Theory of Quantum Path Computing with Fourier Optics and Future Applications for Quantum Supremacy, Neural Networks and Nonlinear Schr\"odinger Equations

Burhan Gulbahar

REVIEW 4 major objections 4 minor 78 references

Theory of Quantum Path Computing with Fourier Optics and Future Applications for Quantum Supremacy, Neural Networks and Nonlinear Schr\"odinger Equations

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper argues that photonic quantum path computing with a classical laser and intensity detection samples the interference of exponentially many paths, producing Riemann theta partial sums that enable quantum-supremacy-scale…

desk verdict Useful LCT wave-optics math buried under a quantum supremacy claim that collapses on the first classical-simulability check. read the letter →

arxiv 1908.02274 v3 pith:W6Q3CHVP submitted 2019-08-06 quant-ph physics.optics

Burhan Gulbahar This is my paper · ORCID

classification quant-phphysics.optics

keywords quantumpathcomputingmulti-planediffractionFourieropticslinearcanonicaltransformRiemannthetafunctionsupremacyneuronnonlinearSchrödingerequation

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

Quantum path computing (QPC) was proposed earlier for diffracting particles through many planes of slits; this paper extends it to ordinary laser light and arbitrary Fourier-optical systems. It claims that the intensity measured on the final plane is a black-box function built from exponentially many interfering paths, and that for lattice-spaced slits this function reduces to the absolute square of a partial sum of a Riemann $\theta$ function. If true, a passive stack of lenses, slits, and a photodetector array could serve as a scalable quantum-computing device, reaching more than $2^{100}$ effective Feynman paths with dozens of planes, while also acting as a quantum neuron and as an analog solver for nonlinear Schrödinger equations.

What carries the argument

The machinery is multi-plane diffraction treated as repeated application of the Fresnel kernel or the more general linear canonical transform (LCT) kernel between planes. Each slit is modeled as a Gaussian amplitude mask, and the propagation through one slit per plane is evaluated iteratively; the resulting Gaussian or Hermite-polynomial forms are what collapse into Riemann theta partial sums. The Wigner-function negative volume and per-path magnitudes are used to measure the growth of interference complexity that the paper claims underlies the computational advantage.

What would settle it

Take a three-plane QPC with a known Gaussian beam, a few slits per plane placed on a lattice, and a camera on the sensor plane; evaluate the predicted intensity from Eq. (39) by direct numerical summation over all paths. If the measured pattern deviates from the predicted theta-sum intensity beyond detector noise when slit masks depart from ideal Gaussians, the central connection between QPC and Riemann theta summation is falsified.

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Extended reading notes

Core claim

The central discovery is a closed iterative model, Eqs. (16), (25), and (27), for the wave function on the sensor plane of a multi-plane diffraction set-up with Gaussian and Hermite-Gaussian laser sources and arbitrary linear-canonical-transform (LCT) optics between planes. The sensor intensity is claimed to be the interference of $N_p$ paths, one for each choice of one slit per plane; the paper shows that when slit positions lie on a lattice and widths are uniform, the normalized intensity equals the squared modulus of a Riemann theta partial sum (Eq. (39)). The paper further claims that this optical oracle can be scaled to quantum-supremacy-size Hilbert spaces, that the same hardware computes a quadratic-form neuron with exponential synaptic chains, and that the theta-sum output can be matched to finite-band solutions of the nonlinear Schrödinger equation.

Load-bearing premise

The derived formulas assume every slit is a perfect Gaussian amplitude mask and that each path visits exactly one slit per plane, never looping between slits on the same plane; the paper itself lists both idealizations as open issues.

Editorial extensions

If this is right

A single passive optical stack—laser, slit planes, lenses, camera—could evaluate the absolute value of Riemann theta partial sums for lattice parameters set by slit positions and widths, giving an optical oracle for problems in number theory and Diophantine approximation.
With a linearly increasing number of slits per plane, the effective number of significant Feynman paths grows exponentially in the number of planes, so the paper estimates over $2^{100}$ effective paths with a few dozen planes even at modest gain values.
The same hardware defines a quantum neuron whose output is a nonlinear function of a quadratic form in the input slit positions; reading the intensity gives the squared output, which can be used as a building block for quantum neural networks.
The paper proposes that, because finite-band solutions of the nonlinear Schrödinger equation are ratios of Riemann theta functions, matching the QPC theta parameters to the Riemann spectrum would let the device return approximate NLSE solutions.

Reading between the lines

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

A concrete testable extension would be to simulate or measure a three-plane set-up with a few slits per plane and compare the sensor intensity to a directly evaluated theta partial sum; agreement would isolate the single-pass Gaussian-slit model from the more speculative supremacy scaling.
If the Gaussian-mask assumption is the bottleneck, the paper's own Gaussian-decomposition argument suggests a natural experiment: use rectangular slits, decompose each mask into a few Gaussian terms, and check whether the predicted sum-of-theta structure still matches; mismatch would quantify the role of exotic same-plane paths.
The neuron construction has a purely classical reading as an analog optical kernel machine whose activation is a quadratic form; whether that yields practical learning advantages over digital computation is not addressed by the paper.

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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

The real content here is a wave-optics modeling exercise: the author derives iterative formulas for multi-plane diffraction with Gaussian and Hermite-Gaussian sources through arbitrary LCTs, and shows numerically that lens-based systems direct more intensity to the slits than free-space propagation. That algebra appears internally consistent and could serve someone building classical simulations of diffractive optical systems.

The advertised quantum supremacy claim does not survive contact with the model. The device is a coherent laser, thin masks, lenses, free space, and intensity detection. That is coherent-state linear optics, whose output statistics are classically simulable in polynomial time via the positive-P representation, regardless of how many Feynman paths you choose to count. The path sum in Eq. (16) is a bookkeeping identity; it is not a computational lower bound. The paper's own QS argument, Eqs. (31)-(35), just multiplies path numbers with arbitrarily chosen attenuation and significance parameters (s, r, m, k) and calls the result 'virtual qubits.' There is no complexity-theoretic hardness, and the paper concedes in Open Issues that verifying QS capability is an open problem. The stress-test note is right: even with perfect Gaussian slits and no looped paths, a classical simulator can compute the intensity by iterating Fresnel propagation over a spatial grid, at cost polynomial in grid size and number of planes.

The other advertised applications are speculative by the author's own admission: the NLSE speedup is a conjecture, and the quantum neuron is a quadratic form with an exponential activation, with no demonstrated learning advantage. The 'quantum path entanglement' resource is imported from the author's earlier papers without independent evidence; at face value it is just the usual interference of single-particle amplitudes.

To be fair, the paper is refreshingly open about its limitations. It lists the Gaussian-slit idealization and exotic paths as open issues, and it does not pretend the NLSE result is proven. But the central computational-advantage claim is a bridge too far, and the paper would be more honest if the QS section were rewritten as a speculative research agenda rather than a path-counting 'feasibility' argument.

Who gets value from this? Someone modeling multi-plane diffraction in Fourier optics, or a reader wanting a map of the author's QPC program. It deserves a serious referee because the LCT math is real, but the referee should be someone who will push back on the complexity claims. I'd recommend send to review, with the expectation of major revision or rejection of the supremacy half.

Referee Report

4 major / 4 minor

Summary. The paper develops a generalized Fourier-optics model of multi-plane diffraction (MPD) for what it calls photonic quantum path computing (QPC). A coherent laser source, Gaussian or Hermite-Gaussian transverse modes, Gaussian slit masks, and free-space/lens (LCT) propagation are modeled iteratively, yielding closed-form expressions for the sensor-plane wavefunction as a superposition over propagation paths (Eqs. (16), (25), and (27)). The paper claims these expressions generalize an earlier electron-based formulation and enable applications: quantum supremacy experiments with more than 2^100 Feynman paths, quantum neuron implementations, and solutions of the nonlinear Schrödinger equation via Riemann theta functions. Numerical examples compare free-space propagation with LCT systems for two diffraction planes.

Significance. If the computational-advantage claims were sound, the paper would be significant: it would propose a simple, room-temperature optical hardware that samples the interference of exponentially many paths while using only classical laser light and intensity detectors. I credit the paper for the systematic iterative algebra of the LCT/Hermite-Gaussian propagation and for explicitly listing the relevant open issues. However, the central advertised capability, quantum supremacy, is not supported: the proposed device belongs to a class of coherent-state linear optics with intensity detection whose output statistics are classically simulable, and the paper's own argument is a path-counting estimate, not a complexity-theoretic lower bound. The remaining applications are sketches or conjectures rather than demonstrated capabilities.

major comments (4)

[Eq. (16) and 'Applications for Quantum Supremacy'] The central quantum-supremacy claim is not supported because the proposed device is classically simulable. A coherent laser field transmitted through Gaussian amplitude masks and free-space or lens propagation is a passive linear-optical system acting on a coherent state; the measured intensity is determined by a propagated complex amplitude, and the output statistics are Poissonian. That amplitude can be computed by iterating the Fresnel/LCT integral plane by plane in about O(N M log M) operations for M spatial samples, independent of the product of slit counts. The path sum in Eq. (16) is therefore a bookkeeping identity, not a computational lower bound. Equations (31)-(35) count Feynman paths with the hand-chosen parameters m, k, s, and r, but counting paths never establishes hardness of sampling; the paper itself states in this section that rigorous complexity characterization is an open issue. The QS claim is nonetheless presented as a primary result, and this gap is load-bearing.
[Eqs. (34)-(35) and Fig. 5] The quantity qpath in Eq. (34) is not a number of qubits and is not shown to be a quantum resource. It is simply log2 of an effective path count after arbitrary attenuation factors s and r are divided out; no tensor-product decomposition into two-level systems, no gate set, and no error model are provided. Calling it a Hilbert-space size and comparing it with the Feynman-path count of Google's experiment conflates path enumeration with computational complexity. Consequently, the claim of 'hundreds of virtual qubits' and the QS feasibility plot in Fig. 5(b) do not establish a transition from classical to quantum advantage.
['Solution of nonlinear Schrödinger equation' (Eqs. (37)-(39))] The claimed speedup for NLSE solutions is explicitly a conjecture and is not supported by any algorithm, resource count, or error analysis. The suggested mapping from QPC intensities to |q(x,t)| requires identifying a Riemann spectrum (Y, k, ω, δ±) realized by a specific QPC setup; the text states only that theoretical modeling and extensive simulations are required. This application is therefore not a demonstrated capability of the architecture.
['Open Issues and Discussion'] The mathematical model assumes perfect Gaussian slit masks and neglects intra-plane 'exotic paths,' and the manuscript acknowledges both as open issues. Since Eq. (16) and the subsequent RTF and QS conclusions rely on exactly these assumptions, the physical relevance of the derived output distributions is not yet established. The proposed superposition-of-Gaussians approximation in Eq. (40) changes the summation structure and is not analyzed, so it does not rescue the ideal-Gaussian model.

minor comments (4)

[Throughout] The manuscript contains numerous LaTeX rendering artifacts (e.g., '/producttext', '/barex', '/summationdisplay') that make equations hard to read; a thorough copyedit is needed.
[Notation, Eqs. (16)-(27)] The notation for path-dependent parameters is heavy even for the N=3 example; a consolidated symbol table separate from Tables 1-3 would improve readability.
[Figs. 9 and 10] The numerical results are limited to N=3 and a single parameter set; they illustrate the formulas but do not test the scaling claims made for quantum supremacy.
[Eq. (40)] The Gaussian decomposition of non-Gaussian slits is cited to Refs. 72-74, but the text does not discuss the resulting growth in the number of terms or its effect on the claimed hardness.

Circularity Check

2 steps flagged · score 4.0 of 10

Propagation and RTF derivations are self-contained, but the QS hardness and QPE-resource premises are load-bearing self-citations from the author's own Refs. 6 and 12.

self citation load bearing [Results, subsection 'Hermite-Gaussian Sources', paragraph after Eq. (25)]
"The complexity of calculating the Gaussian form in (16) is classically hard as thoroughly discussed in Ref. 6 which requires to compute a special form of partial sum of RTF while the complex vector h_N-1,n and the matrix H_N-1,n varying for each path making it much harder compared with the computation of conventional partial sum of RTF. Therefore, the complexity characterization of computing Ψ_N^HG(x_N) is an open issue while it is expected to be significantly hard."

The paper's central computational-advantage claim for quantum supremacy rests on the assertion that evaluating Eq. (16) is classically hard. Instead of proving a lower bound, the paper refers to the author's own prior Ref. 6 ('as thoroughly discussed in Ref. 6') and then extrapolates that path-dependent parameters make it 'much harder'. This is a load-bearing self-citation: Ref. 6 is not an independent, machine-checked, or externally falsified source for the hardness of the present photonic setup. The paper itself concedes in Open Issues that 'it is an open issue to verify QS capability both complexity theoretically and experimentally,' so the hardness premise is unverified rather than derived within the paper.
self citation load bearing [Introduction, third paragraph (definition of QPE)]
"The unique form of temporal correlation freely available among the exponentially increasing number of Feynman paths in the MPD set-up is denoted as quantum path entanglement (QPE) in Ref. 12 as a novel resource to exploit for QC based on the coherence and superposition of the classical light source 13, 14."

The 'quantum' resource that is supposed to make the device a quantum computer rather than a classical optical interference experiment is defined and modeled in the author's own Ref. 12. The present paper adopts QPE from that self-citation and then uses it as the basis for the QC and QNN applications. No independent derivation or external verification of QPE is supplied here; the load-bearing premise that the device exploits 'quantum path entanglement' therefore reduces to a citation to the authors' own prior work.

full rationale

The mathematical core of the paper, Eqs. (15)-(27), is not circular. It starts from standard LCT/Fresnel kernels (Eqs. (6), (45), (47)) and explicit Gaussian/HG source masks, and iteratively integrates plane by plane to obtain path-sum representations; the reduction of the intensity to a partial sum of Riemann theta functions in Eq. (39) is obtained by substituting integer slit coordinates and uniform slit widths into Eq. (16). These are algebraic consequences of the stated optical model, not fits or renamed inputs. The QS path-count formulas (31)-(35) are also an analytic bookkeeping identity for the number of paths under assumed gain and attenuation parameters, not a fitted prediction. The circularity concern is narrower but real: the paper's computational-advantage premise is imported from the same author's earlier Ref. 6 ('classically hard as thoroughly discussed in Ref. 6'), and the 'quantum path entanglement' resource is introduced via the same author's Ref. 12. Both are load-bearing for the claim that this is a quantum computer rather than a classically simulable linear optical interference device. The paper itself concedes that QS capability is an open issue, which confirms that these premises are unverified self-citations rather than established results. The coherent-state/linear-optics simulability objection is a correctness risk, not a circularity, so it does not by itself raise the score; the score reflects the self-citation load-bearing structure above.

Assumptions & free parameters 3 free parameters · 5 assumptions · 2 invented entities

The central model leans on standard Fourier optics, but the computational-advantage claims depend on hand-chosen path-count parameters and on a self-cited notion of temporal entanglement that has no independent experimental handle. The Gaussian-slit and no-looped-paths assumptions are explicitly acknowledged as open issues in the paper itself.

free parameters (3)

Effective path-count model parameters (k, m, s, r or gain G) = chosen by hand for Fig. 5, e.g., G=-5 and G positive cases
Eqs. (31)-(35) define qpath in terms of these parameters; no measurement or first-principles derivation is given for s and r.
Gaussian slit width parameters β_{j,i} = 8 µm for second plane, varied on first plane in numerics
Numerical examples tune slit widths to match source intensity maxima; the central model takes them as given.
Inter-plane distances and lens focal lengths = e.g., L_T=[31.5 30.75 0.9] cm, f=[63 63] mm
Numerical comparison of LCT vs FSP uses these hand-picked values; no optimization or error analysis.

assumptions (5)

domain assumption Fresnel and harmonic-oscillator kernels describe single-photon propagation with classical sources.
Used throughout; justified by citation to Ref. 15, not derived in this paper.
ad hoc to paper Each slit is a Gaussian amplitude mask exp(-(x-X)^2/(2β^2)).
Used from Eq. (15) onward; the author lists perfect Gaussian slits as an open issue.
domain assumption Looped and exotic paths between slits on the same plane are negligible.
Open Issues admits these paths are not included; if significant, Eq. (16) fails.
ad hoc to paper Quantum path entanglement (QPE) as defined in Ref. 12 is a usable non-classical resource.
Self-cited concept; no independent falsifiable evidence is provided in this paper.
domain assumption The equivalent photon mass mλ = ħk/c makes the electron kernel applicable to photons.
Used to convert electron kernels to photons; based on phase-space optics and Ref. 15, not independently validated here.

invented entities (2)

Virtual qubits (qpath)
purpose: Quantify Hilbert-space size of a QS experiment in Eq. (34).
Not a physical qubit; a count of assumed paths, not a demonstrated computational resource.
Quantum path entanglement (QPE)
purpose: Claimed quantum resource for QPC.
Introduced in the author's prior Ref. 12; no measurement or protocol distinguishes it from ordinary wave coherence.

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Cite this review

Pith. "Pith review of Theory of Quantum Path Computing with Fourier Optics and Future Applications for Quantum Supremacy, Neural Networks and Nonlinear Schr\"odinger Equations." pith.science (2026). https://pith.science/paper/W6Q3CHVP

@misc{pith2026190802274,
  author       = {Pith},
  title        = {Pith review of: Theory of Quantum Path Computing with Fourier Optics and Future Applications for Quantum Supremacy, Neural Networks and Nonlinear Schr\"odinger Equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W6Q3CHVP}},
  note         = {Machine review of arXiv:1908.02274}
}

abstract

The scalability, error correction and practical problem solving are important challenges for quantum computing (QC) as more emphasized by quantum supremacy (QS) experiments. Quantum path computing (QPC), recently introduced for linear optic based QCs (LOQCs) as an unconventional design, targets to obtain scalability and practical problem solving. It samples the intensity from the interference of exponentially increasing number of propagation paths obtained in multi-plane diffraction (MPD) of classical particle sources. QPC exploits MPD based quantum temporal correlations of the paths and freely entangled projections a<t different time instants, for the first time, with the classical light source and intensity measurement while not requiring photon interactions or single photon sources and receivers. In this article, photonic QPC is defined, theoretically modeled and numerically analyzed for arbitrary Fourier optical or quadratic phase set-ups while utilizing both Gaussian and Hermite-Gaussian source laser modes. Problem solving capabilities already including partial sum of Riemann theta functions are extended. Important future applications, implementation challenges and open issues such as universal computation and quantum circuit implementations determining the scope of QC capabilities are discussed. The applications include QS experiments reaching more than $2^{100}$ Feynman paths, quantum neuron implementations and solutions of nonlinear Schr\"odinger equation.

Figures

Figures reproduced from arXiv: 1908.02274 by the authors.

**Figure 1.** QPC set-up composed of N −1 diffraction planes, FSP between the planes and a single sensor plane on which exponentially increasing number of propagation paths interfere6 . entanglement resources and quantum circuits are challenging to create due to the difficulty in the interaction of photons with each other. Multi-plane diffraction (MPD) based QC system denoted by quantum path computing (QPC) as shown in [PITH_FUL… view at source ↗

**Figure 2.** MPD based two-qubit state represented with four quantum histories of a single photon with the tensor product structure in time domain in analogy with the entangled state of two spatial qubits of two photons. with the following special form to utilize in solutions of important and classically hard number theoretical problems: fBB[k] ≡ I[k] ≡ [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Photonic QPC architecture composed of classical light source, MPD set-up composed of N −1 diffraction planes with Kj slits on jth plane, general LCT phase-space optics represented with the matrix elements LCTj,j+1 between the planes indexed with j and j +1, and a single sensor plane on which exponentially increasing number of propagation paths interfere. Each LCTj,j+1 is implemented with sections of FSP for the leng… view at source ↗

Figures from the paper (8 more)

**Figure 4.** Figure 4: (a) The set-up with a single thin-lens of focal length 60 (mm), (b) Gaussian source with σ0 = 20 (µm) and λ = 650 (nm), and (c) the distribution of the wave function on the first plane by shifting the lens inside the spatial interval of fixed total length L01 = t01 ×c …

**Figure 5.** Figure 5: (a) The special QPC design with N-1 planes diffracts the quantum wave function expanding with planes consisting of a linearly increasing number of slits. The positions of the planes (dj), the number of slits (k) in the first plane and the linear increment ratio m, i.e.…

**Figure 6.** Figure 6: (a) Classical artificial neuron implementation with the weight and input vectors of w® and x®, respectively, activation function f (.), bias b and the output O. (b) QPC based design of QNNs with quadratic weighting relationship of x® T H x® + x® T h® + b and nonlinear …

**Figure 7.** Figure 7: Photonic QPC set-up with Gaussian and HG classical monochromatic light source with λ = 650 nm, two planes for diffraction with the number of slits K1 = 11 and K2 = 27, respectively, and specific set-up of (a) LCT and (b) FSP design with the only difference of the exist…

**Figure 8.** Figure 8: (a) Gaussian and (b) Hermite-Gaussian (order l = 10) source waveforms with σ0 = 20 µm and W0 = 200 µm, respectively. (c) Slit positions with K1 = 11 and K2 = 27 slits (d) and the widths βej,i for j ∈ [1,2] and i ∈ [1,Kj]. where bΓ ≡ −DHN−1 D/π and the diagonal matrix D…

**Figure 9.** Figure 9: HG source with the order of l = 10 and W0 = 200 µm is utilized where the resulting spatial domain waveforms on the planes with the indices (a) j = 1, (b) j = 2 and (c) j = 3 for FSP, and (e) j = 1, (f) j = 2 and (g) j = 3 for LCT. PE (n, j) for (d) j = 2 and (h) j = 3.…

**Figure 10.** Figure 10: Gaussian source with σ0 = 20 µm is utilized where the resulting spatial domain waveforms on the planes with the indices (a) j = 1, (b) j = 2 and (c) j = 3 for FSP, and (e) j = 1, (f) j = 2 and (g) j = 3 for LCT. PE (n, j) for (d) j = 2 and (h) j = 3. Scaled Wigner dis…

**Figure 11.** Figure 11: Formulation set-up for first type of solution of Rayleigh-Sommerfeld diffraction through a slit Σ 77 . Open Issues and Discussion There are some open issues to best exploit photonic QPC method based on MPD and FO. Mathematical formulation correlating specific set-up p…

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