An Improved Quantum Algorithm of the Multislice Method

Y.C. Wang; Y. Sun; Z.J. Ding

REVIEW 4 major objections 5 minor 42 references

An Improved Quantum Algorithm of the Multislice Method

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper shows that phase-shifting circuits in the quantum multislice algorithm can be rebuilt with only one- and two-qubit gates, and that truncating small Walsh terms cuts the gate count by over an order of magnitude at about 1…

desk verdict Solid modest result: a 2D Walsh-based phase-shifting circuit for the multislice method that removes multi-controlled gates and demonstrably works in simulation; truncation scaling claims are empirical and over-reaching. read the letter →

arxiv 2411.17482 v2 pith:VI2CVW4Q submitted 2024-11-26 quant-ph physics.comp-ph

Y.C. Wang , Y. Sun , Z.J. Ding This is my paper · ORCID

classification quant-phphysics.comp-ph PACS 03.67.Lx61.05.jm

keywords quantumalgorithmmultislicemethodWalsh-Hadamardtransformphase-shiftingcircuittruncationoptimizationelectrondiffractionsimulationtransmissionmicroscopy

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper improves the authors' earlier quantum algorithm for the multislice method, which simulates how high-energy electrons scatter and diffract through a specimen in transmission electron microscopy. The improvement replaces every multi-controlled phase-shifting gate with a cascade of one- and two-qubit gates built from the Walsh-Hadamard expansion of the potential and propagator operators. It then shows that truncating small Walsh coefficients cuts the total gate count by more than an order of magnitude while keeping the average relative error near 1 percent, verified by classical simulation of 12-, 14-, and 16-qubit circuits for 100 keV electrons in gold. If that Walsh sparsity persists at larger sizes and for other materials, the result makes a practical quantum speedup for electron-diffraction simulations substantially more plausible.

What carries the argument

The central object is the phase-shifting quantum circuit built from the Walsh-Hadamard expansion of a diagonal phase operator $U = e^{if(x)}$. Each retained Walsh basis term is implemented by one phase-shifting gate, and the CNOT gates connecting neighboring terms are ordered by Gray code so that successive Walsh indices differ by one bit. The authors extend the one-dimensional construction of [34] to two dimensions by reshaping the $N\times N$ potential matrix into a vector of length $N^2$, which matches the qubit encoding with $n$ qubits per coordinate. A relative truncation threshold $\tau$ removes Walsh coefficients below $\tau$ times the maximum coefficient, and the empirical formula (17) sets the potential threshold as a function of $n$ while the kinetic threshold is fixed to keep all non-zero terms. This machinery is what converts a circuit with many multi-controlled gates into a comparable-size circuit of one- and two-qubit gates and then shrinks it further at controlled cost.

What would settle it

Run the truncated circuit for a 20- or 24-qubit simulation of the same gold specimen, or for another material such as silicon or an oxide, using the threshold formula (17); the claim fails if the remaining-term percentage stops decreasing with qubit count or if the average relative error of the electron density exceeds about 1 percent.

Watch

Extended reading notes

Core claim

The paper's central claim is that the phase-shifting part of the quantum multislice algorithm can be rebuilt exactly using the Walsh-Hadamard expansion of the diagonal phase operator, so that no multi-controlled gates are needed. Each Walsh basis term corresponds to a single phase-shifting gate, and the CNOT gates between them follow Gray-code ordering, giving a circuit whose one- and two-qubit gate count is comparable to the old multi-controlled circuit. The paper further claims that truncating Walsh coefficients below a threshold reduces the total gate count by more than an order of magnitude while holding the average relative error around 1 percent, with the remaining-term percentage falling as the qubit number grows. It supports these claims with classical simulations of 12-, 14-, and 16-qubit circuits for 100 keV electrons in a gold specimen, comparing the exact and truncated circuits against the classical multislice result.

Load-bearing premise

The gate-count reduction depends on the potential's Walsh expansion remaining sparse as the simulation grows and for materials other than the single gold specimen at 100 keV, since the paper's threshold formula is fitted to only three qubit counts (12, 14, and 16).

Editorial extensions

If this is right

The phase-shifting circuit is now implementable with only one- and two-qubit gates, removing the multi-controlled-gate compilation overhead that threatened the earlier algorithm's advantage on real hardware.
Using the paper's threshold formula, the total gate count is reduced by more than one order of magnitude while the average relative error stays around 1 percent.
The retained fraction of Walsh terms decreases as the qubit count grows from 12 to 16, so the truncation benefit is expected to strengthen, not weaken, for larger simulations.
The classical preprocessing step, the fast Walsh-Hadamard transform, costs only $O(N \log N)$ additions and is reusable across slices and input states, so it does not become a new bottleneck.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Reader extension: the threshold formula is fitted to 12-, 14-, and 16-qubit runs for gold at 100 keV; testing other crystals, temperatures, and energies would show whether Walsh-coefficient sparsity is a general property of projected atomic potentials or particular to this specimen.
Reader extension: the kinetic operator's Walsh spectrum is described as having almost all coefficients zero, which suggests an analytic description of its support; deriving one could replace the empirical threshold fit with a rigorous gate-count bound as the qubit number grows.
Reader extension: because truncation acts on the operators rather than on the wave function, the same Walsh-truncation idea transfers to any diagonal-unitary simulation whose generator is sparse in the Walsh basis, such as other scattering or spectral-phase problems.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Joe, quick take on this one. The paper does something genuinely useful: it replaces the multi-controlled phase-shifting gates in the authors' earlier quantum multislice algorithm with a Walsh-decomposition circuit using only one- and two-qubit gates, extending a construction from Welch et al. (which they cite) from 1D to the 2D potential and kinetic operators of the multislice step. In exact form the improved circuit is verified against both their earlier quantum circuit and the classical multislice method, and the gate count is comparable, so no hidden overhead reappears. That part is solid and publishable.

The second contribution, truncation of the Walsh expansion, is more fragile. The authors show an order-of-magnitude gate reduction at roughly 1% average relative error for 12, 14, and 16 qubits on one Au specimen at 100 keV. Their threshold formula, Eq. (17), is an empirical fit to those three sizes, not a derived bound, and the stress-test note is right: nothing in the paper ties the sparsity of the Walsh spectrum to any physical invariant, so extrapolating to larger N or other materials is a claim, not a result. The same caveat applies to their 'speedup improves with size' statement — it rests on three points. This should be tempered in any revision.

Three smaller soft spots. The introduction says the classical cost increases exponentially because of repeated FFTs; that is sloppy. Per slice it's O(N log N), and the exponential growth is in the number of states/configurations, not in the FFT itself. The error metric in Eq. (16) is a full-state average, which can miss localized errors in a diffraction pattern. And the resource comparison leaves out state preparation, sampling to estimate probabilities, and compilation overhead; the paper mentions sampling but does not fold it into the complexity claim. None of this invalidates the exact circuit, which is the real deliverable.

Who should read this: people working on quantum simulation for electron microscopy or on diagonal-unitary circuit synthesis. I'd bring it to a reading group as an example of a well-scoped circuit optimization with a clear empirical component. The exact construction deserves a serious referee; the truncation claims need revision, not rejection. I'd accept it for peer review and push the authors to rewrite the scaling claims more carefully, correct the FFT statement, and ideally test at least one other material or qubit size before claiming asymptotic behavior.

Referee Report

4 major / 5 minor

Summary. The manuscript presents an improved quantum algorithm for the multislice method used in transmission electron microscopy simulations. The authors replace the multi-controlled phase-shifting gates of their previous quantum circuit by a Walsh-Hadamard decomposition, so that the phase-shifting operator is implemented with single-qubit phase gates and CNOT gates only. They then propose a truncation of small Walsh coefficients to reduce the gate count at the cost of a controlled approximation error. The improved exact circuit is verified against the previous quantum algorithm and the classical multislice algorithm for a 100 keV electron beam on a Au specimen with 12 qubits. Truncation thresholds are tested at 12, 14, and 16 qubits, leading to an empirical formula, Eq. (17), intended to keep an average relative error near 1% while reducing the number of gates by more than an order of magnitude. The authors conclude that the speedup from truncation improves as the number of qubits increases.

Significance. If the central claims hold, this is a meaningful step toward making the quantum multislice algorithm more hardware-friendly: the replacement of multi-controlled gates by one- and two-qubit gates is a concrete improvement, and the exact version of the improved circuit is a clean application of the known diagonal-unitary construction by Welch et al. The truncation idea is interesting and the paper provides simulation evidence, including a direct comparison with the classical multislice result. However, the claimed size-improving truncation speedup rests on an empirical fit to three system sizes and a single specimen, and the error metric used for the 1% claim is a full-state average rather than an observable-specific error; these points limit the generality of the conclusions. The paper is therefore of moderate significance pending a strengthening of the scaling evidence.

major comments (4)

[Sec. 4, Eq. (17), Figs. 12-13] The threshold formula and the conclusion that the truncation speedup improves with qubit number rest on three simulated sizes (12, 14, and 16 qubits) and a single Au specimen at 100 keV. The paper offers no physical argument that the Walsh coefficient tail remains sparse as N grows or for other materials; if the tail fills in at larger N, the remaining-term percentage could increase and the order-of-magnitude gate reduction would disappear. This is a load-bearing extrapolation for the central claim, so it needs either a theoretical bound on the coefficient decay or validation on additional system sizes and materials.
[Sec. 4, Eq. (16)] The 1% error claim refers to the average relative error over the full wave function defined in Eq. (16), not to errors in the extracted electron diffraction pattern or other observables. The truncation threshold is tuned to this particular metric, and the paper asserts rather than justifies that this metric is the right proxy for the application. The statement that the error is 'controllable' should be specified with respect to the metric actually controlled.
[Sec. 4, Sec. 5] The complexity statements ('complexity advantage', 'speedup effect may increase') are not accompanied by a formal accounting that includes state preparation, measurement repetitions, and compilation of the improved circuit to hardware-native gates. Figure 15 reports raw gate counts from the simulation; without the omitted overheads, the asymptotic comparison is not quantified and the quantum-advantage claim is not fully supported.
[Sec. 4, Figs. 13 and 15] The reported 'original' term numbers (65,010; 261,057; 1,046,709 for 12, 14, and 16 qubits) appear to count Walsh terms across all 16 potential slices, whereas a single multislice iteration uses only one potential operator. The relationship between these term counts, the per-iteration gate count, and the total simulation gate count is not made explicit, which makes the reported reduction factor ambiguous.

minor comments (5)

[Equations (4), (6), (14), (17)] Several equations contain garbled mathematical expressions (for example, Eq. (17) is printed as '3128 102 V nτ −=×'); the manuscript needs a careful proofreading pass so that all formulas are readable.
[Abstract, Sec. 3] There are typos such as 'multisilce' in the abstract and 'It can be proven' in Sec. 3; these should be corrected to 'multislice' and 'proved'.
[Fig. 13] The caption lists several numbers without a clear legend; a small table giving the original term count, remaining term count, truncation threshold, and error for each qubit count would improve readability.
[Sec. 2] The discussion of initial state preparation cites Ref. [35] for O(N) state preparation and Ref. [36] for sparse states, but these costs are not included in the later complexity comparison; please clarify their role in the overall algorithm.
[Sec. 3] The statement that a phase-shifting gate without control qubits 'will result in a phase shift for half of the basis states' is imprecise: it applies to a single-qubit rotation on one register qubit, and the correspondence to a Walsh function should be stated explicitly.

Circularity Check

1 steps flagged · score 3.0 of 10

The truncation scaling claim rests on Eq. (17), an empirical fit to the same 12/14/16-qubit simulations used to report the 1% error and improving speedup; the circuit reconstruction itself is independently verified.

fitted input called prediction [Sec. 4 (Eq. (17), Figs. 12-13) and Sec. 5 Conclusion]
"Here we provide an empirical formula by the three sample points closest to the dashed line in Fig. 12 to maintain approximately a 1% average relative error: ... Although the truncation threshold is lowered and we keep more terms when the qubit number increases, the percentage of the remaining terms actually decreases. This indicates that the aforementioned problem does not exist; instead, the speedup effect of the truncation will improve as we use more qubits."

Eq. (17) is fit to the same three simulated sizes (12, 14, 16 qubits) whose truncation errors and remaining-term counts appear in Fig. 12. Fig. 13 and the conclusion then apply Eq. (17) at those identical sizes to report roughly 1% error and a decreasing remaining percentage. The 'speedup improves with qubit number' trend is therefore a restatement of the calibration points, not an independent prediction; the ~1% error is achieved by construction because the threshold was chosen to land near the 1% contour of those same runs.

full rationale

Apart from the fitted threshold formula, the paper's core derivation is self-contained: the Walsh-expansion phase-shifting circuit is an exact rewrite, following Welch et al., of the diagonal unitary, and its correctness is verified by comparing no-truncation results with the previous quantum circuit and the classical multislice algorithm. The authors' earlier work is cited for background and for the original phase-shifting circuit, but the new circuit's validity does not depend on an unverified self-citation. The gate-count reduction from truncation is also arithmetically immediate once terms are discarded, though the quantitative claim of 'more than an order of magnitude' is just the measured remaining-term count. The only genuinely circular element is Eq. (17): it is fit to the 12/14/16-qubit data and then used to assert that those same data show ~1% error and an improving speedup with size. That is post-hoc calibration presented as a scalable parameter-setting scheme, which warrants a mild circularity finding but does not undermine the independently verified circuit construction.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central algorithmic construction is grounded in the Walsh-Hadamard transform and the Welch et al. circuit decomposition. The load-bearing empirical content is the truncation threshold formula (Eq. 17), which is fitted to the same data used to verify the error, and the manual choice to keep all nonzero kinetic terms. No new physical entities are introduced.

free parameters (3)

Potential truncation threshold tau_V = 0.002 (12 qubits), 0.001 (14 qubits), 0.0005 (16 qubits)
Fitted to three simulation sizes to maintain approximately 1% average relative error; Eq. (17) interpolates between these tested points.
Kinetic truncation threshold tau_P = 1e-10
Chosen manually to keep all nonzero kinetic Walsh terms; not derived from an error analysis, just a small number that retains all non-negligible terms.
Selection of three sample points for empirical fit = 12, 14, 16 qubit datasets
The points 'closest to the dashed line' in Fig. 12 are chosen to define Eq. (17); this selection is post hoc and affects the extrapolation.

assumptions (5)

domain assumption Multislice iteration formula (Eq. 1) is the correct high-energy approximation for electron propagation through thin slices.
The quantum algorithm simulates this operator; if the multislice approximation were invalid, the simulation target would be wrong. This is established prior literature (Cowley, Ishizuka).
standard math Walsh-Hadamard transform provides a complete orthonormal basis for discrete functions on an N by N grid, and the Welch et al. construction correctly implements each Walsh term as phase gates and CNOT gates.
Invoked in Sec. 3 and verified by simulation, but the circuit construction is taken from ref [34] without re-derivation.
domain assumption The potential and kinetic phase operators are diagonal in coordinate and momentum representations respectively.
Standard multislice property used in Sec. 2 to separate the iteration into diagonal operators and Fourier transforms.
ad hoc to paper The average relative error metric (Eq. 16), computed over the full wave function, is an appropriate proxy for the error in extracted diffraction patterns.
The paper uses this metric in Sec. 4 to choose thresholds, but does not justify that it bounds the error in sampled observables, which is what a quantum computer would actually produce.
domain assumption Classical simulation of the quantum circuit with pyqpanda faithfully represents the gate-level behavior of the algorithm on real hardware.
Needed for the verification in Sec. 4; simulation gates are treated as equivalent to physical gates without modeling hardware noise or compilation constraints.

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Cite this review

Pith. "Pith review of An Improved Quantum Algorithm of the Multislice Method." pith.science (2026). https://pith.science/paper/VI2CVW4Q

@misc{pith2026241117482,
  author       = {Pith},
  title        = {Pith review of: An Improved Quantum Algorithm of the Multislice Method},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VI2CVW4Q}},
  note         = {Machine review of arXiv:2411.17482}
}

read the original abstract

The multisilce method is an important algorithm for electron diffraction and image simulations in transmission electron microscopy. 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 new 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.

Figures

Figures reproduced from arXiv: 2411.17482 by the authors.

**Figure 1.** The flowchart of the wave function iteration between two slices in the multislice method. The black frame represents the coordinate representation, while the gray one represents the momentum representation. In the quantum algorithm of the multislice method, qubits and quantum gates form a quantum circuit. Input information is encoded into the quantum state of qubits and processed by quantum gates to achieve calculat… view at source ↗

**Figure 3.** The previous version of the phase-shifting quantum circuit for 4 qubits, n=2. Apart from QFT circuits and iQFT circuits, there are phase-shifting quantum circuits that apply diagonal phase-shifting operator on the quantum state, satisfying: ifx( ) U x e x = . (5) In the previous work, we used a simple phase-shifting circuit shown in [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Diagram of the quantum circuit required for the evolution of one slice of electron [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

Figures from the paper (4 more)

**Figure 5.** Figure 5: The 1D 8-point Walsh functions in Hadamard order. For 2D situation, we just need to combine the binary expansions of x and y together, and apply the 1D formula. For example, the 2D Walsh basis function when N = 4 are shown in [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

**Figure 8.** Figure 8: (a) The diagram of a side cross section a [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]

**Figure 9.** Figure 9: visually shows the effect of Walsh transform and truncation on the operator matrices. There are 16 different potential operators for the 16 slices per cell, and we choose the one on the atomic plane. Meanwhile, the kinetic operator is the same for every slice. It is cl…

**Figure 10.** Figure 10: The simulation results of the electron p [PITH_FULL_IMAGE:figures/full_fig_p017_10.png]

Discussion (0). Continue with ORCID to comment.

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Pith tools

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