Experimental simulation of Dirac equation in superconducting qubits
Open accessAbstract
The Dirac equation elucidates fundamental physical principles and predicts significant physical phenomena. Its simulation further offers explanations for subsequent physics observations. We map the 3+1-dimensional Dirac equation onto the first excited manifold of four qubits to simulate the internal four-component spinor dynamics. Additionally, we simulate two intriguing and challenging-to-implement physical phenomena: the Zitterbewegung effect and the Sauter-Schwinger effect. Compared to previous simulations, our system provides a larger parameter space, allowing for a more comprehensive exploration of physical properties. This work enhances our understanding of spinor dynamics. The Dirac equation, one of the most fundamental equations in relativistic quantum mechanics, has influenced diverse research fields. The authors demonstrate an analogue simulation of the 3+1-dimensional Dirac equation by applying parametric modulations in superconducting circuits.
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TL;DR
The paper reports an analogue simulation of the 3+1-dimensional Dirac equation using the first-excited manifold of four superconducting qubits, mapping spinor dynamics and measuring the Zitterbewegung and Sauter-Schwinger effects. It demonstrates velocity oscillations and mass-dependent pair production in a parameter space larger than prior 1+1-dimensional simulations.
Key findings
- Free Dirac dynamics were simulated with momentum \(p=(1.3\,\text{MHz},0,0)\) and mass varied from \(0\,\text{MHz}\) to \(2.6\,\text{MHz}\); populations showed single-frequency oscillation, heavier particles oscillated faster, and state \(|3\rangle\) was never populated in the ideal description.
- The Zitterbewegung effect was observed by directly measuring velocities \(v_x\) and \(v_y\) via quantum state tomography; with \(p=(1.3/\sqrt{2}\,\text{MHz},1.3/\sqrt{2}\,\text{MHz},0)\), the square of the ZBW oscillation frequency \(\omega_Z\) increased linearly with \(m^2\), and oscillations appeared only in \(v_y\) when momentum was solely in the \(x\)-direction.
- The Sauter-Schwinger effect was simulated by chirping momentum from \(p_{\text{ini}}=(-2.6\,\text{MHz},0,0)\) to \(p_{\text{fin}}=(2.6\,\text{MHz},0,0)\) with rate \(|\epsilon|=(2.034\,\text{MHz})^2\); pair-production probability decreased with increasing mass, consistent with the Schwinger formula, and the effective electric field reached or surpassed the Sauter-Schwinger limit \(E\approx1.32\times10^{18}\,\text{V}\,\text{m}^{-1}\).
- At large mass, a slight oscillation of \(|3\rangle\) was observed and attributed to deviation of the coupling strength’s phase caused by the shift of the resonant parametric frequency when all four parametric modulations were applied.
Methods
- Four superconducting qubits were arranged in a 2D square, with adjacent qubits connected by a floating tunable coupler; each qubit had individual flux and XY control lines and a coplanar waveguide readout resonator for simultaneous multi-tone readout.
- The 3+1D Dirac Hamiltonian was mapped onto the first-excited manifold of the four qubits using parametric modulation; coupling strengths were set as \(\Omega_{01}e^{-i\phi_1}=p_x-ip_y\), \(\Omega_{02}e^{-i\phi_2}=p_z-im\), \(\Omega_{23}e^{-i\phi_3}=p_x-ip_y\), and \(\Omega_{13}e^{-i\phi_4}=-p_z+im\), followed by a permutation to a supersymmetric representation.
- Measurements used simultaneous readout of all qubits and quantum state tomography with two \(\sqrt{i\text{SWAP}}\)-like gates and four tomography phases \(\phi_m=m\pi/2\) to extract velocity expectation values; for the Schwinger effect, Landau-Zener rates were extracted from measured populations with readout error mitigation.
Limitations
- Stated: velocity oscillations decay due to energy decay and pure dephasing in the superconducting qubits.
- Stated: the slight \(|3\rangle\) oscillation at large mass is caused by coupling-strength phase deviation from the resonant parametric frequency shift when all four parametric modulations are applied.
- Stated: for the Schwinger simulation, when mass exceeds \(1.5\,\text{MHz}\), the population increases because the Landau-Zener rate formula is not applicable within the experimental parameter regime; the deviation is caused by the finite maximum effective coupling strength. Evident from the text: the work is an analogue simulation limited to specific momenta, masses, and a four-qubit manifold, not a direct observation of the real high-energy phenomena.
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Claims the paper makes
1 of 2 cited sources checked · 1 supported · 0 flagged
- Premisecitation§ Experimental simulation of Dirac equation in superconducting qubits Check for updates
Experimental simulations of the Dirac equation in 3+1 dimensions remain lacking.
“Experimental simulations of the Dirac equation in 3+1 dimensions remain lacking.”
Quote in the paperSubstantial claim1 of 1 sources checked- cites [62] Svetitsky, E. & Katz, N. Dirac particle dynamics of a superconducting circuit. Phys. Rev. A 99, 042308 (2019). source
- Findingown data§ Experimental simulation of Dirac equation in superconducting qubits Check for updates
The 3+1-dimensional Dirac equation was simulated using four superconducting qubits in the first-excited manifold, mapping the four-component spinor dynamics.
“We map the 3+1-dimensional Dirac equation onto the first excited manifold of four qubits to simulate the internal fourcomponent spinor dynamics.”
- Also stated in: § Conclusions · boosted
“we have demonstrated an analogue simulation of the 3+1 dimensional Dirac equation using superconducting circuits.”
Quote in the paperSubstantial claim - Also stated in: § Conclusions · boosted
- Findingown data§ Experimental simulation of Dirac equation in superconducting qubits Check for updates
The Sauter-Schwinger effect was simulated by constructing an effective electric field whose strength reaches or surpasses the Sauter-Schwinger limit.
“we demonstrate the Schwinger effect by constructing an effective electric field whose strength reaches or even surpasses the Sauter-Schwinger limit.”
Quote in the paperSubstantial claim - Findingown data
A slight oscillation of |3> is observed with a large mass.
“In practice, we observe a slight oscillation of |3i with a large mass.”
Quote in the paperNot a substantial claim - Findingown data§ Experimental simulation of Dirac equation in superconducting qubits Check for updates
The Zitterbewegung effect was observed in the superconducting qubit simulation.
“Additionally, we simulate two intriguing and challenging-to-implement physical phenomena: the Zitterbewegung effect and the Sauter-Schwinger effect.”
- Also stated in: § Conclusions · plain
“we directly measured the velocities v x and v y using quantum state tomography and observed the ZBW due to the interference between the positive and negative energy components.”
Quote in the paperSubstantial claim - Also stated in: § Conclusions · plain
- Findingown data
In the free Dirac equation experiment, a heavier particle has higher energy, leading to faster oscillation.
“The heavier particle has the higher energy, leading to faster oscillation, as illustrated in Fig. 1a.”
Quote in the paperNot a substantial claim - Findingreasoning
In the free Dirac equation experiment, state |3> is never populated.
“It is worth noting that |3i is never populated; the phenomenon can be explained well when we treat the diamond-shape Hamiltonian as a combination of a Λ and a V configuration as shown in Figure 1b.”
Quote in the paperNot a substantial claim - Findingnone given
The Hamiltonian has two two-fold degeneracies denoted by {|+>_01, |->_23} and {|->_01, |+>_23}.
“The Hamiltonian has two two-fold degeneracies, which are denoted by f|þi 01 ; |Ài 23 g and f|Ài 01 ; |þi 23 g (see Methods).”
Quote in the paperSubstantial claim - Findingown data
The ZBW was demonstrated for both massive and massless free Dirac particles.
“The ZBW was demonstrated for both massive and massless free Dirac particles.”
Quote in the paperSubstantial claim - Findingown data
For a fixed electric field (fixed ramping rate), the pair production probability decreases with increasing mass, consistent with the Schwinger formula.
“As shown in Fig. 3b, for a fixed electric field (fixed ramping rate), the pair production probability decreases with increasing mass, consistent with the Schwinger formula.”
Quote in the paperSubstantial claim - Findingown data
Oscillations are observed only in v_y when momentum is solely in the x-direction.
“Oscillations are observed only in v y when momentum is solely in the x-direction.”
Quote in the paperNot a substantial claim - Findingown data
The square of the ZBW oscillation frequency ω_Z increases linearly with m^2.
“The experiment and simulation data in Fig. 2e indicate that the square of ZBW oscillation frequency ω Z increases linearly with m 2 .”
Quote in the paperSubstantial claim - Findingreasoning
After the permutation transformation, the Dirac Hamiltonian is realized in the supersymmetric representation H_s = ∑_{i=x,y,z} (p_i σ_x ⊗ σ_i) + m σ_y ⊗ I_2.
“After the transformation, H s ¼ U y p HU p , the Dirac Hamiltonian is realized in the supersymmetric representation H s = ∑ i=x,y,z (p i σ x ⊗ σ i ) + mσ y ⊗ I 2”
Quote in the paperSubstantial claim - Findingown data
The experimental results of the populations of four states in the chirping process are in accord with numerical simulations for different masses.
“Experimental results of the populations of four states in the chirping process are demonstrated for different masses, and they are in accord with numerical simulations in (d).”
Quote in the paperSubstantial claim - Interpretationreasoning
The suppressed population in |3> is caused by destructive interference between two transition paths from |0> to |3>.
“The destructive interference between two transition paths from |0i to |3i causes the suppressed population in |3i.”
Quote in the paperSubstantial claim - Interpretationown data
The experimental results indicate that superconducting circuits are excellent platforms for simulating the Dirac equation and other relativistic phenomena that are challenging to observe experimentally, due to their high controllability and scalability.
“Our experimental results indicate that, due to their high controllability and scalability, superconducting circuits are excellent platforms for simulating the Dirac equation and many other relativistic phenomena that are extremely challenging to observe experimentally.”
Quote in the paperSubstantial claim - Interpretationreasoning
Due to energy decay and pure dephasing in superconducting qubits, the velocity oscillation decays in this experiment.
“Due to the energy decay and pure dephasing in superconducting qubits, we see the decay of velocity oscillation in this experiment.”
Quote in the paperSubstantial claim - Interpretationown data
The slight oscillation of |3> with large mass is caused by deviation of the coupling strength's phase resulting from the shift of the resonant parametric frequency when all four parametric modulations are applied.
“The numerical simulation proves this is caused by the deviation of the coupling strength's phase, which results from the shift of the resonant parametric frequency when all four parametric modulations are applied.”
Quote in the paperNot a substantial claim - Interpretationreasoning
The velocity oscillation will never decay without decoherence.
“the interference persists, and the velocity oscillation will never decay without decoherence.”
Quote in the paperSubstantial claim - Interpretationreasoning
The system's dynamics in the Schwinger effect experiment can be described as two independent Landau-Zener transitions in two manifolds.
“Two Landau-Zener transitions can proceed independently in two manifolds.”
- Also stated in: § Parameters mapping · plain
“Therefore, the total dynamics of the system are just two independent Landau-Zener transitions.”
Quote in the paperSubstantial claim - Also stated in: § Parameters mapping · plain
- Study design and conductown data
The velocity was directly measured by quantum state tomography to demonstrate the ZBW effect.
“We directly measure the velocity by quantum state tomography to demonstrate the ZBW effect.”
Quote in the paperSubstantial claim - Interpretationreasoning
At large mass, the deviation from the Landau-Zener transition rate formula is caused by the finite max effective coupling strength.
“Therefore, the deviation from the Landau-Zener transition rate formula in the pink regime is caused by the finite max effective coupling strength.”
Quote in the paperSubstantial claim - Premisenone given
In the model, the mass identifies the avoid-crossing gap.
“In this model, the mass identifies the avoidcrossing gap.”
Quote in the paperSubstantial claim
- May combine two claims. 8
- Classified as probably not a substantial claim. 7
May combine two claims.
Findingown dataWhen the momentum in the y direction is reversed relative to Fig. 2b, the oscillation of v_x shifts by a phase of π, resulting in it being in phase with v_y.
“When the momentum in the y direction is reversed relative to Fig. 2b, the oscillation of v x shifts by a phase of π, resulting in it being in phase with v y .”
Quote in the paperNot a substantial claimMay combine two claims.
Findingnone givenWhen the mass is beyond 1.5 MHz, the population increases because the Landau-Zener rate formula is not applicable within the experiment parameter.
“When the mass is beyond 1.5 MHz, the population increases because the formula of the Landau-Zener rate is not applicable within the experiment parameter.”
Quote in the paperNot a substantial claimMay combine two claims.
Findingown dataThe remaining population in the {|0>,|1>} manifold characterizes the pair production and decreases with increasing mass.
“The remaining population in f|0i; |1ig manifold characterizes the pair production, and it decreases with increasing mass.”
Quote in the paperSubstantial claimMay combine two claims.
Findingnone givenThe momentum in the x-direction can cause ZBW in the y-direction and vice versa.
“the momentum in the x-direction can cause the ZBW in the y-direction and vice versa”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S4 (based).
Interpretationnone given§ Experimental simulation of Dirac equation in superconducting qubits Check for updatesCompared to previous simulations, the system provides a larger parameter space, allowing more comprehensive exploration of physical properties.
“Compared to previous simulations, our system provides a larger parameter space, allowing for a more comprehensive exploration of physical properties.”
- Also stated in: § Zitterbewegung effect · plain
“Our scheme provides larger parameter space than the previous 1+1 dimensional Dirac equation simulation, ensuring we explore more Dirac particle features.”
Quote in the paperNot a substantial claim- Also stated in: § Zitterbewegung effect · plain
May combine two claims.
InterpretationreasoningThe evolution of a free Dirac particle can be described as Rabi oscillation between |0> and the bright state |B_0> in the V system, leading to no population transfer to |3>.
“Therefore, the evolution of free Dirac particle can be described as the Rabi oscillation between |0i and bright state |Bi 0 in V system, leading that there is no population transfer to |3i as shown in the Fig.”
Quote in the paperSubstantial claimClassified as probably not a substantial claim.
Study design and conductnone givenThe qubit frequencies and coupling strength were carefully designed to ensure the Dirac Hamiltonian could be constructed via parametric modulation.
“The qubit frequencies and coupling strength are carefully designed to ensure we can construct the Dirac Hamiltonian via parametric modulation.”
Quote in the paperSubstantial claimClassified as probably not a substantial claim. May combine two claims.
Premisenone given§ Experimental simulation of Dirac equation in superconducting qubits Check for updatesThe Sauter-Schwinger limit requires an electric field E ≈ 1.32 × 10^18 V m^-1 to polarize the vacuum, making direct observation experimentally challenging.
“The Sauter-Schwinger limit indicates that an electric field (E ≈ 1.32 × 10 18 V m -1 ) is required to polarize the vacuum, presenting a significant experimental challenge for direct observation.”
Quote in the paperSubstantial claimClassified as probably not a substantial claim.
Premisenone given§ Experimental simulation of Dirac equation in superconducting qubits Check for updatesDirect experimental observation of Zitterbewegung is exceptionally challenging because its amplitude is on the order of the Compton wavelength (~10^-12 m) and its frequency is ~10^21 Hz.
“ZBW predicts the rapid oscillatory motion of spin-1/2 particles, but the small amplitude (on the order of the Compton wavelength, ~10 -12 m) and high frequency ( ~10 21 Hz) make experimental observation exceptionally challenging.”
Quote in the paperSubstantial claimClassified as probably not a substantial claim.
Study design and conductnone givenThe population measurements were corrected by readout error mitigation.
“The population is corrected by readout error mitigation.(See Supplementary Note 5)”
Quote in the paperSubstantial claimMay combine two claims.
Interpretationown dataThe sum of population |0> and |1> reflects the pair production rate, which corresponds with the Landau-Zener model.
“The sum of the population |0i and |1i reflects the pair production rate, which corresponds with the Landau-Zener model as denoted in Fig. 3e.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
Study design and conductreasoningThe results for different tomography phases were averaged to mitigate the effect of readout error in <σ_z>.
“the data is redundant, so we average the results for different phases to mitigate the effect of readout error in 〈σ z 〉.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
Premisenone givenThe Zitterbewegung effect results from interference between the positive and negative energy solutions of the Dirac equation.
“the predicted super-fast rapid oscillation of relativistic particles, which results from the interference between the positive and negative energy solutions of the Dirac equation.”
Quote in the paperNot a substantial claimMay combine two claims.
Study design and conductcitationThe Landau-Zener rates were extracted from the measured populations and characterize the pair production rate.
“The Landau-Zener rates are extracted from the measured populations 63 , and they characterize the pair production rate.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [63] Shevchenko, S., Ashhab, S. & Nori, F. Landau-Zener-Stückelberg interferometry. Phys. Rep. 492, 1–30 (2010). source
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
Each contradiction between two passages listed here was confirmed by an independent check.
The stated tomography formula for v y is four times too large
MinorThe stated tomography relations make the eight-term numerator equal to four times the earlier expression for v y. Unlike the analogous v x formula, the final v y formula omits division by 4, so it would overstate the measured velocity.
“v y ¼ ðÀhσ z i 0 0 þ hσ z i 2 0 þ hσ z i 0 1 À hσ z i 2 1 À hσ z i 0 2 þ hσ z i 2 2 þ hσ z i 0 3 À hσ z i 2 3 Þ.”
“We can calculate hσ z i m i¼0;1 ¼ jc 2 j 2 þ jc 3 j 2 À ðÀ1Þ i 2Im ðe iφ m c à 0 c 1 Þ and hσ z i m i¼2;3 ¼ jc 0 j 2 þ jc 1 j 2 À ðÀ1Þ i 2 Im ðe iφ m c à 2 c 3 Þ”
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Solinas, P., Amoretti, A., & Giazotto, F. (2021) · Physical Review Letters · doi:10.1103/physrevlett.126.117001 · Cited by 31 · see citing works
Akal, I., Egger, R., Müller, C., et al. (2016) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.93.116006 · Cited by 28 · see citing works
Wave Packet Dynamics in Synthetic Non-Abelian Gauge Fields
Hasan, M., Madasu, C. S., Rathod, K. D., Kwong, C. C., Miniatura, C., Chevy, F., & Wilkowski, D. (2022) · Physical Review Letters · doi:10.1103/physrevlett.129.130402 · Cited by 28 · see citing works
Analog Sauter-Schwinger effect in semiconductors for spacetime-dependent fields
Linder, M. F., Lorke, A., & Schützhold, R. (2018) · Physical review. B./Physical review. B · doi:10.1103/physrevb.97.035203 · Cited by 26 · see citing works
Katsnelson, M. I., & Volovik, G. E. (2012) · Journal of Experimental and Theoretical Physics Letters · doi:10.1134/s0021364012080061 · Cited by 22 · see citing works
Zitterbewegung of Moiré Excitons in Twisted MoS2/WSe2 Heterobilayers
Lavor, I. R., da Costa, D. R., Covaci, L., Miloševıć, M. V., Peeters, F. M., & Chaves, A. (2021) · Physical Review Letters · doi:10.1103/physrevlett.127.106801 · Cited by 20 · see citing works
Observation of interaction-induced phenomena of relativistic quantum mechanics
Zhang, W., Yuan, H., He, W., Zheng, X., Sun, N., Di, F., Sun, H., & Zhang, X. (2021) · Communications Physics · doi:10.1038/s42005-021-00752-8 · Cited by 15 · see citing works
Schwinger pair creation in multilayer graphene
Zubkov, M. A. (2012) · Journal of Experimental and Theoretical Physics Letters · doi:10.1134/s0021364012090135 · Cited by 14 · see citing works
Optical simulation of the free Dirac equation
Silva, T. L., Taillebois, E. R. F., Gomes, R. M., Walborn, S. P., & Avelar, A. T. (2019) · Physical Review A · doi:10.1103/physreva.99.022332 · Cited by 14 · see citing works
Vacuum Particle Creation: Analogy with the Bloch Theory in Solid State Physics
Smolyansky, S. A., Tarakanov, A. V., & Bönitz, M. (2009) · Contributions to Plasma Physics · doi:10.1002/ctpp.200910058 · Cited by 13 · see citing works
Relativistic analogies in direct-gap semiconductors
Hrivnák, Ľ. (1993) · Progress in Quantum Electronics · doi:10.1016/0079-6727(93)90008-w · Cited by 11 · see citing works
Revealing inherent quantum interference and entanglement of a Dirac particle
Ning, W., Zheng, R., Xia, Y., Xu, K., Li, H., Zheng, D., Fan, H., Wu, F., Yang, Z., & Zheng, S. (2023) · npj Quantum Information · doi:10.1038/s41534-023-00770-0 · Cited by 4 · see citing works
Dirac particle dynamics of a superconducting circuit
Svetitsky, E., & Katz, N. (2019) · Physical Review A · doi:10.1103/physreva.99.042308 · Cited by 3 · see citing works
Waveguide array simulation of the Dirac equation for even potentials
Villegas-Martínez, B. M., Soto, F., & Moya-Cessa, H. M. (2022) · Results in Physics · doi:10.1016/j.rinp.2022.105212 · Cited by 1 · see citing works
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Digital Quantum Simulation of the Schwinger Model and Symmetry Protection with Trapped Ions
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Flagged first, then by citations
4 of the 71 references couldn’t be matched to a known paper.
Research works printed but not linked: 1
- Landau, L. To the theory of energy transmission in collissions. II. Phys. Zs. Sowjet 2, 46 (1932).
Cited by
Sorted by citations
Thermal Properties of Graphene Via Dirac Oscillator Model Under Doubly Special Relativity Framework
Boumali, A., & Jafari, N. (2026) · Journal of Low Temperature Physics · doi:10.1007/s10909-026-03376-w · Cited by 2 · see citing works
Boumali, A. (2026) · Nuclear Physics B · doi:10.1016/j.nuclphysb.2026.117506
Frequency collisions in parametrically modulated superconducting circuits
Anonymous, Zhao, P., Tan, X., & Yu, Y. (2026) · Physical Review Applied · doi:10.1103/ptj2-kcpc
“This can be achieved using either first-order qubit-modulated [25] or coupler-modulated [26, 27] schemes, where the modulation parameters (amplitudes, frequencies, and phases) are tuned to realize the desired interactions.”
Self-citation
- Authors
- Zhang, Y., Xu, J., Ma, Z., Zheng, W., Lan, D., Tan, X., & Yu, Y.
- Affiliations
- National Laboratory of Solid State Microstructures
- University of Science and Technology of China (CN)
- Corresponding author
- Lan, Dong
- Tan, Xinsheng
- Yu, Yang
- Venue
- Communications Physics
- Published
- 2025-06-12
- Volume
- 8
- Issue
- 1
- ISSN
- 2399-3650 (ISSN-L)
- OpenAlex
- W4411240065
- Semantic Scholar
- 90efedaa053424503e4f407d7aeb3f925c15c559
- Cited by
- 3
- References
- 71
- Citations by year
- 2026: 3
- FWCI
- 1.28
- Type
- journal article
- Language
- en
- Open access
- gold
- doi.org
- Licence
- Locations
- Communications Physics (publishedVersion, cc-by-nc-nd)
- DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion, cc-by-sa)
- Funders
- National Science Foundation
- National Natural Science Foundation of China (U21A20436, 11890704, 12074179)
- Government of Jiangsu Province (BE2021015-1)
- Natural Science Foundation of Shandong Province (ZR2023LZH002)
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Keywords
- Dirac equation, superconducting qubits, quantum simulation, Zitterbewegung, Schwinger effect, parametric modulation, superconducting circuits, relativistic quantum mechanics
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Claims read by deepseek-v4.1-flash under definition version 3 (extract-v3) on 2026-10-13.
Classified by jev-1.13-20260917 (gold-guideline-v0.2) on 2026-10-13.
Cited sources are checked by gpt-6-sol.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
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