pith. sign in

arxiv: 2302.03249 · v2 · submitted 2023-02-07 · 🪐 quant-ph

Qualitative quantum simulation of resonant tunneling and localization with the shallow quantum circuits

J. L. Shen , P. Wang This is my paper

Pith reviewed 2026-05-24 09:38 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum simulationresonant tunnelinglocalizationTrotter circuitsshallow circuitsspin excitationquantum gatesnear-term quantum computers
0
0 comments X

The pith

Shallow quantum circuits suffice to observe resonant tunneling and localization in spin excitation propagation.

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

The paper establishes that circuits with large Trotter steps and few gates can still display the qualitative signatures of continuous-time quantum evolution for resonant tunneling and localization. These circuits use XY gates, controlled-Rx gates, and Rz gates arranged so that the placement of the Rz gates sets the final distribution of a propagating spin excitation. Resonant tunneling appears with up to four steps and localization persists across dozens of steps. This matters because it shows qualitative quantum features can appear at circuit depths far below those needed for numerical accuracy, opening a route for near-term hardware to exhibit recognizable quantum behavior without error correction.

Core claim

Trotter circuits built from XY gates, controlled-Rx gates, and single-qubit Rz gates, operated with large step size, reproduce the qualitative features of resonant tunneling (visible in up to four steps) and localization (visible across dozens of steps) in the propagation of a spin excitation; the configuration of the Rz gates alone determines the final spatial distribution of the excitation.

What carries the argument

Trotter circuits of XY gates, controlled-Rx gates, and Rz gates with large step size, where Rz-gate placement alone sets the final spin-excitation distribution.

If this is right

  • Resonant tunneling remains visible after at most four Trotter steps.
  • Localization persists through dozens of Trotter steps.
  • The circuit depth needed for qualitative observation is substantially smaller than the depth required for quantitative accuracy.
  • Physics-based analysis of error propagation can be applied directly to these shallow circuits.

Where Pith is reading between the lines

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

  • The same shallow-circuit approach may extend to other continuous-time phenomena such as quantum interference or Bloch oscillations.
  • Near-term devices could be benchmarked by whether they reproduce these qualitative signatures rather than by fidelity to exact numerics.
  • Error-propagation rules derived from the continuous model could guide gate ordering in other Trotter-based simulations.

Load-bearing premise

Large-step discrete Trotter evolution still produces the same qualitative patterns of resonant tunneling and localization that appear in the underlying continuous-time dynamics.

What would settle it

An explicit comparison in which the same spin-excitation initial state is evolved with successively larger Trotter steps and the resonant peaks or localization length disappear once the step size exceeds a threshold set by the continuous-time Hamiltonian.

Figures

Figures reproduced from arXiv: 2302.03249 by J. L. Shen, P. Wang.

Figure 1
Figure 1. Figure 1: (Color online) (a) The schematic diagram of quantum circuit, which includes the initialization, NT Trotter steps, and measurements. The preparation for the initial state is in the red dashed box including a NOT gate (i.e. the X gate). The blue rectangle represents a layer of two-qubit gates. The orange rectangles represent single-qubit Rz gates. (b) The schematic diagram of a layer of two-qubit gates. (c) … view at source ↗
Figure 2
Figure 2. Figure 2: (Color online) (a1)-(d1) The circuit systems. The parameters of Rz gates are marked on the orange squares. For convenience, in (c1) and (d1) we use single green rectangle to denote a layer of two-qubit gates. (a2)-(d2) The quantum wells. The energy levels of the wells are denoted by the horizontal lines in the wells. (a3)- (d3) The tight-binding chain. (a4)-(d4) Numerical result of the discrete-time evolut… view at source ↗
Figure 3
Figure 3. Figure 3: (Color online) (a) 4-qubit circuit. The two-qubit XY gates are replaced by controlled-Rx gates. (b) The probability of finding the spin excitation on the last qubit as function of φ. (c) The probability of observing the spin excitation on the last third of the qubits varies with Trotter step η. (d) The probability distribution of spin excitation at η = 10. (c) and (d) use the same legend and drawing parame… view at source ↗
Figure 4
Figure 4. Figure 4: (Color online) (a) The IPRη varies with Trotter number η, the drawing parameters are N = 15, NT = 80, θ = φ = π/2. The red and blue lines represent the ordered and disordered case respectively. The horizontal lines are the average value of IPRη, i.e. IPRave. (b) IPRave as function of the degree of randomness. For a fixed R, we have 20 data. The blue line is the average value of the 20 data, and error bar i… view at source ↗
read the original abstract

In a circuit-based quantum computer, the computing is performed via the discrete-time evolution driven by quantum gates. Accurate simulation of continuoustime evolution requires a large number of quantum gates and therefore suffers from more noise. In this paper, we find that shallow quantum circuits are sufficient to qualitatively observe some typical quantum phenomena in the continuous-time evolution limit, such as resonant tunneling and localization phenomena. We study the propagation of a spin excitation in Trotter circuits with a large step size. The circuits are formed of two types of two-qubit gates, i.e. XY gates and controlled- Rx gates, and single-qubit Rz gates. The configuration of the Rz gates determines the distribution of the spin excitation at the end of evolution. We demonstrate the resonant tunneling with up to four steps and the localization phenomenon with dozens of steps in Trotter circuits. Our results show that the circuit depth required for qualitative observation of some significant quantum phenomena is much smaller than that required for quantitative computation, suggesting that it is feasible to apply qualitative observations to near-term quantum computers. We also provide a way to use the physics laws to understand the error propagation in quantum circuits.

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

2 major / 2 minor

Summary. The paper claims that shallow Trotter circuits with large time steps, composed of XY, controlled-Rx, and Rz gates, suffice to qualitatively reproduce continuous-time phenomena such as resonant tunneling (demonstrated with up to four steps) and localization (with dozens of steps) in the propagation of a spin excitation, without requiring deep circuits for quantitative accuracy. It further suggests a physics-law approach to analyzing error propagation.

Significance. If the large-step discrete evolution faithfully captures the qualitative signatures of resonant tunneling and localization, the result would indicate that near-term quantum hardware can observe important coherent quantum effects at depths far below those needed for quantitative simulation, thereby lowering the barrier for NISQ-era experiments. The physics-based error analysis is a secondary positive feature.

major comments (2)
  1. [Abstract / resonant-tunneling results] Abstract and resonant-tunneling demonstration: the central claim that large-step Trotter evolution reproduces the qualitative features of continuous-time resonant tunneling (energy-matched transmission via coherent interference) is not accompanied by any side-by-side comparison against small-dt Trotterization or exact continuous-time evolution at identical parameters. Without such a check, observed transmission peaks could arise from discretization artifacts rather than the underlying resonance condition.
  2. [Localization results] Localization demonstration (dozens of steps): no quantitative fidelity metrics, error bars, or comparison to the exact continuous-time dynamics are supplied, so it is impossible to assess whether the reported localization survives the O(dt²) phase and coherence errors inherent to the large-step regime.
minor comments (2)
  1. [Abstract] The abstract contains the run-on word 'continuoustime'; insert a hyphen.
  2. [Methods] Explicit matrix representations or circuit diagrams for the XY, controlled-Rx, and Rz gates would clarify the Trotter decomposition in the methods section.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help clarify the presentation of our qualitative simulation results. We address each major comment below and indicate planned revisions.

read point-by-point responses

  1. Referee: [Abstract / resonant-tunneling results] Abstract and resonant-tunneling demonstration: the central claim that large-step Trotter evolution reproduces the qualitative features of continuous-time resonant tunneling (energy-matched transmission via coherent interference) is not accompanied by any side-by-side comparison against small-dt Trotterization or exact continuous-time evolution at identical parameters. Without such a check, observed transmission peaks could arise from discretization artifacts rather than the underlying resonance condition.

    Authors: The referee correctly notes the absence of direct comparisons. Our demonstration shows transmission peaks only when Rz parameters satisfy the energy-matching condition taken from the continuous-time Hamiltonian; this parameter dependence is the signature we use to link the result to resonance. Nevertheless, adding explicit side-by-side comparisons with small-dt Trotterization and exact dynamics at the same parameters would remove any ambiguity about discretization artifacts. We will include such comparisons in the revised manuscript. revision: yes

  2. Referee: [Localization results] Localization demonstration (dozens of steps): no quantitative fidelity metrics, error bars, or comparison to the exact continuous-time dynamics are supplied, so it is impossible to assess whether the reported localization survives the O(dt²) phase and coherence errors inherent to the large-step regime.

    Authors: We agree that quantitative fidelity metrics and direct comparisons to exact dynamics are not provided. The manuscript instead relies on the persistence of localization over dozens of steps together with the physics-law error-propagation analysis to argue that the qualitative feature remains observable. To strengthen the claim, we will add fidelity metrics (with error bars where applicable) and comparisons to exact continuous-time evolution in the revised version. revision: yes

Circularity Check

0 steps flagged

No circularity: forward simulation of Trotter circuits

full rationale

The paper executes explicit Trotterized quantum circuits (XY gates, controlled-Rx, Rz) on a spin excitation and reports the resulting propagation patterns for resonant tunneling (up to 4 steps) and localization (dozens of steps). No parameters are fitted to data and then re-predicted; no self-citations supply load-bearing uniqueness theorems or ansatzes; the configuration of Rz gates is an explicit input that determines the output distribution. The central claim is therefore an empirical observation of circuit output rather than a derivation that reduces to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities; the central claim rests on the unstated premise that large-step Trotterization preserves qualitative continuous-time features.

pith-pipeline@v0.9.0 · 5730 in / 1052 out tokens · 52262 ms · 2026-05-24T09:38:16.837552+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

57 extracted references · 57 canonical work pages · 1 internal anchor

  1. [1]

    R. P. Feynman, Simulating physics with computers, Int J Theor Ph ys 21, 467–488 (1982)

    work page 1982

  2. [2]

    M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantu m Information, Cambridge University Press, Cambridge, (2010)

    work page 2010

  3. [3]

    Arute et al., Quantum supremacy using a programmable superc onducting processor, Nature 574, 505 (2019)

    F. Arute et al., Quantum supremacy using a programmable superc onducting processor, Nature 574, 505 (2019)

    work page 2019

  4. [4]

    Lloyd, Universal quantum simulators, Science 273, 1073 (1996)

    S. Lloyd, Universal quantum simulators, Science 273, 1073 (1996)

    work page 1996

  5. [5]

    A. M. Childs, D. Maslov, Y. Nam, N. J. Ross, and Y. Su, Toward the first quantum simulation with quantum speedup, Proceedings of the National Academy of Sc iences 115, 9456 (2018)

    work page 2018

  6. [6]

    A. M. Childs, Y. Su, M. C. Tran, N. Wiebe, and S. Zhu, Theory of tr otter error with commutator scaling, Phys. Rev. X 11, 011020 (2021)

    work page 2021

  7. [7]

    Yi, Robustness of discretization in digital adiabatic simulation, a rXiv:2107.06404 (2021)

    C. Yi, Robustness of discretization in digital adiabatic simulation, a rXiv:2107.06404 (2021)

    work page arXiv 2021

  8. [8]

    Preskill, Quantum computing in the NISQ era and beyond, Quant um 2, 79 (2018)

    J. Preskill, Quantum computing in the NISQ era and beyond, Quant um 2, 79 (2018)

    work page 2018

  9. [9]

    Landauer, Is quantum mechanics useful? Phil

    R. Landauer, Is quantum mechanics useful? Phil. Tran. R. Soc. L ond. 353, 367 (1995); R. Landauer, The physical nature of information, Phys. Lett. A 217, 188 (1996)

    work page 1995

  10. [10]

    W. G. Unruh, Maintaining coherence in quantum computers, Phy s. Rev. A 51, 992 (1995)

    work page 1995

  11. [11]

    Haroche and J

    S. Haroche and J. M. Raimond, Quantum computing: dream or nig htmare? Phys. Today 49, 51 (1996)

    work page 1996

  12. [12]

    P. W. Shor, Scheme for reducing decoherence in quantum comp uter memory. Phys. Rev. A 52, R2493 (1995)

    work page 1995

  13. [13]

    A. M. Steane, Error Correcting Codes in Quantum Theory. Phy s. Rev. Lett. 77, 793 (1996)

    work page 1996

  14. [14]

    A. R. Calderbank and P. W. Shor, Good Quantum Error Correct ing codes exist, Phys. Rev. A 54, 1098 (1996)

    work page 1996

  15. [15]

    Laamme, C

    R. Laamme, C. Miquel, J. P. Paz, and W. H. Zurek, Perfect Quan tum Error Correcting Code. Phys. Rev. Lett. 77, 198 (1996)

    work page 1996

  16. [16]

    C. H. Bennett, D. P. DiVincenzo J. A. Smolin, and W. K. Wooters, Mixed State Entanglement and Quantum Error Correction, Phys. Rev. A 54, 3824, (1996)

    work page 1996

  17. [17]

    A. M. Steane, Multiple particle interference and quantum error correction, Proc. Royal Society of London A 452, 2551 (1996)

    work page 1996

  18. [18]

    A. M. Steane, Simple quantum error correcting codes, Phys. R ev. A 54, 4741 (1996)

    work page 1996

  19. [19]

    Knill and R

    E. Knill and R. Laamme, A Theory of Quantum Error Correcting C odes. Phys. Rev. Lett. 84, 2525 (2000) Qualitative quantum simulation of resonant tunneling and l ocalization with the shallow quantum circuits 15

    work page 2000

  20. [20]

    P. W. Shor, Fault-Tolerant quantum computation, Proc. 37th IEEE Symp. on Foundations of Computer Science, 56-65 (1996)

    work page 1996

  21. [21]

    Gottesman, Fault-Tolerant Quantum Computation with Loca l Gates, J

    D. Gottesman, Fault-Tolerant Quantum Computation with Loca l Gates, J. Mod. Opt 47, 333 (2000)

    work page 2000

  22. [22]

    Knill, R

    E. Knill, R. Laamme, and W. H. Zurek, Accuracy threshold for Qu antum Computation (1996)

    work page 1996

  23. [23]

    Holmes, M

    A. Holmes, M. R. Jokar, G. Pasandi, Y. Ding, M. Pedram, and F. T . Chong, Nisq+: Boosting quantum computing power by approximating quantum error correc tion, arXiv:2004.04794 (2020)

    work page arXiv 2004

  24. [24]

    Y. Ueno, M. Kondo, M. Tanaka, Y. Suzuki, and Y. Tabuchi, Qeco ol: On-line quantum error correction with a superconducting decoder for surface code, ar Xiv:2103.07526 (2021)

    work page arXiv 2021

  25. [25]

    P. Das, A. Locharla, and C. Jones, Lilliput: A lightweight low-laten cy lookup-table based decoder for near-term quantum error correction, arXiv:2108.06569 (202 1)

    work page arXiv

  26. [26]

    Li and S

    Y. Li and S. Benjamin, Practical Quantum Error Mitigation for N ear-Future Applications, Phys. Rev. X 8, 031027 (2018)

    work page 2018

  27. [27]

    S. Endo, Z. Cai, S. C. Benjamin, and X. Yuan, Hybrid quantum-c lassical algorithms and quantum error mitigation, J. Phys. Soc. Jpn 90. 032001 (2021)

    work page 2021

  28. [28]

    Kandala, K

    A. Kandala, K. Temme, A. D. C´ orcoles, A. Mezzacapo, J. M. Ch ow, and J. M. Gambetta, Error mitigation extends the computational reach of a noisy quantum pro cessor, Nature 567, 209–212 (2019)

    work page 2019

  29. [29]

    J. J. Wallman and J. Emerson, Randomized compiling with twirling gat es, Phys. Rev. A 94, 052325 (2016)

    work page 2016

  30. [30]

    Strikis, D

    A. Strikis, D. Y. Qin, Y. Z. Chen, S. C. Benjamin, and Y. Li, Learn ing-based quantum error mitigation, PRX Quantum 2, 040330 (2021)

    work page 2021

  31. [31]

    Wang , Y

    Z. Wang , Y. Z. Chen , Z. X. Song, D. Qin, H. K. Li, Q. J. Guo, H. Wa ng, C. Song, and Ying Li, Scalable evaluation of quantum-circuit error loss using Clifford samplin g, Phys. Rev. Lett. 126, 080501 (2021)

    work page 2021

  32. [32]

    V. N. Premakumar and R. Joynt, Error mitigation in quantum com puters subject to spatially correlated noise, arXiv:1812.07076 (2018)

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  33. [33]

    Temme, S

    K. Temme, S. Bravyi, and J. M. Gambetta, Error mitigation for s hort-depth quantum circuits, Phys. Rev. Lett. 119, 180509 (2017)

    work page 2017

  34. [34]

    C. Song, J. Cui, H. Wang, J. Hao, H. Feng, and Ying Li, Quantum c omputation with universal error mitigation on a superconducting quantum processor, Sci. Ad v 5, eaaw5686 (2019)

    work page 2019

  35. [35]

    Kandala, K

    A. Kandala, K. Temme, A. D. C orcoles, A. Mezzacapo, J. M. Cho w, and J. M. Gambetta, Error mitigation extends the computational reach of a noisy quantum pro cessor, Nature 567, 491-495 (2019)

    work page 2019

  36. [36]

    Otten and S

    M. Otten and S. K. Gray, Recovering noise-free quantum obse rvables, Phys. Rev. A 99, 012338 (2019)

    work page 2019

  37. [37]

    Tranter, P

    A. Tranter, P. J. Love, F. Mintert, N. Wiebe, and P. V. Covene y, Ordering of Trotterization: Impact on Errors in Quantum Simulation of Electronic Structure, En tropy 21, 1218 (2019)

    work page 2019

  38. [38]

    Yi, Success of digital adiabatic simulation with large Trotter st ep, Phys

    C. Yi, Success of digital adiabatic simulation with large Trotter st ep, Phys. Rev. A 104, 052603 (2021)

    work page 2021

  39. [39]

    P. J. J. O’Malley et al., Scalable Quantum Simulation of Molecular Ener gies, Phys. Rev. X 6, 031007 (2016)

    work page 2016

  40. [40]

    A. M. Childs, Y. Su, M. C. Tran, N. Wiebe, and S. Zhu, Theory of T rotter Error with Commutator Scaling, Phys. Rev. X 11, 011020 (2021)

    work page 2021

  41. [41]

    Poulin, M

    D. Poulin, M. B. Hastings, D. Wecker, N. Wiebe, A. C. Doherty, a nd M. Troyer, The trotter step size required for accurate quantum simulation of quantum Chemistr y, Quantum Information & Computation 15, 361 (2015)

    work page 2015

  42. [42]

    M. Heyl, P. Hauke, P. Zoller, Quantum localization bounds Trotte r errors in digital quantum simulation, Sci Adv 5, eaau8342 (2019)

    work page 2019

  43. [43]

    L. M. Sieberer, T. Olsacher, A. Elben, M. Hey, P. Hauke, F. Haa ke and P. Zoller, Digital quantum simulation, Trotter errors, and quantum chaos of the kicked top, NPJ quantum iniformation 5, Qualitative quantum simulation of resonant tunneling and l ocalization with the shallow quantum circuits 16 1-11 (2019)

    work page 2019

  44. [44]

    J. T. Edwards and D. J. Thouless, Numerical studies of localizat ion in disordered systems, J. Phys. C: Solid State Phys. 5 807 (1972)

    work page 1972

  45. [45]

    Segev, Y

    M. Segev, Y. Silberberg, and D. N. Christodoulides, Anderson lo calization of light, Nature Photon 7, 197–204 (2013)

    work page 2013

  46. [46]

    Jordan and E

    P. Jordan and E. Wigner, ¨Uber das Paulische ¨Aquivalenzverbot, Z. Phys. 47, 631 (1928)

    work page 1928

  47. [47]

    J. G. Wang, S. J. Yang, Ultracold bosons in a one-dimensional op tical lattice chain: Newton’s cradle and Bose enhancement effect, Phys. Lett. A 381, 1665 (2017)

    work page 2017

  48. [48]

    A. S. Buyskikh , L. Tagliacozzo, D. Schuricht, C. A. Hooley, D. P ekker, and A. J. Daley, Resonant two-site tunneling dynamics of bosons in a tilted optical superlattice , Phys. Rev. A 100, 023627 (2019)

    work page 2019

  49. [49]

    Scherg, T

    S. Scherg, T. Kohlert, P. Sala, F. Pollmann, B. H. Madhusudhan a, I. Bloch, and M. Aidelsburger, Observing non-ergodicity due to kinetic constraints in tilted Fermi-H ubbard chains, Nat. Commun 12, 4490 (2021)

    work page 2021

  50. [50]

    Tsu and L

    R. Tsu and L. Esaki, Tunneling in a finite superlattice, Appl. Phys. Lett. 22, 562 (1973)

    work page 1973

  51. [51]

    T. C. L. G. Sollner, W. D. Goodhue, P. E. Tannenwald, C. D. Park er, and D. D. Peck, Resonant tunneling through quantum wells at frequencies up to 2.5 THz, Appl. P hys. Lett. 43, 588 (1983)

    work page 1983

  52. [52]

    P. W. Anderson, Absence of Diffusion in Certain Random Lattices , Phys. Rev. 109, 1492 (1957)

    work page 1957

  53. [53]

    D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, Localizat ion of light in a disordered medium, Nature 390, 671–673 (1997)

    work page 1997

  54. [54]

    B. L. Altshuler, D. Khmel’nitzkii, A. I. Larkin, and P. A. Lee, Magn etoresistance and Hall effect in a disordered two-dimensional electron gas, Phys. Rev. B 22, 5142 (1980)

    work page 1980

  55. [55]

    Vollhardt, P

    D. Vollhardt, P. W¨ olfle, Scaling Equations from a Self-Consistent Theory of Anderson Localization, Phys. Rev. Lett. 48, 699 (1982)

    work page 1982

  56. [56]

    A. N. Poddubny, M. V. Rybin, M. F. Limonov, and Y. S. Kivshar, F ano interference governs wave transport in disordered systems, Nat. Commun 3, 914 (2012)

    work page 2012

  57. [57]

    Haake, Quantum Signatures of Chaos (Springer, 2010)

    F. Haake, Quantum Signatures of Chaos (Springer, 2010)

    work page 2010