The multislice method is an important algorithm for electron diffraction and image simulations in transmission electron microscopy (TEM). We have proposed a quantum algorithm of the multislice method based on quantum circuit model previously. In this work, we have developed an improved quantum algorithm. We reconstruct the phase-shifting quantum circuit without using the multi-controlled quantum gates, thereby significantly improve the computation efficiency. The improved quantum circuit also allows further gate count reduction at the cost of a controllable error. We have simulated the quantum circuit on a classical supercomputer and analyzed the result to prove the feasibility and correctness of the improved quantum algorithm. We also provide proper parameter settings through testing, allowing the minimization of the necessary number of quantum gates while limiting the relative error within 1%. This work demonstrates the potential of applying quantum computing to electron diffraction simulations and achieving quantum advantages.
The multislice method is an important algorithm for electron diffraction and image simulations in transmission electron microscopy (TEM). We have proposed a quantum algorithm of the multislice method based on quantum circuit model previously. In this work, we have developed an improved quantum algorithm. We reconstruct the phase-shifting quantum circuit without using the multi-controlled quantum gates, thereby significantly improve the computation efficiency. The improved quantum circuit also allows further gate count reduction at the cost of a controllable error. We have simulated the quantum circuit on a classical supercomputer and analyzed the result to prove the feasibility and correctness of the improved quantum algorithm. We also provide proper parameter settings through testing, allowing the minimization of the necessary number of quantum gates while limiting the relative error within 1%. This work demonstrates the potential of applying quantum computing to electron diffraction simulations and achieving quantum advantages.
Multislice method is widely used to simulate the elastic interaction of an electron beam with matter in electron microscopy. In this paper, we propose a quantum algorithm of multislice method. With the use of quantum circuit this quantum version of multislice method costs less storage space and less runtime than the classical algorithm by taking advantage of the feature of quantum computing. We have used a classical supercomputer to simulate a quantum circuit, and successfully carried out the quantum multislice computation and obtained the expected result on a virtual quantum circuit, which proves the feasibility and correctness of the quantum algorithm. This work demonstrates the potential application prospect of quantum computing to the research of computational physics as a good example.
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.
Image formation in transmission electron microscopy (TEM) is governed by the coherent evolution of the electron wavefunction through the specimen and the objective lens. This physics maps naturally onto the gate model of quantum computation. We present QuScope, an open-source Python framework that expresses the complete TEM image-formation pipeline as quantum circuits. The N×N electron wavefunction is amplitude-encoded in 2log2N qubits, and every optical element, including phase-grating transmission, Fresnel propagation between specimen slices, and the aberrated objective lens, is implemented as a diagonal unitary conjugated by quantum Fourier transforms. On this foundation, QuScope v0.2.0 implements validated imaging pipelines, covering conventional TEM under the phase-object approximation, full multislice CTEM and STEM for thick specimens. All quantum results reported here come from exact, noise-free statevector simulation of the circuits on classical hardware, and every result is validated against a classical twin implementation. Quantum and classical multislice exit waves agree to unit fidelity, and all physical constants are verified against standard references. We provide transpiled quantum-resource estimates for each algorithm, an analysis of the diagonal-synthesis bottleneck that governs near-term hardware execution, and a fully tested, documented, and pip-installable package. QuScope v0.2.0 is available at https://github.com/QuScope/QuScope.
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.
In the field of microscopy and microanalysis, technological advancements continually push the boundaries of what is possible in data analysis and simulation. Quantum computing, with its ability to perform complex calculations at unprecedented speeds, represents a promising frontier in this ongoing evolution. By harnessing the principles of quantum mechanics, such as superposition and entanglement, quantum computing offers a novel approach to processing and analyzing vast and complex datasets more efficiently than traditional computing methods [1,2]. In this context, our work focuses on leveraging quantum algorithms to enhance the processing, simulation, and analysis of image and spectroscopic data in microscopy. The primary goal is to utilize the unique capabilities of quantum computing to achieve not only faster but also more precise analysis of microscopy data. Our work primarily involves developing and applying quantum machine learning (QML) [3,4] techniques that are specifically tailored to address the challenges faced in microscopic analysis, such as handling complex data structures and improving the accuracy of simulations. Through this approach, we aim to open new pathways for advancements in microscopy, contributing to more detailed and faster analysis processes that can support a wide range of scientific investigations. Figure 1A illustrates the key parts of QML designed for the analysis and processing of microscopy data. It outlines the steps from data acquisition through preprocessing, quantum feature extraction, and quantum-enhanced machine learning. In addition, we have explored the potential incorporation of quantum algorithms into traditional image and electron diffraction pattern simulations. This approach aims to harness the unique capabilities of quantum computing to perform calculations at exponentially faster rates than classical computing. To this end, Figure 1B illustrates the methodologies we have developed, providing a visual representation of how quantum algorithms can be integrated into the image simulation process, leading to significant improvements in both accuracy and efficiency. This diagram outlines the sequential steps involved in simulating electron microscopy images and diffraction patterns through the application of quantum computing. Starting from the initialization of sample and beam parameters, it progresses through quantum state preparation, the application of quantum algorithms (including Hamiltonian evolution and Quantum Fourier Transform), measurement and data retrieval, post-processing and analysis, and concludes with the visualization of simulation results. A critical aspect of this work is the performance comparison between classical and quantum methodologies. Benchmarking against traditional methods indicates significant performance improvements when utilizing quantum algorithms. These improvements are not limited to computational speed but also include enhanced accuracy and the ability to handle complex data sets, which are often challenging for classical computing methods. Figure 2 summarizes key quantum algorithms, their associated subroutines, the theoretical speedup they offer over classical algorithms, and the main equations underlying each algorithm. These algorithms represent advanced quantum computing techniques aimed at enhancing data processing, analysis, and machine learning tasks, showcasing the potential for exponential and quadratic speedups in specific computational domains. Furthermore, we have conducted quantum cost estimations to compare the resource efficiency of quantum simulations with classical simulations, particularly in the context of electron scattering simulations. This analysis provides valuable insights into the scalability and practicality of quantum computing in microscopy, laying the groundwork for future developments in quantum-enhanced microscopy. In summary, our work demonstrates the feasibility and benefits of integrating quantum computing into the analysis pipeline of microscopy data. By leveraging the power of quantum algorithms, including quantum machine learning, we open the door to new capabilities in microscopy data analysis, promising significant advancements in the field [5-6] A) Quantum Machine Learning (QML) workflow for microscopy data analysis. B) Outlines the steps involved in simulating electron microscopy images and diffraction patterns through the application of quantum computing Overview of quantum algorithms and their applications in electron microscopy data analysis
The QCAD which has been technologically advanced during last few years to support basic transformations, simulating multi-dimensional quantum devices and circuits. It is most important to demonstrate the theoretical operation of quantum computer using the simulator. Besides of describing various noise models in terms of quantum gates, a direct support of quantum error correction codes. Using the quantum error correction codes, the effects of applying single bit or phase flip error to the quantum circuits are reported along with the error recovery method for different qubit codes. The results are generated using the QCAD simulator. A simulation model for evaluating single qubit error in the proposed quantum circuit is also given. The working software prototype of an extremely efficient quantum computer simulator which is an attractive and versatile tool for both research and education.