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Floquet-Enhanced Swaps

Haifeng Qiao,1 Yadav P. Kandel,1 John S. Van Dyke,2 Saeed Fallahi,3, 4 Geoffrey C. Gardner,4, 5 Michael J. Manfra,3, 4, 5, 6 Edwin Barnes,2 and John M. Nichol1, 7 1Department of Physics and Astronomy, University of Rochester, Rochester, NY, 14627 USA 2Department of Physics, Virginia Tech, Blacksburg, Virginia, 24061, USA 3Department of Physics and Astronomy, Purdue University, West Lafayette, IN, 47907 USA 4Birck Nanotechnology Center, Purdue University, West Lafayette, IN, 47907 USA 5School of Materials Engineering, Purdue University, West Lafayette, IN, 47907 USA 6School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN, 47907 USA 7Corresponding author: [email protected] Abstract. The transfer of information between quantum systems is essential for quantum com- munication and computation. In quantum computers, high connectivity between can improve the efficiency of algorithms, assist in error correction, and enable high-fidelity readout. However, as with all quantum gates, operations to transfer information between qubits can suffer from errors associated with spurious interactions and disorder between qubits, among other things. Here, we harness interactions and disorder between qubits to improve a swap operation for spin eigenstates in gate-defined quantum-dot spins. We use a system of four electron spins, which we configure as two exchange-coupled singlet-triplet qubits. Our approach, which relies on the physics underlying discrete , enhances the quality factor of spin-eigenstate swaps by up to an order of magnitude. Our results show how interactions and disorder in multi- systems can stabilize non-trivial quantum operations and suggest potential uses for non-equilibrium quan- tum phenomena, like time crystals, in processing applications. Our results also confirm the long-predicted emergence of effective Ising interactions between exchange-coupled singlet-triplet qubits.

INTRODUCTION system predicted to exhibit DTC behavior [4]. Experi- mental signatures of DTC behavior have been observed in Over the decades, quantum information proces- many systems [9–12], but nearest-neighbor Ising-coupled sors have undergone remarkable progress, culminating in spin chains have yet to be experimentally investigated in recent demonstrations of their astonishing power [1]. As this regard. quantum information processors continue to scale up in Our system of two singlet-triplet qubits is clearly not size and complexity, new challenges come to light. In par- a DTC in the strict sense, because it is not a many-body ticular, maintaining the performance of individual qubits system [13]. However, this system does exhibit some of and high connectivity are both essential for continued im- the key characteristics of DTC behavior, including ro- provement in large systems [2]. bustness against interactions, noise, and pulse imperfec- At the same time, developments in non-equilibrium tions [13, 14]. We also find that the required experimental many-body physics have yielded insights into many-qubit conditions for observing the quality-factor enhancement phenomena, which feature, in some sense, improved per- are identical to some of the theoretical conditions for the formance of many-body quantum systems when disorder DTC in infinite spin chains. In total, these ob- and interactions are included. Chief among these phe- servations suggest the Floquet-enhanced spin-eigenstate nomena are many-body localization [3] and time crys- swaps in our device are closely related to discrete time- tals [4–8]. Although these phenomena are interesting in translation . their own right, applications of these concepts are only Our results also illustrate how non-equilibrium many- beginning to emerge. body phenomena could potentially be used for quan- In this work, we exploit discrete-time- (DTC) tum information processing. On the one hand, we ob- arXiv:2006.10913v2 [cond-mat.mes-hall] 5 Mar 2021 physics to demonstrate Floquet-enhanced spin-eigenstate serve Floquet-enhanced π rotations in two singlet-triplet swaps in a system of four quantum-dot electron spins. qubits. But on the other hand, these singlet-triplet π When we harness interactions and disorder in our sys- rotations correspond to spin-eigenstate swaps, when we tem, the quality factor of spin-eigenstate swaps improves view the system as four single spins. The enhanced spin- by nearly an order of magnitude. As we discuss in detail eigenstate swaps are not coherent SWAP gates, but in- further below, this system of four exchange-coupled sin- stead are “projection-SWAP” gates [15]. Because of the gle spins undergoing repeated SWAP pulses maps onto critical importance of such operations for reading out lin- a system of two Ising-coupled singlet-triplet qubits un- ear qubit arrays, these results may point the way toward dergoing repeated π pulses. Periodically-driven Ising- the use of non-equilibrium quantum phenomena in quan- coupled spin chains are the prototypical example of a tum information processing applications, especially for 2

a gether with a shelving mechanism [31] for high readout fidelity. Further details about the device can be found in Methods. The four-spin array is governed by the following Hamil- tonian:

3 4 h h z z H = Ji(σi σi+1) + B σ , (1) 200 nm 4 · 2 i i i=1 i=1 X X b where Ji is the tunable exchange coupling strength (with x y z units of frequency), σi = [σi , σi , σi ] is the Pauli vec- J2 J1 J3 tor describing the components of spin i, h is Planck’s z constant, and Bi is the z component of the magnetic field (also with units of frequency) experienced by spin ST Qubit 1 ST Qubit 2 z i. Bi includes both a large 0.5-T external magnetic field c and the smaller hyperfine field. The exchange couplings 1 Floquet step J1, J2, and J3 are controlled by pulsing virtual barrier gate voltages [32]. We model the dependence of the ex- change couplings on the virtual barrier gate voltages in S1 S2 S1 S2 the Heitler-London framework [32, 33]. The model allows

Initilization S S int int Measurement us to predict the required barrier gate voltages for a set of desired exchange couplings. In our device, we estimate Time the residual exchange coupling at the idling tuning of the device to be a few MHz. FIG. 1. Experimental setup. (a) Scanning electron micro- graph of the quadruple quantum dot device. The locations Heisenberg exchange coupling does not naturally en- of the electron spins are overlaid. (b) Schematic showing the able the creation of a DTC phase [8]. Additional con- two-qubit Ising system in a four-spin Heisenberg chain. (c) trol pulses can convert the Heisenberg interaction into The pulse sequence used in the experiments. an Ising interaction [8], which permits the emergence of a DTC phase. A DTC can also be created using a suffi- ciently strong magnetic field gradient instead of applying initialization, readout, and information transfer. More- extra pulses [34]. Here, we introduce a new method for over, recent theoretical work shows how entangled states generating DTC behavior that does not require compli- can be preserved, and robust single-, and two-qubit gates cated pulse sequences or large field gradients, but instead can be implemented, within this framework [16]. Our re- relies only on periodic exchange pulses. sults are also significant because they provide experimen- To see how we can still obtain an effective Ising inter- tal evidence of the predicted Ising coupling that emerges action in this case, it helps to view each pair of spins as between exchange-coupled singlet-triplet qubits [17]. an individual singlet-triplet (ST) qubit [Fig. 1(b)] [27]. Specifically, consider the scenario where the joint spin- state of each pair is confined to the subspace spanned by RESULTS S = 1 ( ) and T = 1 ( + ). Ac- | i √2 |↑↓i − |↓↑i | 0i √2 |↑↓i |↓↑i cording to Ref. [17], when J1 = J3 = 0 and J2 > 0, the Device and Hamiltonian effective Hamiltonian of the system is

We fabricate a quadruple quantum dot array in a h h H = ∆ + B¯ σ˜z + ∆ + B¯ σ˜z GaAs/AlGaAs heterostructure with overlapping gates eff 2 12 1 2 34 2 [Fig. 1(a)] [18–20]. The confinement potentials of the  h J σ˜zσ˜z . (2) dots are controlled through “virtual gates” [21–24]. Two − 4 2 1 2 extra quantum dots are placed nearby and serve as fast z charge sensors [25, 26]. We configure the four-spin array Here,σ ˜k is the Pauli z-operator for ST qubit k, ∆ij = into two pairs (“left” and “right”) for initialization and Bz Bz is the intraqubit gradient between spins i and i − j readout. Each pair of spins can be prepared in a product j, and B¯ is the effective global magnetic field gradi- state ( or ) via adiabatic separation of a singlet ent, which depends on ∆ij and Ji [17]. In this sys- in the|↑↓i hyperfine|↓↑i gradient [27–29]. We can also initialize tem, all magnetic gradients result from the hyperfine either pair as T+ = by exchanging electrons with interaction between the electron and nuclear spins [35]. the reservoirs [28,| i 30].|↑↑i Both pairs are measured through The gradients are quasistatic on typical qubit manipu- spin-to-charge conversion via Pauli spin blockade [27], to- lation timescales [36]. The basis for the singlet-triplet 3

P1 P2 P1 P2 a g b g e g f g 0.9 0.9 0.9 0.9 250 250 400 400 0.8 200 0.8 0.8 200 0.8 300 300 150 0.7 150 0.7 (MHz) 0.7 (MHz) (MHz) 0.7 (MHz) 1 200 3 1 200 3 J J 100 0.6 J J 100 0.6 0.6 0.6 100 100 50 0.5 50 0.5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 J (MHz) J (MHz) J (MHz) J (MHz) 2 2 2 2

P1 P2 P1 P2 c g d g g g h g 0.9 0.9 0.9 0.9 0.5 0.5 0.5 0.5 0.8 0.8 0.8 0.8

0.7 0.7 0.7 0.7

0 0 0 0 0.6 0.6 0.6 0.6

0.5 0.5 0.5 0.5 -0.5 -0.5 -0.5 -0.5 0.4 0.4 0.4 0.4 0 2 4 0 2 4 0 2 4 0 2 4 J (MHz) J (MHz) J (MHz) J (MHz) 2 2 2 2

FIG. 2. Floquet-enhanced π rotations. (a-b) Measured ground state return probabilities of (a) ST qubit 1 and (b) ST qubit π π 2, after four Floquet steps, with interaction time τ = 1.4 µs. The ranges of J1 and J3 center around J1 and J3 . 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. k The experimental data in (a,b,e,f) are averaged over 8192 realizations. In all figures, Pg indicates the ground state return probability for ST qubit k.

z eff i h z x qubit operatorσ ˜ is , provided that J2 and S = exp t2 (∆34σ˜ + J3σ˜ ) . In writing 2 ~ 2 2 2 B B [17, 36, 37].{|↑↓i In our|↓↑i} experiments, the typical eff eff − | 2 − 3| S1 and S2 , we have ignored overall energy shifts J1/4 value of J2 is a few MHz, while the typical value of the and J3/4 of the single-qubit Hamiltonians, because the magnetic field gradient in the device is tens of MHz. Now system dynamics do not depend on these shifts. Assum- let us define ing J ∆ and J ∆ , when t J = t J = 0.5, 1  12 3  34 1 1 2 3 4 these two operators implement SWAP gates between i h h S = exp τ J (σ σ ) + Bzσz , (3) spins 1-2 and 3-4. Equivalently, they induce nominal int − 4 2 2 · 3 2 i i " ~ i=1 !# π pulses about the x axis of each singlet-triplet qubit. X The presence of the intraqubit gradients ∆12 and ∆34 where τ is an interaction time. Within the S , T0 subspace of each pair, this operator is equivalent{| i | i} to slightly tilts the rotation axis towards the z axis for each eff i singlet-triplet qubit, introducing uncontrolled errors to S = exp τHeff , and it describes the of int ~ the two Ising-coupled− qubits [17]. Systems of exchange- the π pulses. We can also manually introduce additional   pulse errors by changing J1 and J3 while fixing t1 and coupled singlet-triplet qubits have been the focus of sig- π t2. We represent the error as , with J1 = J1 (1 + ) nificant theoretical research [17, 38–40]. Until now, such π π π a system has evaded implementation. and J3 = J3 (1 + ). Here J1 and J3 are the interaction strengths that yield π pulses. In the case when J2 = 0, but when J1,J3 > 0, the over- all Hamiltonian describes two uncoupled singlet-triplet qubits. Thus, let us define Floquet-Enhanced Spin Swaps 4 i h h z z S1 = exp t1 J1(σ1 σ2) + Bi σi , (4) − 4 · 2 We define a Floquet operator U = Sint S2 S1 " ~ i=1 !# · · X [Fig. 1(c)], and we repeatedly apply this operator to our 4 i h h system of four spins. As discussed above, U implements S = exp t J (σ σ ) + Bzσz . (5) 2 − 2 4 3 3 · 4 2 i i spin SWAP gates between spins 1-2 and 3-4 followed by " ~ i=1 !# X a period of exchange interaction between spins 2 and 3. In the S , T0 subspace of each pair, these operators Equivalently, U implements π pulses on both ST qubits {| i | i} eff i h z x are equivalent to S = exp t1 (∆12σ˜ + J1σ˜ ) and then a period of Ising coupling between them. One 1 − ~ 2 1 1   4 might naively imagine that the highest fidelity SWAP due to the imperfect calibration of the exchange coupling operations between spins should occur when J2 = 0 and J2 [32]. In particular, the presence of the hyperfine field τ = 0, given the presence of intraqubit hyperfine gra- gradient makes it difficult to measure and control the ex- dients. In this case, as we have discussed in Ref. [29], change couplings with sub-MHz resolution. If our mod- repeated SWAP operations are especially susceptible to eling of the exchange coupling were more precise, then errors from the hyperfine gradients ∆ij. we would expect the periodicity of the patterns However, by allowing J2 > 0 and τ > 0, we find specific to be closer to 1 and the quality-factor enhancement to conditions in which we observe a significant enhancement occur closer to J2τ = 0.5 in the experimental data. of the spin-eigenstate-swap quality factor [Fig. 2]. To ex- We can interpret our data using a semiclassical model plore this phenomenon, we prepare each ST qubit in inspired by Choi et al. in Ref. [10] to explain DTC be- |↑↓i or . (The specific state is governed by the sign of ∆12 havior (see Methods). In brief, an initial state of ST |↓↑i iφ0 and ∆34, which are random quasistatic gradients result- qubit 1, ψ0 = cos(θ0/2) g + e sin(θ0/2) e evolves z x z x | i iφ1σ /2 iθσ |/2i iφ2σ /2 iθσ /2| i ing from the nuclear hyperfine interaction.) We apply to ψf = e− e− e− e− ψ0 after multiple instances of the Floquet operator U to the sys- two| Floqueti steps. Here θ π indicates a| nominali ≈ tem and measure the ground state return probabilities π pulse, and φ1 = 2π(J2/2 + ∆12 + B¯)τ, and φ2 = for both ST qubits. 2π( J /2 + ∆ + B¯)τ. In this semiclassical model, the − 2 12 First we set the interaction time τ = 1.4 µs and SWAP effect of ST qubit 2 on ST qubit 1 is to generate the pulse t1 = t2 = 5 ns, and apply four Floquet steps. πJ2τ term in the operator that switches sign after each We sweep J2 linearly from 0.05 MHz to 5 MHz [Fig. 2(a- Floquet step, because ST qubit 2 undergoes a nominal b)]. (Setting J2 < 0.05 MHz would require large nega- π pulse. As emphasized in Ref. [10], the change in sign tive voltage pulses applied to the barrier gate due to the of this part between Floquet steps is entirely a result of residual exchange, which could disrupt the tuning of the interactions in the system. The resulting single-qubit ro- device.) We also sweep J1 from 80 MHz to 460 MHz, tations in this semiclassical model are reminiscent of dy- and J3 from 50 MHz to 260 MHz. The ranges of J1 and namical decoupling [10]. We have numerically simulated π π J3 roughly center around J1 and J3 , respectively. Away the semiclassical single-qubit evolution over two Floquet from the center, J1 and J3 induce pulse errors. The ex- steps for our system (see Methods). The black lines in π π perimental values of t1J1 and t2J3 are much larger than Figs. 2(a) and (b) indicate the regions where approximate 0.5, because the voltage pulses experienced by the qubits ST-qubit eigenstates are also exactly eigenstates of the have rise times of about 1 ns (see Methods and Supple- evolution operator over two steps, i.e., they are exactly mentary Fig. 1). To compensate for the pulse rise times preseved by the 2 Floquet steps [10]. The size of these π (which are slightly different for each qubit), t1J1 and regions confirms that interactions are essential for the ef- π t2J3 must be larger than 0.5 in order to properly induce fects we observe. Exactly the same enhancement regions π pulses. are expected for end spins in longer chains, because our Clear, bright diamond patterns are visible in the data system is a nearest-neighbor Ising spin chain. Simula- [Fig. 2(a-b)]. These bright regions correspond to im- tions for an eight-site Ising spin chain at late times show proved spin-eigenstate-swap quality factors. Note that DTC behavior in exactly these regions (see Methods and the brightest regions correspond to configurations when Supplementary Figure 2). J2 > 0. Note also that the are approximately Next, we also sweep J3 from 220 MHz to 430 MHz. In π periodic in J2τ, as expected for a Floquet operator. We this case, the range of J3 roughly centers around 2J3 . repeat the same experiments with τ = 1 µs, and we ob- The interaction time is τ = 1.4 µs and the ranges of J1 serve similar diamond patterns, although they have an and J2 remain the same. Again we apply four Floquet increased period in J2 [Fig. 2(e-f)]. The diamond pat- steps and measure the ground state return probabilities. terns of ST qubit 2 appear narrower due to the large This time the data do not show diamond patterns [Fig. 3], hyperfine gradient ∆34, which causes larger pulse errors and the return probability of ST qubit 1 is lower than the and reduces the size of the qualit-factor-enhancement re- Floquet-enhanced return probability shown in Fig. 2(a). gion. These data from an effective two-qubit system re- This indicates that the Floquet enhancement is no longer semble predicted DTC phase diagrams of a true nearest- . In fact, if either of the Floquet operators S1 or neighbor many-body system (see Methods and Supple- S2 fails to induce approximately a π rotation, then the mentary Figs. 2 and 3) [7, 8]. Floquet enhancement does not appear. Our simulations agree well with the data [Fig. 2(c-d), On the one hand, this effect is striking, when one con- (e-h)] (see Methods). In the simulations, the diamond siders the individual spins themselves. Recall that the pattern is periodic in J2τ with the periodicity of exactly ST qubit splittings ∆12 and ∆34 are generated by the hy- 1, and the strongest quality-factor enhancement occurs perfine interaction between the Ga and As nuclei in the at J2τ = 0.5. In the experimental data, however, the semiconductor heterostructure and the electron spins in periodicity is slightly larger than 1, and the strongest the quantum dots. Although ∆12 and ∆34 are quasistatic quality-factor enhancement occurs at J2τ > 0.5. This is on millisecond timescales, they each independently fluc- 5

a P1 b P2 g g and β are fit parameters. We also investigate the qual- 0.9 0.9 400 400 ity factor of the qubits under non-enhanced regular π 0.8 0.8 pulses. Here we use the same interaction time τ = 1.4 300 350 0.7 µs, but we turn off the interaction strength J2 by setting (MHz) 0.7 (MHz)

1 3 300 200 J J 0.6 the barrier-gate pulse to zero. To further eliminate any 0.6 250 100 effects associated with Floquet enhancement, we only ap- 0.5 1 2 3 4 5 1 2 3 4 5 ply π pulses to one qubit while the other qubit remains J (MHz) J (MHz) 2 2 idle after initialization. Again, we apply 50 π pulses and measure the ground state return probability, and we fit FIG. 3. Absence of Floquet enhancement due to the omission the data with the same decaying sinusoidal function. By of a π pulse. (a-b) Measured ground state return probabilities comparing the fit parameter Q, we can obtain the ratio of (a) ST qubit 1 and (b) ST qubit 2, after four Floquet steps, between the quality factors of the qubits under Floquet- with interaction time τ = 1.4 µs. The ranges of J1 and J3 π π enhanced and non-enhanced π rotations. center around J1 and J3 , respectively. The values of J1 and J3 are swept simultaneously. The data are averaged over 8192 We find a 3-fold quality-factor improvement on qubit realizations. 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 tuate randomly, and can change sign, over the duration ∆34 in qubit 2, which causes an exceptionally low quality of a typical data-taking run, which is about one hour. factor for non-enhanced π rotations. The quality-factor Each of the 8192 different realizations for each pixel in enhancement is striking in this case. To extract an es- the data of Fig. 2 likely contain instances where both ST timated uncertainty, we repeat the same experiment 30 qubits have the same or different ground-state spin ori- times and calculate the mean and the variance of the entations. (The ground state of each ST qubit is either quality factor ratio, as shown in the first row of Table I. or , depending on the sign of the instantaneous |↑↓i |↓↑i So far, we have initialized both ST qubits in their hyperfine gradient.) ground states. We can also initialize either ST qubit Thus, the data of Fig. 2 likely include realizations with in its excited state by applying an extra π pulse to the all possible combinations of the orientations of spins 2 qubit immediately before the first Floquet step. We run and 3 before the interaction period. Despite the random the same experiment with different initial states and ex- orientations of spins 2 and 3, the Floquet enhancement tract the quality factors by fitting the data [Fig. 4(b-c)]. still appears. It might therefore seem that whether or Again, for each initial state, we repeat the experiment 30 not spins 1-2 or 3-4 undergo a SWAP before the interac- times and calculate the mean and the variance of the tion period should not affect the behavior of the system. quality-factor ratio, which are listed in Table I. The However, as shown in Fig. 3, implementing a 2π rota- quality-factor improvements of both qubits are consis- tion, as opposed to a π rotation, on one of the ST qubits tent across different initial states. eliminates the Floquet enhancement. We also initialize the right pair as T+ = On the other hand, when one considers the semiclassi- and measure the quality-factor improvement| oni qubit|↑↑i 1 cal picture described above, the absence of a π pulse on [Fig. 4(d)]. We notice that the quality-factor ratio is one of the ST qubits spoils the semiclassical decoupling much lower when the right pair is initialized in T+ . This evolution discussed above and in Ref. [10]. In this case, is not surprising since the effective Ising interaction| i be- ST-qubit eigenstates are no longer eigenstates of two in- tween qubit 1 and qubit 2 (Eq. 2) is only valid when stances of the Floquet operator, and the enhancement no both qubits are restricted to the Sz = 0 subspace. The longer occurs. We have now determined the optimal conditions for the Floquet enhancement. For the remainder of the pa- Quality-factor enhancement Initialization per, we set J1 = 270 MHz and J3 = 150 MHz with Qubit 1 Qubit 2 t = t = 5 ns for the SWAP operators S and S , respec- 1 2 1 2 g g 3.60 0.89 8.47 3.29 tively, and we set τ = 1.4 µs and J2 = 0.41 MHz for the | i ⊗ | i ± ± e g 3.24 0.94 9.33 2.96 Ising interaction. To quantify the Floquet enhancement, | i ⊗ | i ± ± we evolve the system for 50 Floquet steps and measure g e 3.15 0.79 9.10 2.87 the ground state return probabilities for both qubits af- | i ⊗ | i ± ± g T+ 1.92 0.27 N/A ter each step. The results are shown in Fig. 4(a). Note | i ⊗ | i ± that the system exhibits a clear subharmonic response TABLE I. Quality-factor enhancements of both qubits for dif- to the Floquet operator. We extract a swap quality Q ferent initial states. Here g and e represent the ground by fitting the data with a decaying sinusoidal function state and the excited state| i of the| STi qubit, respectively. Pg(n) = α exp( n/Q) cos(nπ) + β, where Pg(n) denotes Thirty sets of data are taken for each initialization, from the return probability− at the nth Floquet step, and Q, α, which the means and the standard deviations are calculated. 6

a b c d

1 1 1 1

0.8 0.8 0.8 0.8

0.6 0.6 0.6 0.6 1 1 1 1 g g g g P P 0.4 0.4 P 0.4 P 0.4

0.2 0.2 0.2 0.2

0 0 0 0 1 1 1 1

0.8 0.8 0.8 0.8

0.6 0.6 0.6 0.6 2 2 2 2 g g g g P P 0.4 0.4 P 0.4 P 0.4

0.2 0.2 0.2 0.2

0 0 0 0 0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50 Floquet Steps Floquet Steps Floquet Steps Floquet Steps

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| arei shown| i 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 reason why we still see a 2-fold quality-factor improve- our device demonstrates a clear subharmonic response as ment instead of no improvement∼ at all is likely because well as a robustness against pulse errors, both expected of the imperfect T+ preparation due to thermal popula- as defining signatures of the DTC. Our experiments also tion of excited states| i [30]. Load errors will cause the right indicate the necessity of two essential ingredients for re- pair to occupy the ST-qubit ground or excited states a alizing the Floquet-enhanced π pulses: 1) an effective small fraction of the time. In these cases, the Floquet Ising interaction, and 2) global π pulses. If either of enhancement of the left-pair ST qubit is expected to oc- the components is missing, we no longer observe the sig- cur. Thus, the overall quality factor should appear to nificant quality-factor enhancement [Figs. 3 and 4(d)]. improve slightly, because of the imperfect initialization. These two components both ensure that the semiclassical Correspondingly, it is likely that imperfect initialization dynamical decoupling can occur. In the thermodynamic limits the quality factor enhancement when both qubits limit, these components would ensure that eigenstates of are initialized in ST-qubit eigenstates. the Floquet operator are long-range correlated, which is Finally, we emphasize that a Floquet drive, i.e., re- required for discrete time-translation symmetry break- peated SWAP gates, is required to realize the enhance- ing [5]. We have also shown that the quality-factor en- ment shown in Fig. 4. Based on the data of Fig. 4, the hancement does not depend on the eigenstate into which first SWAP gate is not substantially enhanced by the either ST qubit is initialized (provided that the effective protocol. It is only subsequent SWAP gates that are en- Ising coupling is maintained), which is another key fea- hanced. This is consistent with the requirement for a ture of the DTC [13]. In the , implementing these periodic drive in a DTC. As we discuss below, this peri- experiments in larger spin chains could lead to a verifi- odic drive is also useful for constructing quantum gates. cation that these effects in fact originate from the DTC phase. We emphasize that we have observed Floquet enhance- DISCUSSION ment associated with ST-qubit eigenstates undergoing π pulses. In the language of single spins, we observed Flo- Strictly speaking, a DTC only occurs in the thermody- quet enhancement associated with swaps between spin namic limit [13]. Nonetheless, we argue the quality-factor eigenstates, when the total z component of angular mo- enhancement we observe relies on the essential elements mentum for both spins vanishes. This observation is of DTC physics. The disordered Ising-coupled system in qualitatively consistent with expectations for qubits in 7

a a true many-body DTC, where the components of the 1 qubits oriented along the direction defined by the Ising coupling are preserved [8]. While not a coherent SWAP gate, a spin-eigenstate swap (projection-SWAP), has sig- 0.5 J=0 nificant potential to aid in readout for large qubit ar- J=0.25/ rays [41]. Return Probability 0 The Floquet enhancement we observe can immediately 0 5 10 15 20 Period be leveraged to perform additional quantum-information b processing tasks of significant importance. For exam- 1 ple, recent theoretical work shows that entangled states CZ gate 0.75 of single spins (or superposition states of ST qubits) Initial State can be preserved using Floquet operators identical to 0.5 Expected Post-CZ State

what we have demonstrated [16]. The same work also Joint Probability Joint shows that single-qubit gates can be incorporated into 0.25 this framework, and even two-qubit CZ gates can be im- 0 5 10 15 20 25 30

Time (¡ s) plemented [16]. A significant potential advantage of these operations, compared with conventional single- and two- FIG. 5. Preserving and generating entangled states. (a) qubit gates, is that dynamical decoupling is an essential Return probability for the singlet state of an ST qubit defined component of these operations, as discussed above. on sites 3 and 4 of an L = 6 spin chain. The two remaining ST qubits are initialized in the state . (b) Two-qubit As an illustration of the above capabilities, we perform | ↑↓i simulations that show the preservation of the singlet state probabilities before and after the execution of a two-qubit CZ of an ST qubit by periodic driving under the evolution gate using the modified DTC protocol (for a chain of length L = 4). The initial state of ST qubit 1 is the triplet T0 and U = S S , where S represents the execution of S | i int · 12 12 1 of qubit 2 is the singlet. The x coordinate of each point is and S2 in parallel. Fig. 5(a) shows the return probability the total time of all the pulse sequences. The y coordinate of for the singlet state of an ST qubit defined on sites 3 and 4 each point is the joint two-qubit probability. The “Expected of an L = 6 site spin chain. The ST qubits defined on the Post-CZ State” is an entangled state of the two ST qubits. pairs of sites (1,2) and (5,6) are initialized to the product The results in both panels are averaged over 4096 realizations. state . When the interaction J between neighboring |↑↓i ST qubits (the generalization of J2 in Eq. 3) is turned off, iπ/4 i(π/4)˜σz i(π/4)˜σz i(π/4)˜σz σ˜z the maximum return probability decreases to 0.75 at rotations: CZ = e e− 1 e− 2 e 1 2 [42]. long times. In contrast, when τJ = 0.25, with τ∼= 1.4 µs Applying the necessary rotations by appropriately timed as before, the maxima of the return probability remain SWAP pulses during an additional evolution step with higher than the non-interacting case out to 20 periods of trot = 4 µs executes a full CZ gate [16], which we then evolution. In this case, the return probability shows a 4T preserve for another 8 periods using the DTC protocol. periodicity, as the interactions produce a relative phase Unlike the case of the Floquet-enhanced SWAP, here the between the basis states and such that the gate itself is necessarily produced by the inter-ST qubit original state is recovered| only ↑↓i after| four ↓↑i periods (after coupling, and so it is not enhanced but rather enabled two periods this phase yields the T0 state, and the S by them. Since the CZ gate and arbitrary single-qubit return probability vanishes) [16].| Wei note that the cal-| i unitaries form a universal gate set, this suggests that the culated return probability p = Tr[ρ(3,4) S S ] accounts DTC-inspired methods presented here yield a promising for the fact that interactions will entangle| ih the| (3,4) pair direction for spin-based . with its neighbors through the use of the reduced density The experimental investigation of all of these ideas re- matrix ρ(3,4). The optimal value of J is reduced by half mains an important subject of future work. We expect compared to the previous simulations and experiments, that these phenomena can readily be explored in Si spin due to the presence of two neighbors in the interior of the qubits. Barrier-controlled exchange coupling between Si spin chain [34]. The possibility of stabilizing superposi- spin qubits is now routine [43]. The operation of Si ST tion states of ST qubits, such as singlets, also highlights qubits in the regime where magnetic gradients exceed ex- the possibility of interleaving arbitrary single-qubit op- change couplings has also been demonstrated [15, 44]. erations within this Floquet framework. Note that to observe the Floquet enhancement, or to The condition τJ = 0.25 also serves to implement perform any of the protocols described in Ref. [16], mul- a two-qubit CZ gate, for which simulations are shown tiple Sz = 0 electron pairs undergoing the same Flo- in Fig. 5(b) for the L = 4 chain. Here, the ordinary quet operators are typically required. As we have shown DTC protocol with τJ = 0.5 is applied for 8 periods, fol- above, one ST qubit alone cannot experience the Floquet lowed by two periods with the reduced value τJ = 0.25. enhancement without the other. This notion is consistent Since the effective interaction between ST qubits is of with expectations for many-body DTCs, which are true Ising form, this yields a CZ gate up to single-qubit z many-body phenomena. One can view the “extra” qubits 8 undergoing repeated instances of the Floquet operator as METHODS the resource required to implement an improved opera- tion on a specific qubit. It is also interesting to note that Device DTC-like behavior can emerge in systems with as few as two qubits in this nearest-neighbor-coupled system, as The quadruple quantum dot device is fabricated on a we have shown. Thus, only a relatively small number of GaAs/AlGaAs heterostructure substrate with three lay- qubits is required in order to realize the benefits of Flo- ers of overlapping Al confinement gates and a final Al quet enhancement, highlighting its potential for further top gate. The Al gates are patterned and deposited using use in quantum information processing applications. E-beam lithography and thermal evaporation, and each In summary, we have demonstrated Floquet-enhanced layer is isolated from the other layers by a few nanome- spin-eigenstate swaps in a four-spin two-qubit Ising chain ters of native oxide. The top gate covers the main device in a quadruple quantum-dot array. The system shows area and is grounded during the experiments. It likely a subharmonic response to the driving frequency, and it smooths anomalies in the quantum dot potentials. The also shows an improvement in swap quality factor even in two-dimensional electron resides at the GaAs and Al- the presence of pulse imperfections. We have also shown GaAs interface, 91 nm below the semiconductor surface. that the necessary conditions for this quality-factor en- The device is cooled in a dilution refrigerator with base hancement are identical to some key components for real- temperature of approximately 10 mK. A 0.5-T external izing discrete time crystals. Our results also confirm the magnetic field is applied parallel to the device surface prediction of an effective Ising coupling that emerges be- and perpendicular to the axis connecting the quantum tween two exchange-coupled single-triplet qubits. This dots. work indicates the possibility of realizing discrete time crystals using extended Heisenberg spin chains in semi- conductor quantum dots, and suggests potential uses for Pulse rise times quantum information processing applications. π π The experimental values of t1J1 and t2J3 are much larger than 0.5, because the voltage pulses experienced by the qubits have rise times of about 1 ns. Supplemen- tary Fig. 1 shows measured exchange oscillations for both ACKNOWLEDGMENTS ST qubits vs. evolution time. The observable frequency chirp at early evolution times demonstrates the effects of This research was sponsored by the Defense Ad- rise times in our system and shows that the π-pulse times vanced Research Projects Agency under Grant No. yield tJ > 0.5. D18AC00025, the Army Research Office under Grant Nos. W911NF16-1-0260 and W911NF-19-1-0167, and the National Science Foundation under Grant Nos. Simulation DMR-1941673 and DMR-2003287. The views and con- clusions contained in this document are those of the We simulate the Floquet-enhanced phenomena by authors and should not be interpreted as representing evolving a four-spin array according to the Floquet oper- the official policies, either expressed or implied, of the ator U = Sint S2 S1, as defined in the main text. We set Army Research Office or the U.S. Government. The U.S. · · π π t1 = t2 = 2 ns, and J = J = 250 MHz for the SWAP Government is authorized to reproduce and distribute 1 3 operators S1 and S2 to give π pulses. While t1 and t2 reprints for Government purposes notwithstanding any are chosen to be 5 ns in the experiments, we expect the copyright notation herein. realistic SWAP times to be approximately 2 3 ns due to the pulse rise and fall times of about 1 ns.∼ We include the π-pulse errors by adjusting the exchange couplings as π π J1 = J1 (1 + ) and J3 = J3 (1 + ), where  represents the fractional error in the rotation angle of the π pulse AUTHOR CONTRIBUTIONS applied to ST qubits 1 and 2. For better comparison with the experimental data, the J.VD., E.B., and J.M.N conceptualized the experi- simulations take into account all known error sources, ment. H.Q, Y.P.K., and J.VD. conducted the investi- including state preparation, readout, charge noise, and gation. H.Q, Y.P.K., J.VD., E.B., and J.M.N. analyzed hyperfine field noise. The initial state of each ST qubit the data. S.F., G.C.G., and M.J.M. provided resources. is prepared as All authors participated in writing. J.M.N. supervised the effort. ψi = s1 g + s2 e + s3 T+ + s4 T , (6) | i | i | i | i | −i 9 where g and e are the ground state and the excited noise and hyperfine field fluctuations. The values of the | i | i z state in the , basis. The exact spin orien- exchange couplings Ji and the local hyperfine fields Bi tation for the{|↑↓i ground|↓↑i} state is determined by the hy- are randomly sampled from a normal distribution for perfine gradient. The coefficient s 2 = f represents each simulation run. We set the standard deviation for | 1| g the ground state preparation fidelity, and we assume Ji to be Ji/(√2πQ), where Q = 21 is the exchange oscil- 2 2 2 1 s2 = s3 = s4 = 3 (1 fg) for simplicity. We esti- lation quality factor. We set the standard deviation for | | | | | | − z mate fg to be 0.9 for ST qubit 1 and 0.95 for ST qubit Bi to be σBz =18 MHz, and we assume the mean values 2 in our device. The preparation fidelity assumes errors to be [0, 20, 0, 50] MHz plus a uniform magnetic field of from both the singlet loading and the charge separation. 3.075 GHz (which accounts for the 0.5-T external mag- The readout errors are included by calculating the final netic field). The simulated data in Fig. 2 in the main ground state return probability as text are obtained by averaging over 128 realizations. The simulations of Fig. 5 do not include exchange cou- pling noise or state preparation and measurement errors. P˜g = (1 r 2q)Pg + r + q , (7) − − We neglected these errors to clearly illustrate the mech- anisms underlying the singlet-state preservation and CZ where P = g ψ 2 is the true ground state return g f gate. Hyperfine fluctuations with σBz =18 MHz were probability. Here| h | r =i | 1 exp( t /T ) is the probability − − m 1 included, and the magnetic field values at the locations of the excited state relaxing to the ground state during of each dot were 3.075 GHz, to account for the external measurements, with tm being the measurement time and magnetic field. To simulate the CZ gate of Fig. 5(b) in T1 being the relaxation time. Also q = 1 fm is the prob- the main text, we simulate two periods of Floquet opera- ability of misidentifying the ground state− as the excited tor U = Sint S12, where S12 represents the execution of state due to random noise. We set t = 4 µs, T = 60 µs, · m 1 S1 and S2 in parallel, with τJ = 0.25. These two periods and fm = 0.99 for ST qubit 1, and tm = 6 µs, T1 = 50 implement the CZ gate, up to single-qubit rotations. In µs, and fm = 0.95 for ST qubit 2. the simulation, we evolve the system for two additional We use a Monte-Carlo method to incorporate charge periods with the operator U = U U , where rot 1 ⊗ 2

i T 2T tr h i T 2T + tr h U = S exp − g − s − 1 ∆ σ˜z S exp − g − s 1 ∆ σ˜z 1 1 · 2 2 12 1 · 1 · 2 2 12 1  ~     ~    i T 2T tr h i T 2T + tr h U = S exp − g − s − 2 ∆ σ˜z S exp − g − s 2 ∆ σ˜z 2 2 · 2 2 34 2 · 2 · 2 2 34 2  ~     ~   

where Tg is the overall time of the operation (Urot lasts and denote a Floquet operator for a duration Tg), and Ts is duration of the SWAP gate. We define the rotation time i U(τ) = exp HI τ −~ × π/(2∆ ), if ∆ > 0   r 12 12 N t1 = (8) i h h 3π/(2 ∆12 ), if ∆12 < 0 x z z R ( exp (1 + ) Jπσi + Bi σi Ti , | | −~ 2 2 and 1     Y (11)

r π/(2∆34), if ∆34 > 0 t2 = . (9) where  is a pulse error, and T R = 1/(2 J 2 + (Bz)2), (3π/(2 ∆34 ), if ∆34 < 0 i π i | | with J = 250 MHz. Supplementary Fig. 2 shows the π p In total, Urot which implements a π/2 rotation about the results of simulations for an N = 2 Ising chain, after z z axis of both ST qubits via a spin-echo-like sequence, 4 Floquet steps, with τ = 1.4 µs, Bi = [20, 50] MHz, such that the overall operation over the four periods is and σBz = √2 18 MHz. These simulation conditions an exact CZ gate [16]. correspond to the× data of Fig. 2 in the main text, and To confirm that the behavior we report in the two ST they agree with the data of that figure. This agreement qubits corresponds to that of an effective Ising spin chain, provides additional strong evidence of the effective Ising we also simulate an Ising spin chain. Define coupling between ST qubits in our system.

N 1 N Supplementary Fig. 3(a) shows the results of an N = 8 h − z z h z z z H = Jσ σ + B σ , (10) Ising spin chain after four Floquet steps, with Bi = 20 I 4 i i+1 2 i i i=1 i=1 MHz. These results agree with the two-site data of Sup- X X 10

+ plementary Fig. 2(a) and Fig. 2 in the main text, provid- and Sint− and Sint describe the effect of interactions, de- ing evidence that the behavior we observe in a 2-qubit pending on the state of ST qubit 2. The total Floquet + system corresponds to the expected behavior for larger operator over 2 steps is U = SintS1Sint− S1. To see a ro- systems. Supplementary Fig. 3(b) shows the simulated bust period doubling, we require that ψ0 = U ψ0 for behavior after 1024 Floquet steps. The predicted semi- initial states with θ 0. | i | i 0 ≈ classical phase diagram discussed further below and in We numerically calculate the eigenvectors of U for the the main text is overlaid. The close agreement between different interaction strengths and pulse errors  we dis- the semiclassical phase diagram and the regions of state cuss in the manuscript. We will say that when the preservation provide additional confirmation of the link Floquet eigenstate ψ0 has a ground-state probability between the semiclassical phase diagram and the discrete P = g ψ 2 = cos| 2(θi /2) > 0.9, the system can enter g | h | 0i | 0 time crystal (DTC) phase. the DTC-like phase. For each pulse error and J2 con- figuration, we compute the eigenvectors for 256 different hyperfine and charge realizations. For each realization, Semiclassical Phase Diagram Calculation 2 we compute the value of cos (θ0/2), and then we aver- 2 age the values of cos (θ0/2) for all noise realizations for In Ref. [10], Choi et al., explain the DTC-like be- the same values of J2 and pulse error. The phase dia- havior of their system with a semiclassical model. In- grams obtained in this way are shown in Fig. 2 in the spired by their work, we present a related semiclassical main text. To relate the phase diagram to our data in model for our system. Let us consider an initial state Fig. 2 in the main text, we rescale the values of the inter- iφ0 of ST qubit 1: ψ0 = cos(θ0/2) g + e sin(θ0/2) e . qubit coupling J we used in the simulation by 0.54/0.5, | i | i | i 2 Now, in the ideal case, we can imagine that after as discussed in the main text. The need for this correc- two Floquet steps, this initial state evolves to ψf = tion occurs because of the error in our calibration of the z x z x iφ1σ /2 iθσ /2 iφ2σ /2 iθσ /2 | i e− e− e− e− ψ0 . Here θ π in- interqubit coupling. | i ≈ ¯ dicates a nominal π pulse, φ1 = 2π(J2/2 + ∆12 + B)τ, This construction clearly illustrates that without in- and φ = 2π( J /2 + ∆ + B¯)τ. The key assumption 2 − 2 12 teractions or global π-pulses, the robust period-doubling in this model is that the net effect of ST qubit 2 on ST will not be observed, as discussed in Ref. [10]. In this qubit 1, is to generate the πJ2τ term in the propagator case, the eigenstates of U are the same as the eigen- that switches sign after each Floquet step, because ST states of a single Floquet step, and there is no symmetry qubit 2 undergoes a nominal π pulse. The change in sign breaking. Without a global π-pulse, initial states with of this part between Floquet steps is entirely a result of θ0 0 can only be approximately preserved after two interactions in the system. Figure 3(d) of Ref. [10] shows Floquet≈ steps (they are not exactly preserved), unlike a single-qubit trajectory for this type of evolution. One the case with interactions, where these states are exactly can immediately see the relationship between this semi- preserved. classical approach and dynamical decoupling. This semiclassical calculation is also valid for an end- In order to see a period doubling in the system, even spin of a longer spin chain, because we are considering in the presence of errors, we require that ψ = ψ . In | f i | 0i a nearest-neighbor Ising chain. The argument we have general, we can pick a θ0 and a φ0 to ensure that this is provided applies to the first spin in the chain, and the the case for a given θ. To see a robust period doubling interaction part depends on the state of the second spin. for θ = π, we should see that approximately the same 6 In our model, spin 2 is assumed to undergo perfect π ψ0 is also unchanged under this evolution, even as we pulses. In a long spin chain, this assumption becomes |allowi θ = π. 6 more accurate, because one can view the effect of the To write down the actual evolution operator for our third and first spins in the chain as stabilizing the π- system, set rotations on spin 2, and so on.

i h z x S1 = exp t1 (∆12σ + J1σ ) , (12) −~ 2   DATA AVAILABILITY and let us define The datasets generated during and/or analysed during i h J2 z S− = exp τ ∆12 + B¯ σ (13) the current study are available from the corresponding int − 2 − 2  ~    author on reasonable request. i h J S+ = exp τ ∆ + B¯ + 2 σz . (14) int − 2 12 2  ~    In these definitions, we have suppressed the tildes, al- though the Pauli operators refer to the ST qubits. As [1] F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. before, S1 describes a nominal π pulse about the x axis, Bardin, R. Barends, R. Biswas, S. Boixo, F. G. S. L. 11

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Wegscheider, and L. M. K. Vandersypen, Nature Supplementary Information for Floquet-Enhanced Spin Swaps

Haifeng Qiao,1 Yadav P. Kandel,1 John S. Van Dyke,2 Saeed Fallahi,3, 4 Geoffrey

4, 5 3, 4, 5, 6 2 1, C. Gardner, Michael J. Manfra, Edwin Barnes, and John M. Nichol ∗

1Department of Physics and Astronomy, University of Rochester, Rochester, NY, 14627 USA 2Department of Physics, Virginia Tech, Blacksburg, Virginia, 24061, USA 3Department of Physics and Astronomy, Purdue University, West Lafayette, IN, 47907 USA 4Birck Nanotechnology Center, Purdue University, West Lafayette, IN, 47907 USA 5School of Materials Engineering, Purdue University, West Lafayette, IN, 47907 USA 6School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN, 47907 USA

a b 1 10 20 0.9 10 a.u. 5 a.u. 0.8 0.8 0 0 0 100 200 0 100 200 0.7 f f 2 g

1 g (MHz) (MHz)

P 0.6 P 0.6

0.5 0.4 0.4 0.3 0.2 0 10 20 30 0 10 20 30 T (ns) T (ns)

Supplementary Figure 1. (a) Exchange oscillations vs. evolution time (T ) measured on the left pair. (b) Exchange oscillations vs. T measured on the right pair. In both panels, the red line arXiv:2006.10913v2 [cond-mat.mes-hall] 5 Mar 2021 indicates the expected π-pulse time based on the frequency of exchange oscillations. The frequency is chosen as the frequency giving the peak in the absolute value of the FFT of the data (insets). In both cases, the actual π-pulse time is longer than the expected value, because of the ns-level rise time in the pulse, leading to the observed frequency chirp.

∗ [email protected] 2

a P1 b P2 -0.5 -0.5

0.9 0.9

0.8 0.8

0.7 0.7 0 0 0.6 0.6

0.5 0.5

0.4 0.4 0.5 0.5 0 2 4 0 2 4 J (MHz) J (MHz)

Supplementary Figure 2. Numerical simulation of an N = 2 Ising spin chain after 4 Floquet steps. The initial state is . (a) Spin-down probability of the left spin. (b) Spin-up probability of the |↓↑i right spin. The simulations are averaged over 128 charge and hyperfine realizations.

a P1 b P1 -0.5 -0.5

0.9 0.9

0.8 0.8 0.7 0 0 0.7 0.6 0.6 0.5

0.4 0.5 0.5 0.5 0 2 4 0 2 4 J (MHz) J (MHz)

Supplementary Figure 3. Numerical simulation of an N = 8 Ising spin chain after (a) 4 and (b) 1024 Floquet steps. Both panels plot the probability to find the first spin in the state. |↑i The initial state is . In panel (b), the theoretical semiclassical DTC phase diagram is |↑↓↑↓↑↓↑↓i overlaid. The simulations are averaged over 128 charge and hyperfine realizations.