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.
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28 pages, 9 figures (including appendices); supplementary code available at https://github.com/QuScope/examples-applications/blob/main/notebooks/Quantum_ctem_paper_full_example.ipynb
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.
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.
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.
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.
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.
The first-principles quantum simulation of the full electronic Hamiltonian of a material model of realistic size appears daunting even in a future era of fully fault-tolerant quantum computers. In contrast, classical simulations of material excited states rely on heuristic quasiparticle frameworks, such as the GW approximation and Bethe-Salpeter equation, where only a few quasiparticles are explicitly considered. Here we give a detailed description of quasiparticle quantum simulation for multi-excitonic states within the BSE. We develop an explicit first-quantized, orbital-based block encoding of the multi-exciton BSE Hamiltonian for qubitization, and reduce its cost using integral factorization and crystal symmetries. For a materials model of size N, and a fixed number of quasiparticles, the resulting algorithms require O(N) Toffoli gates and O(logN) logical qubits, or, using space-time tradeoffs, O(N) Toffoli gates and O(N) logical qubits, achieving up to a sixth power reduction in Toffoli count, or up to exponential reduction in qubit count, versus standard full Hamiltonian simulation techniques that have been explored in materials simulation. Resource estimates for the classically challenging problem of triexcitonic eigenvalue estimation across a range of semiconductor models of a realistic physical size of O(103) atoms, show that our Toffoli-optimized implementation reduces the Toffoli count by 106−107 relative to full Hamiltonian simulation estimates, while a qubit-optimized implementation requires only a few hundred logical qubits. More broadly, these results show that classical heuristic frameworks will be essential in the design of future fault-tolerant quantum algorithms for materials.
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.