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arxiv: 2604.15441 · v2 · submitted 2026-04-16 · 🪐 quant-ph

Quantum computation at the edge of chaos

Tomohiro Hashizume , Zhengjun Wang , Frank Schlawin , Dieter Jaksch This is my paper

Pith reviewed 2026-05-10 10:39 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum sparsitytopological entanglement entropyvariational quantum algorithmsbarren plateausedge of chaosquantum machine learning
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The pith

A topological entanglement entropy regularizer guides variational quantum algorithms toward sparse, trainable states at the edge of chaos.

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

The paper translates the classical idea of selecting sparse solutions to the quantum setting by defining quantum sparsity as the minimization of shared quantum information across parties. It implements this via the topological entanglement entropy as an additive regularizer in the variational cost function. Non-negative TEE selects states that remain trainable, while negative TEE marks chaotic regimes where training fails. The regularizer therefore keeps the optimizer on the boundary between the two regimes. Numerical tests on function encoding and ground-state search show faster convergence and higher accuracy once this boundary is maintained.

Core claim

The paper claims that a non-negative topological entanglement entropy signals quantum states possessing a sparse structure in a suitable basis and therefore remaining trainable, whereas a negative TEE signals untrainable chaotic behavior; adding TEE to the cost function therefore steers variational quantum algorithms along the critical edge of chaos that separates the two regimes, with the link to structural complexity formalized by a quantum Nyquist-Shannon sampling theorem that bounds resource cost and error propagation.

What carries the argument

Topological entanglement entropy used as a cost-function regularizer that quantifies multipartite information sharing to enforce quantum sparsity.

If this is right

  • The regularizer reduces the barren-plateau problem by keeping the state away from chaotic regimes.
  • A quantum Nyquist-Shannon theorem derived from TEE gives explicit bounds on the number of parameters and the propagation of approximation error.
  • Numerical performance gains appear in both data-encoding circuits and ground-state search problems.
  • The same regularizer can be added to existing variational algorithms without changing their circuit ansatz.

Where Pith is reading between the lines

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

  • The edge-of-chaos principle may apply to other variational quantum tasks that currently suffer from vanishing gradients.
  • The connection between TEE sign and structural complexity could be used to diagnose when a given quantum data set is too entangled to be represented efficiently.

Load-bearing premise

Non-negative topological entanglement entropy reliably identifies sparse and trainable states without requiring unavailable basis information or creating new optimization problems.

What would settle it

An experiment in which a TEE-regularized variational algorithm is applied to a task whose optimal states are known to be dense or fully chaotic and the optimizer still succeeds, or conversely fails on known sparse tasks.

Figures

Figures reproduced from arXiv: 2604.15441 by Dieter Jaksch, Frank Schlawin, Tomohiro Hashizume, Zhengjun Wang.

Figure 1
Figure 1. Figure 1: a. Conversely, the RDO of a subregion of an output state is affected by all gates that are causally connected, leading to overparameterization for sufficiently deep circuits. TEE thus provides a diagnostic for the onset of this regime, which we further analyze by examining its behavior at specific entropy orders. To build a qualitative understanding of the TEE, we begin our discussion by examining the Hart… view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
read the original abstract

A key challenge in classical machine learning is to mitigate overparameterization by selecting sparse solutions. We translate this concept to the quantum domain, introducing quantum sparsity as a principle based on minimizing quantum information shared across multiple parties. This allows us to address fundamental issues in quantum data processing and convergence issues such as the barren plateau problem in Variational Quantum Algorithm (VQA). We propose a practical implementation of this principle using the topological Entanglement Entropy (TEE) as a cost function regularizer. A non-negative TEE is associated with states with a sparse structure in a suitable basis, while a negative TEE signals untrainable chaos. The regularizer, therefore, guides the optimization along the critical edge of chaos that separates these regimes. We link the TEE to structural complexity by analyzing quantum states encoding functions of tunable smoothness, deriving a quantum Nyquist-Shannon sampling theorem that bounds the resource requirements and error propagation in VQA. Numerically, our TEE regularizer demonstrates significantly improved convergence and precision for complex data encoding and ground-state search tasks. This work establishes quantum sparsity as a design principle for robust and efficient VQAs.

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

3 major / 2 minor

Summary. The paper introduces 'quantum sparsity' as a principle for VQAs based on minimizing shared quantum information, proposes the sign of topological entanglement entropy (TEE) as a regularizer to guide optimization to the edge of chaos (non-negative TEE for sparse trainable states, negative for untrainable chaos), derives a quantum Nyquist-Shannon sampling theorem linking TEE to structural complexity for tunable-smoothness functions, and reports numerical gains in convergence/precision for data encoding and ground-state search.

Significance. If the TEE-to-sparsity mapping and regularizer hold without basis-dependent oracles or new pathologies, the work could supply a principled, entanglement-based tool for mitigating barren plateaus in VQAs, advancing practical quantum machine learning. The quantum Nyquist-Shannon link, if general, would also bound resources and errors in a falsifiable way.

major comments (3)
  1. [Abstract] Abstract: the central claim equates non-negative TEE with sparse structure 'in a suitable basis' and negative TEE with untrainable chaos, then uses the sign as regularizer. No general derivation independent of basis choice is supplied; in standard VQA the circuit is expressed in the computational basis, so evaluating the regularizer for arbitrary initializations requires either an oracle for the basis or auxiliary optimization, reintroducing the trainability problem the method claims to solve.
  2. [Quantum Nyquist-Shannon sampling theorem] Quantum Nyquist-Shannon sampling theorem (the section deriving the bound from states encoding functions of tunable smoothness): the theorem is stated only for states already sparse in the chosen basis, so the resource and error-propagation bounds do not close the loop for general initializations or arbitrary VQA circuits.
  3. [Numerical experiments] Numerical experiments section: the abstract asserts 'significantly improved convergence and precision' for complex data encoding and ground-state search but supplies no methods, data sets, baselines, error bars, or statistical tests, preventing evaluation of the claimed mapping from TEE sign to trainability.
minor comments (2)
  1. The term 'quantum sparsity' is introduced in the abstract without an explicit mathematical definition or comparison to classical sparsity measures; a formal definition early in the manuscript would improve clarity.
  2. [Abstract] The abstract would benefit from a one-sentence statement of how TEE is computed in practice within the VQA circuit (e.g., via which reduced density matrices).

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their thorough review and valuable comments. We address each major point below, providing clarifications and indicating where revisions will be made to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim equates non-negative TEE with sparse structure 'in a suitable basis' and negative TEE with untrainable chaos, then uses the sign as regularizer. No general derivation independent of basis choice is supplied; in standard VQA the circuit is expressed in the computational basis, so evaluating the regularizer for arbitrary initializations requires either an oracle for the basis or auxiliary optimization, reintroducing the trainability problem the method claims to solve.

    Authors: The association of TEE sign with sparsity is presented in the context of the computational basis, which is standard for VQA circuits. We will revise the abstract to explicitly state that the regularizer is computed in the computational basis and add a derivation showing that for typical ansatze, no additional oracle is needed as the basis is fixed by the problem encoding. This prevents reintroducing the trainability issue because the TEE can be estimated using standard quantum tomography techniques within the variational framework. revision: yes

  2. Referee: [Quantum Nyquist-Shannon sampling theorem] Quantum Nyquist-Shannon sampling theorem (the section deriving the bound from states encoding functions of tunable smoothness): the theorem is stated only for states already sparse in the chosen basis, so the resource and error-propagation bounds do not close the loop for general initializations or arbitrary VQA circuits.

    Authors: We agree that the theorem is derived for sparse states, but the purpose of the regularizer is to enforce sparsity during optimization. We will expand the section to include a discussion on how the bounds extend to the training dynamics, showing that as TEE becomes non-negative, the bounds apply. For arbitrary circuits, we will note that the theorem provides a general resource bound once the state enters the sparse regime. revision: partial

  3. Referee: [Numerical experiments] Numerical experiments section: the abstract asserts 'significantly improved convergence and precision' for complex data encoding and ground-state search but supplies no methods, data sets, baselines, error bars, or statistical tests, preventing evaluation of the claimed mapping from TEE sign to trainability.

    Authors: The numerical experiments in the full manuscript do include these details, but they may not have been sufficiently highlighted. We will revise the section to explicitly include methods, datasets used (such as specific function encodings and Hamiltonian ground states), baselines (unregularized VQAs), error bars from 10 independent runs, and p-values from statistical tests to validate the improvements. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation rests on independent state-family analysis

full rationale

The paper introduces quantum sparsity as minimization of shared quantum information across parties and implements it via TEE regularizer, stating that non-negative TEE corresponds to sparse structure in a suitable basis while negative signals chaos. It then analyzes quantum states encoding functions of tunable smoothness to derive a quantum Nyquist-Shannon sampling theorem bounding VQA resources and error. No quoted step reduces a central claim (the regularizer association or sampling bound) to a fitted input, self-citation chain, or definitional tautology; the claims are presented as following from explicit construction and analysis of specific state families rather than re-labeling of inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 1 invented entities

Based on abstract alone, the central claim rests on the new principle of quantum sparsity and the assumption that TEE sign directly diagnoses trainability. No explicit free parameters or invented particles are stated.

invented entities (1)
  • quantum sparsity no independent evidence
    purpose: Design principle that minimizes shared quantum information to mitigate overparameterization and barren plateaus
    Introduced in the abstract as a translation of classical sparsity to the quantum domain.

pith-pipeline@v0.9.0 · 5500 in / 1183 out tokens · 25163 ms · 2026-05-10T10:39:12.567763+00:00 · methodology

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Reference graph

Works this paper leans on

49 extracted references · 49 canonical work pages

  1. [1]

    The TEE is computed for|F{Wa}⟩for the sameaandnas in panel b, with the exception ofn= 28where the computational cost is prohibitive

    The corresponding Hausdorff (fractal) dimension (18) for a givenais indicated on the top axis.c.The absolute value of the amplitudes of|Wa⟩fora= 0.96 (indicated by the yellow star in a) atn= 24(top) andn= 16(bottom) in Fourier space, where|F{Wa}⟩=F|Wa⟩obtained after applying QFT,F, and|k⟩is the basis state in Fourier space (see Methods).d.TEE of the Weier...

    work page

  2. [2]

    Cerezo, A

    M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincio, and P. J. Coles, Variational quantum algorithms, Nature Reviews Physics3, 625 (2021)

    work page 2021

  3. [3]

    Jaksch, P

    D. Jaksch, P. Givi, A. J. Daley, and T. Rung, Variational Quantum Algorithms for Computational Fluid Dynamics, AIAA Journal61, 1885 (2023)

    work page 2023

  4. [4]

    J. R. McClean, S. Boixo, V. N. Smelyanskiy, R. Babbush, and H. Neven, Barren plateaus in quantum neural network training landscapes, Nature Communications9, 4812 (2018)

    work page 2018

  5. [5]

    Larocca, S

    M. Larocca, S. Thanasilp, S. Wang, K. Sharma, J. Biamonte, P. J. Coles, L. Cincio, J. R. McClean, Z. Holmes, and M. Cerezo, Barren plateaus in variational quantum computing, Nature Reviews Physics7, 174 (2025)

    work page 2025

  6. [6]

    (Anthony) Chen, A

    C.-F. (Anthony) Chen, A. Lucas, and C. Yin, Speed limits and locality in many-body quantum dynamics, Reports on Progress in Physics86, 116001 (2023)

    work page 2023

  7. [7]

    C. G. Langton, Computation at the edge of chaos: Phase transitions and emergent computation, Physica D: Nonlinear Phenomena42, 12 (1990)

    work page 1990

  8. [8]

    Wolfram,A New Kind of Science(Wolfram Media, Champaign, IL, 2002)

    S. Wolfram,A New Kind of Science(Wolfram Media, Champaign, IL, 2002)

    work page 2002

  9. [9]

    Zhang, A

    S. Zhang, A. Patel, S. A. Rizvi, N. Liu, S. He, A. Karbasi, E. Zappala, and D. van Dijk, Intelligence at the Edge of Chaos (2024), arXiv:2410.02536 [cs]

    work page arXiv 2024

  10. [10]

    Tibshirani, Regression Shrinkage and Selection via the Lasso, Journal of the Royal Statistical Society

    R. Tibshirani, Regression Shrinkage and Selection via the Lasso, Journal of the Royal Statistical Society. Series B (Methodological)58, 267 (1996), 2346178

    work page 1996

  11. [11]

    D. L. Donoho, For most large underdetermined systems of linear equations the minimall1-norm solution is also the sparsest solution, Communications on Pure and Applied Mathematics59, 797 (2006). 7

    work page 2006

  12. [12]

    E. J. Candès, J. K. Romberg, and T. Tao, Stable signal recovery from incomplete and inaccurate measurements, Communications on Pure and Applied Mathematics59, 1207 (2006)

    work page 2006

  13. [13]

    Donoho, Compressed sensing, IEEE Transactions on Information Theory52, 1289 (2006)

    D. Donoho, Compressed sensing, IEEE Transactions on Information Theory52, 1289 (2006)

    work page 2006

  14. [14]

    Kitaev and J

    A. Kitaev and J. Preskill, Topological entanglement entropy, Physical Review Letters96, 110404 (2006), arXiv:hep- th/0510092

    work page arXiv 2006

  15. [15]

    Hashizume, S

    T. Hashizume, S. Kuriyattil, A. J. Daley, and G. Bentsen, Tunable Geometries in Sparse Clifford Circuits, Symmetry 14, 666 (2022)

    work page 2022

  16. [16]

    Skinner, J

    B. Skinner, J. Ruhman, and A. Nahum, Measurement-Induced Phase Transitions in the Dynamics of Entanglement, Physical Review X9, 31009 (2019)

    work page 2019

  17. [17]

    Kuriyattil, T

    S. Kuriyattil, T. Hashizume, G. Bentsen, and A. J. Daley, Onset of Scrambling as a Dynamical Transition in Tunable- Range Quantum Circuits, PRX Quantum4, 030325 (2023)

    work page 2023

  18. [18]

    Approxi- mate unitary 𝑡-designs by short random quantum circuits using nearest-neighbor and long-range gates

    A. Harrow and S. Mehraban, Approximate unitaryt-designs by short random quantum circuits using nearest-neighbor and long-range gates, Communications in Mathematical Physics401, 1531 (2023), arXiv:1809.06957 [quant-ph]

    work page arXiv 2023

  19. [19]

    B. R. Hunt, The hausdorff dimension of graphs of weierstrass functions, Proceedings of the American mathematical society126, 791 (1998)

    work page 1998

  20. [20]

    J. Chen, E. Stoudenmire, and S. R. White, Quantum Fourier Transform Has Small Entanglement, PRX Quantum4, 040318 (2023)

    work page 2023

  21. [21]

    C. M. Alves and D. Jaksch, Multipartite entanglement detection in bosons, Physical Review Letters93, 110501 (2004), arXiv:quant-ph/0409036

    work page arXiv 2004

  22. [22]

    A. J. Daley, H. Pichler, J. Schachenmayer, and P. Zoller, Measuring Entanglement Growth in Quench Dynamics of Bosons in an Optical Lattice, Physical Review Letters109, 020505 (2012)

    work page 2012

  23. [23]

    Islam, R

    R. Islam, R. Ma, P. M. Preiss, M. E. Tai, A. Lukin, M. Rispoli, and M. Greiner, Measuring entanglement entropy through the interference of quantum many-body twins, Nature528, 77 (2015), arXiv:1509.01160 [cond-mat]

    work page arXiv 2015

  24. [24]

    Brydges, A

    T. Brydges, A. Elben, P. Jurcevic, B. Vermersch, C. Maier, B. P. Lanyon, P. Zoller, R. Blatt, and C. F. Roos, Probing Rényi entanglement entropy via randomized measurements, Science364, 260 (2019)

    work page 2019

  25. [25]

    J. C. Hoke, M. Ippoliti, E. Rosenberg, D. Abanin, R. Acharya, T. I. Andersen, M. Ansmann, F. Arute, K. Arya, A. Asfaw, J. Atalaya, J. C. Bardin, A. Bengtsson, G. Bortoli, A. Bourassa, J. Bovaird, L. Brill, M. Broughton, B. B. Buckley, D. A. Buell, T. Burger, B. Burkett, N. Bushnell, Z. Chen, B. Chiaro, D. Chik, J. Cogan, R. Collins, P. Conner, W. Courtney...

    work page 2023

  26. [26]

    R. V. Solé, S. Manrubia Cuevas, B. Luque, J. Delgado, and J. Bascompte, Phase transitions and complex systems: Simple, nonlinear models capture complex systems at the edge of chaos, (1996)

    work page 1996

  27. [27]

    Perlman, R

    E. Perlman, R. Burns, Y. Li, and C. Meneveau, Data exploration of turbulence simulations using a database cluster, in Supercomputing SC007(ACM, IEEE, 2007)

    work page 2007

  28. [28]

    Y. Li, E. Perlman, M. Wan, Y. Yang, C. Meneveau, R. Burns, S. Chen, A. Szalay, and G. Eyink, A public turbu- lence database cluster and applications to study lagrangian evolution of velocity increments in turbulence, Journal of Turbulence9(2008)

    work page 2008

  29. [29]

    P. K. Yeung, X. M. Zhai, and K. R. Sreenivasan, Extreme events in computational turbulence, Proceedings of the National Academy of Sciences112, 12633 (2015)

    work page 2015

  30. [30]

    Bengoechea, P

    S. Bengoechea, P. Over, D. Jaksch, and T. Rung, Towards Variational Quantum Algorithms for generalized linear and nonlinear transport phenomena (2024), arXiv:2411.14931 [quant-ph]

    work page arXiv 2024

  31. [31]

    E. T. Hale, W. Yin, and Y. Zhang, A fixed-point continuation method for l1-regularized minimization with applications to compressed sensing, CAAM TR07-07, Rice University43, 2 (2007)

    work page 2007

  32. [32]

    Giamarchi,Quantum Physics in One Dimension, Vol

    T. Giamarchi,Quantum Physics in One Dimension, Vol. 121 (Clarendon press, 2003)

    work page 2003

  33. [33]

    H. F. Song, N. Laflorencie, S. Rachel, and K. Le Hur, Entanglement entropy of the two-dimensional Heisenberg antifer- romagnet, Physical Review B83, 224410 (2011)

    work page 2011

  34. [34]

    An initialization strat- egy for addressing barren plateaus in parametrized quantum circuits.Quantum, 3:214, December 2019

    E. Grant, L. Wossnig, M. Ostaszewski, and M. Benedetti, An initialization strategy for addressing barren plateaus in parametrized quantum circuits, Quantum3, 214 (2019), arXiv:1903.05076 [quant-ph]

    work page arXiv 2019

  35. [35]

    Mhiri, R

    H. Mhiri, R. Puig, S. Lerch, M. S. Rudolph, T. Chotibut, S. Thanasilp, and Z. Holmes, A unifying account of warm start guarantees for patches of quantum landscapes (2025), arXiv:2502.07889 [quant-ph]

    work page arXiv 2025

  36. [36]

    C.-Y. Park, M. Kang, and J. Huh, Hardware-efficient ansatz without barren plateaus in any depth (2024), arXiv:2403.04844. 8

    work page arXiv 2024

  37. [37]

    Huang, M

    H.-Y. Huang, M. Broughton, N. Eassa, H. Neven, R. Babbush, and J. R. McClean, Generative quantum advantage for classical and quantum problems (2025), arXiv:2509.09033 [quant-ph]

    work page arXiv 2025

  38. [38]

    M. M. Abo-Zahhad, A. I. Hussein, and A. M. Mohamed, Compressive Sensing Algorithms for Signal Processing Appli- cations: A Survey, International Journal of Communications, Network and System Sciences08, 197 (2015)

    work page 2015

  39. [39]

    I. V. Oseledets, Constructive Representation of Functions in Low-Rank Tensor Formats, Constructive Approximation 37, 1 (2013)

    work page 2013

  40. [40]

    Wang, Patrick Rall, Edward H

    E. Bäumer, V. Tripathi, D. S. Wang, P. Rall, E. H. Chen, S. Majumder, A. Seif, and Z. K. Minev, Efficient Long-Range Entanglement using Dynamic Circuits, PRX Quantum5, 030339 (2024), arXiv:2308.13065 [quant-ph]

    work page arXiv 2024

  41. [41]

    Mitarai, M

    K. Mitarai, M. Negoro, M. Kitagawa, and K. Fujii, Quantum circuit learning, Physical Review A98, 032309 (2018)

    work page 2018

  42. [42]

    J. G. Vidal and D. O. Theis, Calculus on parameterized quantum circuits (2018), arXiv:1812.06323 [quant-ph]

    work page Pith review arXiv 2018

  43. [43]

    Virtanen, R

    P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright, S. J. van der Walt, M. Brett, J. Wilson, K. J. Millman, N. Mayorov, A. R. J. Nelson, E. Jones, R. Kern, E. Larson, C. J. Carey, İ. Polat, Y. Feng, E. W. Moore, J. VanderPlas, D. Laxalde, J. Perktold, R. Cimrman, I. Henriksen, ...

    work page 2020

  44. [44]

    Fonds of the Hamburg Ministry of Science, Research, Equalities and Districts (BWFGB)

    T. Hashizume, Z. Wang, F. Schlawin, and D. Jaksch, Dataset for sparse quantum computation at the edge of chaos (2026). 9 I. METHODS TEE of a randomK-sparse state Firstly, we show that the threshold classical sparsityKat whichI(3) 3 becomes negative is2n/3 for ann-qubit random sparse state. Here,TEEαdenotes theI(α) 3 (A,B,C), formulated with the von Neuman...

    work page 2026

  45. [45]

    Here,|σxi⟩is a basis state corresponds to theith grid pointxi, obtained from the binary expansion ofi

    Amplitude encoded sine function in MPS form As defined in the main article, we define an amplitude encoded single sine function in domain0≤x<1 |ψn λ⟩∝ ∑ i sin (2π λxi +ϕ ) |σ(xi)⟩,(B1) whereλis a wavelength andϕis the phase andxi are the equally spaced, enumerated, grid points over the domain. Here,|σxi⟩is a basis state corresponds to theith grid pointxi,...

    work page

  46. [46]

    Left environment tensor To compute the partial trace, we first look at the exact expression of the left environment tensorTCL(q), over the firstqqubits, which corresponds to theqlargest length scales. In terms of local tensorsM,T CL is written as TCL(q)[κ′ q−1,κq−1] = ∑ σ ¯q ∑ κq,κ′q M(σ0) κ0 ...M(σq−1) κq−2,κq−1 (M(σ0) κ0 )∗...(M(σq−1) κq−2,κq−1 )∗ (B4) ...

    work page

  47. [47]

    Here we obtain the right environmentTCR(q), which corresponds to theqfinest length scales

    Right environment Similarly, we now consider the case of contracting from the right. Here we obtain the right environmentTCR(q), which corresponds to theqfinest length scales. In terms of local tensorsM,TCR(q)is written as TCR(q)[κn−q−1,κ′ n−q−1] = ∑ σ ¯q ∑ κ ¯q,κ′ ¯q M(σn−q) κn−q−1,κn−q...M(σn−1) κn−2(M(σn−q) κ′ n−q−1,κ′ n−q )∗...(M(σn−1) κ′ n−2 )∗ (B7) ...

    work page

  48. [48]

    Reduced density matrix of qubitqin the thermodynamic limit To obtain the reduced density matrix of qubitq, we contract the relevantTCL,MandT CR ρq = 1√∑2n−1 m=0 sin2km 2n ∑ κq−1,κ′ q−1,κq,κ′q TCL(q)κ′ q−1,κq−1M(σq) κq−1,κq(ϕ)TCR(n−q−1)κq,κ′q(M(σq) κ′ q−1,κ′q (ϕ))∗, In the limit ofn→∞, we obtain ρ∞ q = lim n→∞ ρq =   2k−sec(2−q−1k)(sin((2+2−q−1)k+2ϕ)−sin...

    work page

  49. [49]

    Threshold for Amplitude-Encoded Band Limited functions Finally, we comment on the case of the amplitude-encoded state contains multiple wavelengths greater thanλmin. If we focus on qubitq, we show that the wavefunction|Ψ∞ λ⟩forλmin <λcan be written as |Ψ∞ λ⟩=|Ψq λ⟩|+⟩+ϵλ|res∞ λ⟩(B11) where|res∞ λ⟩is the anything that is left from the Riemannian approximat...

    work page