Quantum Circuits Simulate Electron Microscope Images Accurately

Scientists are developing a novel quantum algorithm framework to simulate phase-contrast transmission electron microscopy (CTEM) image formation, potentially revolutionising image processing and analysis. Sean D. Lam from the Department of Physics and Department of Chemistry & Biochemistry at Colorado College, working with Roberto dos Reis from the Department of Materials Science & Engineering and The NUANCE Center at Northwestern University, have detailed a method utilising a gate-based circuit model to represent the electron wavefield and simulate imaging processes. This collaborative research establishes a physics-grounded mapping between CTEM theory and quantum circuits, offering potential advantages for specific tasks like Fourier-space queries and the extraction of global image statistics, which are challenging for classical methods. By validating the framework against established multislice simulations using molybdenum disulphide, the team demonstrates its accuracy and provides crucial resource estimates, paving the way for more complex simulations incorporating multislice and inelastic scattering models.

This breakthrough promises to accelerate materials discovery by enabling rapid, detailed modelling of how materials interact with electron beams. Ultimately, it could unlock insights currently limited by the constraints of classical simulation techniques.

This work addresses a critical bottleneck in quantitative electron microscopy: the computational cost of accurately modelling image contrast, particularly for complex materials and high-resolution imaging. Classical multislice simulations, the current standard, struggle with the exponential increase in computational demands as image size and specimen thickness grow, limiting exhaustive parameter sweeps and real-time analysis.

The research team recast the CTEM imaging process as a quantum circuit, amplitude-encoding the electron wavefield onto a register of qubits and implementing key steps like free-space propagation and lens aberrations using quantum Fourier transforms and phase operators. This innovative approach establishes a direct link between established CTEM theory and the emerging field of fault-tolerant quantum computing.

By representing the electron wave as a quantum state, the simulation bypasses some of the limitations inherent in classical methods, offering a pathway towards accelerating complex image formation models. The framework was validated against classical multislice simulations using molybdenum disulfide (MoS2) over experimentally relevant parameters, confirming its accuracy in projecting potentials, predicting contrast transfer function behaviour, and reproducing image contrast trends.

While obtaining complete intensity images still presents challenges, the study demonstrates the potential for quantum advantage in specific tasks. Specifically, the framework excels at Fourier-space queries, extracting global image statistics, and accessing phase-coherent observables, information inaccessible through conventional intensity-only detection methods.

Although full image reconstruction does not immediately benefit from quantum speedup, the ability to efficiently analyse these other image characteristics opens new avenues for materials characterisation and analysis. The framework uses a 2log2 N-qubit register to represent an N × N grid, and resource estimates are provided to determine end-to-end runtime. This work not only provides a physics-grounded mapping from CTEM theory to quantum circuits but also establishes a crucial baseline for future extensions incorporating more complex models, such as full multislice and inelastic scattering.

Quantum and classical potentials reproduce MoS2 atomic column structure

Projected potentials for MoS2, obtained via quantum-encoded simulations, directly correspond to classical multislice inputs generated with ABTEM, demonstrating a close agreement across a 128×128 grid. The potential accurately resolves molybdenum and sulfur columns, exhibiting the expected hexagonal symmetry, with molybdenum sites displaying higher peak values than sulfur sites.

Relative intensities of the threefold sulfur motifs surrounding each molybdenum column are also reproduced, aligning with classical ABTEM simulations performed under identical conditions. After applying element-specific scaling factors to align absolute phase shifts, the quantum and classical projected potentials agree to within a few percent across the field of view.

The full CTEM imaging pipeline was evaluated through complete quantum simulation, revealing a 3 × 2 MoS2 supercell on a 128 × 128 grid that resolves the Mo and S columns with the anticipated hexagonal symmetry. The resulting quantum CTEM image, generated at 80kV with an underfocus of −800 Å and a C3 aberration coefficient of 1.3mm, demonstrates strong phase contrast and clearly resolved atomic columns arranged in the characteristic hexagonal lattice structure of MoS2.

Bright spots at the Mo and S column positions, exhibiting clear hexagonal symmetry and relative intensities, confirm the correct implementation of phase-to-amplitude conversion via lens aberrations within the quantum circuit. Substantial underfocus generates strong oscillatory contrast transfer function modulation, producing pronounced atomic contrast consistent with the expected behaviour of weak phase objects under negative defocus conditions.

Agreement between projected potentials and resulting image contrast reaches within a few percent of classical multi-slice simulations using ABTEM, validating the faithful reproduction of standard CTEM image formation physics. A 4 × 4 instance of the CTEM circuit was successfully implemented on the ibm_torino superconducting device, showing high-fidelity agreement between hardware and ideal simulations.

Systematic comparisons between statevector simulations and classical multislice evaluation on grids ranging from 8 × 8 to 128 × 128 confirmed numerical identity, with a correlation coefficient of 1.000000 and a mean-squared error of approximately 10−24 limited by floating-point roundoff. This verifies the circuit’s faithful implementation of the weak phase object approximation propagator without algorithmic errors.

Beyond replicating classical intensity images, the quantum CTEM framework represents the full complex image-plane wavefunction, enabling access to phase-sensitive observables inaccessible to conventional intensity-only detectors. The framework lifts a fundamental phase-sign ambiguity inherent in classical CTEM, where weak phase objects differing only by a sign flip in the projected potential produce indistinguishable intensity images. This is achieved by coherently interfacing the CTEM circuit with an ancilla qubit prepared in the |+⟩a state.

Quantum Wavefield Propagation and Specimen Interaction Modelling

A 2D grid representing the electron wavefield serves as the foundation for this work, with its amplitude directly encoded into a 16-qubit register. Free-space propagation and objective-lens aberrations are meticulously simulated using two-dimensional Quantum Fourier Transforms (QFTs), a quantum analogue of the classical Fast Fourier Transform, and diagonal phase operators applied in reciprocal space.

This choice leverages the inherent efficiency of QFTs in manipulating wavefield amplitudes, offering a potential speedup over classical FFT-based methods. Specimen interaction is modelled using the weak phase object approximation (WPOA), representing the sample as a position-dependent phase grating that modifies the electron wavefield. The research team constructed explicit arithmetic circuits for these specimen potentials, encoding them from Kirkland/PAW parameterizations, extending beyond previous approaches that relied on abstract oracles.

This gate-level synthesis was then validated against established classical codes, ensuring the quantum simulation accurately reflects known physical behaviour. Projected potentials, contrast transfer function (CTF) behaviour, and image contrast trends were quantitatively validated against classical multislice simulations for molybdenum disulfide (MoS2) across experimentally relevant parameters, achieving exact numerical agreement at floating-point precision.

Resource estimates, including gate counts, ancilla requirements, and measurement complexity, were meticulously calculated for both near-term noisy intermediate-scale quantum (NISQ) devices and fault-tolerant implementations. A detailed analysis revealed an O(N2/ε2) measurement bottleneck for full-image reconstruction, highlighting the challenges of extracting complete intensity images.

However, the framework also identifies opportunities for quantum advantage in tasks requiring Fourier-space queries, global image statistics, or phase-coherent observables, which are inaccessible to classical intensity-only detection methods. This focus on phase-sensitive observables represents a key methodological innovation, shifting the emphasis from full image reconstruction to specific, quantum-accessible features.

Quantum computation emulates electron propagation for advanced materials imaging

The persistent challenge of simulating materials at the atomic scale has encountered a potentially transformative approach. For decades, accurately modelling how electrons interact with matter, crucial for interpreting transmission electron microscopy images, has been limited by the computational power needed to handle the wave-like nature of those electrons.

This work doesn’t offer a faster classical algorithm, but a fundamentally different pathway using quantum computing to bypass those limitations. What makes this notable is the shift from simply trying to simulate electron behaviour on conventional computers, to mapping that behaviour directly onto the principles of quantum mechanics. By encoding the electron wavefield into qubits, and using quantum Fourier transforms to mimic how electrons propagate and interact with a sample, researchers have created a framework that, in theory, scales far more efficiently than classical methods.

While full image reconstruction remains a distant goal, the ability to perform specific calculations, particularly those involving the phase of the electron wave, offers immediate advantages. This isn’t about replacing existing electron microscopy anytime soon. The number of stable, error-corrected qubits required remains a significant hurdle. However, the framework’s modular design, allowing for hybrid classical-quantum approaches, suggests a pragmatic path forward.

Pre-processing certain aspects of the simulation on conventional computers, while offloading the most computationally intensive parts to a quantum processor, could yield benefits even with near-term quantum hardware. The longer-term implications extend beyond materials science. Any field reliant on wave-based simulations, from seismic analysis to medical imaging, could potentially benefit from this physics-grounded mapping to quantum circuits. The real test will be translating these theoretical advantages into tangible results, and demonstrating that the promise of quantum simulation can overcome the formidable engineering challenges that lie ahead.