Quantum Algorithm Framework for Phase-Contrast Transmission Electron Microscopy Image Simulation

TL;DR

The paper develops a complete quantum algorithmic framework for simulating CTEM image formation under the weak phase object approximation by amplitude-encoding the $N\times N$ wavefield into $n=2\log_2 N$ qubits and applying 2D QFT-based propagation together with diagonal phase operators for the specimen and lens. It rigorously validates the quantum circuit against classical multislice simulations for MoS$_2$, showing exact numerical agreement within floating-point precision for the WPOA model, and provides a detailed resource and measurement cost analysis that identifies the dominant bottleneck as full-image readout with $O(N^2/\epsilon^2)$ shots. The work highlights three avenues for quantum advantage: (i) Fourier-space queries and global statistics, (ii) phase-coherent observables accessible via ancilla-assisted measurements, and (iii) extended physics beyond WPOA such as inelastic scattering and many-body effects. It establishes a physics-grounded mapping from CTEM theory to gate-based quantum circuits, offers a baseline for extending toward full multislice and DFT-informed potentials, and outlines practical near-term demonstrations on NISQ devices alongside fault-tolerant scaling strategies toward production-scale quantum advantage.

Abstract

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 $N\times N$ grid is amplitude-encoded into a $2\log_2 N$-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 MoS$_2$ over experimentally relevant parameters, and provide resource estimates and key assumptions that determine end-to-end runtime. While extracting complete $N\times N$ intensity images requires $O(N^2/ε^2)$ 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.

Figures (14)

• Figure 1: Simplified CTEM column schematic (left) and mapping between the classical CTEM image-formation model and the quantum circuit representation (right). The specimen acts as a weak phase grating, free-space propagation and lens aberrations are implemented via Fourier-space phase factors or QFT-based diagonal unitaries, and the recorded intensity corresponds to $\lvert \phi_{\mathrm{img}} \rvert^2$ or $\lvert \psi_{ij} \rvert^2$ at the detector.
• Figure 2: Example of quantum CTEM circuit implementing state preparation, specimen phase grating $U_{\mathrm{obj}}$, two-dimensional QFT $U_{\mathrm{QFT}}$, Fresnel propagator $U_P$, lens phase $U_{\chi}$, and inverse QFT, followed by measurement of the image-plane intensity. The schematic is shown here for a $16\times 16$ grid ($n=8$ qubits).
• Figure 3: Quantum-computed contrast transfer function for varying spherical aberration at 80 kV. The CTF is shown for four representative values: (a) aberration-corrected $C_3 = 0.0$ mm at Scherzer defocus $\Delta f = -500$ Å (first CTF zero at 0.02 Å; practical information limit $\sim$1.5 Å with coherence envelopes) Eq. \ref{['eq:ctf_effective']}-\ref{['eq:chromatic_envelope']}), (b) $C_3 = 0.5$ mm at $\Delta f = -539$ Å (first CTF zero at 2.81 Å; practical limit $\sim$2.5 Å),(c) $C_3 = 1.3$ mm at $\Delta f = -869$ Å (first CTF zero at 3.57 Å; practical limit $\sim$3.0 Å), and(d) $C_3 = 2.0$ mm at $\Delta f = -1078$ Å (first CTF zero at 3.97 Å; practical limit $\sim$3.5 Å). The idealized coherent CTF (Eq. \ref{['eq:ctf_equation']}) assumes perfect spatial and temporal coherence; realistic envelope functions from finite source size and energy spread damp high-frequency transfer, restricting the practical information limit to approximately 1.5 times the first CTF zero for typical field-emission gun conditions.
• Figure 4: Fault-tolerant resource scaling for quantum CTEM simulation of MoS$_2$ (18 atoms, 5 Gaussians/atom). (a) Logical qubit requirements: data qubits (blue) grow logarithmically as $2\log_2 N$, ancilla (green) remain approximately constant at $\sim 30$-50, and total logical qubits (red) stay below 100 for grids up to $1024\times 1024$. Horizontal dashed line marks a representative NISQ-limit estimate ($\sim 50$ logical qubits). (b) Non-Clifford (T) gate count breakdown: QFT layers (green), specimen operator $U_{\text{obj}}$ (blue), lens CTF $U_\chi$ (red), and total (black line). Gate counts increase modestly from $\sim 5\times10^5$ to $7\times10^5$ across grid sizes, dominated by $U_{\text{obj}}$ for this fixed atomic structure (90 Gaussian terms). The near-constant behavior confirms polylogarithmic scaling in grid size $N$ for fixed specimen complexity. (c) Measurement cost: shot count for full $N\times N$ image reconstruction at $\epsilon = 0.01$ relative error scales as $O(N^2/\epsilon^2)$ (purple shaded region), reaching $10^{10}$ circuit evaluations for $1024\times 1024$. This measurement overhead dominates the single-shot circuit cost (panels a-b) by four to six orders of magnitude, representing the primary end-to-end runtime bottleneck for full-image reconstruction tasks and precluding quantum advantage over classical GPU codes for this baseline application.
• Figure 5: Fully quantum CTEM simulation of a $\mathrm{MoS}_2$ supercell on a $128\times 128$ grid at 80 kV, $\Delta f = -800$ Å, and $C_3 = 1.3$ mm. (a) Projected potential $V(x,y)$ from the Kirkland parameterization used to construct the specimen phase operator $U_{\mathrm{obj}}$. (b) Quantum CTEM image intensity $I(x,y)$ obtained from the amplitude-encoded circuit, showing the expected hexagonal lattice with bright atomic columns under strong underfocus phase contrast.