Device and Hamiltonian

We fabricate a quadruple quantum dot array in a GaAs/AlGaAs heterostructure with overlapping gates (Fig. 1a)^18–20. The confinement potentials of the dots are controlled through “virtual gates”^21–24. Two extra quantum dots are placed nearby and serve as fast charge sensors^25,26. We configure the four-spin array into two pairs (“left” and “right”) for initialization and readout. Each pair of spins can be prepared in a product state ($$ ∣\uparrow \downarrow ⟩$$ or $$ ∣\downarrow \uparrow ⟩$$) via adiabatic separation of a singlet in the hyperfine gradient^27–29. We can also initialize either pair as $$ ∣T_{+}⟩=∣\uparrow \uparrow ⟩$$ by exchanging electrons with the reservoirs^28,30. Both pairs are measured through spin-to-charge conversion via Pauli spin blockade²⁷, together with a shelving mechanism³¹ for high readout fidelity. Further details about the device can be found in “Methods”.

Fig. 1

Experimental setup.

a Scanning electron micrograph of the quadruple quantum dot device. The locations of the electron spins are overlaid. b Schematic showing the two-qubit Ising system in a four-spin Heisenberg chain. c The pulse sequence used in the experiments.

The four-spin array is governed by the following Hamiltonian:

Heisenberg exchange coupling does not naturally enable the creation of a DTC phase⁸. Additional control pulses can convert the Heisenberg interaction into an Ising interaction⁸, which permits the emergence of a DTC phase. A DTC can also be created using a sufficiently strong magnetic field gradient instead of applying extra pulses³⁴. Here, we introduce a new method for generating DTC behavior that does not require complicated pulse sequences or large field gradients, but instead relies only on periodic exchange pulses.

To see how we can still obtain an effective Ising interaction in this case, it helps to view each pair of spins as an individual ST qubit (Fig. 1b)²⁷. Specifically, consider the scenario where the joint spin state of each pair is confined to the subspace spanned by $$ ∣S⟩=\frac{1}{\sqrt{2}}\left(∣\uparrow \downarrow ⟩-∣\downarrow \uparrow ⟩\right)$$ and $$ ∣T_0⟩=\frac{1}{\sqrt{2}}\left(∣\uparrow \downarrow ⟩+∣\downarrow \uparrow ⟩\right)$$. According to ref. ¹⁷, when J₁ = J₃ = 0 and J₂ > 0, the effective Hamiltonian of the system is

In the case when J₂ = 0, but when J₁, J₃ > 0, the overall Hamiltonian describes two uncoupled ST qubits. Thus, let us define

In the $$ \left\{∣S⟩,∣T_0⟩\right\}$$ subspace of each pair, these operators are equivalent to $$ S_1^{\mathrm{eff}}=\exp \left[-\frac{i}{\hslash }{t}_1\frac{h}{2}\left({\mathrm{\Delta }}_{12}{\tilde{\sigma }}_1^z+J_1{\tilde{\sigma }}_1^x\right)\right]$$ and $$ S_2^{\mathrm{eff}}=\exp \left[-\frac{i}{\hslash }{t}_2\frac{h}{2}\left({\mathrm{\Delta }}_{34}{\tilde{\sigma }}_2^z+J_3{\tilde{\sigma }}_2^x\right)\right]$$. In writing $$ S_1^{\mathrm{eff}}$$ and $$ S_2^{\mathrm{eff}}$$, we have ignored overall energy shifts J₁/4 and J₃/4 of the single-qubit Hamiltonians, because the system dynamics do not depend on these shifts. Assuming J₁ ≫ Δ₁₂ and J₃ ≫ Δ₃₄, when t₁J₁ = t₂J₃ = 0.5, these two operators implement SWAP gates between spins 1–2 and 3–4. Equivalently, they induce nominal π pulses about the x-axis of each ST qubit. The presence of the intraqubit gradients Δ₁₂ and Δ₃₄ slightly tilts the rotation axis toward the z-axis for each ST qubit, introducing uncontrolled errors to the π pulses. We can also manually introduce additional pulse errors by changing J₁ and J₃, while fixing t₁ and t₂. We represent the error as ϵ, with $$ J_1=J_1^{\pi }\left(1+ϵ\right)$$ and $$ J_3=J_3^{\pi }\left(1+ϵ\right)$$. Here $$ J_1^{\pi }$$ and $$ J_3^{\pi }$$ are the interaction strengths that yield π pulses.

Floquet-enhanced spin swaps

We define a Floquet operator U = S_int ⋅ S₂ ⋅ S₁ (Fig. 1c), and we repeatedly apply this operator to our system of four spins. As discussed above, U implements spin SWAP gates between spins 1–2 and 3–4 followed by a period of exchange interaction between spins 2 and 3. Equivalently, U implements π pulses on both ST qubits and then a period of Ising coupling between them. One might naively imagine that the highest fidelity SWAP operations between spins should occur when J₂ = 0 and τ = 0, given the presence of intraqubit hyperfine gradients. In this case, as we have discussed in ref. ²⁹, repeated SWAP operations are especially susceptible to errors from the hyperfine gradients Δ_ij.

However, by allowing J₂ > 0 and τ > 0, we find specific conditions in which we observe a significant enhancement of the spin-eigenstate-swap quality factor (Fig. 2). To explore this phenomenon, we prepare each ST qubit in $$ ∣\uparrow \downarrow ⟩$$ or $$ ∣\downarrow \uparrow ⟩$$. (The specific state is governed by the sign of Δ₁₂ and Δ₃₄, which are random quasistatic gradients resulting from the nuclear hyperfine interaction.) We apply multiple instances of the Floquet operator U to the system and measure the ground-state return probabilities for both ST qubits.

Fig. 2

Floquet-enhanced π rotations.

a, b Measured ground-state return probabilities of (a) ST qubit 1 and (b) ST qubit 2, after four Floquet steps, with interaction time τ = 1.4 μs. The ranges of J₁ and J₃ center around $$ J_1^{\pi }$$ and $$ J_3^{\pi }$$. The values of J₁ and J₃ are swept simultaneously. In both figures, the red cross marks the condition for the Floquet-enhanced π rotations. The black ovals are the semiclassical phase boundaries. c, d Simulated return probabilities of (c) ST qubit 1 and (d) ST qubit 2, corresponding to the data in (a) and (b), respectively. e, f Measured ground-state return probabilities of (e) ST qubit 1 and (f) ST qubit 2, after four Floquet steps, with interaction time τ = 1.0 μs. J₁ and J₃ values are the same as in a and b. g, h Simulated return probabilities of (g) ST qubit 1 and (h) ST qubit 2, corresponding to the data in (e) and (f), respectively. The experimental data in (a, b, e, f) are averaged over 8192 realizations. In all figures, $$ P_g^k$$ indicates the ground-state return probability for ST qubit k.

First, we set the interaction time τ = 1.4 μs and SWAP pulse times t₁ = t₂ = 5 ns, and apply four Floquet steps. We sweep J₂ linearly from 0.05 to 5 MHz (Fig. 2a, b). (Setting J₂ < 0.05 MHz would require large negative voltage pulses applied to the barrier gate due to the residual exchange, which could disrupt the tuning of the device.) We also sweep J₁ from 80 to 460 MHz, and J₃ from 50 to 260 MHz. The ranges of J₁ and J₃ roughly center around $$ J_1^{\pi }$$ and $$ J_3^{\pi }$$, respectively. Away from the center, J₁ and J₃ induce pulse errors. The experimental values of $$ t_1{J}_1^{\pi }$$ and $$ t_2{J}_3^{\pi }$$ are much larger than 0.5, because the voltage pulses experienced by the qubits have rise times of ~1 ns (see “Methods” and Supplementary Fig. 1). To compensate for the pulse rise times (which are slightly different for each qubit), $$ t_1{J}_1^{\pi }$$ and $$ t_2{J}_3^{\pi }$$ must be larger than 0.5 in order to properly induce π pulses.

Clear, bright diamond patterns are visible in the data (Fig. 2a, b). These bright regions correspond to improved spin-eigenstate-swap quality factors. Note that the brightest regions correspond to configurations when J₂ > 0. Note also that the diamonds are approximately periodic in J₂τ, as expected for a Floquet operator. We repeat the same experiments with τ = 1μs, and we observe similar diamond patterns, although they have an increased period in J₂ (Fig. 2e, f). The diamond patterns of ST qubit 2 appear narrower due to the large hyperfine gradient Δ₃₄, which causes larger pulse errors and reduces the size of the quality-factor-enhancement region. These data from an effective two-qubit system resemble predicted DTC phase diagrams of a true nearest-neighbor many-body system (see “Methods” and Supplementary Figs. 2 and 3)^7,8.

Our simulations agree well with the data (Fig. 2c, d, g, h; see “Methods”). In the simulations, the diamond pattern is periodic in J₂τ with the periodicity of exactly 1, and the strongest quality-factor enhancement occurs at J₂τ = 0.5. In the experimental data, however, the periodicity is slightly larger than 1, and the strongest quality-factor enhancement occurs at J₂τ > 0.5. This is due to the imperfect calibration of the exchange coupling J₂ (ref. ³²). In particular, the presence of the hyperfine field gradient makes it difficult to measure and control the exchange couplings with sub-MHz resolution. If our modeling of the exchange coupling were more precise, then we would expect the periodicity of the diamond patterns to be closer to 1 and the quality-factor enhancement to occur closer to J₂τ = 0.5 in the experimental data.

We can interpret our data using a semiclassical model inspired by Choi et al. in ref. ¹⁰ to explain DTC behavior (see “Methods”). In brief, an initial state of ST qubit 1, $$ ∣{\psi }_0⟩=\cos \left({\theta }_0/2\right)∣g⟩+e^{i{\varphi }_0}\sin \left({\theta }_0/2\right)∣e⟩$$ evolves to $$ ∣{\psi }_f⟩=e^{-i{\varphi }_1{\sigma }^z/2}{e}^{-i\theta {\sigma }^x/2}{e}^{-i{\varphi }_2{\sigma }^z/2}{e}^{-i\theta {\sigma }^x/2}∣{\psi }_0⟩$$ after two Floquet steps. Here θ ≈ π indicates a nominal π pulse, and $$ {\varphi }_1=2\pi \left(J_2/2+{\mathrm{\Delta }}_{12}+\overline{B}\right)\tau $$, and $$ {\varphi }_2=2\pi \left(-J_2/2+{\mathrm{\Delta }}_{12}+\overline{B}\right)\tau $$. In this semiclassical model, the effect of ST qubit 2 on ST qubit 1 is to generate the πJ₂τ term in the operator that switches sign after each Floquet step, because ST qubit 2 undergoes a nominal π pulse. As emphasized in ref. ¹⁰, the change in sign of this part between Floquet steps is entirely a result of interactions in the system. The resulting single-qubit rotations in this semiclassical model are reminiscent of dynamical decoupling¹⁰. We have numerically simulated the semiclassical single-qubit evolution over two Floquet steps for our system (see “Methods”). The black lines in Fig. 2a, b indicate the regions where approximate ST-qubit eigenstates are also exactly eigenstates of the evolution operator over two steps, i.e., they are exactly preseved by the two Floquet steps¹⁰. The size of these regions confirms that interactions are essential for the effects we observe. Exactly the same enhancement regions are expected for end spins in longer chains, because our system is a nearest-neighbor Ising spin chain. Simulations for an eight-site Ising spin chain at late times show DTC behavior in exactly these regions (see “Methods” and Supplementary Fig. 2).

Next, we also sweep J₃ from 220 to 430 MHz. In this case, the range of J₃ roughly centers around $$ J_3^{2\pi }$$. The interaction time is τ = 1.4 μs and the ranges of J₁ and J₂ remain the same. Again, we apply four Floquet steps and measure the ground-state return probabilities. This time the data do not show diamond patterns (Fig. 3), and the return probability of ST qubit 1 is lower than the Floquet-enhanced return probability shown in Fig. 2a. This indicates that the Floquet enhancement is no longer present. In fact, if either of the Floquet operators S₁ or S₂ fails to induce approximately a π rotation, then the Floquet enhancement does not appear.

Fig. 3

Absence of Floquet enhancement due to the omission of a π pulse.

a, b Measured ground-state return probabilities of (a) ST qubit 1 and (b) ST qubit 2, after four Floquet steps, with interaction time τ = 1.4 μs. The ranges of J₁ and J₃ center around $$ J_1^{\pi }$$ and $$ J_3^{2\pi }$$, respectively. The values of J₁ and J₃ are swept simultaneously. The data are averaged over 8192 realizations.

On the one hand, this effect is striking, when one considers the individual spins themselves. Recall that the ST-qubit splittings Δ₁₂ and Δ₃₄ are generated by the hyperfine interaction between the Ga and As nuclei in the semiconductor heterostructure and the electron spins in the quantum dots. Although Δ₁₂ and Δ₃₄ are quasistatic on millisecond timescales, they each independently fluctuate randomly, and can change sign, over the duration of a typical data-taking run, which is ~1 h. Each of the 8192 different realizations for each pixel in the data of Fig. 2 likely contain instances, where both ST qubits have the same or different ground-state spin orientations. (The ground state of each ST qubit is either $$ ∣\uparrow \downarrow ⟩$$ or $$ ∣\downarrow \uparrow ⟩$$, depending on the sign of the instantaneous hyperfine gradient.)

Thus, the data of Fig. 2 likely include realizations with all possible combinations of the orientations of spins 2 and 3 before the interaction period. Despite the random orientations of spins 2 and 3, the Floquet enhancement still appears. It might therefore seem that whether or not spins 1–2 or 3–4 undergo a SWAP before the interaction period should not affect the behavior of the system. However, as shown in Fig. 3, implementing a 2π rotation, as opposed to a π rotation, on one of the ST qubits eliminates the Floquet enhancement.

On the other hand, when one considers the semiclassical picture described above, the absence of a π pulse on one of the ST qubits spoils the semiclassical decoupling evolution discussed above and in ref. ¹⁰. In this case, ST-qubit eigenstates are no longer eigenstates of two instances of the Floquet operator, and the enhancement no longer occurs.

We have now determined the optimal conditions for the Floquet enhancement. For the remainder of the paper, we set J₁ = 270 MHz and J₃ = 150 MHz with t₁ = t₂ = 5 ns for the SWAP operators S₁ and S₂, respectively, and we set τ = 1.4 μs and J₂ = 0.41 MHz for the Ising interaction. To quantify the Floquet enhancement, we evolve the system for 50 Floquet steps and measure the ground-state return probabilities for both qubits after each step. The results are shown in Fig. 4a. Note that the system exhibits a clear subharmonic response to the Floquet operator. We extract a swap quality Q by fitting the data with a decaying sinusoidal function $$ P_g\left(n\right)=\alpha \exp \left(-n/Q\right)\cos \left(n\pi \right)+\beta $$, where P_g(n) denotes the return probability at the nth Floquet step, and Q, α, and β are fit parameters. We also investigate the quality factor of the qubits under non-enhanced regular π pulses. Here, we use the same interaction time τ = 1.4 μs, but we turn off the interaction strength J₂ by setting the barrier gate pulse to zero. To further eliminate any effects associated with Floquet enhancement, we only apply π pulses to one qubit, while the other qubit remains idle after initialization. Again, we apply 50 π pulses and measure the ground-state return probability, and we fit the data with the same decaying sinusoidal function. By comparing the fit parameter Q, we can obtain the ratio between the quality factors of the qubits under Floquet-enhanced and non-enhanced π rotations.

Fig. 4

Floquet-enhanced spin swaps.

a–d Quality-factor enhancement of spin-eigenstate swaps for different initial states. In each figure, the top panel shows the measurements of ST qubit 1, and the bottom panel shows the measurements of ST qubit 2. The initial states are shown on the top, where $$ ∣g⟩$$ and $$ ∣e⟩$$ represent the ground state and the excited state of the ST qubit, respectively. The Floquet-enhanced π-pulse data are shown in blue, and the non-enhanced regular π-pulse data are shown in red. The fitted exponential decay envelopes are overlaid as dashed lines for all data except for the bottom panel in (d). The data are averaged over 4096 realizations.

We find a ~3-fold quality-factor improvement on qubit 1, and ~9-fold improvement on qubit 2. The significant discrepancy between the quality-factor improvements of the two qubits is likely due to the large hyperfine gradient Δ₃₄ in qubit 2, which causes an exceptionally low quality factor for non-enhanced π rotations. The quality-factor enhancement is striking in this case. To extract an estimated uncertainty, we repeat the same experiment 30 times, and calculate the mean and the standard deviation of the quality-factor ratio, as shown in the first row of Table 1.

Table 1

Quality-factor enhancements of both qubits for different initial states.

Initialization	Quality-factor enhancement
	Qubit 1	Qubit 2
$$ ∣g⟩\otimes ∣g⟩$$	3.60 ± 0.89	8.47 ± 3.29
$$ ∣e⟩\otimes ∣g⟩$$	3.24 ± 0.94	9.33 ± 2.96
$$ ∣g⟩\otimes ∣e⟩$$	3.15 ± 0.79	9.10 ± 2.87
$$ ∣g⟩\otimes ∣T_{+}⟩$$	1.92 ± 0.27	N/A

So far, we have initialized both ST qubits in their ground states. We can also initialize either ST qubit in its excited state by applying an extra π pulse to the qubit immediately before the first Floquet step. We run the same experiment with different initial states and extract the quality factors by fitting the data (Fig. 4b, c). Again, for each initial state, we repeat the experiment 30 times and calculate the mean and the variance of the quality-factor ratio, which are listed in Table 1. The quality-factor improvements of both qubits are consistent across different initial states.

We also initialize the right pair as $$ ∣T_{+}⟩=∣\uparrow \uparrow ⟩$$ and measure the quality-factor improvement on qubit 1 (Fig. 4d). We notice that the quality-factor ratio is much lower when the right pair is initialized in $$ ∣T_{+}⟩$$. This is not surprising since the effective Ising interaction between qubit 1 and qubit 2 (Eq. (2)) is only valid when both qubits are restricted to the S_z = 0 subspace. The reason why we still see a ~2-fold quality-factor improvement instead of no improvement at all is likely because of the imperfect $$ ∣T_{+}⟩$$ preparation due to thermal population of excited states³⁰. Load errors will cause the right pair to occupy the ST-qubit ground or excited states a small fraction of the time. In these cases, the Floquet enhancement of the left-pair ST qubit is expected to occur. Thus, the overall quality factor should appear to improve slightly, because of the imperfect initialization. Correspondingly, it is likely that the imperfect initialization limits the quality-factor enhancement when both qubits are initialized in ST-qubit eigenstates.

Finally, we emphasize that a Floquet drive, i.e., repeated SWAP gates, is required to realize the enhancement shown in Fig. 4. Based on the data of Fig. 4, the first SWAP gate is not substantially enhanced by the protocol. It is only subsequent SWAP gates that are enhanced. This is consistent with the requirement for a periodic drive in a DTC. As we discuss below, this periodic drive is also useful for constructing quantum gates.

Device

The quadruple quantum dot device is fabricated on a GaAs/AlGaAs heterostructure substrate with three layers of overlapping Al confinement gates and a final Al top gate. The Al gates are patterned and deposited using E-beam lithography and thermal evaporation, and each layer is isolated from the other layers by a few nanometers of native oxide. The top gate covers the main device area and is grounded during the experiments. It likely smooths anomalies in the quantum dot potentials. The two-dimensional electron gas resides at the GaAs and AlGaAs interface, 91 nm below the semiconductor surface. The device is cooled in a dilution refrigerator with base temperature of ~10 mK. A 0.5-T external magnetic field is applied parallel to the device surface and perpendicular to the axis connecting the quantum dots.

Pulse rise times

The experimental values of $$ t_1{J}_1^{\pi }$$ and $$ t_2{J}_3^{\pi }$$ are much larger than 0.5, because the voltage pulses experienced by the qubits have rise times of ~1 ns. Supplementary Fig. 1 shows measured exchange oscillations for both ST qubits vs. evolution time. The observable frequency chirp at early evolution times demonstrates the effects of rise times in our system and shows that the π-pulse times yield tJ > 0.5.

Simulation

We simulate the Floquet-enhanced phenomena by evolving a four-spin array according to the Floquet operator U = S_int ⋅ S₂ ⋅ S₁, as defined in the main text. We set t₁ = t₂ = 2 ns, and $$ J_1^{\pi }=J_3^{\pi }=250$$ MHz for the SWAP operators S₁ and S₂ to give π pulses. While t₁ and t₂ are chosen to be 5 ns in the experiments, we expect the realistic SWAP times to be ~2–3 ns due to the pulse rise and fall times of ~1 ns. We include the π-pulse errors by adjusting the exchange couplings as $$ J_1=J_1^{\pi }\left(1+ϵ\right)$$ and $$ J_3=J_3^{\pi }\left(1+ϵ\right)$$, where ϵ represents the fractional error in the rotation angle of the π pulse applied to ST qubits 1 and 2.

For better comparison with the experimental data, the simulations take into account all known error sources, including state preparation, readout, charge noise, and hyperfine field noise. The initial state of each ST qubit is prepared as

We use a Monte-Carlo method to incorporate charge noise and hyperfine field fluctuations. The values of the exchange couplings J_i and the local hyperfine fields $$ B_i^z$$ are randomly sampled from a normal distribution for each simulation run. We set the standard deviation for J_i to be $$ J_i/\left(\sqrt{2}\pi Q\right)$$, where Q = 21 is the exchange oscillation quality factor. We set the standard deviation for $$ B_i^z$$ to be $$ {\sigma }_{B^z}=$$18 MHz, and we assume the mean values to be [0, 20, 0, 50] MHz plus a uniform magnetic field of 3.075 GHz (which accounts for the 0.5-T external magnetic field). The simulated data in Fig. 2 in the main text are obtained by averaging over 128 realizations.

The simulations of Fig. 5 do not include exchange coupling noise or state preparation and measurement errors. We neglected these errors to clearly illustrate the mechanisms underlying the singlet-state preservation and CZ gate. Hyperfine fluctuations with $$ {\sigma }_{B^z}=$$18 MHz were included, and the magnetic field values at the locations of each dot were 3.075 GHz, to account for the external magnetic field. To simulate the CZ gate of Fig. 5b in the main text, we simulate two periods of the Floquet operator U = S_int ⋅ S₁₂, where S₁₂ represents the execution of S₁ and S₂ in parallel, with τJ = 0.25. These two periods implement the CZ gate, up to single-qubit rotations. In the simulation, we evolve the system for two additional periods with the operator U_rot = U₁ ⊗ U₂, where

To confirm that the behavior we report in the two ST qubits corresponds to that of an effective Ising spin chain, we also simulate an Ising spin chain. Define

Supplementary Fig. 3a shows the results of an N = 8 Ising spin chain after four Floquet steps, with $$ B_i^z=20$$ MHz. These results agree with the two-site data of Supplementary Fig. 2a and Fig. 2 in the main text, providing evidence that the behavior we observe in a two-qubit system corresponds to the expected behavior for larger systems. Supplementary Fig. 3b shows the simulated behavior after 1024 Floquet steps. The predicted semiclassical phase diagram discussed further below and in the main text is overlaid. The close agreement between the semiclassical phase diagram and the regions of state preservation provide additional confirmation of the link between the semiclassical phase diagram and the DTC phase.

Semiclassical phase diagram calculation

In ref. ¹⁰, Choi et al., explain the DTC-like behavior of their system with a semiclassical model. Inspired by their work, we present a related semiclassical model for our system. Let us consider an initial state of ST qubit 1: $$ ∣{\psi }_0⟩=\cos \left({\theta }_0/2\right)∣g⟩+e^{i{\varphi }_0}\sin \left({\theta }_0/2\right)∣e⟩$$. Now, in the ideal case, we can imagine that after two Floquet steps, this initial state evolves to $$ ∣{\psi }_f⟩=e^{-i{\varphi }_1{\sigma }^z/2}{e}^{-i\theta {\sigma }^x/2}{e}^{-i{\varphi }_2{\sigma }^z/2}{e}^{-i\theta {\sigma }^x/2}∣{\psi }_0⟩$$. Here θ ≈ π indicates a nominal π pulse, $$ {\varphi }_1=2\pi \left(J_2/2+{\mathrm{\Delta }}_{12}+\overline{B}\right)\tau $$, and $$ {\varphi }_2=2\pi \left(-J_2/2+{\mathrm{\Delta }}_{12}+\overline{B}\right)\tau $$. The key assumption in this model is that the net effect of ST qubit 2 on ST qubit 1, is to generate the πJ₂τ term in the propagator that switches sign after each Floquet step, because ST qubit 2 undergoes a nominal π pulse. The change in sign of this part between Floquet steps is entirely a result of interactions in the system. Figure 3d of ref. ¹⁰ shows a single-qubit trajectory for this type of evolution. One can immediately see the relationship between this semiclassical approach and dynamical decoupling.

In order to see a period doubling in the system, even in the presence of errors, we require that $$ ∣{\psi }_f⟩=∣{\psi }_0⟩$$. In general, we can pick a θ₀ and a ϕ₀ to ensure that this is the case for a given θ. To see a robust period doubling for θ ≠ π, we should see that approximately the same $$ ∣{\psi }_0⟩$$ is also unchanged under this evolution, even as we allow θ ≠ π.

To write down the actual evolution operator for our system, set

In these definitions, we have suppressed the tildes, although the Pauli operators refer to the ST qubits. As before, S₁ describes a nominal π pulse about the x-axis, and $$ S_{\mathrm{int}}^{-}$$ and $$ S_{\mathrm{int}}^{+}$$ describe the effect of interactions, depending on the state of ST qubit 2. The total Floquet operator over two steps is $$ U=S_{\mathrm{int}}^{+}{S}_1{S}_{\mathrm{int}}^{-}{S}_1$$. To see a robust period doubling, we require that $$ ∣{\psi }_0⟩=U∣{\psi }_0⟩$$ for initial states with θ₀ ≈ 0.

We numerically calculate the eigenvectors of U for the different interaction strengths and pulse errors ϵ we discuss in the manuscript. We will say that when the Floquet eigenstate $$ ∣{\psi }_0⟩$$ has a ground-state probability $$ P_g=\mid ⟨g\mid {\psi }_0⟩{\mid }^2={\cos }^2\left({\theta }_0/2\right)>0.9$$, the system can enter the DTC-like phase. For each pulse error and J₂ configuration, we compute the eigenvectors for 256 different hyperfine and charge realizations. For each realization, we compute the value of $$ {\cos }^2\left({\theta }_0/2\right)$$, and then we average the values of $$ {\cos }^2\left({\theta }_0/2\right)$$ for all noise realizations for the same values of J₂ and pulse error. The phase diagrams obtained in this way are shown in Fig. 2 in the main text. To relate the phase diagram to our data in Fig. 2 in the main text, we rescale the values of the interqubit coupling J₂ we used in the simulation by 0.54/0.5, as discussed in the main text. The need for this correction occurs because of the error in our calibration of the interqubit coupling.

This construction clearly illustrates that without interactions or global π-pulses, the robust period doubling will not be observed, as discussed in ref. ¹⁰. In this case, the eigenstates of U are the same as the eigenstates of a single Floquet step, and there is no symmetry breaking. Without a global π-pulse, initial states with θ₀ ≈ 0 can only be approximately preserved after two Floquet steps (they are not exactly preserved), unlike the case with interactions, where these states are exactly preserved.

This semiclassical calculation is also valid for an end-spin of a longer spin chain, because we are considering a nearest-neighbor Ising chain. The argument we have provided applies to the first spin in the chain, and the interaction part depends on the state of the second spin. In our model, spin 2 is assumed to undergo perfect π pulses. In a long spin chain, this assumption becomes more accurate, because one can view the effect of the third and first spins in the chain as stabilizing the π-rotations on spin 2, and so on.