Abstract
A fundamental challenge for quantum dot spin qubits is to extend the strength and range of qubit interactions while suppressing their coupling to the environment, since both effects have electrical origins. Key tools include the ability to take advantage of physical resources in different regimes, and to access optimal working points, sweet spots, where dephasing is minimized. Here, we explore an important resource for singlettriplet qubits: a transverse sweet spot (TSS) that enables transitions between qubit states, a strong dipolar coupling, and leadingorder protection from electrical fluctuations. Of particular interest is the possibility of transitioning between the TSS and symmetric operating points while remaining continuously protected. This arrangement is ideal for coupling qubits to a microwave cavity, because it combines tunability of the coupling with noise insensitivity. We perform simulations with \(1/f\)type electrical noise, demonstrating that twoqubit gates mediated by a resonator can achieve fidelities >99% under realistic conditions.
Introduction
Recent advances in semiconducting spin qubits^{1,2} have enabled singlequbit gates with high fidelities^{3,4,5,6,7}, and twoqubit exchangebased gates^{8,9,10,11,12} with fidelities >94%^{13}. While these exchange gates are relatively fast, their interaction range is limited—typically to nearest neighbors. One method for increasing the interaction range is to insert an intermediary coupler, such as a superconducting microwave cavity^{14,15,16,17,18,19,20}. However, strong qubitresonator couplings have been difficult to realize, due to the small magnetic dipole of the spins^{21,22,23}, which results in slow qubit gates. A common strategy for enhancing this coupling involves hybridizing the spin and charge degrees of freedom via the spinorbit interaction, which arises naturally in GaAs, and can be induced by micromagnets in Si^{24,25}. In this way, strong coupling has been achieved in both GaAs and Si^{26,27,28}. However the gates are still slow and susceptible to electrical (charge) noise, motivating a search for alternative methods to enhance the qubit charge dipole, as well as sweet spots to suppress the effects of noise.
For singlettriplet spin qubits^{29,30,31}, a useful sweet spot has been identified in the \(S\)\({T}_{}\) subspace^{32}. For the \(S\)\({T}_{0}\) qubit, recent attention has focused on a sweet spot known as the symmetric operating point (SOP), due to its favorable coherence properties^{33,34}. The position of the SOP—as far as possible from the (2, 0)(1, 1) or (1, 1)(0, 2) charging transitions—reduces its sensitivity to charge noise, but also suppresses its charge dipole moment. In this regime, the weak dipole coupling is mainly longitudinal in form^{35,36}, enabling \(Z\) rotations and twoqubit CPHASE gates. In contrast, the charge dipole increases near a charging transition—particularly its transverse component, enabling \(X\) rotations and twoqubit iSWAP gates. In this regime, when the interdot Zeeman energy difference (or gradient) \(\Delta B\,=\,g{\mu }_{B}({B}_{L}{B}_{R})\) is larger than the tunnel coupling, the transverse coupling can dominate over the longitudinal coupling^{37,38,39,40}; however, the qubit also becomes more sensitive to charge noise.
In this work, we investigate a family of sweet spots, with strong transverse couplings, located far from the SOP. We show that these transversely coupled sweet spots (TSS) represent an interesting working regime for singlettriplet qubits that can be exploited to perform highfidelity singlequbit gates with AC electrical driving fields, or to enable capacitively coupled twoqubit gates. (Here, we focus on twoqubit gates mediated by a superconducting cavity.) We describe protocols for one and twoqubit gate operations that provide constant noise protection, even while transitioning between operating points. This allows us to take advantage of the resources available in different working regimes, and greatly enhances the toolbox for operating singlettriplet qubits.
Results
TSS and SOP sweet spots
We initially assume that the global magnetic field \(B\) is large enough that the polarized triplet states may be ignored. We include the polarized triplets later; however, the simpler model serves to illustrate the key physics. We restrict our analysis to Si parameters. In this case, the Hubbard Hamiltonian of a singlettriplet qubit in the basis \(\{\leftS(1,1)\right\rangle ,\left{T}_{0}(1,1)\right\rangle ,\leftS(0,2)\right\rangle ,\leftS(2,0)\right\rangle \}\) is given by^{41}
where \(\tau\) is the tunnel coupling between the two sides of the double dot (Fig. 1), \(\varepsilon\) is the detuning between dots, and \(U\) is the charging energy for doubly occupied states. Here we define \(\varepsilon =0\) as the position of the \(\leftS(1,1)\right\rangle\)–\(\leftS(0,2)\right\rangle\) charging transition, with the SOP located at \(\varepsilon =U\).
A typical energy level diagram for \({H}_{{\rm{ST}}}\) is shown in Fig. 2a for \(\tau\,\gtrsim\,\Delta B\) (solid lines); here we have assumed large values of \(\tau\) and \(\Delta B\) to help visualize the key features of the plot. In this regime, the qubit energy splitting, \(\hslash {\omega }_{q}\,=\,{E}_{1}{E}_{0}\), has positive curvature and there is only one sweet spot, located at the SOP (Fig. 2b). Here, the qubit states are largely unperturbed from \(\leftS(1,1)\right\rangle\) and \(\left{T}_{0}(1,1)\right\rangle\). In contrast, when \(\tau\) falls below a critical value, \({\tau }_{{\rm{crit}}}\,\approx\,1.37\Delta B\), a dip emerges in the energy dispersion near \(\varepsilon\,=\,0\), representing a sweet spot—the TSS. In this case, the energy levels become strongly hybridized and bent (dashed lines in Fig. 2a), yielding eigenstates that resemble \(\left\uparrow \downarrow \right\rangle\) and \(\left\downarrow \uparrow \right\rangle\). (Note that mirrorsymmetric features are also observed near \(\varepsilon\,=\,2U\). However, since we focus here on the regime near \(\varepsilon\,=\,0\), the magnitude of \(U\) and the presence of \(\leftS(2,0)\right\rangle\) have almost no effect on the results reported below. For convenience, we therefore set \(U\,=\,3\) meV and ignore it for the remainder of this work.) When \(\tau\,\approx\,{\tau }_{{\rm{crit}}}\), it is not obvious which type of behavior will dominate: \(\tau\)like behavior (positive dispersion curvature) or \(\Delta B\)like behavior (a TSS). Interestingly, for a small range of \(\tau\,\le\,{\tau }_{{\rm{crit}}}\), both features are present, resulting in the emergence of an additional peak feature in the energy dispersion (Fig. 2b, lower inset), which we refer to as the alternative transverse sweet spot (ATSS). As its name indicates, the ATSS also has a transverse coupling, and its position on the \(\varepsilon\) axis may occur anywhere between the SOP and the TSS, depending on the value of \(\tau\) (Fig. 2b, upper inset). At a second critical value of \(\tau\), \({\tau }_{{\rm{SSS}}}\,\approx\,\sqrt{3/2}\Delta B\ \approx \ 1.22\Delta B\) (the super sweet spot), the ATSS merges with the SOP. For \(\tau\,<\,{\tau }_{{\rm{SSS}}}\), the curvature of the energy dispersion at the SOP becomes negative, and only two sweet spots remain—the SOP and the TSS.
The extent to which the TSS, ATSS, and SOP sweet spots are protected from charge noise depends on the flatness of the energy dispersion, which is determined in part by the order of the sweet spot: a sweet spot is classified as \({n}{{\rm{th}}}\)order if \({\partial }^{m}{\omega }_{q}/\partial {\varepsilon }^{m}\,=\,0\) for all \(m\,\le\,n\). The SOP is a firstorder sweet spot. However in Supplementary Note 1 (See Supplementary Materials for details), we show that higher derivatives of the energy dispersion can be very small, in terms of the parameter \({(\Delta B/U)}^{m}\,\ll\,1\), yielding an approximate ninthorder sweet spot when \(\tau ={\tau }_{{\rm{SSS}}}\) (and an exact thirdorder sweet spot), which accounts for the extreme flatness of the energy dispersion.
While singlequbit gates can be performed at the SOP, using the tunnel coupling \(\tau\) as a control parameter^{33,34}, the absence of a charge dipole moment makes it more difficult to implement resonatormediated gates^{36}. On the other hand, for the same reason, the SOP makes a useful idling point for qubits coupled to a cavity. At the special point, \(\tau\,=\,{\tau }_{{\rm{SSS}}}\), the extreme flatness of the energy dispersion makes the SOP an excellent idling point. The dipole moment of the ATSS is also small, as discussed below, and the sweet spot is relatively broad, making it an alternative candidate for idling. The position of the ATSS varies rapidly as function of \(\tau\) (Fig. 2b, upper inset), which could present a challenge for controlling the qubit; however recent experiments have demonstrated fast and accurate control over both \(\tau\) and \(\varepsilon\)^{9,33,34,42}. The TSS forms a narrower sweet spot (Fig. 2b), and its charge dipole is large, which increases its sensitivity to charge noise, but makes it a good candidate for performing gate operations and coupling to a cavity. In principle, it is possible to adiabatically transition between the TSS and the ATSS, and then the SOP, by simultaneously adjusting the parameters \(\tau\) and \(\varepsilon\) (Fig. 2b, upper inset), even while \(\Delta B\) remains fixed, as is typical in a given experiment. We now explore these possibilities in greater detail.
Characterizing the TSS
The position of the TSS in detuning space, \({\varepsilon }_{{\rm{SS}}}\), depends on all the parameters of the Hamiltonian, but generally occurs near \(\varepsilon\,=\,0\) (Fig. 2b, upper inset). As shown below, the location of the operating point plays a key role in determining the qubit behavior, which has two basic types. (1) When \(\tau\,\gtrsim\,\Delta B\) (Fig. 2c), we mainly find that \({\varepsilon }_{{\rm{SS}}}\,<\,0\); in this case, the energy splitting of the lowest nonlogical state \(\leftS(0,2)\right\rangle\) is approximately resonant with the qubit frequency, resulting in enhanced leakage. (2) When \(\tau\,<\,\Delta B\) (Fig. 2d), we have \({\varepsilon }_{{\rm{SS}}}\,> \,0\); in this case, leakage is suppressed, but the TSS is very narrow, and the qubit is chargelike. More generally, any qubit property (e.g., decoherence, coupling, or gate fidelity) depends on the specific control parameters. We now evaluate and compare these properties, first for an isolated qubit, then for a qubit coupled capacitively to a microwave resonator.
We first consider singlequbit gate operations in isolated qubits. The gates are performed by applying an AC drive to the detuning parameter. In the presence of charge noise \(\delta \varepsilon (t)\), the timedependent detuning is given by \(\Delta \varepsilon (t)\,=\,{\varepsilon }_{{\rm{AC}}}\cos (\omega t)\,+\delta \varepsilon (t)\). From Eq. (1), the resulting interaction is given by
Since the states \(\leftS(0,2)\right\rangle\) or \(\leftS(2,0)\right\rangle\) generate the charge dipole in this system, \({H}_{{\rm{int}}}\) is proportional to the dimensionless dipole operator, \(\,\,{\hat{\!\!d}}=\partial {H}_{{\rm{ST}}}/\partial \varepsilon\). In general, \({H}_{{\rm{int}}}\) can have longitudinal and transverse components; however at a sweet spot, the longitudinal component vanishes, by definition.
We begin by solving the total Hamiltonian, defined as \(H={H}_{{\rm{ST}}}\,+\,{H}_{{\rm{int}}}\). First, we ignore the state \(\leftS(2,0)\right\rangle\) in Eq. (1), since it is very high in energy. We then evaluate \(H\) in the \(\{\left0\right\rangle ,\left1\right\rangle ,\leftL\right\rangle \}\) eigenbasis, which diagonalizes \({H}_{{\rm{ST}}}\), obtaining
Here \({E}_{n}\) are the eigenvalues of \({H}_{{\rm{ST}}}\), \({\sigma }_{nm}\,=\,\leftn\right\rangle \left\langle m\right\), where \(n,m\in \{0,1,L\}\), and \({d}_{nm}\,=\,\left\langle n\right\,\,\hat{\!\!d}\leftm\right\rangle\), where \(\Delta \varepsilon \ {d}_{01}\) is the transverse dipole coupling induced by \({H}_{{\rm{int}}}\). In Fig. 3a, we plot numerical solutions for \({d}_{01}\), evaluated at the TSS, as a function of \(\tau\) and \(\Delta B\). The large white triangle in the lowerright portion of the plot corresponds to \(\tau\,> \,{\tau }_{{\rm{crit}}}(\Delta B)\), where no TSS solutions exist. The general features of the plot can be understood as follows. When \(\tau\,\ll\,\Delta B\), the TSS occurs near the charging transition, which causes \(\leftS(0,2)\right\rangle\) and \(\leftS(1,1)\right\rangle\) to strongly hybridize, and yields an effective charge qubit, for which \({d}_{01}\,\approx\,0.5\) at the sweet spot. When \(\tau \to {\tau }_{{\rm{crit}}}\), the TSS moves away from the charging transition, resulting in a suppressed dipole, \({d}_{01} \approx 1{0}^{3}\)–\(1{0}^{4}\), which vanishes completely at the SOP. To a good approximation, \({d}_{01}\) depends only on the ratio \(\tau /{\tau }_{{\rm{crit}}}\propto \tau /\Delta B\) over the entire plot range of Fig. 3a, yielding a radial plot. The large\({d}_{01}\) (small\(\tau /\Delta B\)) operating regime is preferential for boosting gate speeds; however we now show that the dephasing rate \(1/{T}_{\varphi }\) also grows in this regime.
We define \({T}_{\varphi }\) as the decay time of the \({\rho }_{01}\) component of the qubit density matrix, which we estimate by simulating its freeinduction decay. As described in Methods section, we introduce \(\delta \varepsilon (t)\) fluctuations into the simulations, sampling from a gaussian distribution with \(1/f\) spectral correlations. We then average over a large number of chargenoise realizations to obtain the results shown in Fig. 3b. Since the TSS is a sweet spot, it is protected from small \(\delta \varepsilon\) fluctuations, to lowest order. The main contribution to dephasing therefore occurs at order \(\delta {\varepsilon }^{2}\), and its behavior correlates with the width of the sweet spot. In Supplementary Note 2 we consider two additional noise mechanisms that could potentially contribute to \({T}_{\varphi }\): tunnelcoupling noise and Rabifrequency fluctuations. In summary, tunnelcoupling noise is found to have a stronger effect on twoqubit gates, where it is comparable to detuning noise. Rabifrequency fluctuations are a strongdriving effect, which can become important for fast singlequbit gates. In the following discussion, we include all such dephasing mechanisms in our fidelity simulations. However, we do not include direct magnetic field fluctuations arising from nuclear spin dynamics, since we assume these can be suppressed by isotropic purification of the Si/SiGe heterostructure.
In the \(\tau\,\lesssim\,\Delta B\) regime (Fig. 2d), the TSS is well separated from other features in the energy dispersion; its shape therefore does not depend on \(\Delta B\), which only determines the splitting between states \(\left\uparrow \downarrow \right\rangle\) and \(\left\downarrow \uparrow \right\rangle\). Hence, the qubit is chargelike, and the width of the sweet spot is determined by \(\tau\) rather than \(\Delta B\). This is consistent with Fig. 2b where the sweet spot is quite narrow for small \(\tau\). It is also consistent with Fig. 3b where \({T}_{\varphi }\) approaches 100 ns in the limit \(\tau\,\ll\,\Delta B\), and becomes independent of \(\Delta B\). On the other hand, for \(\tau\,\to\,{\tau }_{{\rm{crit}}}\), we observe a wider sweet spot in Fig. 2b. In this regime, the presence of the leakage state actually helps to flatten the energy dispersion, yielding \({T}_{\varphi }\) approaching 10 \(\mu\)s.
Using the same simulations, we also compute \({T}_{L}\), defined here as the decay time of \({\rho }_{00}(t)\,+\,{\rho }_{11}(t)\), due to leakage. The results, which are plotted in Fig. 3c, exhibit a similar range of timescales as Fig. 3b; however, the trends are very different. This is easy to understand because leakage is caused by the hybridization of logical and nonlogical states, which occurs near the resonance condition \({E}_{L}{E}_{1}\,=\,{E}_{1}{E}_{0}\), causing a dip in \({T}_{L}\) when \(\tau\,\approx\,\Delta B\).
Finally, we note that phononmediated decay processes have not been considered in the current analysis, although they also contribute to \({T}_{1}\)type relaxation. For GaAsbased devices, such processes are expected to reduce \({T}_{1}\) to a few nanoseconds for the large magnetic field gradients considered here, due to the presence of piezoelectric phonons^{43}. In the current proposal, we have therefore focused on Sibased devices, where piezoelectric phonons are absent, and the phononmediated \({T}_{1}\) is generally much longer than any time scale relevant to our analysis^{43,44}. For this system, we therefore conclude that \({T}_{1}\) is dominated by leakage.
To summarize the results of this section, the behaviors of \({T}_{\varphi }\) and \({T}_{L}\) exhibit opposite trends as a function of \(\tau\) when \(\varepsilon\) is tuned to a TSS; the best working points must therefore be determined via optimization. We address this problem below, by computing the fidelities of onequbit and twoqubit gates.
Singlequbit gate fidelity
In the previous section, we studied free induction. Here we consider resonantly driven, singlequbit \({X}_{\pi }\) gate operations performed at a TSS. We consider singlequbit interactions mediated by ACdriven gates, which are generally expected to be faster than singlequbit gates mediated by a resonator. However, the twoqubit gates in the following section are mediated by a resonator, with a capacitive coupling that cannot be turned off, as indicated in Fig. 1; we therefore include this interaction in the present analysis. In our simulations, we further assume that the cavity resonant frequency \({\omega }_{r}\) cannot be tuned. However, we note that the cavityqubit detuning \({\Delta}_{0}\,=\,\hslash ({\omega }_{r}{\omega }_{q})\) can be varied, because the qubit frequency depends on the parameters \(\Delta B\) and \(\tau\).
We model the qubitresonator system with the Hamiltonian
where \({a}^{\dagger }(a)\) is the photon creation (annihilation) operator, \({g}_{0}\,=\,e{V}_{0}\) is the bare capacitive coupling between the qubit and resonator, \({V}_{0}\,=\,\sqrt{\hslash {Z}_{r}}\ {\omega }_{r}\) is the amplitude of the resonator voltage antinode, and \({Z}_{r}\) is the resonator impedance^{36}. The effective qubitcavity coupling, \(g\,=\,{g}_{0}{d}_{01}\), is proportional to the transverse dipole moment, which is maximized near the charging transition. As noted above, the coupling can be turned off (\({d}_{01}\,=\,0\)) at the SOP, while \({d}_{01}\,\approx\,0.5\) for large \(\Delta B\).
We perform simulations of Eq. (4) for a range of \(\Delta B\) and \(\tau\). For each pair of values, \((\Delta B,\tau )\), we tune the intraqubit detuning parameter to a TSS [\(\varepsilon\,=\,{\varepsilon }_{{\rm{SS}}}(\Delta B,\tau )\)] to improve the gate fidelity, and apply an AC drive at the qubit resonant frequency: \(\Delta \varepsilon (t)\,=\,{\varepsilon }_{{\rm{AC}}}\cos ({\omega }_{q}t)\). Since we do not limit the simulations to the weakdriving regime, the \({X}_{\pi }\) gate times must be determined numerically; we do this by evolving over many Rabi oscillations, to more accurately locate the initial peak. The simulations are computationally expensive, compared to Fig. 3a, b, since they include photon basis states. Therefore, we do not explicitly include either charge noise or photon decay at the Hamiltonian level. Instead, we solve a master equation based on Eq. (4), in which dephasing effects are included phenomenologically through the dephasing rate \(1/{T}_{\varphi }\) and leakage effects are included through the decay rate \(1/{T}_{L}\), which were both obtained as functions of \(\Delta B\) and \(\tau\) in the previous section. Resonator photon decay is included through a constant decay rate, \(\kappa\). We then compute the gate fidelity, obtaining the results shown in Fig. 3d. See Methods section for details of these calculations.
We observe the following behavior. First, gate fidelities are generally found to be high, except very near the resonance condition \({\Delta }_{0}\,=\,0\) (dashed line), where excited photons in the resonator form leakage levels that naturally suppress the singlequbit gate fidelity. For larger values of \(\kappa\), the fidelity is further suppressed near the resonance condition. On the other hand, \({\Delta }_{0}\) increases quickly as we move away from this line, suppressing this effect. Even further away from the resonance condition, the gate fidelity is slightly suppressed for small \(\tau\), due to strong dephasing (Fig. 3b), or near the line \(\tau\,=\,{\tau }_{{\rm{crit}}}\), due to enhanced leakage (Fig. 3c) and smaller charge dipoles (Fig. 3a). The best fidelities are therefore obtained midway between the resonance condition and \(\tau\,=\,{\tau }_{{\rm{crit}}}\), at larger values of \(\Delta B\). For the physically realistic simulation parameters used in Fig. 3d, the fidelities can be quite high, approaching 99.85%, and are limited by Rabifrequency fluctuations due to strong driving. Finally, we note that closer inspection of the resonance condition in Fig. 3d reveals weak oscillations. As discussed in Supplementary Note 3, these can be understood as a combination of leakage and strongdriving effects.
Twoqubit gate fidelity
We consider twoqubit gates mediated by a cavity, with a setup similar to Fig. 1, and with both qubits positioned at voltage antinodes. Simulations are performed analogously to the previous section, but with a twoqubit Hamiltonian given by
where the subscript \(i\) refers to qubits \(a\) or \(b\). The native gate for \({H}_{qqr}\) is iSWAP, with gate times determined analogously as for singlequbit gates. The twoqubit gate can be switched off by tuning either of the qubits to its SOP. To simplify the following analysis, we set \({\Delta }_{0a} = {\Delta }_{0b}\equiv {\Delta }_{0}\), \({d}_{nm,a}={d}_{nm,b}\equiv {d}_{nm}\), \(\Delta {B}_{a} = \Delta {B}_{b}\equiv \Delta B\), \({\tau }_{a} = {\tau }_{b} \equiv \tau\), and \({\varepsilon }_{a} = {\varepsilon }_{b} \equiv \varepsilon\), to reduce the number of independent control parameters. The gate fidelities are computed by solving the master equation associated with Eq. (5), including the decoherence rates \(1/{T}_{L}\), \(1/{T}_{\varphi }\), and \(\kappa\), as before, and comparing the result to an ideal iSWAP gate. Our results are shown in Fig. 4a, using the same simulation parameters as Fig. 3d.
Although similar physics determines the fidelities of one and twoqubit gates, the trends observed in Figs. 3d and 4a are very different. In particular, the fidelity dip along the resonance line in Fig. 3d becomes a double peak in Fig. 4a. This is because the singlequbit gates are driven, with the resonator acting only as a leakage channel. For twoqubit gates, the cavity mediates the interaction, and the fidelity is generally enhanced near the resonance condition, \({\Delta }_{0}\,=\,0\), where the effective qubitqubit coupling is maximized^{45}. (The same is true for singlequbit gates mediated by a resonator, although we do not explore that possibility here.) Very near the resonance, however, spontaneous excitation of the qubits by the cavity (the Purcell effect) suppresses the gate fidelity (i.e., increases the infidelity), causing maxima to form on either side of this line. The same process also reduces the individual qubit lifetimes. In cases where the Purcell effect dominates the fidelity, we note that an alternative approach would be to replace the cavity with a direct capacitive coupling^{11}, although we do not explore that possibility here.
Far from the resonance condition, twoqubit gate fidelities are typically low, because offresonant gates tend to be slow, and therefore susceptible to charge noise. (This is not a problem for singlequbit gates, which can be strongly driven.) However, fidelities are found to increase for larger \(\Delta B\), due to stronger qubitcavity couplings and reduced leakage (Fig. 3). To exploit this trend, we note that nanomagnets in recent doubledot experiments have already achieved \(\Delta B\) values as large as 80 mT^{46} (\(=2.2\) GHz), corresponding to a maximum fidelity of \(98.5\)% in Fig. 4a. Finally, we note that small fidelity oscillations are observed near the resonance condition, which are reminiscent of those in Fig. 3d, and can also be attributed to leakage and strong driving (see Supplementary Note 4).
Leakage induced by polarized triplets
Up to this point, we have not considered the polarized spin triplet states, \(\left\uparrow \uparrow \right\rangle\) and \(\left\downarrow \downarrow \right\rangle\), which present new leakage channels. In this case, hybridization with the qubit states is caused by a transverse magnetic field gradient. It is reduced, however, when the levels are split off by a large global field; further details of these calculations are presented in Supplementary Note 5. To estimate the effect of such leakage on twoqubit gate fidelities, we first extend Eq. (5) to include a global \(B\) field and a transverse field gradient \(\Delta {B}_{\perp }\). Since \(\Delta {B}_{\perp }\) and \(\Delta B\) are expected to be similar in size^{46}, we simply set \(\Delta {B}_{\perp }\,=\,\Delta B\). We then compute the iSWAP gate fidelity for a fixed \(\Delta B\), and determine its maximum as a function of \(\tau\). Repeating this procedure as a function of \(\Delta B\), for several values of \(B\), yields the results shown in Fig. 4b. As expected, we find that fidelities improve uniformly as a function of \(B\). However, the dependence on \(\Delta B\,=\,\Delta {B}_{\perp }\) is nonmonotonic: the fidelity initially increases (infidelity decreases) by the same mechanism as Fig. 4a; for larger \(\Delta {B}_{\perp }\), this behavior saturates, and leakage eventually dominates the fidelity. For the range of \(\Delta B\) plotted here, we find that \(B\,\ge\,2\) T is sufficient for avoiding most leakage. More generally, \(B\,\ge\,0.8\) T yields fidelities >99%, when \(\Delta B\,> \,100\) mT.
Discussion
We have shown that qubit coherence and onequbit and twoqubit gate fidelities are strongly affected by the operating points in a control space spanned by the parameters \(B\), \(\Delta B\), \(\Delta {B}_{\perp }\), \(\tau\), \(\varepsilon\), \({\varepsilon }_{{\rm{AC}}}\), \({g}_{0}\), and \({\omega }_{r}\), as well as the noise characteristics of the qubits and the resonator. The transverse sweet spots (TSS) studied in this work make good working points, because they provide protection against environmental noise while offering a strong coupling to external driving fields or a microwave resonator.
To achieve highfidelity gates at a TSS, it is important to provide a large gradientinduced Zeeman splitting, \(\Delta B\), and a nearly resonant coupling between the qubit and cavity. Since \(\hslash {\omega }_{q}\,\le\, 2\Delta B\) for a TSS, we therefore require that \(2\Delta B\,\gtrsim\,\hslash {\omega }_{r}\), while noting that neither \(\Delta B\) nor \({\omega }_{r}\) is easy to change after a device is fabricated. Fortunately, recent work shows that it is possible to form highkineticinductance resonators with low resonant frequencies, \({\omega }_{r}/2\pi\,\approx\,2.8\) GHz^{47}, while maintaining a high cavity \(Q\,> \,1{0}^{5}\) in the presence of a large inplane field \(B=6\) T. Moreover, as noted above, large gradients, \(\Delta B/h\,\approx\,2.2\) GHz (\(=80\) mT), have already been achieved in the lab^{46}, indicating that the requirements for a TSS have already been met.
Adopting the values for \(\Delta B\) and \({\omega }_{r}\) from the previous paragraph, and choosing \(\Delta {B}_{\perp }=\Delta B\), \(B=2\) T, a resonator coupling of \({g}_{0}/2\pi\,=\,0.05\) GHz, and a realistic driving field of \({\varepsilon }_{{\rm{AC}}}/h\,=\,0.5\) GHz, we obtain the following results at a TSS. Singlequbit gates are found to be fairly fast, with a gate time of \({t}_{1{\rm{Q}}}\,\approx\,7\) ns, yielding a gate fidelity of 99.3% for an optimal tunnel coupling of \(\tau /h\,=\) 2.1 GHz. Twoqubit gates are slightly slower, with \({t}_{2{\rm{Q}}}\,\approx\,50\) ns, yielding a gate fidelity of 98.2% for the optimal tunnel coupling \(\tau /h=\) 1.5 GHz. Our simulations also show that when \(2\Delta B\,\simeq\,\hslash {\omega }_{r}\), leakage tends to dominate the infidelity, while for smaller values of \(\Delta B\), dephasing is the dominant problem. We find that, as a rule of thumb, \(1.5\Delta B\,\simeq\, \hslash {\omega }_{r}\) provides a good balance for obtaining higher fidelities, which explains our choice of \(\Delta B\) and \({\omega }_{r}\) values in the simulations. However, better fidelities can be achieved by increasing both of these parameters simultaneously. Theoretical calculations suggest that larger \(\Delta B\,\ge\,150\) mT = 17 \(\mu\)eV values should be possible for nearterm experiments^{48,49}. Repeating our simulations with this \(\Delta B\) and \({\omega }_{r}/2\pi\,=\,5.2\) GHz gives optimal fidelities of 99.85% and 99.2% for one and twoqubit gates, respectively.
The results described above exploit optimized TSS working points for singlettriplet qubits, but reveal that these points differ for single and twoqubit gate operations. In addition, resonatormediated gates require an idling point, where the effective coupling to the resonator is turned off. We have identified two good candidates for idling points: the ATSS, where \({d}_{01}\) is very small, or the SOP where \({d}_{01}\,=\,0\), particularly when \(\tau\,=\,{\tau }_{{\rm{SSS}}}\). Interestingly, for a fixed value of \(\Delta B\), we can navigate between gating and idling points while maintaining a TSS, by simultaneously tuning \(\tau\) and \(\varepsilon\) such that \(\varepsilon\,=\,{\varepsilon }_{{\rm{SS}}}(\tau )\). In Supplementary Note 7, we estimate the time scales for adiabatically transitioning between these working points.
Longitudinal and transverse couplings can be viewed as distinct, physical resources, with unique advantages and disadvantages for quantum computing. It is therefore important to compare their attributes^{50}; the singlettriplet qubit provides a testbed for doing so in a single experimental setting. In this work, we have focused on the TSS, which has a purely transverse coupling and can be formed over a continuous range of parameters. In fact, the TSS and ATSS are the only tunings with purely transverse couplings for singlettriplet qubits. The SOP is the only tuning with a purely longitudinal, curvaturetype coupling (see below), which can be formed over a continuous range of \(\tau\) when \(\varepsilon\,=\,U\). All other operating points have both transverse and longitudinal components. Such mixing reduces the response to AC driving for singlequbit gates, and yields complicated behavior for twoqubit gates, which may be undesirable from a control perspective. These mixed operating points also do not correspond to sweet spots, and should therefore experience faster decoherence. The TSS coupling is particularly strong because the qubit’s charge character is maximized. In contrast, at the SOP, the charge dipole vanishes, resulting in a weaker, secondorder curvature coupling^{35}, which is consistent with slower gates that are well protected by a highorder sweet spot. Alternatively, gate speeds at the SOP may be enhanced by employing AC driving techniques^{36}. Using this method, we can simulate gates performed at the SOP, under noise conditions similar to those considered above, at the TSS. As described in Supplementary Note 6, we adopt realistic experimental parameters for the SOP^{33} and apply an AC drive to the tunnel coupling with a driving amplitude equal to 1/10 of its average value, obtaining a singlequbit \({X}_{\pi }\) gate fidelity of 99.6% and a CZ gate fidelity of 93.6%. These results are limited by tunnelcoupling noise, which dominates at the SOP because the effects of detuning noise are suppressed.
Finally, we note that readout of ST qubits can be challenging in the presence of a large magnetic field gradient^{51}. Two methods to overcome this problem are (1) mapping the qubit onto different spins states, as described in ref. ^{51}, or (2) tuning the tunnel barrier to the more conventional regime for readout, where \(\tau\,\gg\,\Delta B\).
Methods
Overview
In this work, we perform two types of numerical simulations: (i) free induction of single qubits, and (ii) one and twoqubit gate operations. The simulations employ different theoretical methods, and are repeated for cases with and without charge noise. All numerical calculations use the QuTiP software package^{52}.
Freeinduction simulations
These are performed after adding timedependent charge noise to the detuning parameter in Eq. (3), with \(\Delta \varepsilon\,=\,\delta \varepsilon (t)\). A similar procedure is used to model noise in the tunnelcoupling parameter, as discussed in Supplementary Note 2. Noise sequences are generated following the method described in refs. ^{4,53}: we first generate random white noise \(\delta \varepsilon (t)\) over a discrete time sequence. This sequence is then Fourier transformed and scaled in frequency space by the noise power spectrum \(\sqrt{S(\omega )}\), where
and \({\omega }_{l}/2\pi\,=\)100 kHz and \({\omega }_{h}/2\pi\,=\)20 GHz are lower and upper frequency cutoffs. We choose a noise strength of \({c}_{\varepsilon }=0.56\) µeV, corresponding to a standard deviation of \({\sigma }_{\varepsilon }\,=\,{c}_{\varepsilon }{\left[2\mathrm{ln}(\sqrt{2\pi }{c}_{\varepsilon }/\hslash {\omega }_{l})\right]}^{1/2}\approx 2\) µeV (see ref. ^{53}) for noise integrated over the entire frequency spectrum, as consistent with several recent experiments^{54,55,56}. We note that, for a given value of \({c}_{\varepsilon }\), lowering \({\omega }_{l}\) increases the noise in the system. The resulting frequency sequence is Fourier transformed back to the time domain, yielding the desired noise sequence. For each point in the \(\Delta B\)\(\tau\) plots shown in Fig. 3b, c, we average the density matrix \(\rho (t)\) over 10,000 different noise realizations, with initial states \(\rho (0) = \lefti\right\rangle \left\langle i\right\), where \(\lefti\right\rangle = (\left0\right\rangle\,+\,\left1\right\rangle )/\sqrt{2}\). We use the same simulations to obtain the density matrix \({\rho }_{{\rm{leak}}}(t)\), using \({\rho }_{{\rm{leak}}}(0) = (\left0\right\rangle \left\langle 0\right+\left1\right\rangle \left\langle 1\right)/2\), to obtain purely leakage errors. \({T}_{\varphi }\) and \({T}_{L}\) are obtained by fitting the averaged results to^{57}
and
where \(\rho\,=\,{\bf{1}}/3\) represents the fully mixed state in the \(\{\left0\right\rangle ,\left1\right\rangle ,\leftL\right\rangle \}\) basis. Here, we only assume coupling to the dominant leakage state associated with \(\leftS(0,2)\right\rangle\). \(\beta\) is left as a fitting parameter in Eq. (7) to account for the fact that nondephasing, leakage processes can dominate the decoherence in some cases. As discussed in the main text, \({T}_{1}\) is dominated by \({T}_{L}\) in Si. We may therefore extract the pure dephasing time, \({T}_{\varphi }^{\varepsilon }\), through the relation \(1/{T}_{2}^{* }\,=\,1/{T}_{\varphi }^{\varepsilon }\,+\,1/2{T}_{L}\), where the superscript \(\varepsilon\) indicates that this quantity arises from noise in the detuning parameter.
Fidelity estimations
For one and twoqubit gates, we incorporate the freeinduction results into our simulations of the qubitcavity master equation, defined as^{45}
where \(H\) represents the appropriate onequbit (\(j\,=\,a\)) or twoqubit (\(j\,=\,a,b\)) Hamiltonian in the lab frame, as presented in Eqs. (4) or (5) of the main text, \(\kappa\) is the cavity decay rate, \({T}_{\varphi }\) and \({T}_{L}\) are computed as functions of \(\Delta B\) and \(\tau\), as described above, \({\sigma }_{z,i}\,\equiv\,{\sigma }_{11,i}{\sigma }_{00,i}\) is the dephasing operator for qubit \(i\), and \({\sigma }_{L,i}\equiv {\sum }_{n\,=\,0,1}({\sigma }_{nL,i}\,+\,{\sigma }_{Ln,i})\) is the operator associated with leakage between the logical subspace of qubit \(i\) and its leakage state \(L\). In Eq. (9), we include noise from both the detuning and tunnel coupling parameters through the relation \(1/{T}_{\varphi }\,=\,1/{T}_{\varphi }^{\varepsilon }\,+\,1/{T}_{\varphi }^{\tau }\). In addition, for singlequbit Rabi simulations, we include the term \((1/2{T}_{\varphi }^{R}){\sum }_{j}({\sigma }_{+,j}\rho {\sigma }_{,j}\,+\,{\sigma }_{,j}\rho {\sigma }_{+,j}\rho )\) on the righthand side of Eq. (9) to account for Rabi frequency fluctuations as described in Supplementary Note 2, where \({\sigma }_{+,j}\,\equiv\,{\sigma }_{10,j}\) and \({\sigma }_{,j}\,\equiv\, {\sigma }_{01,j}\). Although including \({T}_{\varphi }\) as we have done in Eq. (9) is common practice^{45}, it may be argued that the Markovian nature of the master equation is inconsistent with the nonMarkovian origins of \({T}_{\varphi }\). We have addressed this question in Supplementary Note 8 by performing corresponding master equation and quasistatic simulations of freeinduction decay, obtaining nearly identical results for \({T}_{\varphi }\) in either case.
The initial states for the master equation simulations in Eq. (9) are taken to be \(\rho (0)\,=\,\lefti\right\rangle \left\langle i\right\), where \(\lefti\right\rangle\,=\,{\left0\right\rangle }_{c}(\left0\right\rangle +\left1\right\rangle )/\sqrt{2}\) for singlequbit gates (here, \({\left0\right\rangle }_{c}\) represents the zerophoton state of the cavity), or \(\lefti\right\rangle\,=\,{\left0\right\rangle }_{c}{\lefte\right\rangle }_{a}{\leftg\right\rangle }_{b}\) for twoqubit gates, where \({\lefte\right\rangle }_{a}\) and \({\leftg\right\rangle }_{b}\) represent the ground and excited qubit eigenstates for qubits \(a\) and \(b\), respectively (e.g., see ref. ^{45}). The corresponding gate fidelities are then computed from
where \({t}_{g}\) is the appropriate gate time, and the ideal density matrix is computed in the absence of noise or leakagestate couplings.
Data availability
Data sharing is not applicable to this article as no data sets were generated or analyzed during this study.
Code availability
The simulation results reported may be obtained by following the computational scheme described in Methods section. All numerical calculations use the QuTiP software package^{51}. The Mathematica code for the characterization of the SOP (Supplementary Note 1) is provided as Supplementary Software 1.
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Acknowledgements
We are grateful to Mark Gyure, Joseph Kerckhoff, Thaddeus Ladd, and Emily Pritchett for illuminating discussions. This work was supported in part by the Army Research Office (W911NF1710274) and the Vannevar Bush Faculty Fellowship program sponsored by the Basic Research Office of the Assistant Secretary of Defense for Research and Engineering and funded by the Office of Naval Research through Grant No. N000141510029. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein.
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J.C.A.U. performed numerical simulations and analytical calculations. J.C.A.U., M.E., S.N.C. and M.F. analyzed the results and prepared the paper. This work was carried out under the supervision of S.N.C. and M.F.
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AbadilloUriel, J.C., Eriksson, M.A., Coppersmith, S.N. et al. Enhancing the dipolar coupling of a ST_{0} qubit with a transverse sweet spot. Nat Commun 10, 5641 (2019). https://doi.org/10.1038/s4146701913548w
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DOI: https://doi.org/10.1038/s4146701913548w
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