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arxiv: 2605.19570 · v1 · pith:RRG2DIWLnew · submitted 2026-05-19 · ⚛️ physics.optics

Energy-efficient programmable integrated photonics via optimized Euler rotations

Pablo Mart\'inez-Carrasco Romero , Andr\'es Macho-Ort\'iz , Jos\'e Roberto Rausell-Campo , Francisco Javier Fraile-Pel\'aez , Jos\'e Capmany Francoy This is my paper

Pith reviewed 2026-05-20 02:19 UTC · model grok-4.3

classification ⚛️ physics.optics
keywords programmable integrated photonicsenergy optimizationEuler rotationsBloch sphereunitary matricessilicon photonicsphotonic circuitsquantum gates
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The pith

Equivalent unitary matrices in photonic meshes can be realized with far lower energy by selecting the shortest Euler rotation trajectories on the Bloch sphere.

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

The paper establishes that any given N by N unitary transformation can be built from many different sequences of 2 by 2 interferometers, each sequence corresponding to a different set of rotation paths on the Bloch sphere. Because the energy cost of setting the phase shifters scales with the total angular length of those paths, some sequences are markedly cheaper than others even though they produce exactly the same overall matrix. By systematically choosing the shortest available trajectory for every interferometer, the authors obtain lower-energy implementations without altering the circuit function. This matters for scalability because actuation energy is now a primary barrier to large programmable photonic processors used in signal processing, neural networks, and quantum gates. The method is shown to work across feedforward meshes, hexagonal lattices, and several silicon chip demonstrations.

Core claim

Equivalent implementations of the same N×N unitary matrix (N ≥ 2) can be realized by different concatenations of Euler rotations on the Bloch sphere, each concatenation having a different total angular length. Since physical energy consumption is taken to be proportional to this length, choosing the minimal-length trajectories for every 2×2 block produces a lower-energy circuit that performs the identical transformation. The approach is validated numerically and experimentally in multiple silicon programmable integrated photonics architectures.

What carries the argument

Representation of each 2×2 unitary matrix as a product of three Euler rotations on the Bloch sphere, with energy minimization achieved by selecting the shortest angular-length decomposition among equivalent options.

If this is right

  • Minimum-energy configurations exist for any N×N unitary and can be found by exhaustive or heuristic search over rotation choices.
  • The savings apply without hardware modification to feedforward meshes, hexagonal meshes, neural-network accelerators, and photonic quantum-gate circuits.
  • Large-scale programmable integrated photonics becomes more feasible because total energy scales with the sum of selected path lengths rather than with a fixed worst-case value.
  • The geometric selection leaves the performed unitary unchanged, so classical and quantum functionality is preserved while energy drops.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same shortest-path principle could be applied to other platforms that represent operations as trajectories on the Bloch sphere, such as superconducting qubits or trapped ions.
  • When real devices have trajectory-dependent losses or crosstalk, the truly minimal-energy choice may shift away from the pure angular-length minimum.
  • Combining this geometric optimizer with existing sparsity or pruning techniques for unitary matrices could produce still larger energy reductions.

Load-bearing premise

The energy cost of actuating each interferometer is strictly proportional to the angular length of its chosen Euler rotation trajectory, with no extra overheads from hardware specifics or from switching between equivalent paths.

What would settle it

Implement two different but mathematically equivalent decompositions of the same 4×4 unitary matrix on the same photonic chip and measure whether the version with shorter total rotation length indeed requires less total phase-shifter actuation energy.

Figures

Figures reproduced from arXiv: 2605.19570 by Andr\'es Macho-Ort\'iz, Francisco Javier Fraile-Pel\'aez, Jos\'e Capmany Francoy, Jos\'e Roberto Rausell-Campo, Pablo Mart\'inez-Carrasco Romero.

Figure 1
Figure 1. Figure 1: Programmable integrated photonic (PIP) circuits for unitary matrix trans [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Geometrical interpretation of universal 2 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Comparison between dual- and single-phase-shifter implementations of tun [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Experimental validation and scalability analysis of the proposed energy op [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Power-consumption optimization in optical-computing applications. a [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Comparison of angular consumption per phase shifter in the decompositions as a function of matrix size (N = 2, 4, 8, 16) and the number of optimized phase shifters. 8 [PITH_FULL_IMAGE:figures/full_fig_p025_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Comparison between the standard building block, consisting of an MZI with two internal and two external phase shifters enabling the implementation of both positive and negative angles, and its equivalent realization in a hexagonal mesh. Figure S.8 presents the measurement results for implementing arbitrary unitary matrices on a hexagonal mesh for 2×2, 3×3, and 4×4 cases. It shows that the improvement in po… view at source ↗
Figure 8
Figure 8. Figure 8: Power consumption for implementing random matrices of varying sizes on a hexagonal mesh. 12 [PITH_FULL_IMAGE:figures/full_fig_p029_8.png] view at source ↗
read the original abstract

Programmable integrated photonics (PIP) has emerged as a powerful on-chip platform for optical signal processing and computing, enabling the implementation of reconfigurable N$\times$N unitary matrix transformations through meshes of tunable interferometers, which realize 2$\times$2 unitary matrices. However, the energy consumption associated with phase-shifter actuation is becoming a major limitation to the scalability of PIP platforms. Here, we introduce a geometric framework for energy optimization in PIP circuits by exploiting the representation of 2$\times$2 unitary matrices as concatenations of basic Euler rotations on the Bloch sphere. We show that equivalent implementations of the same N$\times$N unitary matrix (N $\geq$ 2) can exhibit markedly different energy costs depending on the rotation trajectories on the Bloch sphere implemented by each interferometer. Leveraging this insight, we identify minimum-energy configurations by systematically selecting the shortest rotation trajectories. We experimentally and numerically validate the proposed approach in diverse silicon PIP architectures, including feedforward and multipurpose hexagonal meshes, neural network accelerators, and photonic quantum-gate implementations. These results establish a general route toward more energy-efficient large-scale PIP processors for classical and quantum signal processing and computing applications.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript claims that equivalent N×N unitary transformations in programmable integrated photonics can be realized with markedly different energy costs depending on the specific Euler rotation trajectories chosen on the Bloch sphere for each 2×2 interferometer. By systematically selecting the shortest trajectories, minimum-energy configurations are identified and shown to reduce actuation energy; the approach is validated experimentally and numerically across feedforward meshes, hexagonal meshes, neural-network accelerators, and photonic quantum gates in silicon platforms.

Significance. If the energy-to-trajectory mapping holds, the geometric framework supplies a hardware-agnostic, parameter-light method for lowering the dominant energy bottleneck in large-scale PIP without redesigning the mesh topology or phase-shifter technology. The multi-architecture experimental and numerical validation, together with the explicit use of Bloch-sphere geometry rather than empirical fitting, constitutes a concrete advance toward scalable optical computing and quantum signal processing.

major comments (1)
  1. [§3] §3 (Energy Optimization Framework), paragraph following Eq. (2): the central ranking of equivalent decompositions and the declaration of minimum-energy configurations rest on the assumption that actuation energy scales linearly with the angular length of the chosen Euler rotation trajectory. This proportionality is load-bearing for the claimed savings, yet the text provides no derivation or measurement showing that real silicon phase-shifter energetics (thermo-optic power, electro-optic voltage swing, or MEMS displacement) are monotonic or linear in that geometric angle; non-linear phase-voltage curves, thermal crosstalk, and fixed per-actuation overheads are not addressed.
minor comments (2)
  1. [Figure 4] Figure 4 caption and associated text: the reported energy-reduction percentages are given without error bars or details on the number of independent calibrations; adding these would strengthen the experimental claim.
  2. [§2.1] Notation: the symbol E_p is introduced without an explicit definition in the main text before its first use in the optimization algorithm; a one-line definition would remove ambiguity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. The comment regarding the energy scaling assumption is well taken, and we have revised the manuscript to include a detailed physical derivation and supporting discussion of the linear approximation for silicon phase shifters. This strengthens the framework without altering the core claims. We address the point below.

read point-by-point responses

  1. Referee: [§3] §3 (Energy Optimization Framework), paragraph following Eq. (2): the central ranking of equivalent decompositions and the declaration of minimum-energy configurations rest on the assumption that actuation energy scales linearly with the angular length of the chosen Euler rotation trajectory. This proportionality is load-bearing for the claimed savings, yet the text provides no derivation or measurement showing that real silicon phase-shifter energetics (thermo-optic power, electro-optic voltage swing, or MEMS displacement) are monotonic or linear in that geometric angle; non-linear phase-voltage curves, thermal crosstalk, and fixed per-actuation overheads are not addressed.

    Authors: We agree that the original manuscript did not explicitly derive or validate the linear scaling assumption. In the revised version, we have added a new paragraph and supporting derivation immediately following Eq. (2) in §3. For thermo-optic phase shifters (the dominant technology in the silicon platforms considered), steady-state power P is proportional to the local temperature rise ΔT (P = κ ΔT, where κ is the thermal conductance), while the phase shift φ = (2π L / λ) (dn/dT) ΔT is linearly proportional to ΔT. Consequently, the required actuation power (and thus energy for a given settling time) scales linearly with φ. The Euler rotation angle on the Bloch sphere directly maps to this differential phase φ in the 2×2 interferometer, so the trajectory length corresponds to the total phase actuation. We include device-level measurements from our silicon test structures confirming near-linear power-vs-phase behavior over the relevant range (0 to 2π). Non-linearities in electro-optic or MEMS devices are acknowledged as platform-specific; our framework remains applicable by substituting the appropriate monotonic mapping. Thermal crosstalk and fixed overheads are mesh-dependent but affect all trajectories equally for a fixed topology, preserving the relative ranking of shortest paths. These additions are now in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity: optimization selects shortest trajectories under an explicit geometric energy model

full rationale

The paper posits that energy consumption scales with the angular length of Euler rotation trajectories on the Bloch sphere for 2x2 interferometers and then selects the shortest such trajectories to minimize energy for equivalent N x N unitary decompositions. This is an explicit modeling assumption rather than a quantity fitted to or derived from the target result itself. No equations reduce the claimed minimum-energy configurations to the same data by construction, no parameters are fitted and then renamed as predictions, and no load-bearing self-citations or uniqueness theorems imported from prior author work are required for the central geometric selection rule. Experimental and numerical validation is invoked to support the model, keeping the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The approach relies on the standard mathematical representation of 2x2 unitaries as Euler rotations and on an implicit linear mapping from rotation angle to actuation energy; no new entities are postulated.

free parameters (1)
  • energy scaling factor per rotation angle
    The model implicitly assumes or calibrates how much energy corresponds to each unit of rotation angle in the physical phase shifters.
axioms (1)
  • standard math Any 2x2 unitary matrix can be decomposed into Euler rotations on the Bloch sphere
    This is a standard fact from quantum optics used to map interferometer settings to sphere trajectories.

pith-pipeline@v0.9.0 · 5768 in / 1315 out tokens · 41619 ms · 2026-05-20T02:19:27.692834+00:00 · methodology

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Reference graph

Works this paper leans on

55 extracted references · 55 canonical work pages

  1. [1]

    Nature , volume=

    Programmable photonic circuits , author=. Nature , volume=. 2020 , publisher=

    work page 2020

  2. [2]

    Nature , volume=

    Limits on fundamental limits to computation , author=. Nature , volume=. 2014 , publisher=

    work page 2014

  3. [3]

    Advances in Optics and Photonics , volume=

    Optimization methods for integrated and programmable photonics in next-generation classical and quantum smart communication and signal processing systems , author=. Advances in Optics and Photonics , volume=. 2025 , publisher=

    work page 2025

  4. [4]

    2020 , publisher=

    Programmable integrated photonics , author=. 2020 , publisher=

    work page 2020

  5. [5]

    Nature Communications , volume=

    Fusion-based quantum computation , author=. Nature Communications , volume=. 2023 , publisher=

    work page 2023

  6. [6]

    Nature , volume=

    Quantum computational advantage with a programmable photonic processor , author=. Nature , volume=. 2022 , publisher=

    work page 2022

  7. [7]

    Science , volume=

    Multidimensional quantum entanglement with large-scale integrated optics , author=. Science , volume=. 2018 , publisher=

    work page 2018

  8. [8]

    Nature photonics , volume=

    Integrated photonic quantum technologies , author=. Nature photonics , volume=. 2020 , publisher=

    work page 2020

  9. [9]

    Nanophotonics , volume=

    Progress in neuromorphic photonics , author=. Nanophotonics , volume=. 2017 , publisher=

    work page 2017

  10. [10]

    Nature photonics , volume=

    Deep learning with coherent nanophotonic circuits , author=. Nature photonics , volume=. 2017 , publisher=

    work page 2017

  11. [11]

    Nature , volume=

    Inference in artificial intelligence with deep optics and photonics , author=. Nature , volume=. 2020 , publisher=

    work page 2020

  12. [12]

    Nature communications , volume=

    Multipurpose silicon photonics signal processor core , author=. Nature communications , volume=. 2017 , publisher=

    work page 2017

  13. [13]

    Nature Communications , volume=

    General-purpose programmable photonic processor for advanced radiofrequency applications , author=. Nature Communications , volume=. 2024 , publisher=

    work page 2024

  14. [14]

    Laser & Photonics Reviews , volume=

    A review of silicon-based integrated optical switches , author=. Laser & Photonics Reviews , volume=. 2023 , publisher=

    work page 2023

  15. [15]

    Optics express , volume=

    Photonic switching in high performance datacenters , author=. Optics express , volume=. 2018 , publisher=

    work page 2018

  16. [16]

    Science , volume=

    Universal linear optics , author=. Science , volume=. 2015 , publisher=

    work page 2015

  17. [17]

    Optica , volume=

    Linear programmable nanophotonic processors , author=. Optica , volume=. 2018 , publisher=

    work page 2018

  18. [18]

    Nature , volume=

    Parallel convolutional processing using an integrated photonic tensor core , author=. Nature , volume=. 2021 , publisher=

    work page 2021

  19. [19]

    Light: Science & Applications , volume=

    Unscrambling light—automatically undoing strong mixing between modes , author=. Light: Science & Applications , volume=. 2017 , publisher=

    work page 2017

  20. [20]

    Nature Photonics , volume=

    Single-chip photonic deep neural network with forward-only training , author=. Nature Photonics , volume=. 2024 , publisher=

    work page 2024

  21. [21]

    Laser & Photonics Reviews , volume=

    Analog Programmable-Photonic Computation , author=. Laser & Photonics Reviews , volume=. 2023 , publisher=

    work page 2023

  22. [22]

    Laser & Photonics Reviews , volume=

    Optical implementation of 2 2 universal unitary matrix transformations , author=. Laser & Photonics Reviews , volume=. 2021 , publisher=

    work page 2021

  23. [23]

    Physical review letters , volume=

    Experimental realization of any discrete unitary operator , author=. Physical review letters , volume=. 1994 , publisher=

    work page 1994

  24. [24]

    Optica , volume=

    Optimal design for universal multiport interferometers , author=. Optica , volume=. 2016 , publisher=

    work page 2016

  25. [25]

    ACS Photonics , volume=

    Scalability of large-scale photonic integrated circuits , author=. ACS Photonics , volume=. 2023 , publisher=

    work page 2023

  26. [26]

    Photonics Research , volume=

    Lowering the energy consumption in silicon photonic devices and systems , author=. Photonics Research , volume=. 2015 , publisher=

    work page 2015

  27. [27]

    Journal of Lightwave Technology , volume=

    Integrated photonic tensor processing unit for a matrix multiply: a review , author=. Journal of Lightwave Technology , volume=. 2023 , publisher=

    work page 2023

  28. [28]

    arXiv preprint arXiv:2506.05988 , year=

    Resource-efficient crosstalk mitigation for the high-fidelity operation of photonic integrated circuits with induced phase shifters , author=. arXiv preprint arXiv:2506.05988 , year=

    work page arXiv

  29. [29]

    Nanophotonics , volume=

    Fundamental Building Blocks for Scalable Photonic Programmable Processors , author=. Nanophotonics , volume=. 2026 , publisher=

    work page 2026

  30. [30]

    Nature Communications , volume=

    Asymptotically fault-tolerant programmable photonics , author=. Nature Communications , volume=. 2022 , publisher=

    work page 2022

  31. [31]

    APL Photonics , volume=

    Toward the information-theoretic limit of programmable photonics , author=. APL Photonics , volume=. 2025 , publisher=

    work page 2025

  32. [32]

    Nature Communications , volume=

    Heavy tails and pruning in programmable photonic circuits for universal unitaries , author=. Nature Communications , volume=. 2023 , publisher=

    work page 2023

  33. [33]

    Nature Photonics , pages=

    Programmable integrated quantum photonics , author=. Nature Photonics , pages=. 2026 , publisher=

    work page 2026

  34. [34]

    Physical Review A , volume=

    Universal programmable photonic architecture for quantum information processing , author=. Physical Review A , volume=. 2020 , publisher=

    work page 2020

  35. [35]

    Proceedings of the IEEE , volume=

    Gradient-based learning applied to document recognition , author=. Proceedings of the IEEE , volume=. 2002 , publisher=

    work page 2002

  36. [36]

    An introduction to Cartan’s KAK decomposition for QC programmers (2005) , author=

    work page 2005

  37. [37]

    2013 , publisher =

    Matti Hirvensalo , title =. 2013 , publisher =

    work page 2013

  38. [38]

    IBM Quantum Platform , year =

    work page

  39. [39]

    Harris, N. C. and Ma, Y. and Mower, J. and Baehr-Jones, T. and Englund, D. and Hochberg, M. and Galland, C. , title =. Optics Express , volume =. 2014 , doi =

    work page 2014

  40. [40]

    Advanced Photonics Nexus , volume=

    Silicon thermo-optic phase shifters: a review of configurations and optimization strategies , author=. Advanced Photonics Nexus , volume=. 2024 , publisher=

    work page 2024

  41. [41]

    and Takabayashi, A

    Quack, N. and Takabayashi, A. Y. and Sattari, H. and Edinger, P. and Jo, G. and Bleiker, S. J. and others , title =. Microsystems & Nanoengineering , volume =

    work page

  42. [42]

    and Kristinsson, K

    Edinger, P. and Kristinsson, K. and Errando-Herranz, C. and Takabayashi, A. Y. and Sattari, H. and Quack, N. and others and Gylfason, K. B. , title =. CLEO: Science and Innovations , pages =. 2021 , month = may, publisher =

    work page 2021

  43. [43]

    , journal=

    Quack, Niels and Sattari, Hamed and Takabayashi, Alain Yuji and Zhang, Yu and Verheyen, Peter and Bogaerts, Wim and Edinger, Pierre and Errando-Herranz, Carlos and Gylfason, Kristinn B. , journal=. MEMS-Enabled Silicon Photonic Integrated Devices and Circuits , year=

    work page

  44. [44]

    High-Speed Non-Volatile Barium Titanate Field Programmable Photonic Gate Array , year =

    Catal. High-Speed Non-Volatile Barium Titanate Field Programmable Photonic Gate Array , year =

    work page

  45. [45]

    Baets , title =

    Abdul Rahim and Artur Hermans and Benjamin Wohlfeil and Despoina Petousi and Bart Kuyken and Dries Van Thourhout and Roel G. Baets , title =. Advanced Photonics , number =

    work page

  46. [46]

    Applied Physics Reviews , volume =

    Wu, Bo and Zhou, Hailong and Dong, Jianji and Zhang, Xinliang , title =. Applied Physics Reviews , volume =. 2024 , month =

    work page 2024

  47. [47]

    Science Advances , volume =

    Yiwei Xie and Xiyuan Ke and Shihan Hong and Yuxin Sun and Lijia Song and Huan Li and Pan Wang and Daoxin Dai , title =. Science Advances , volume =. 2025 , abstract =

    work page 2025

  48. [48]

    IEEE Journal of Selected Topics in Quantum Electronics , volume=

    Column-row addressing of thermo-optic phase shifters for controlling large silicon photonic circuits , author=. IEEE Journal of Selected Topics in Quantum Electronics , volume=. 2020 , publisher=

    work page 2020

  49. [49]

    Advanced Photonics , volume=

    Taking silicon photonics modulators to a higher performance level: state-of-the-art and a review of new technologies , author=. Advanced Photonics , volume=. 2021 , publisher=

    work page 2021

  50. [50]

    Nanophotonics , volume=

    Recent breakthroughs in carrier depletion based silicon optical modulators , author=. Nanophotonics , volume=. 2014 , publisher=

    work page 2014

  51. [51]

    IEEE Journal of Quantum Electronics , volume=

    MEMS-enabled silicon photonic integrated devices and circuits , author=. IEEE Journal of Quantum Electronics , volume=. 2019 , publisher=

    work page 2019

  52. [52]

    Nature , volume=

    A large-scale microelectromechanical-systems-based silicon photonics LiDAR , author=. Nature , volume=. 2022 , publisher=

    work page 2022

  53. [53]

    Nature materials , volume=

    Large Pockels effect in micro-and nanostructured barium titanate integrated on silicon , author=. Nature materials , volume=. 2019 , publisher=

    work page 2019

  54. [54]

    Nature Photonics , volume=

    A ferroelectric multilevel non-volatile photonic phase shifter , author=. Nature Photonics , volume=. 2022 , publisher=

    work page 2022

  55. [55]

    APL Photonics , volume=

    Programming universal unitary transformations on a general-purpose silicon photonic platform , author=. APL Photonics , volume=. 2025 , publisher=

    work page 2025