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arxiv: 2607.01624 · v1 · pith:NVJQCIESnew · submitted 2026-07-02 · 🪐 quant-ph · cond-mat.quant-gas· cond-mat.str-el

Shallow Unitary Circuits for Kramers-Wannier Dualities

Yanting Cheng , Shang Liu This is my paper

Pith reviewed 2026-07-03 12:59 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.quant-gascond-mat.str-el
keywords Kramers-Wannier dualityunitary circuitsshort-range entangled stateslong-range entangled statesZ2 symmetrynonlocal gatesquantum duality maps
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0 comments X

The pith

Logarithmic depth nonlocal unitary circuits realize exact Z2 Kramers-Wannier dualities in one and two dimensions.

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

The paper constructs unitary circuits that use nonlocal connections and run in logarithmic depth to perform the Kramers-Wannier duality. This duality maps short-range entangled states to long-range entangled ones. The circuits implement the full duality map exactly within the symmetric sector for arbitrary inputs, not just fixed-point states. The construction extends to arbitrary Zn symmetries. This supplies a coherent method to study dualities and phase transitions without tailoring algorithms to individual target states.

Core claim

Explicit constructions of logarithmic-depth, spatially nonlocal unitary circuits realize the exact Z2 KW dualities in both one and two spatial dimensions and generalize to arbitrary Zn. Within the symmetric sector these circuits map arbitrary non-fixed-point short-range entangled states to their corresponding long-range entangled duals, implementing complete duality transformations rather than preparing specific states.

What carries the argument

Spatially nonlocal unitary circuits of logarithmic depth that implement the complete KW duality maps.

If this is right

  • The duality transformation can be performed in logarithmic rather than linear depth.
  • Arbitrary non-fixed-point SRE states are mapped exactly to LRE duals inside the symmetric sector.
  • A coherent pathway exists for exploring phase transitions and topological dualities on quantum hardware.
  • The same circuit style extends directly to Zn KW dualities.

Where Pith is reading between the lines

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

  • Platforms supporting native long-range gates could run duality-based simulations in much shorter time than local circuits allow.
  • Analogous log-depth constructions may exist for other symmetry or duality maps in many-body systems.
  • Small-system experiments could directly test the mapping by preparing an SRE state and measuring entanglement growth after circuit execution.

Load-bearing premise

Nonlocal connectivity is available on the platform and the circuits remain exact when restricted to the symmetric sector for arbitrary SRE inputs.

What would settle it

Apply the constructed circuit to a product state (an SRE input) and verify whether the output exhibits the long-range entanglement signature required by the dual LRE state.

Figures

Figures reproduced from arXiv: 2607.01624 by Shang Liu, Yanting Cheng.

Figure 1
Figure 1. Figure 1: Let us now take care of more general values of N. Sup￾pose 2p < N < 2 p+1 for some positive integer p. Let M = 2p and M′ = N − 2 p < M. We may first generate [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The realization of two-dimensional [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
read the original abstract

The quantum Kramers-Wannier (KW) duality is a fundamental transformation mapping short-range entangled (SRE) states to long-range entangled (LRE) states. While spatially local unitary circuits require linear-in-system-size depth to implement this duality, the ultimate speed limit for purely unitary circuits equipped with nonlocal connectivity remains an open question. Here, we explicitly construct logarithmic depth, spatially nonlocal unitary circuits that realize the exact $\mathbb{Z}_2$ KW dualities in both one and two spatial dimensions. We further generalize the construction to arbitrary $\mathbb{Z}_n$ KW dualities. Unlike algorithms tailored to prepare specific target states, our circuits implement complete duality maps. Within the symmetric (charge-neutral) sector, these dualities exactly transform arbitrary non-fixed-point SRE states into their corresponding LRE duals. Consequently, our results establish an efficient, purely coherent pathway for exploring phase transitions and topological dualities on modern quantum platforms.

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

0 major / 3 minor

Summary. The manuscript claims to explicitly construct logarithmic-depth, spatially nonlocal unitary circuits realizing the exact Z_2 Kramers-Wannier dualities in one and two spatial dimensions, with a generalization to arbitrary Z_n. These circuits are asserted to implement complete duality maps that, restricted to the symmetric (charge-neutral) sector, exactly transform arbitrary non-fixed-point short-range entangled (SRE) states into their long-range entangled (LRE) duals.

Significance. If the constructions and exactness claims hold, the work establishes an efficient, purely unitary and coherent route to duality maps beyond fixed-point states, which could enable direct exploration of phase transitions and topological dualities on quantum hardware supporting nonlocal connectivity. The log-depth scaling and complete-map property (as opposed to state-preparation algorithms) represent a clear technical advance over linear-depth local-circuit approaches.

minor comments (3)
  1. The abstract and introduction would benefit from a brief statement of the assumed gate set (e.g., whether arbitrary two-qubit gates or a specific universal set is used) and the precise notion of 'spatially nonlocal' (e.g., all-to-all vs. limited-range long-range interactions).
  2. Figure captions and circuit diagrams (presumably in §§3–4) should explicitly label the depth scaling and confirm that the circuits remain exact when restricted to the symmetric subspace for generic SRE inputs, not merely for fixed-point states.
  3. A short discussion of the overhead in the symmetric-sector projection (if any) and how the construction avoids leakage outside the charge-neutral sector would improve clarity for experimental implementation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the recommendation of minor revision. The provided summary correctly captures the central claims regarding the construction of logarithmic-depth nonlocal unitary circuits for exact Z_2 and Z_n Kramers-Wannier dualities.

Circularity Check

0 steps flagged

Explicit construction; no circularity detected

full rationale

The paper's central claim is an explicit construction of log-depth nonlocal unitary circuits realizing exact Z2 (and Zn) KW dualities on the symmetric sector. No fitted parameters, no predictions derived from data subsets, and no load-bearing self-citations or ansatzes imported from prior author work appear in the provided abstract or description. The derivation is self-contained as a direct circuit design rather than a reduction to its own inputs or external self-referential theorems.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only abstract available; no free parameters, axioms, or invented entities are described.

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discussion (0)

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Reference graph

Works this paper leans on

32 extracted references · 19 canonical work pages · 11 internal anchors

  1. [1]

    Fracton Topological Order, Generalized Lattice Gauge Theory and Duality

    S. Vijay, J. Haah, and L. Fu, Fracton topological order, generalized lattice gauge theory, and duality, Phys. Rev. B94, 235157 (2016), arXiv:1603.04442 [cond-mat.str-el]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  2. [2]

    J. B. Kogut, An introduction to lattice gauge theory and spin systems, Rev. Mod. Phys.51, 659 (1979)

    1979

  3. [3]

    Bravyi, M

    S. Bravyi, M. B. Hastings, and F. Verstraete, Lieb- robinson bounds and the generation of correlations and topological quantum order, Phys. Rev. Lett.97, 050401 (2006)

    2006

  4. [4]

    X. Chen, A. Dua, M. Hermele, D. T. Stephen, N. Tan- tivasadakarn, R. Vanhove, and J.-Y. Zhao, Sequential quantum circuits as maps between gapped phases, Phys. Rev. B109, 075116 (2024), arXiv:2307.01267 [cond- mat.str-el]

    work page arXiv 2024

  5. [5]

    Tantivasadakarn, R

    N. Tantivasadakarn, R. Thorngren, A. Vishwanath, and R. Verresen, Long-range entanglement from mea- suring symmetry-protected topological phases, arXiv e-prints , arXiv:2112.01519 (2021), arXiv:2112.01519 [cond-mat.str-el]

    work page arXiv 2021

  6. [6]

    Bravyi, I

    S. Bravyi, I. Kim, A. Kliesch, and R. Koenig, Adap- tive constant-depth circuits for manipulating non- abelian anyons, arXiv e-prints , arXiv:2205.01933 (2022), arXiv:2205.01933 [quant-ph]

    work page arXiv 2022

  7. [7]

    Tantivasadakarn, A

    N. Tantivasadakarn, A. Vishwanath, and R. Verresen, Hierarchy of Topological Order From Finite-Depth Uni- taries, Measurement, and Feedforward, PRX Quantum 4, 020339 (2023), arXiv:2209.06202 [quant-ph]

    work page arXiv 2023

  8. [8]

    Lootens, C

    L. Lootens, C. Delcamp, D. Williamson, and F. Ver- straete, Low-Depth Unitary Quantum Circuits for Duali- ties in One-Dimensional Quantum Lattice Models, Phys. Rev. Lett.134, 130403 (2025), arXiv:2311.01439 [quant- ph]

    work page arXiv 2025

  9. [9]

    Barredo, S

    D. Barredo, S. de L´ es´ eleuc, V. Lienhard, T. Lahaye, and A. Browaeys, An atom-by-atom assembler of defect-free arbitrary two-dimensional atomic arrays, Science354, 1021 (2016)

    2016

  10. [10]

    Labuhn, D

    H. Labuhn, D. Barredo, S. Ravets, S. de L´ es´ eleuc, T. Macr` ı, T. Lahaye, and A. Browaeys, Tunable two- dimensional arrays of single rydberg atoms for realizing quantum ising models, Nature534, 667 (2016)

    2016

  11. [11]

    Bernien, S

    H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Om- ran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner, V. Vuleti´ c, and M. D. Lukin, Probing many- body dynamics on a 51-atom quantum simulator, Nature 551, 579 (2017)

    2017

  12. [12]

    S. J. Evered, D. Bluvstein, M. Kalinowski, S. Ebadi, T. Manovitz, H. Zhou, S. H. Li, A. A. Geim, T. T. Wang, N. Maskara, H. Levine, G. Semeghini, M. Greiner, V. Vuleti´ c, and M. D. Lukin, High-fidelity parallel entan- gling gates on a neutral-atom quantum computer, Nature 622, 268 (2023)

    2023

  13. [13]

    Pichard, D

    G. Pichard, D. Lim, E. Bloch, J. Vaneecloo, L. Boura- chot, G.-J. Both, G. M´ eriaux, S. Dutartre, R. Hostein, J. Paris, B. Ximenez, A. Signoles, A. Browaeys, T. La- haye, and D. Dreon, Rearrangement of individual atoms in a 2000-site optical-tweezer array at cryogenic temper- atures, Phys. Rev. Appl.22, 024073 (2024)

    2000

  14. [14]

    Bluvstein, S

    D. Bluvstein, S. J. Evered, A. A. Geim, S. H. Li, H. Zhou, T. Manovitz, S. Ebadi, M. Cain, M. Kali- nowski, D. Hangleiter, J. P. Bonilla Ataides, N. Maskara, I. Cong, X. Gao, P. Sales Rodriguez, T. Karolyshyn, G. Semeghini, M. J. Gullans, M. Greiner, V. Vuleti´ c, and M. D. Lukin, Logical quantum processor based on reconfigurable atom arrays, Nature626, 58 (2024)

    2024

  15. [15]

    Q. Xu, J. P. Bonilla Ataides, C. A. Pattison, N. Raveen- dran, D. Bluvstein, J. Wurtz, B. Vasi´ c, M. D. Lukin, L. Jiang, and H. Zhou, Constant-overhead fault-tolerant quantum computation with reconfigurable atom arrays, Nature Physics20, 1084 (2024)

    2024

  16. [16]

    Bluvstein, A

    D. Bluvstein, A. A. Geim, S. H. Li, S. J. Evered, J. P. Bonilla Ataides, G. Baranes, A. Gu, T. Manovitz, M. Xu, M. Kalinowski, S. Majidy, C. Kokail, N. Maskara, E. C. Trapp, L. M. Stewart, S. Hollerith, H. Zhou, M. J. Gullans, S. F. Yelin, M. Greiner, V. Vuleti´ c, M. Cain, and M. D. Lukin, A fault-tolerant neutral-atom architec- ture for universal quantu...

    2026

  17. [17]

    J. M. Pino, J. M. Dreiling, C. Figgatt, J. P. Gaebler, S. A. Moses, M. S. Allman, C. H. Baldwin, M. Foss-Feig, D. Hayes, K. Mayer, C. Ryan-Anderson, and B. Neyen- huis, Demonstration of the trapped-ion quantum ccd computer architecture, Nature592, 209 (2021)

    2021

  18. [18]

    S. A. Moses, C. H. Baldwin, M. S. Allman, R. An- cona, L. Ascarrunz, C. Barnes, J. Bartolotta, B. Bjork, P. Blanchard, M. Bohn, J. G. Bohnet, N. C. Brown, N. Q. Burdick, W. C. Burton, S. L. Campbell, J. P. Campora, C. Carron, J. Chambers, J. W. Chan, Y. H. Chen, A. Chernoguzov, E. Chertkov, J. Colina, J. P. Curtis, R. Daniel, M. DeCross, D. Deen, C. Delan...

    2023

  19. [19]

    Quantum Circuits with Mixed States

    D. Aharonov, A. Kitaev, and N. Nisan, Quantum Circuits with Mixed States, arXiv e-prints , quant-ph/9806029 (1998), arXiv:quant-ph/9806029 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  20. [20]

    D. T. Stephen, A. Dua, A. Lavasani, and R. Nandk- ishore, Nonlocal Finite-Depth Circuits for Constructing Symmetry-Protected Topological States and Quantum Cellular Automata, PRX Quantum5, 010304 (2024), arXiv:2212.06844 [quant-ph]

    work page arXiv 2024

  21. [21]

    Quantum Circuit Depth Lower Bounds For Homological Codes

    D. Aharonov and Y. Touati, Quantum Circuit Depth Lower Bounds For Homological Codes, arXiv e-prints , arXiv:1810.03912 (2018), arXiv:1810.03912 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  22. [22]

    D. Cruz, R. Fournier, F. Gremion, A. Jeannerot, K. Komagata, T. Tosic, J. Thiesbrummel, C. L. Chan, N. Macris, M.-A. Dupertuis, and C. Javerzac-Galy, Ef- ficient quantum algorithms forGHZandWstates, and implementation on the IBM quantum computer, arXiv e-prints , arXiv:1807.05572 (2018), arXiv:1807.05572 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  23. [23]

    Moore and M

    C. Moore and M. Nilsson, Parallel Quantum Compu- tation and Quantum Codes, arXiv e-prints , quant- ph/9808027 (1998), arXiv:quant-ph/9808027 [quant-ph]

    work page arXiv 1998

  24. [24]

    A class of quantum many-body states that can be efficiently simulated

    G. Vidal, Class of Quantum Many-Body States That Can Be Efficiently Simulated, Phys. Rev. Lett.101, 110501 (2008), arXiv:quant-ph/0610099 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  25. [25]

    Entanglement renormalization and topological order

    M. Aguado and G. Vidal, Entanglement Renormaliza- tion and Topological Order, Phys. Rev. Lett.100, 070404 (2008), arXiv:0712.0348 [cond-mat.str-el]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  26. [26]

    Exact entanglement renormalization for string-net models

    R. K¨ onig, B. W. Reichardt, and G. Vidal, Exact entangle- ment renormalization for string-net models, Phys. Rev. B79, 195123 (2009), arXiv:0806.4583 [cond-mat.str-el]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  27. [27]

    Entanglement renormalization and wavelets

    G. Evenbly and S. R. White, Entanglement Renormaliza- tion and Wavelets, Phys. Rev. Lett.116, 140403 (2016), arXiv:1602.01166 [cond-mat.str-el]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  28. [28]

    Rigorous free fermion entanglement renormalization from wavelet theory

    J. Haegeman, B. Swingle, M. Walter, J. Cotler, G. Even- bly, and V. B. Scholz, Rigorous Free-Fermion Entan- glement Renormalization from Wavelet Theory, Physical Review X8, 011003 (2018), arXiv:1707.06243 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  29. [29]

    We say that an operatorOtransforms asO7→O ′ under a linear mapfiff O=O ′f

  30. [30]

    D. J. Williamson and M. Cheng, Designer non-Abelian fractons from topological layers, Phys. Rev. B107, 035103 (2023), arXiv:2004.07251 [cond-mat.str-el]

    work page arXiv 2023

  31. [31]

    Coupled-Layer Construction of Quantum Product Codes

    S. Zhang, T.-C. Wei, and N. Tantivasadakarn, Coupled- Layer Construction of Quantum Product Codes, arXiv e-prints , arXiv:2603.08711 (2026), arXiv:2603.08711 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  32. [32]

    Constructing Bulk Topological Orders via Layered Gauging

    S. Liu, Constructing Bulk Topological Orders via Lay- ered Gauging, arXiv e-prints , arXiv:2604.27363 (2026), arXiv:2604.27363 [cond-mat.str-el]

    work page internal anchor Pith review Pith/arXiv arXiv 2026