Revealing inherent quantum interference and entanglement of a Dirac particle

It may state two separate claims as one.

Zitterbewegung can also appear in some classical wave systems, so it is not a unique character of Dirac particles.

“which itself, however, is not a unique character of Dirac particles as similar phenomena can also appear in some classical wave systems [35][36][37][38][39][40].”

• cites [35] Zhang, X. Observing Zitterbewegung for photons near the Dirac point of a two-dimensional photonic crystal. Phys. Rev. Lett. 100, 113903 (2008). sourceSupported
• cites [36] Otterbach, J., Unanyan, R. G. & Fleischhauer, M. Confining stationary light: Dirac dynamics and Klein tunneling. Phys. Rev. Lett. 102, 063602 (2009). sourceSupported
• cites [37] Longhi, S. Photonic analog of Zitterbewegung in binary waveguide arrays. Opt. Lett. 35, 235 (2010). sourceSupported
• cites [38] Chen, Y. et al. Non-abelian gauge field optics. Nat. Commun. 10, 3125 (2019). sourceSupported
• cites [39] Dreisow, F. et al. Classical simulation of relativistic Zitterbewegung in photonic lattices. Phys. Rev. Lett. 105, 143902 (2010). sourceNot checked — no full text of the source
• cites [40] Silva, T. L., Taillebois, E. R. F., Gomes, R. M., Walborn, S. P. & Avelar, A. T. Optical simulation of the free Dirac equation. Phys. Rev. A 99, 022332 (2019). sourceSupported

Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.

The resulting cat-like state was previously predicted as a solution of a relativistic spin-1/2 charged particle in an external magnetic field, but has not been characterized in previous simulations.

“The resulting cat-like state was previously predicted to exist as a solution of a relativistic spin-1/2 charged particle in an external magnetic field [49], but has not been characterized in previous simulations [50,51].”

• cites [49] Bermudez, A., Martin-Delgado, M. A. & Solano, E. Mesoscopic superposition states in relativistic Landau levels. Phys. Rev. Lett. 99, 123602 (2007). sourceSupported
• cites [50] Gerritsma, R. et al. Quantum simulation of the Klein paradox with trapped ions. Phys. Rev. Lett. 106, 060503 (2011). sourceCould not verify

  The cited paper describes simulations of relativistic scattering in electrostatic potentials and reconstruction of particle wave packets (pp. 1–3). Our copy does not identify a cat-like state or address whether such a state had been characterized in previous simulations. The magnetic-field prediction is attributed to a different citation, [49]. (LLM, conf=0.72)

  Experimental — accuracy not yet measured
• cites [51] Salger, T., Grossert, C., Kling, S. & Weitz, M. Klein tunneling of a quasirelativistic Bose-Einstein condensate in an optical lattice. Phys. Rev. Lett. 107, 240401 (2011). sourceSupported

The quote only partly states this claim.

The numerical results with the full Hamiltonian agree well with the experimental simulations shown in Fig. 2 of the main text.

“All these numerical results agree well with the experimental simulations shown in Fig. 2 of the main text.”

It may state two separate claims as one.

The measured Wigner functions and entanglement entropy agree well with theoretical predictions.

“The measured WFs and entanglement entropy agree well with theoretical predictions.”

It may state two separate claims as one.

The measured time evolutions of ⟨x⟩ and ⟨p⟩ imply that the two wavepackets have the same momentum at each moment but move in opposite directions.

“The measured results imply that the two wavepackets have the same momentum at each moment, but move along the opposite directions.”

It may state two separate claims as one.

Even when the particle remains in the positive branch, there still exists phase space quantum interference, though ZB disappears.

“We further note that even when the particle remains in the positive branch, there still exists phase space quantum interference, though ZB disappears.”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

There is a fixed resonator frequency shift induced by the non-resonant coupling.

“There is a fixed resonator frequency shift induced by the non-resonant coupling, thus we adjust the microwave frequency to about 0.75 MHz to get better results.”

It may state two separate claims as one.

For the present Rabi model, the effective frequency of the qubit and the effective coupling strength are ω/(2π)=2.2 MHz and η=2π×0.78 MHz, respectively.

“For the present Rabi model, the effective frequency of the qubit and the effective coupling strength are ω/(2π) = 2.2 MHz and η = 2π× 0.78 MHz, respectively.”

Probably not a substantial claim. It may state two separate claims as one.

The Wigner functions correlated with the measurement outcomes |g⟩ and |e⟩, and the result irrespective of the test qubit's state, were measured after an evolution time of 330 ns.

“The WFs correlated with the measurement outcomes |g⟩ and |e⟩ are presented in the upper panels of Figs.2(a) and 2(b), and the result irrespective of the test qubit's state is displayed in Fig. 2(c), all measured after an evolution time 330 ns.”

It may state two separate claims as one.

In the far-relativistic regime |p| ≫ mc, A ≪ h/2⟨p⟩ ≪ δx, so ZB cannot be observed.

“In the far-relativistic regime |p| ≫ mc, A ≪ h/2 ⟨p⟩ ≪ δx, where δx = h/2δp is the limitation of precision attainable for any position measurement imposed by the Heisenberg uncertainty relation.”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

The presence of a dispersion coupling between test qubit and bus resonator is λ1^2/Δ ≈ 2π × 1.25 MHz.

“During the preparation of the initial state in the ZB simulation experiment, the presence of a dispersion coupling (form of σ z a † a) between the test qubit and the bus resonator is λ 2 1 /∆ ≈ 2π × 1.25 MHz.”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

The phase difference caused by dispersive interaction during Wigner tomography (time scale 60 ns) is 2λ1^2/Δ × 60 ns ∼ 0.3.

“Note θ e and θ g are slightly different because the dispersive interaction during Wigner tomography (time scale 60 ns, causing phase difference 2λ 2 1 /∆ × 60 ns ∼ 0.3).”

It may state two separate claims as one.

The linear potential drags the phase space evolution down along the p-axis by an amount √2 ε t, but does not affect motion along the x-axis.

“As expected, the linear potential drags the phase space evolution down along the p-axis by an amount √ 2ϵt, but does not affect the motion along the x-axis.”

Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.

In the experiment, P_a^g(0) ≤ 0.05, τ is taken every 2 ns from 0 to 200 ns.

“In the experiment, P a g (0) ≤ 0.05, τ is taken every 2 ns from 0 to 200 ns.”

It may state two separate claims as one.

In the non-relativistic regime |p| ≪ mc, the ZB amplitude A ≃ λ_c/2, and consequently ZB cannot be observed.

“In the non-relativistic regime |p| ≪ mc, the ZB amplitude A ≃ λ c /2, where λ c = h/mc is the Compton wavelength, which sets the lower bound for the uncertainty of the position, and consequently, the ZB cannot be observed.”

It may state two separate claims as one.

For a specific momentum p, H_D has eigenvalues ±E_p = ±√(p^2 c^2 + m^2 c^4) and corresponding eigenstates |φ_+(p)⟩ and |φ_-(p)⟩.

“For a specific value of p, H D has two eigenvalues ±E p = ± p 2 c 2 + m 2 c 4 , with the corresponding eigenstates |ϕ + (p)⟩ = (cos ϕ p , i sin ϕ p ) T and”

It may state two separate claims as one.

The experiment simulates Klein tunneling in a linear potential field and observes mesoscopic superpositions of two separated wavepackets in phase space.

“Furthermore, we simulate the Klein tunneling [44,45] in a linear potential field and observe mesoscopic superpositions of two separated wavepackets in phase space.”

• cites [44] Klein, O. Die Reflexion von Elektronen an einem Potentialsprung nach der relativistischen Dynamik von Dirac. Zeit. Phys. 53, 157 (1929). sourceNot checked — no full text of the source
• cites [45] Dombey, N. Seventy years of the Klein paradox. Phys. Rep. 315, 41 (1999). sourceNot checked — no full text of the source

Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds. It may state two separate claims as one.

The photon number capability of the present system is about 20, resulting in the boundary of phase space in the main text.

“Ergo, the photon number capability of the present system is about 20, resulting in the boundary of phase space in the main text.”

It may state two separate claims as one.

The quantum interference signatures appear in the phase space spanned by position and momentum, but disappear when the momentum is traced out.

“The quantum interference signatures appear in the phase space spanned by position and momentum, but disappear when the momentum is traced out.”

It may state two separate claims as one.

The average position evolution under the Dirac Hamiltonian includes a Zitterbewegung term, and ZB is observable only in the intermediate regime where mc is comparable with p.

“The ZB, manifested by the last term, is observable only in the intermediate regime where mc is comparable with p.”

• AssumptionThe authors state this as an assumption: The system is initially in the product state |ψ(0)⟩ = ∫ dp ξ_p |p⟩ |X⟩.

It may state two separate claims as one.

The Dirac particle evolves with a time-dependent quantum interference pattern, hidden in the entangled internal-spatial state, and expresses itself in the position-momentum quasiprobability distribution correlated to the internal state.

“These results unambiguously demonstrate that the Dirac particle evolves with a time-dependent quantum interference pattern, which is hidden in the entangled internal-spatial state, and expresses itself in the position-momentum quasiprobability distribution, correlated to the internal state.”

It may state two separate claims as one.

Under the Dirac time evolution, the populations of the two states |±X⟩ become increasingly balanced, and the entropy tends to 1.

“Under the time evolution, the populations of two states |±X⟩ become increasingly balanced, and the entropy tends to 1.”

• AssumptionThe authors state this as an assumption: The system is initially in the product state |ψ(0)⟩ = ∫ dp ξ_p |p⟩ |X⟩.

Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.

The qubit-state-dependent resonator frequency shift in the dynamical process is still unavoidable; therefore, additional correction of the results is necessary.

“The qubit-state-dependent resonator frequency shift in the dynamical process is still unavoidable; therefore, additional correction of the results is necessary so as to make the results more intuitive.”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

We use the same method as in Ref. [9].

“We use the same method as in Ref. [9].”

• cites [9] Krekora, P., Su, Q. & Grobe, R. Relativistic electron localization and the lack of Zitterbewegung. Phys. Rev. Lett. 93, 043004 (2004). sourceNot checked — no full text of the source

It may state two separate claims as one.

Phase space quantum interference is a universal inherent characteristic of Dirac particles, which can manifest itself even without negative components.

“Although ZB appears only when the system is in a superposition of positive and negative components, phase space quantum interference is actually a universal inherent characteristic of Dirac particles, which can manifest itself even without negative components.”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

The simulation is performed without the second parametric modulation so that the system dynamics are analogous to a massless particle penetrating a linear potential.

“The simulation is performed without the second parametric modulation so that the system dynamics are analogous to a massless particle penetrating a linear potential.”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

Up to now, together with the initial phase of the microwave pulse, we can implement arbitrary displacement operators D(γ) for Wigner measurements.

“Up to now, together with the initial phase of the microwave pulse, we can implement arbitrary displacement operators D(γ) for Wigner measurements.”

Probably not a substantial claim.

The experiment uses a calibration matrix to reconstruct readout results and correct errors caused by decoherence.

“To calibrate these errors, we use a calibration matrix, defined as”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

At t=0 and 76 ns, the positive and negative wavepackets cannot be fully distinguished, so the data for these two mean positions ⟨x⟩ are abandoned in Fig. 3(d) of the main text.

“At t = 0 and 76 ns, the positive and negative wavepackets cannot be fully distinguished, so the data for these two mean positions ⟨x⟩ are abandoned in Fig. 3(d) of the main text.”

It may state two separate claims as one.

With further increase of photon number, entanglement between auxiliary qubit Q2 and resonator can not be ignored, and photon number and its distribution in resonator can not be obtained accurately.

“However, with the further increase of the photon number, the entanglement between the auxiliary qubit Q 2 and the resonator can not be ignored, and the photon number and its distribution in the resonator can not be obtained accurately.”

Probably not a substantial claim.

For a one-dimensional Dirac particle, the Hamiltonian reduces to H_D = cσ_y p + mc^2 σ_z.

“H D = cσ y p + mc 2 σ z .(1)”

It may state two separate claims as one.

The circuit experiment realizes the Dirac Hamiltonian in a rotating frame, with effective light speed c* = √2η and effective mass m*c*^2 = ε2/4.

“Then the resulting effective Hamiltonian reduces to the form of Eq. ( 1), with the correspondences σ y = i|g⟩⟨e| -i|e⟩⟨g|, σ z = |e⟩⟨e| -|g⟩⟨g|, c * = √ 2η and m * c * 2 = ε 2 /4”

• AssumptionThe authors state this as an assumption: The superconducting qubit can be approximated as a two-level system when the coupling λ and transverse field strength Ω are relatively small against the anharmonicity.
• AssumptionThe authors state this as an assumption: ν2 = 2K ≫ η, ε2/2.

Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.

For the selection of the time point, we try to make the population of states |e⟩ and |g⟩ of the test qubit close to 0.5, making the normalized data of the auxiliary qubit accurate and convenient to fit the photon number distribution of the resonator.

“For the selection of the time point, we try to make the population of states |e⟩ and |g⟩ of the test qubit close to 0.5, making the normalized data of the auxiliary qubit accurate and convenient to fit the photon number distribution of the resonator.”

It may state two separate claims as one.

Phase space quantum interference effects were previously predicted for some special states with both positive and negative components, but their presence without ZB had not been revealed.

“We note that the phase space quantum interference effects were previously predicted for some special states with both positive and negative components [11], but the presence of such effects without ZB has not been revealed.”

• cites [11] Pedernales, J. S., Di Candia, R., Ballester, D. & Solano, E. Quantum simulations of relativistic quantum physics in circuit QED. New J. Phys. 15, 055008 (2013). sourceNot checked yet

The quote only partly states this claim.

When restricted to the positive branch, the internal degrees of freedom are necessarily entangled with the momentum except for a plane wave with δp → 0.

“This implies that the internal degrees freedom is necessarily entangled with the momentum except for a plane wave with δp → 0.”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

The desired modulation of ωq(t) can be realized by mapping it to the modulation of Z bias.

“The desired modulation of ωq(t) can be realized by mapping it to the modulation of Z bias.”

Probably not a substantial claim.

The phases of the driving pulses are optimized to make the experimental dynamics closer to the Dirac dynamics.

“making the experiment dynamics closer to the Dirac dynamics.We traverse these two phases (ϕ 1 , ϕ 2 ) between 0 and 2π”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

The superconducting bus resonator possesses a bosonic mode, as demonstrated in prior work.

“the superconducting bus resonator, characterized by a quantized microwave field, was demonstrated [1,[3][4][5][6][7] to possess a bosonic mode”

• cites [1,[3][4][5][6][7]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim.

The present platform can be utilized to emulate the external degrees of freedom of a Dirac particle.

“which can be utilized to emulate the external degrees of freedom of a Dirac particle.”

It may state two separate claims as one.

With the assumption ν2 = 2K ≫ η, ε2/2, the system Hamiltonian can be well approximated by H''_I = -η e^{-iθ} a† σ_θ + H.c. + ωσ_z, and when θ = π/2 this effective Hamiltonian has the same form as the 1+1 Dirac equation.

“With the assumption ν 2 = 2K ≫ η, ε 2 /2 and under the further transformation exp(iKσ θ t), the system Hamiltonian can be well approximated by”

• Also stated in: § Figure 4 · plain

  “When θ = π/2, this effective Hamiltonian has the same form as the 1+1 Dirac equation of a spin-1/2 particle, with the correspondence c * = √ 2η and m * c * 2 = ω.”

• AssumptionThe authors state this as an assumption: The superconducting qubit can be approximated as a two-level system when the coupling λ and transverse field strength Ω are relatively small against the anharmonicity.
• AssumptionThe authors state this as an assumption: ν2 = 2K ≫ η, ε2/2.

It may state two separate claims as one.

For x > 0, the momentum of the Dirac particle is given by p^2 = (E - V) 2m^2 (c=1), with group velocity v_g = dE/dp = p/(E - V); when a massless particle moves from left to right with E < V, p is assigned its negative solution so that v_g is positive.

“For x > 0, the momentum of the Dirac particle is given by p 2 = (E -V ) 2m 2 (c = 1), with the group velocity v g = dE/dp = p/(E -V ) [45].When the massless particle moves from left to right with E < V , p is assigned with its negative solution, so that v g is positive.”

• cites [45] Dombey, N. Seventy years of the Klein paradox. Phys. Rep. 315, 41 (1999). sourceNot checked — no full text of the source

Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.

Both θ_e and θ_g are around 1.3, offsetting the resonator phase induced by the frame difference, demonstrating that the rotation of our data is reasonable.

“While both θ e and θ g are around 1.3, offsetting the resonator phase induced by the frame difference, and therefore demonstrating that the rotation of our data is reasonable.”

Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds. It may state two separate claims as one.

This limitation can be solved by increasing the detuning between the qubit and cavity (>2π × 800 MHz), which requires improvement of microwave signal control hardware.

“This limitation can be solved by increasing the detuning between the qubit and cavity (>2π × 800 MHz), which requires the improvement of microwave signal control hardware in the present experimental platform.”

Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.

A larger number of photons requires Wigner tomography within a larger phase space region.

“As mentioned in the main text, a larger number of photons requires Wigner tomography within a larger phase space region.”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

Due to hardware-limited pulse amplitude, we can increase the size of |γ| by increasing the length of the tomography pulse, at the cost of increasing the dispersive interaction time.

“Due to the hardware-limited pulse amplitude, we can increase the size of |γ| by increasing the length of the tomography pulse, at the cost of increasing the dispersive interaction time.”

It may state two separate claims as one.

Due to the finite detuning Δ = 2π × 470 MHz between the auxiliary qubit Q2 and bus resonator R, as the number of photons increases, ancilla qubit Q2 inevitably interacts with the resonator during the Dirac dynamics, the ground state population P_a^g(0) may not be 1 at τ = 0.

“It is worth noting that due to the finite detuning ∆ = 2π × 470 MHz between the auxiliary qubit Q 2 and bus resonator R, as the number of photons increases, ancilla qubit Q 2 inevitably interacts with the resonator during the Dirac dynamics, the ground state population P a g (0) may not be 1 at τ = 0.”

It may state two separate claims as one.

The Zitterbewegung phenomenon appears as the integral of the weighted mixture of the two Wigner functions over momentum, which reflects the classical probability distribution but does not manifest the underlying quantum nature.

“The ZB phenomenon appears as the integral of the weighted mixture of the two WFs over the momentum, which reflects the classical probability distribution, but does not manifest the underlying quantum nature.”

It may state two separate claims as one. The quote only partly states this claim.

The negativity arises because each component accumulates a momentum- and mass-dependent nonlinear phase, a non-Gaussian operation, and the two rods sprouting from the distorted wavepacket interfere, resulting in negative quasiprobability between them.

“Under the Dirac Hamiltonian, each component accumulates a phase that nonlinearly depends on the”

It may state two separate claims as one.

Quantum interference is a universal inherent characteristic of Dirac particles, which can manifest itself even without negative components.

“As stated in the main text, quantum interference is actually a universal inherent characteristic of Dirac particles, which can manifest itself even without negative components.”

Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.

Proper phase space rotations performed to compensate for Stark shifts produced by discarded off-resonant couplings do not change the quantum effects in any way.

“To compensate for the difference due to the Stark shifts produced by the discarded off-resonant couplings, we perform proper phase space rotations, which, however, do not change the quantum effects in any way.”

Probably not a substantial claim.

All the WF values in the main text are deduced from the photon number distribution.

“All the WF values in the main text are deduced from the photon number distribution.”

Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.

Referring to [10], the potentially slight excitation can be ignored and thus P_a^g(0) is introduced to make the fit more accurate.

“Referring to [10], the potentially slight excitation can be ignored and thus P a g (0) is introduced to make the fit more accurate.”

• cites [10] Wang, Z.-Y. & Xiong, C.-D. Zitterbewegung by quantum field-theory considerations. Phys. Rev. A 77, 045402 (2008). sourceNot checked — no full text of the source

It may state two separate claims as one.

Zitterbewegung itself does not manifest quantum effects, but is closely related to quantum entanglement between the internal and spatial degrees of freedom produced by their coupling.

“As we have noted, ZB itself does not manifest quantum effects, but is closely related to quantum entanglement between the internal and spatial degrees of freedom, produced by their coupling.”

Probably not a substantial claim.

In this way, we can get all the information about the resonator state.

“In this way, we can get all the information about the resonator state.”

Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.

Current experiments are limited by the accuracy of measuring too many photons (n>20) in the microwave bosonic field.

“Nevertheless, current experiments are limited by the accuracy of measuring too many photons (n>20) in the microwave bosonic field.”

It may state two separate claims as one.

The time-evolving mean position inferred from the measured Wigner function coincides with the simulation, confirming that Zitterbewegung has a deeper root that is of pure quantum characteristic.

“This evolution, inferred from the measured WF, is presented in Fig. 2(e), which coincides with the simulation (green curve), confirming that ZB has a deeper root that is of pure quantum characteristic.”

Probably not a substantial claim.

The photon number distribution P_n is inferred by the Rabi oscillation signal [14].

“The photon number distribution P n is inferred by the Rabi oscillation signal [14].”

• cites [14] Vaishnav, J. Y. & Clark, C. W. Observing Zitterbewegung with ultracold atoms. Phys. Rev. Lett. 100, 153002 (2008). sourceNot checked yet

Probably not a substantial claim. The paper’s conclusions may not depend on it.

Considering the finite detuning between the auxiliary qubit and the resonator, our ZB simulation ends at 330 ns.

“Considering the finite detuning between the auxiliary qubit and the resonator, our ZB simulation ends at 330 ns.”

Probably not a substantial claim. It may be too vague to check against evidence. The paper’s conclusions may not depend on it.

The demonstrated nonclassicality can serve as a resource for quantum-enhanced sensing.

“In addition to fundamental interest, the demonstrated nonclassicality can serve as a resource for quantum-enhanced sensing [52].”

• cites [52] Kwon, H., Tan, K. C., Volkoff, T. & Jeong, H. Nonclassicality as a quantifiable resource for quantum metrology. Phys. Rev. Lett. 122, 040503 (2019). sourceNot checked yet

  Checked before — its earlier verdict is held back until a new check runs.

Probably not a substantial claim. It may state two separate claims as one.

With the setting ω_r = ω0 + 2ν1, the resonator interacts with the qubit at the second sideband associated with the first modulation, while the microwave drive works at the carrier.

“With this setting, the resonator interacts with the qubit at the second sideband associated with the first modulation, while the microwave drive works at the carrier.”

• AssumptionThe authors state this as an assumption: ω_r = ω0 + 2ν1.

It may state two separate claims as one.

During evolution, the wavepacket is continually deformed by competition between momentum-dependent and static energy terms, leading to nonlinear energy dependence on momentum; this non-Gaussian process manifests pseudospin-dependent quantum interference signatures.

“During the evolution, the wavepacket is continually deformed under the competition between the momentum-dependent and static energy terms in the Dirac Hamiltonian, which leads to a nonlinear dependence of the energy on the momentum.This nonlinear process evolves an initial Gaussian wavepacket to a non-Gaussian one, manifesting pseudospin-dependent quantum interference signatures.”

Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.

Although ancilla qubits can be biased further down to increase detuning in Dirac dynamics, performance of qubits deteriorates and frequency changes more widely, affecting measurement results.

“It is also worth noting that although ancilla qubits can be biased further down to increase the detuning in the Dirac dynamics, the performance of the qubits deteriorates and the frequency changes more widely, affecting the measurement results.”

It may state two separate claims as one.

The experimental data are in good agreement with the ideal ones, which proves the correctness of our method.

“In this case, the optimized population is shown in Supplementary Figure 8, and the experimental data are in good agreement with the ideal ones, which proves the correctness of our method.”