens with atomic resolution. Following a recent suggestion that interaction-free measurements are possible with electrons, we now analyze the difficulties of actually building an atomic resolution interaction-free electron microscope, or "quantum electron microscope". A quantum electron microscope would require a number of unique components not found in conventional transmission electron microscopes. These components include a coherent electron beam-splitter or two-state-coupler, and a resonator structure to allow each electron to interrogate the specimen multiple times, thus supporting high success probabilities for interaction-free detection of the specimen. Different system designs are presented here, which are based on four different choices of two-state-couplers: a thin crystal, a grating mirror, a standing light wave and an electro-dynamical pseudopotential. Challenges for the detailed electron optical design are identified as future directions for development. While it is concluded that it should be possible to build an atomic resolution quantum electron microscope, we have also identified a number of hurdles to the development of such a microscope and further theoretical investigations that will be required to enable a complete interpretation of the images produced by such a microscope.
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.
A transmission electron microscope that takes advantage of superconducting quantum circuitry is proposed. The microscope is designed to improve the image contrast of radiation-sensitive weak-phase objects, in particular, biological specimens. The objective in this setting is to measure the phase shift of the probe electron wave to a precision \ensuremath{\Delta}\ensuremath{\theta} within the number of electrons N that does not destroy the specimen. In conventional electron microscopy \ensuremath{\Delta}\ensuremath{\theta} scales as \ensuremath{\sim}1/N1/2, which falls short of the Heisenberg limit \ensuremath{\sim}1/N. To approach the latter by using quantum entanglement, we propose a design that involves a Cooper-pair box placed on the surface of an electrostatic electron mirror in the microscope. Significant improvement could be attained if inelastic scattering processes are sufficiently delocalized.
In electron microscopy, the pursuit of atomic-scale resolution for imaging biological [1] samples and soft-matter systems [2] is perpetually constrained by beam-induced sample degredation. While classical approaches attempt to optimize the trade-off between dose efficiency and image quality, they remain fundamentally limited by the physical interactions between electrons and the sample. Quantum technologies, however, present a transformative opportunity by enhancing information extraction per electron without increasing exposure [3], thereby breaking through classical constraints. In this work, we integrate for the first time a trapped-ion qubit into a transmission electron microscope, leveraging the strong Coulomb interaction between swift electrons and Ca⁺ ions [4], along with the precise control and readout capabilities of trapped Ca⁺ ions [5]. By harnessing quantum coherence and controlled interactions, this hybrid approach seeks to advance the capabilities of low-dose imaging, improving the analysis of radiation-sensitive samples while minimizing damage. For this purpose, we customize a transmission electron microscope to provide the required ultra-high-vacuum integrity and adapt the Ca⁺ trap design, including laser cooling and state readout/manipulation, to the geometric constraints of the microscope’s pole-piece gap.
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.