Cavity-mediated entanglement of parametrically driven spin qubits via sidebands

We consider a pair of quantum dot-based spin qubits that interact via microwave photons in a superconducting cavity, and that are also parametrically driven by separate external electric fields. For this system, we formulate a model for spin qubit entanglement in the presence of mutually off-resonant qubit and cavity frequencies. We show that the sidebands generated via the driving fields enable highly tunable qubit-qubit entanglement using only ac control and without requiring the qubit and cavity frequencies to be tuned into simultaneous resonance. The model we derive can be mapped to a variety of qubit types, including detuning-driven one-electron spin qubits in double quantum dots and three-electron resonant exchange qubits in triple quantum dots. The high degree of nonlinearity inherent in spin qubits renders these systems particularly favorable for parametric drive-activated entanglement. We determine multiple common resonance conditions for the two driven qubits and the cavity and identify experimentally relevant parameter regimes that enable the implementation of entangling gates with suppressed sensitivity to cavity photon occupation and decay. The parametrically driven sideband resonance approach we describe provides a promising route toward scalability and modularity in spin-based quantum information processing through drive-enabled tunability that can also be implemented in micromagnet-free electron and hole systems for spin-photon coupling.

Building on the promise of long coherence times for spins in silicon [15][16][17][29][30][31], strong spin-photon coupling [32][33][34][35][36] as well as coherent photon-mediated interaction of two single-electron silicon spin qubits [37,38] have now been achieved. While these results provide a path to scalability for spin-based quantum information processing, tuning and scaling challenges remain for applying this approach to more than two qubits. For resonant cavity-mediated qubit-qubit coupling [37], all qubit frequencies must be tuned into simultaneous reso- * vsriniv@uri.edu nance with the cavity frequency, and the micromagnets required in silicon for achieving sufficient spin-charge coupling must be precisely positioned for each qubit. In the standard dispersive approach to cavity-mediated coupling [21,23,35,[38][39][40], these constraints are partially relaxed as the cavity virtually mediates the interaction and its frequency is distinct from those of the qubits. However, an entangling interaction between the two qubits still requires their (cavity-shifted) frequencies to be tuned into mutual resonance. To allow for increased flexibility in achieving qubit-qubit entanglement, off-resonant coupling approaches have been developed for cQED systems with superconducting qubits [23,26,[41][42][43][44][45][46][47] and related two-qubit gates have also recently been explored in the context of semiconductor qubits [48][49][50][51][52][53][54]. These approaches enable qubits to be fixed at optimal operation points where decoherence is minimized, while interactions are activated via external microwave driving of either the coupling or one or more qubits.
In this work, we consider a pair of qubits based on electron spins in quantum dots that interact via microwave photons in a superconducting cavity, and that are also parametrically driven by classical external electric fields (Fig. 1). For this system, we formulate a model for entanglement between the two qubits that incorporates mutually off-resonant qubit and cavity frequencies and makes use of the Mollow triplet sidebands of the driven qubits [55,56] to effectively provide multiple qubit transition frequencies for cavity-mediated coupling. This approach enables highly tunable qubit-cavity photon interactions and qubit-qubit entanglement using ac control through  Figure 1. Schematic illustration of system for cavity-mediated coupling of two parametrically driven quantum dot spin qubits via sidebands. The qubits have transition frequencies ω1 and ω2 and are coupled to the fundamental mode of a microwave transmission line resonator (cavity), which has frequency ωc, with strengths g1 and g2, respectively. The qubits are also driven by external microwave fields with frequencies ω d 1 and ω d 2 . The approach in this work can be applied to multiple types of qubits that can be sinusoidally driven, including one-electron spin qubits in double dots and three-electron resonant exchange (RX) qubits in triple dots.
the applied driving fields, without requiring dc tuning of the qubit frequencies. The spin qubits can therefore be fixed at optimal operation points that allow for maximal coherence times. The model we develop can be mapped to a variety of qubit types. Here, we illustrate this mapping for both single-electron spin qubits in double quantum dots [57,58] and three-electron resonant exchange (RX) qubits in triple quantum dots [59,60] in the driven resonant regime [61].
We determine common resonance conditions for the two driven qubits and the cavity and identify parameters for implementing multiple entangling gates. In contrast to the sideband-based gates obtained for the driven resonant regime in prior work [61,62], entangling gates do not require sequences of multiple sideband pulses and additionally exhibit suppressed sensitivity to cavity photon occupation, as we verify through two-qubit gate fidelity calculations. The enhanced spectral flexibility inherent in the approach we describe provides a promising route toward scalability and modularity in spin-based quantum information processing through drive-enabled tunable entanglement that can also be implemented in micromagnet-free systems for spin-photon coupling [34][35][36].

II. THEORETICAL FRAMEWORK
We now develop a general theoretical description of the sideband-based, cavity-mediated entangling gates between driven qubits discussed in this work. To illustrate the approach in more concrete terms, we consider two specific types of quantum dot-based electron spin qubits for which strong spin-photon coupling has been realized ( Fig. 1 and Appendix A): (1) One-electron spin qubits in double quantum dots with micromagnets for spincharge coupling [32,33,38,57]; and (2) three-electron RX qubits in triple quantum dots, which couple directly to photons through their intrinsic electric dipole moments [34,35,[59][60][61]63]. Both types of qubits, as well as several other classes of spin qubits such as two-electron singlettriplet qubits [64][65][66][67][68][69], quantum dot hybrid qubits [70], and hole spin qubits in silicon and germanium double quantum dots [36,71], can be manipulated electrically by parametrically driving the detuning j between the (outer) two dots of qubit j [58][59][60]72]. We write this driving field as in terms of a dc operation point 0,j , as well as the amplitude 2F j , frequency ω d j , and phase φ j of an ac drive that sinusoidally modulates the detuning about 0,j as a function of time. Here, we fix the qubits at the "sweet spot" detuning operation points 0,j = 0 for j = 1, 2, where the first derivative of the qubit frequency vanishes for both one-electron spin and symmetric RX qubits, enabling leading-order protection from charge noise [57][58][59][60][72][73][74][75][76][77][78]. The operation point 0,j = 0 also maximizes the effective spin-photon coupling strength for one-electron spin qubits [58,72], while optimization of the RX qubitphoton coupling strength involves multiple parameters (see Appendix A for more details). In what follows, all sums are over the qubit index j = 1, 2 unless otherwise noted.
As shown in Appendix A, for both one-electron spin qubits and RX qubits driven according to Eq. (1), the system Hamiltonian including the cavity and driving fields with 0,j = 0 can be written in the qubit basis as ( = 1) where a † and a are photon creation and annihilation operators for the fundamental mode of the cavity with frequency ω c , σ α j with α = x, y, z and σ z j ≡ |1 j 1| − |0 j 0| are Pauli operators and ω j is the transition frequency for spin qubit j, g j is the strength of the coupling between spin qubit j and photons in the fundamental cavity mode, and 2Ω j is the effective driving amplitude for spin qubit j. We see from Eq. (2) that, for both types of qubits, the detuning drive [Eq. (1)] acting on the electron charge degrees of freedom is translated into an effective transverse (σ x ) drive on the qubit. Note that we have redefined the phases φ j of the drives with respect to Eq. (1) to take into account sign changes occuring in the derivation of H p (see Appendix A for details).
To derive sideband-mediated entangling interactions from Eq. (2), we first transform H p to a frame rotating at the frequencies of both drives and the cavity via which is equivalent to an interaction picture for resonant driving of the qubits, ω d j = ω j . Defining the cavity-drive detunings ∆ j ≡ ω c − ω d j and the qubit-drive detunings δ j ≡ ω j − ω d j , and making a rotating wave approximation for |∆ j | ω c +ω d j , 2ω d j , we drop rapidly oscillating terms ∼ e ±i(ωc+ω d j )t , ∼ e ±2iω d j t and find where we assume g j 2Ω j and take V (t) as a timedependent perturbation to the other terms in Eq. (4).
We initially take the limit g j → 0 and diagonalize H 0 . Choosing the phases of the driving fields to be φ j = 0 for j = 1, 2 to simplify the analysis, we find where we have for convenience re-expressed H 0 in terms of W j and θ j in the last line of Eq. (5), which serve to redefine the Pauli operators σ z j ≡ |e j e|−|g j g| and σ x j in terms of the dressed qubit basis |e j , |g j . We can then diagonalize this Hamiltonian via a rotation around the y axis for each qubit, described by where tan θ j = 2Ω j /δ j , which yields with the dressed qubit frequencies W j ≡ δ 2 j + 4Ω 2 j . Applying the same rotation U q [Eq. (6)] to the qubit-cavity coupling perturbation V (t) then yields the full Hamiltonian H q ≡ H 0,q + V q (t) in the dressed qubit basis, where V q (t) ≡ U † q V (t) U q . Finally, we transform to a second frame rotating at the dressed qubit frequencies W j for the driven qubits via which is equivalent to the interaction picture with respect to H 0,q [see Eq. (7)]. In this frame, we find the Hamiltonian In Eq. (9), we have defined which represents a sum of time-dependent qubit operators. We note that A (t) , and therefore V I (t) , have the three characteristic frequencies ∆ j and ∆ j ±W j for qubit j, which correspond to the Mollow triplet frequencies [55] consisting of the center and sideband frequencies of each driven qubit. In the present case, these frequencies are uniformly shifted by the cavity frequency ω c due to the rotating-frame transformation U 1 [Eq. (3)]. Driving the qubits on resonance such that ω d j = ω j gives δ j = 0 and W j = 2Ω j , so that sin θ j = 1 and cos θ j = 0. In this case, Eq. (10) reduces to and we find that the three characteristic frequencies for driven qubit j become ∆ j , ∆ j ± 2Ω j . In order to determine the gate operations generated by the time-dependent Hamiltonian V I (t) in Eq. (9), we use the Magnus expansion [79] up to second order to approximate the time evolution operator. For g j W j such that λ ≡ g j /W j is a small parameter, we write U (τ ) ≈ e −iH eff τ , where H eff (τ ) = λH 1 (τ ) + λ 2H 2 (τ ) represents an effective Hamiltonian to O λ 2 with We first consider the term λH 1 (τ ) . From Eqs. (9) and (10), we see that the integral in Eq. (12) is evaluated via the corresponding integrals of A (t) and its Hermitian conjugate. The first-order term in the effective Hamiltonian is then given by where we have defined the integral with frequencies µ = ∆ j , ∆ j ± W j . The first-order term λH 1 (τ ) describes the direct (∼ g j ) interaction of each qubit with the cavity and includes both red (∼ σ + j a, σ − j a † ) and blue (∼ σ + j a † , σ − j a) sideband terms. These interaction terms can be used to generate entanglement via sequences of multiple sideband pulses [23,42,43,61,62,80].
We now set ∆ j = p j η and W j = q j η with p j , q j integers and η ≡ 2π/τ. We also assume µ = 0. In this case, we find that µ = r j η with r j = p j , p j ± q j = 0 also an integer, so that all integrals f (µ) in Eq. (14) vanish and λH 1 (τ ) = 0. Thus, when ∆ j and W j are both integer multiples of the same frequency η, we can completely eliminate the first-order sideband interaction terms [Eq. (14)] from H eff (τ ) .
To O λ 2 , the effective interaction Hamiltonian generating the gate operation is now given entirely by the second-order term (16) Using Eqs. (9) and (10), we can write the commutator in Eq. (16) as where [V I , V I ] 1 contains terms involving only one-qubit operators and [V I , V I ] 2 contains all two-qubit operator terms along with some additional one-qubit operator terms, as we show below. We note in particular that [V I , V I ] 2 does not involve any photon operators. The form of the qubit-qubit interaction terms in the effective Hamiltonian, and thus the generated two-qubit gate, is determined by For j, k = 1, 2, we can write this commutator as The terms A † j A j − A † j A j in [V I , V I ] 2 lead to one-qubit terms in H eff (τ ) , while the qubit-qubit interaction terms arise entirely from the last line of Eq. (18). In Appendix B, we show that for ∆ j , W j = 0 and W j = |∆ j | , |2∆ j | , the complete one-qubit contribution to H eff (τ ) appearing at second order and originating from [V I , V I ] 1 and [V I , V I ] 2 reduces to These terms represent parametric driving-induced dispersive shifts that can be tuned via the drive amplitudes and frequencies, and are well defined in the absence of decay provided W j = |∆ j | . Such shifts can be harnessed for drive-tunable qubit measurement [81]. Two specific cases of interest are δ j = 0 and δ j = 0, corresponding to resonant and off-resonant driving of the qubits, respectively. For δ j = 0 and W j = 0, we find from Eq. (19) that Λ = 0 and the dispersive shift terms persist. In this case, the qubit frequencies are W j = 2Ω j and the dispersive shift terms in Eq. (19) become As we show in Appendix D for multiple example cases, the effects of Λ r on the dynamics for δ j = 0 can effectively be eliminated in specific situations of interest by an appropriate choice of parameters and operation times. On the other hand, for δ j = 0, we can choose ∆ j and W j such that Λ = 0 (see Appendix D for further details). The effective Hamiltonian H eff (τ ) then consists purely of qubit-qubit interaction terms, which arise from the terms with j = k in Eq. (18).

III. DRIVE-TAILORED ENTANGLING GATES VIA SIDEBAND RESONANCES
We next focus on the qubit-qubit interaction terms in H eff (τ ), which are given by Using Eqs. (10) and (B3) in Appendix B to write Eq. (21) in terms of functions h (µ 1 , µ 2 ) , h (µ 2 , µ 1 ) , and their complex conjugates, we can identify resonance conditions µ 1 = µ 2 that each give rise to specific qubit-qubit terms in V qq (τ ) . Since there are nine resonance conditions. Each condition corresponds to resonance between a center or sideband frequency of qubit 1 and a center or sideband frequency of qubit 2. These conditions and the corresponding qubit-qubit terms appearing in the effective Hamiltonian H eff (τ ) are derived in Appendix C and summarized in Table I, where we have defined ∆ ± j ≡ ∆ j ± W j . We now specifically consider the case of resonant qubit driving (δ j = 0 or ω d j = ω j ), so that W j = 2Ω j . The Hamiltonian for this case is given by V I (t) [Eq. (9)] with A (t) = A r (t) as given in Eq. (11). The characteristic frequencies are therefore ∆ j , ∆ j + 2Ω j , and ∆ j − 2Ω j , corresponding to center, red sideband, and blue sideband frequencies, respectively, for driven qubit j (shifted with respect to the cavity frequency ω c ). Assuming that we choose parameters such that the effects of the driveinduced dispersive shift terms Λ r in Eq. (20) can be neglected (see Appendix D for details), the evolution generated by the effective Hamiltonian H eff (τ ) reduces to that generated by the qubit-qubit interaction V qq (τ ) in Eq. (21). Thus, we consider only the dynamics generated by V qq (τ ) in what follows.
For each resonance condition, Table I gives the form of the interaction V qq ≡ V qq (τ ) for resonant qubit driving (second-to-last column), assuming that the dressed qubit frequencies W 1 and W 2 satisfy the associated constraints (third column). These constraints are based on Eq. (B3) and are obtained for each resonance condition by applying the condition µ 1 = µ 2 to the remaining resonance conditions in Table I, so that all qubit-qubit interaction terms except for the specified interaction V qq vanish (see Appendix C). We define a corresponding ideal two-qubit operation generated by V qq as U m ≡ U (τ m ) = e −iVqqτm , where τ m ≡ mτ = 2πm/η with m = 0, 1, 2, . . . represents the corresponding gate time and is an integer multiple of τ . By adjusting the drive amplitudes 2Ω j and frequencies ω d j to tune to a particular resonance condition and set W 1 and W 2 appropriately, it is then possible to select desired qubit-qubit interaction terms and two-qubit entangling gates. Examples of universal entangling gates are given in the last column of Table I. The drives can also be used to switch off a given interaction by tuning the sidebands away from the corresponding resonance condition. We emphasize that: (1) multiple two-qubit interaction terms exist even with mutually off-resonant frequencies (i.e., for j = k, ω j = ω k = ω c ); and (2) the cavity photon operators a, a † do not appear explicitly in V qq , suggesting suppression of errors due to cavity photon decay in the sideband-based entangling gate approach we propose in this work. Our analysis of expected gate performance in Sec. IV quantitatively demonstrates the presence of this suppressed cavity photon sensitivity.
To more concretely illustrate the approach, we now consider the effective qubit-qubit interaction and entangling gates generated between resonantly driven qubits (δ j = 0, W j = 2Ω j ) for the specific resonance conditions 7 and 9 in Table I. First, we consider ∆ − 1 = ∆ − 2 (resonance condition 7), which is equivalent to ∆ 1 − W 1 = ∆ 2 − W 2 . As described in Appendix C, the resonance condition can also be written as ω 1 + 2Ω 1 = ω 2 + 2Ω 2 for resonantly driven qubits and describes resonance between the blue sidebands of both qubits [ Fig. 2(a)]. Setting ∆ j = p j η and W j = q j η with p j , q j integers leads to the equivalent condition p 1 − q 1 = p 2 − q 2 . We also define the integer Table I, the constraints associated with resonance condition 7 are W 1 = W 2 , 2W 2 , W 2 /2. Assuming these constraints are satisfied, the qubit-qubit interaction takes the form This interaction can be used to generate the iSWAP gate, which together with single-qubit rotations constitutes a universal set of quantum gates [82] and also represents a key element of recently proposed improvements to the implementation of quantum error correction using the surface code [83]. Since V qq is independent of the photon operators a, a † , U m ≡ e −iVqqτm acts nontrivially only on the qubits and we can work in a subspace of fixed photon number n. Thus, we now project all states and operators into the subspace defined by P n ≡ |n n| . For notational simplicity, we use U m to denote the evolution operators within the n-photon twoqubit subspace {|ee, n , |eg, n , |ge, n , |gg, n } in the re- Qubit sidebands in resonance but highly detuned from cavity Qubit sidebands in resonance and near cavity  Table I). The diagram shown depicts a case in which the resonant sidebands are highly detuned from the cavity frequency ωc. (b) Blue-red sideband resonance, described by ∆ = ∆ − 1 = ∆ + 2 (resonance condition 9 in Table I). The diagram depicts a case in which the resonant sidebands are near the cavity frequency ωc. In both cases, the qubit frequencies ω1,2 and the cavity frequency ωc are mutually off-resonant, i.e., ω1 = ω2 = ωc. mainder of this work unless otherwise specified. Defining Σ x ≡ |eg ge| + |ge eg| (where we have suppressed the photon number state |n for convenience), we can write V qq = −J Σ x and U m = e iJ τmΣx . In the full two-qubit dressed basis {|ee , |eg , |ge , |gg } , U m takes the form In order to obtain an iSWAP gate U iSW , we set τ m = mτ = π/2J . We choose the initial state |ψ i = |eg for our analysis. To cancel the dynamics due to the dispersive shift terms in Eq. (20) for this case, we also set χ 1 = χ 2 . As shown in Appendix D, both the generation of the iSWAP gate and the drive-induced dispersive shift cancellation can be achieved for resonance condition 7 by choosing parameters that satisfy the constraints in Eqs.
(D9) and (D10). These relations are satisfied for multiple sets of parameters. For the analysis in this work, we choose the set of parameters shown for resonance condition 7 in Table II. The evolution time unit in the Magnus expansion becomes τ = 2π/η = 20 ns, yielding an iSWAP gate time τ m = mτ = 800 ns. The ideal gate evolution generated by V qq at time τ m for resonance condition 7 and these parameters approximates well the numerical evolution directly due to Eq. (9) according to the timedependent Schrodinger equation and in the absence of qubit and cavity decay, with a fidelity F 0 ≈ 0.998 (we choose the subspace with n = 0, 1, 2 and the initial state |ψ i = |eg, 0 for the numerical analysis; see Appendix D). Resonance condition 9 in Table I is given by For resonantly driven qubits, this condition can also be written as ω 1 + 2Ω 1 = ω 2 − 2Ω 2 (see Appendix C) and describes resonance between the blue sideband of qubit 1 and the red sideband of qubit 2 [ Fig. 2(b)]. For ∆ j = p j η and W j = q j η, the resonance condition can also be expressed as We again note that V qq is independent of the photon operators a, a † and generates the gate U m ≡ e −iVqqτm within the n-photon subspace at time τ m = mτ. In terms of Σ x ≡ |ee gg| + |gg ee| , we find V qq = J Σ x and U m = e −iJ τmΣx , yielding in the full two-qubit dressed basis. The gate at τ m = mτ = −π/2J is analogous to an iSWAP gate but acts in the subspace spanned by {|ee , |gg } . We denote this gate, which we refer to as the double-excitation gate, by U iDE . As the gate is related to U iSW via a rotation of qubit 2, U iDE together with single-qubit rotations also constitutes a universal set of quantum gates. For our analysis, we choose the initial state |ψ i = |ee and also set χ 1 = −χ 2 to cancel the dynamics due to the dispersive shift terms in Eq. (20) (Appendix D). Simultaneous generation of the gate U iDE and cancellation of the driveinduced dispersive shifts for resonance condition 9 is possible by choosing parameters that satisfy the constraints in Eqs. (D16) and (D17). As in the case of resonance condition 7, these relations are satisfied for multiple sets of parameters. Here, we choose the parameters specified Table II. Parameter values used in the effective interaction Hamiltonian and entangling gate analysis for resonance conditions 7 and 9 in Table I, assuming resonantly driven qubits (δj ≡ ωj − ω d j = 0 for j = 1, 2) such that Wj = 2Ωj. Each set of parameters satisfies the associated resonance condition with Wj = 2Ωj = qjη, ∆j ≡ ωc − ω d j = pjη, ∆ ± j ≡ ∆j ± Wj, J = g1g2/4∆, χj as given in Eq. (20), and the constraints in Eqs. (D9) and (D10) [Eqs. (D16) and (D17)] for resonance condition 7 (9).

Parameter
Resonance condition 7:  Table II for resonance condition 9. The evolution time unit in the Magnus expansion is again τ = 2π/η = 20 ns, yielding a gate time τ m = mτ = 200 ns for U iDE . Comparison of the ideal gate evolution generated by V qq at time τ m for resonance condition 9 and these parameters with the numerical evolution directly due to Eq. (9) according to the time-dependent Schrodinger equation and in the absence of qubit and cavity decay again yields a fidelity F 0 ≈ 0.998 (as described in Appendix D, we again choose the subspace with n = 0, 1, 2 along with the initial state |ψ i = |ee, 0 for the numerical analysis). We therefore find that, for both resonance conditions 7 and 9 and appropriately selected parameters, the dynamics due to V I with cavity photon operators explicitly included are well approximated by the dynamics generated by just the two-qubit interaction V qq , from which cavity photon operators are absent. In the next section, we show that this absence of explicit cavity photon dependence in the effective Hamiltonian is manifested in the full dynamics as suppressed sensitivity of these sidebandbased entangling gates to cavity photon decay.

IV. SIDEBAND GATE PERFORMANCE IN THE PRESENCE OF QUBIT AND CAVITY DECAY
To evaluate the performance of the sideband resonance-based gates U iSW and U iDE and quantitatively illustrate the reduced dependence of the dynamics on cavity photons, we use a master equation analysis and numerically calculate the gate fidelity for resonance conditions 7 and 9 in the presence of qubit and cavity decay. Here, we assume that dephasing in the original qubit basis with rate γ j for qubit j and cavity photon loss with rate κ are the dominant sources of decoherence, as is relevant for silicon quantum dots [61]. We write the master equation as [23,61] with H p given by Eq. (2). Following steps similar to those used to obtain the interaction-picture Hamiltonian V I [Eq. (9)], we transform the master equation in Eq. (24) to the interaction picture. Moving to a rotating frame via U 1 [Eq. (3)], making a rotating wave approximation  Table II. (b) Error in double excitation gate UiDE, generated by a blue-red sideband resonance [ Fig. 2(b)] with the initial state |ψi = |ee, 0 and parameters for resonance condition 9 given in Table II. Lines are guides for the eye. (6)] to change to the dressed qubit basis, and moving to the interaction picture via U 2 yields, after setting δ j = 0 for resonant qubit driving and dropping rapidly oscillating terms ∼ e ±2iWj t , where (25) is the master equation describing the dynamics in the interaction picture. For the numerical calculations, we again work in the photon subspace with n = 0, 1, 2 and set γ 1 = γ 2 ≡ γ for simplicity. To analyze the effects of qubit and cavity decay on the performance of the entangling gates, we calculate the fidelity where ρ I (τ m ) denotes the final state at time τ m for the evolution obtained by numerically integrating Eq. (25) and ρ (0) I (τ m ) denotes the final state for the ideal evolution given by γ = κ = 0. We calculate this fidelity as a function of γ and κ for the resonance conditions 7 and 9 using the parameter sets in Table II and the initial states chosen above for the ideal gates U iSW and U iDE . The initial state is ρ I (0) = |ψ i ψ i | = |eg, 0 eg, 0| for resonance condition 7 and ρ I (0) = |ψ i ψ i | = |ee, 0 ee, 0| for resonance condition 9.
In Fig. 3, we plot the error 1 − F (τ m ) for the two resonance conditions and corresponding two-qubit entangling gates. We find theoretical entangling gate fidelities F > 0.995 for the full range of κ shown (up to κ = 100 kHz) and γ 1 kHz (γ 10 kHz) for resonance condition 7 (9). While γ and κ are varied over three orders of magnitude in both cases, the error for both resonance conditions depends more strongly on qubit decay γ than on cavity decay κ. This reduced dependence on κ is expected for the dispersive regime, in which the cavity virtually mediates the qubit-qubit interaction in the absence of direct qubit-photon interaction, and was also found in the results of Ref. [61] for the dispersive regimes relative to the driven resonant regime of cavity-mediated coupling. We note, however, that the gates derived here are based on interactions via sideband resonances, as in the driven resonant regime.
For resonance condition 7, given by ∆ − Fig. 3(a)], the resonant blue sidebands of the driven qubits are highly detuned from the cavity frequency . We see that the error varies over approximately three orders of magnitude with γ but over less than one order of magnitude with κ. The suppressed sensitivity to κ is expected given the large detuning between the qubit sidebands and the cavity. On the other hand, for resonance condition 9, given by ∆ − 1 = ∆ + 2 [ Fig. 3(b)], the resonant sidebands -the blue sideband of qubit 1 and the red sideband of qubit 2 -are close to ω c . Here, Fig. 2(b)]. In this case, we see that the error again varies over approximately three orders of magnitude with γ, but varies over less than two orders of magnitude with κ. The increased variation with κ relative to resonance condition 7 reflects the smaller detuning ∆ of the resonant qubit sidebands from the cavity. However, even in this case, we find that the sensitivity of the error to κ is suppressed relative to the sideband-based two-qubit gates in the driven resonant regime (compare Fig. 7 in Ref. [61]).
While the reduced sensitivity to cavity decay is consistent with the dispersive regime, it also reflects the absence of explicit cavity dependence in the effective interaction Hamiltonian V qq generating the two-qubit entan-gling gates [Eq. (21) and Table I]. We have seen (Sec. III) that for appropriately chosen parameters, the gates generated by V qq closely approximate the dynamics due to the full interaction-picture Hamiltonian V I in Eq. (25), where a and a † are explicitly present in general [see Eq. (9)]. Thus, the effective Hamiltonian we derive here illustrates that tuning the parametric drive frequencies and amplitudes with the remaining parameters set appropriately effectively enables suppression of the sensitivity to cavity decay.

V. CONCLUSIONS
In this work, we have developed an approach for achieving long-range interactions between a pair of driven spin qubits via cavity-mediated coupling combined with sideband resonances. Our approach is applicable to a variety of qubit types that can be controlled via parametric driving, including one-electron spin qubits in double quantum dots, three-electron RX qubits in triple quantum dots, and hole spin qubits, and enables highly tunable qubit-qubit interactions whose nature can be tailored via the driving fields. The interactions can also be switched on and off using only ac control, without requiring dc tuning of the qubits away from optimal operation points and thus allowing for improved qubit coherence relative to resonant and standard dispersive approaches for cavity-mediated qubit coupling.
We note that the approach we describe here is based on the driving of inherently nonlinear effective two-level systems (i.e., qubits), whereas commonly used off-resonant coupling approaches for weakly anharmonic superconducting qubits such as transmons effectively involve driving multilevel systems [26]. As transmon anharmonicities are typically no more than a few hundred MHz, rapid and high-fidelity gates in these systems are limited by leakage for sufficiently large drive amplitudes and pulse bandwidths such that transitions to states outside the qubit space can be excited, in the absence of additional techniques that compensate for weak anharmonicity [84][85][86]. Limits to the entangling fidelity and scalability of cross-resonance [44,45] and FLICFORQ [23,41] gates also exist due to small anharmonicities, required qubit frequency spacing, available bandwidth for control, and spurious interaction terms ∼ σ z 1 σ z 2 that cannot be eliminated for conventional transmons [26]. By contrast, spin qubit systems such as those considered in this work are characterized by highly nonlinear spectra in which differences of > 1 GHz in transition frequencies are routinely realized, including within hybrid cQED systems [32][33][34]. Therefore, spin qubits should in principle allow for greater flexibility in the choice of amplitudes, frequencies, and gate times for achieving desired highfidelity gates via parametric driving, without requiring the added complexity of low-anharmonicity compensation techniques.
For our parametrically driven, sideband-based ap-proach, we expect that limits on the driving amplitudes and entangling rates for implementing high-fidelity gates will instead arise primarily from the requirements that ∆ j and W j are both integer multiples of the same frequency η = 2π/τ to eliminate the first-order interaction in Eq. (14), which sets a lower bound on τ and thus τ m since η ≤ |∆ j | , W j , together with the conditions g j W j required for the validity of the effective Hamiltonian, the tuning of the drive amplitudes and frequencies to a desired resonance condition and interaction (Table I), and the constraints for eliminating other interaction terms and dynamics due to the parametric drive-induced dispersive shifts. As we have shown in this work, multiple sets of experimentally relevant [87] parameters exist for which these requirements can be simultaneously satisfied (see, e.g., Table II) in order to select desired and eliminate undesired interaction terms.
We have analyzed specific examples of sideband-based entangling gates that include a 200-ns double-excitation gate U iDE , which is generated via a blue-red sideband resonance and does not exist for the standard dispersive regime in the absence of driving fields [21,23,35,38]. Furthermore, the rates of the entangling gates described in this work are set by J ∝ ∆ −1 , in contrast to the typical ∼ ∆ −1 j scaling for standard dispersive entangling gate rates (where ∆ j = ω c − ω j for resonant qubit driving). As it is possible to have ∆ < ∆ j for multiple sideband resonance conditions (see, e.g., Table II), the corresponding entangling gates can potentially be more rapid than those in the standard dispersive regime. As the gates do not require sequences of multiple qubit-cavity sideband pulses, the potential also exists for gate speed improvements relative to the sideband-based gates in the driven resonant regime considered in prior work [61,62].
Unlike the resonant and standard dispersive approaches, realizing cavity-mediated entangling interactions via the sideband resonance method we describe here does not rely on simultaneous mutual resonance among multiple qubit and cavity photon frequencies. Instead, several sideband resonance conditions are available for generating entanglement between dressed qubits even with all qubit and cavity frequencies mutually offresonant, providing enhanced spectral flexibility. As a result, the sideband resonance-based approach represents a potential alternative to the challenging tuning required to bring single-spin qubit frequencies into simultaneous resonance via precisely oriented micromagnets that has been essential to spin-spin coupling demonstrations in silicon to date [37,38]. Together with the suppressed sensitivity to cavity decay expected from our analysis of example entangling gates, these features render the approach we present in this work favorable for scaling and promising for the implementation of modular quantum information processing with spin qubits.

Supported by Army Research Office Grants W911NF-15-1-0149 and W911NF-23-1-0104.
Appendix A: Hamiltonian for cavity-coupled driven spin qubits The Hamiltonian H p in Eq. (2) describes parametrically driven qubits coupled via the fundamental mode of microwave cavity photons and forms the basis for the sideband-based cavity-mediated entangling gates derived in this work. Here, we show how we obtain H p for the two specific examples of driven spin qubits illustrated in Fig. 1.

Driven one-electron spin qubits in double quantum dots
We first consider two one-electron spin qubits in double quantum dots (DQDs) coupled via a microwave cavity [37,38]. In the following analysis, we take into account only the lowest-energy orbital level in each dot. The charge dipole of the electron in each DQD couples to the electric field of a microwave cavity photon [20], and the spin of the electron couples to the charge via spin-orbit coupling and/or a magnetic field gradient [88]. We focus on electrons occupying the lowest-energy valley states within silicon quantum dots, for which spin-charge coupling is achieved through gradient fields produced by a micromagnet [32,33,37,38,57].
Assuming coupling to only the fundamental cavity photon mode with frequency ω c , we write the system Hamiltonian including parametrically driven detuning as where τ z j ≡ |L j L| − |R j R| with |L j and |R j the lowest-energy orbital in the left and right dots of DQD j, respectively, s z j ≡ |↑ j ↑| − |↓ j ↓| is the Pauli z operator for the electron spin in DQD j, and the other Pauli orbital (charge) and spin operators are defined accordingly. The remaining parameters in Eqs. (A1) and (A2) are the detuning j between the orbital levels in the left and right dot, the interdot tunnel coupling 2t j , the Zeeman splittings B z j and B x j due to the net magnetic field components along the z and x axes, respectively (due to both the external and micromagnet fields [32]), and the charge-cavity coupling strength g c,j for DQD j.
We describe sinusoidal (ac) driving of the detuning for each DQD via Eq. (1) and choose the "sweet spot" detuning operation points 0,j = 0 for j = 1, 2 where the first derivative of the charge qubit frequency vanishes, enabling leading-order protection from charge noise [57,58,72,89]. As in the main text, all sums are over the qubit index j = 1, 2 unless otherwise noted. We first apply the rotation to the charge subspace. The system Hamiltonian H s becomes Writing the transformed DQD Hamiltonian H d in the rotated charge-spin product basis 2 are the double-dot charge eigenstates for 0,j = 0, reveals a block-diagonal structure with two decoupled subspaces that we label as H a,j and H b,j and that are spanned by |+, ↑ j , |−, ↓ j and |+, ↓ j , |−, ↑ j , respectively [57,88].
Full diagonalization of the DQD low-energy space including spin for 0,j = 0 is then achieved by applying whereα y j ≡ −i |+, ↑ j −, ↓| − |−, ↓ j +, ↑| ,β y j ≡ −i |+, ↓ j −, ↑| − |−, ↑ j +, ↓| , and tan Φ a(b),j = B x j / 2t j ± B z j . For 2t j − B z j 2t j + B z j , tan Φ a,j tan Φ b,j and the degree of mixing between |−, ↓ j and |+, ↑ j is much smaller than that between |−, ↑ j and |+, ↓ j . The eigenstates in the subspace H a,j can then be approximated as [57] |0 j ≈ |−, ↓ j and |3 j ≈ |+, ↑ j , and the corresponding eigenvalues (for = 1) are ω 0,j = −W j /2 and ω 3, convenience, the eigenstates in the subspace H b,j are given by and the corresponding eigenvalues are ω 1,j = −V j /2 and Defining the operators σ kl j ≡ |k j l| , we find in the DQD eigenstate basis where we have defined The Hamiltonian in Eq. (A9) describes each DQD in terms of a multilevel picture in which the ground state of the electron is |0 j ≈ |−, ↓ j and the highest excited state is |3 j ≈ |+, ↑ j , while the dominant chargespin character of |1 j and |2 j for 0,j = 0 depends on the relative magnitudes of 2t j and B z j [57]. To reduce the DQD description to an approximate two-level picture corresponding to a spin qubit, we choose the case 2t j > B z j (illustrated in Fig. 2(a) of Ref. [57]) such that |1 j ≈ |−, ↑ j is the first excited state and |2 j ≈ |+, ↓ j is the second excited state for each DQD. We also assume ω 1,j − ω 0,j ω 2,j − ω 1,j , which is equivalent to To first order in these small parameters, Eq. (A9) can be written in the form where we have defined σ z j ≡ |1 j 1| − |0 j 0| and the effective spin qubit frequencies ω j ≡ ω 1,j − ω 0,j = (W j − V j ) /2, 2Ω j ≡ F j |sin (Φ j /2)| is the effective amplitude of the drive acting on the spin qubit, g j ≡ g c,j |sin (Φ j /2)| is the effective spin-photon coupling strength, and the sign of sin (Φ j /2) has been taken into account by making the replacement φ j → φ j and redefining the phase of a accordingly in Eq. (A9). The Hamiltonian H (1) p is identical in form to Eq. (2). Finally, we note that the Hamiltonian for DQD charge qubits [89] at the operation points 0,j = 0 [described by setting B z j = B x j = 0 in Eq. (A2)] can also be written in a form analogous to Eq. (A11), with σ z,x j → τ z,x j , ω j → 2t j , 2Ω j → F j , and g j → g c,j .

Three-electron resonant exchange qubits in triple quantum dots
We now consider two resonant exchange (RX) qubits, each defined by the spin states of three electrons in a triple quantum dot [59,60], that interact via a microwave cavity [61]. In contrast to one-electron spin qubits, each RX qubit couples directly to the electric field of microwave cavity photons via an intrinsic electric dipole moment. This dipole moment arises from the admixture of polarized charge states in the qubit states, without requiring spin-orbit coupling, magnetic gradients, or the fabrication of additional device elements such as micromagnets. The Hamiltonian given in Eq. (29) of Ref. [61] that is used to describe RX qubits coupled to a microwave cavity in the driven resonant regime has the same general form as Eq. (2). Here, we briefly summarize the theory used to derive this Hamiltonian for the case of RX qubits and adapt it to the case of silicon triple quantum dots.
An effective Hamiltonian for each RX qubit can be obtained from a Hubbard model for electrons confined within a linear triple quantum dot [60,61]. This model can be used to calculate a charge stability diagram ( Fig. 1(b) of Ref. [60]) that describes the triple dot in terms of the lowest-energy charge configuration (n 1 , n 2 , n 3 ) (where n i denotes the occupation number for dot i and the occupation numbers are ordered from the left dot to the right dot) as a function of the gate voltages applied to the three dots, which set the on-site energies (−ε 1 , −ε 2 , −ε 3 ) . For fixed V tot ≡ i ε i , the lowestenergy charge configuration depends on both the detuning ≡ (ε 3 − ε 1 ) /2 and the relative middle dot on-site energy V m ≡ −ε 2 + 1 2 (ε 1 + ε 3 ) . Distinct charge configurations are coupled via the left-center and center-right interdot tunneling amplitudes t l and t r , respectively.
References [60] and [61] consider a three-electron system in the subspace of the charge configurations (1, 1, 1) , (2, 0, 1) , and (1, 0, 2) . The relevant operation point is illustrated in Fig. 1(b) of Ref. [60]. Here, we focus on a silicon triple dot and assume that a sufficiently large ( 100 mT [59]) static magnetic field is applied to the triple dot. Furthermore, we assume that excited valley states are well-separated in energy from the lowest-energy valley manifold (by a valley splitting energy E V 100 µeV [12,[90][91][92]), such that a singleorbital picture is valid. We can then define a spin qubit using three-electron states in the lower-energy S = 1/2 spin subspace, which have a spin quantum number for the total z component m s = −1/2 due to the positive electron g-factor of silicon. The relevant subspace is spanned by the (1, 1, 1) states together with the (2, 0, 1) and (1, 0, 2) states where |vac denotes the vacuum.

and the exchange between the center and left (right) dots is
Here, ∆ is defined in terms of Hubbard model parameters and V m [60,61]. Diagonalizing H (3) eff yields H 0 = ωσ z /2 with σ z ≡ |1 1| − |0 0| , where the eigenstates |0 and |1 define the RX qubit and ω ≡ J 2 + 3j 2 is the qubit energy splitting (here, we set = 1). As the exchange interactions J l and J r between dots are controlled using only voltages applied to the triple quantum dot, the RX qubit is a spin qubit that can be fully manipulated using electric fields alone [59,60,93,94].
In addition to being fully controllable via dc electric field pulses, the RX qubit couples directly to microwave photons by virtue of the small admixture of the polarized charge states (2, 0, 1) and (1, 0, 2) in the logical qubit states [60,61]. This feature enables full resonant control of the RX qubit via electric fields, in direct analogy to resonant control of individual electronic and nuclear spins via magnetic fields in electron spin resonance (ESR) and nuclear magnetic resonance (NMR). The same charge admixture also enables coupling of the qubit to the electric field of photons in a microwave cavity, with a strength characterized by the charge admixture parameter ξ ≡ t/∆ (here, t ≡ t l = t r ). The parameter ξ is a measure of the electric dipole moment of the qubit and is inversely proportional to ∆, which sets the width of the (1, 1, 1) region and is tunable via V m .
We can write the Hamiltonian for the RX qubit including electric dipole coupling as H RX = H 0 + H int , where H int is the dipole interaction in the RX qubit basis. The operation point ( 0 , ∆ 0 ) for the RX qubit determines the qubit frequency ω and the specific form of the electric dipole moment. Variations in both and ∆ about this operation point enable coupling to microwave cavity photons [34,60,61,63,78] and are implemented via gate voltage control of the on-site energies −ε i . Here, we focus on electric dipole coupling for small variations in the detuning [60,61]. For this case, the electric dipole moment of the RX qubit is along the triple dot axis and can be described in terms of the operator d = ew0 2 (n 1 − n 3 ) , where e is the magnitude of the electron charge and w 0 is the size of the triple dot (i.e., the distance between the centers of the outer dots). The operation point = 0 for the RX qubit is chosen such that the coupling to variations in the detuning [see, e.g., Eq. (1)] are perpendicular to the quantization axis of the RX qubit. In the basis of the RX qubit states, the dipole interaction of the triple dot with the fundamental cavity mode then takes the form [61] where the effective qubit-photon coupling strength is given by and g c is the charge-cavity coupling strength (an expression for g c is derived from a circuit model in Ref. [61]). For the qubit operation point 0 = 0 chosen in this work, t l = t r ≡ t and g = √ 3ξ 2 g c /2. Thus, maximizing ξ maximizes the coupling strength g.
In the driven resonant regime described in Ref. [61], two RX qubits interact with microwave cavity photons, whose frequency we here denote as ω c , in the presence of an external driving field of frequency ω d ≡ ν applied to the cavity. A displacement transformation in the regime of large driving field amplitude and large cavity-drive detuning ω c − ω d g, ω − ω d then effectively transfers the drive from the cavity to the qubit and leads to Eq. (29) in Ref. [61] for each qubit. We can write this Hamiltonian for two RX qubits as where we have redefined the phase φ j to match the form of H p in Eq. (2). We see that, like H has a form identical to H p . We also note that this form for the Hamiltonian can be obtained for cavity-coupled RX qubits without a displacement transformation by directly driving the detuning of the RX qubits [59,60] as described by Eq. (1). Thus, the sideband-based interactions and associated entangling gates that we derive from H p can be implemented using both RX qubits and one-electron spin qubits.
Appendix B: One-qubit second-order terms in effective Hamiltonian In this appendix, we analyze in more detail the onequbit terms appearing at second order in the effective Hamiltonian H eff (τ ) [Eq. (16)], which originate from the commutators [V I , V I ] 1 and [V I , V I ] 2 defined in Eqs. (17) and (18). The one-qubit terms arising from [V I , V I ] 1 are given by and those arising from [V I , V I ] 2 are where A j ≡ A j (t) and A j ≡ A j (t ) . We note that the cavity photon operators a, a † appear in each term of Λ 1 but are absent from Λ 2 .
Substituting for A j and A j using Eq. (10) shows that evaluating the double integrals in Eqs. (B1) and (B2) amounts to evaluating double integrals of products of exponentials of the form Here, µ, µ ∈ {±∆ j , ± (∆ j + W j ) , ± (∆ j − W j )} . Since we set ∆ j = p j η and W j = q j η with integers p j and q j , we have µ j = r j η and µ j = r j η where r j and r j are also integers. Equation (B1) then becomes and Eq. (B2) becomes Using Eqs. (B4) and (B5) together with Eq. (B3), we can obtain the resonance conditions µ = µ under which h (µ, µ ) = 0 and particular one-qubit terms appear (i.e., are nonzero) in the effective Hamiltonian H eff (τ ).
Note that these conditions are identical for h (µ, µ ) and h * (µ, µ ) . The one-qubit resonance conditions and corresponding forms of the effective Hamiltonian terms are summarized in Table III. We see that the a 2 and a †2 terms occur only for W j = |2∆ j | or ∆ j = 0. Assuming that we choose an operation point such that these conditions are not satisfied, the a 2 and a †2 terms can be eliminated from the Hamiltonian. Additionally, the case W j = 0 corresponds to a vanishing gap for dressed qubit j and thus does not have well-defined physical meaning. We can therefore also ignore the σ + j a † a, σ − j a † a, σ + j , and σ − j terms. As a result, the only remaining one-qubit terms appearing at second order in the effective Hamiltonian are those of the form σ z j a † a in Eq. (B4) and σ z j in Eq. (B5). These terms exist for any values of W j and ∆ j . Finally, we note that Eq. (B3) is well defined only for µ, µ = 0, which leads to the constraints ∆ j = 0 and W j = |∆ j | . The full one-qubit contribution to H eff (τ ) for ∆ j , W j = 0 and W j = |∆ j | , |2∆ j | is thus

Resonance condition Hamiltonian terms
where we have used cos θ j = δ j /W j [see Eq. (5)]. Equation (B6) represents the dispersive shift terms induced by the parametric driving [81].

Appendix C: Qubit-qubit interaction terms in effective Hamiltonian
The qubit-qubit interaction terms in H eff (τ ) [Eq. (16)], and therefore the generated two-qubit gates U (τ ) ≈ e −iH eff τ , are given by Eq. (21). Here, we derive the form of V qq (τ ) for each of the nine resonance condi- As noted in the main text, these resonance conditions are obtained by substituting Eq. (10) into V qq (τ ) , which yields terms with frequencies µ 1 − µ 2 , and subsequently applying Eq. (B3) to identify nonzero terms. In what follows, we use the abbreviated notation ∆ ± j ≡ ∆ j ± W j and refer to the resonance conditions by the numbers given in Table I.
First, we consider the resonance condition ∆ 1 = ∆ 2 (resonance condition 1), which is equivalent to ω d 1 = ω d 2 and therefore describes the driving of both qubits at the same frequency. In order to select a specific form for the two-qubit interaction, we also assume that the constraints W 1 = ±W 2 are satisfied so that terms in V qq (τ ) with frequencies µ 1 − µ 2 = ∆ 1 − ∆ 2 remain nonzero and vanish according to Eq. (B3). In terms of the resonant detuning ∆ ≡ ∆ 1 = ∆ 2 , we find that Eq. (21) becomes where we have defined J ≡ g 1 g 2 /4∆. For resonant qubit driving such that δ 1 = δ 2 = 0, W 1 = 2Ω 1 , W 2 = 2Ω 2 , and sin θ 1 = sin θ 2 = 1. In this case, ∆ 1 = ∆ 2 corresponds to ω 1 = ω 2 . In other words, since the detunings of both qubit frequencies from the cavity frequency ω c are aligned, the center frequencies of the driven qubits are also resonant. Equation (C1) then reduces to which can be used to generate a controlled-phase gate (see Appendix D).

Appendix D: Elimination of drive-induced dispersive shift dynamics from effective Hamiltonian
As shown in Appendix B, the parametric drive-induced dispersive shift terms Λ [Eq. (B6)] in general exist for any W j and ∆ j (provided W j = |∆ j |). Nevertheless, it is possible to effectively eliminate the dynamics generated by Λ in specific cases by appropriately choosing parameters and operation times such that the gate in the presence of Λ is equivalent to that for Λ = 0. Here, we first illustrate this method for the resonant qubit driving case δ j = 0 and the resonance conditions 7 and 9 in Table  I and Fig. 2. We then consider the effects of Λ for additional resonance conditions, as well as the off-resonant qubit driving case δ j = 0 for which we can choose parameters such that Λ = 0.
We first consider resonant qubit driving, described by δ j = 0 such that W j = 2Ω j , Λ = Λ r , and χ j = 0. From Eqs. (16), (20), and (21), the effective Hamiltonian for ∆ j = p j η and W j = q j η with p j , q j integers, ∆ j , W j = 0, and W j = |∆ j | , |2∆ j | is given by As discussed in the main text, the desired ideal twoqubit evolution is generated by the qubit-qubit interaction term V qq according to U m ≡ e −iVqqτm , where τ m ≡ mτ = 2πm/η is the corresponding gate time. We now also define an evolution operator with the dispersive shift terms included as U m ≡ e −iH eff τm , where H eff is given by Eq. (D1). From Eq. (21), we note that V qq is independent of photon operators a, a † . We also see from Eq. (20) that Λ r is diagonal in the photon number n. We can therefore write where Λ n ≡ P n Λ r P n = n| Λ r |n P n is the operator describing the dispersive shift terms within the n-photon subspace and P n ≡ |n n| is the corresponding subspace projector, and work in a subspace of fixed n. Noting that V qq (τ ) consists only of qubit operators [see Eq. (21)], we find H (n) eff ≡ P n H eff P n = P n (V qq + Λ r ) P n = V qq + Λ n , where we now use V qq to denote the n-photon subspace operator. In the basis {|ee, n , |eg, n , |ge, n , |gg, n } , Λ n takes the form while the form of V qq depends on the particular resonance condition according to Table I. The evolution with the dispersive shift terms included is described by P n U m P n = e −iH (n) eff τm = e −i(Vqq+Λn)τm . As in the main text, we use U m and U m to denote the evolution operators within the n-photon two-qubit subspace and suppress the photon number state |n in what follows.
We first consider resonance condition 7, which is defined by In the main text, we noted that the ideal evolution within the n-photon subspace can be written as U m = e iJ τmΣx , where Σ x ≡ |eg ge| + |ge eg|. The ideal operation U m therefore acts nontrivially only within the two-dimensional subspace {|eg , |ge } , and Σ x has an action analogous to the Pauli operator σ x in this subspace.
From Eq. (D3) and the form of V qq for resonance condition 7 [Eq. (C8)], we find that H (n) eff is block-diagonal. In this case, we can write H where we have defined the two-qubit subspace projectors P ≡ |ee ee| + |gg gg| , Q ≡ |eg eg| + |ge ge| . (D4) We can then characterize the effect of Λ n on the dynamics for a given initial state |ψ i via a fidelity where we have used U m = P U m P + QU m Q and U m = P U m P + QU m Q.
For resonance condition 7, we find P H (n) eff P = n + where we have defined the operator Σ z ≡ |eg eg| − |ge ge| in analogy to the Pauli operator σ z . For an initial state |ψ i in the subspace associated with Q, evolution for Λ n = 0 remains within this space such that U m |ψ i = QU m Q |ψ i . In the main text, we consider the initial state |ψ i = |eg . For the iSWAP gate U iSW , which is equivalent to U m at time τ m = π/2J [see Eq. (22)], the ideal final state is We now set n = 0 for simplicity [in what follows, generalization of the expressions to any n using Eq. (D3) is straightforward and amounts to the replacements (χ 1 ± χ 2 ) /2 → n + 1 2 (χ 1 ± χ 2 )]. In the presence of Λ 0 and defining the unit vectorû ≡ [( Using Eqs. (D5) and (D7) along with P U m P |ψ i = P U m P |ψ i = 0, we find which oscillates with the modified frequencyΩ instead of J in the ideal case. Nevertheless, we can recover the ideal dynamics and obtain F s = 1 for the iSWAP gate time τ m = π/2J by choosing parameters for which χ 1 = χ 2 , such that QH (n) eff Q = V qq and QU m Q = QU m Q. The constraints that must be satisfied by the parameters for resonance condition 7 (in addition to those listed in Table  I) are therefore (a) τ m = π/2J , (b) χ 1 = χ 2 , and (c) w ≡ p 1 − q 1 = p 2 − q 2 from the resonance condition itself From constraint (a) and using ∆ = wη, we find τ m ≡ 2πm/η = π/2J = 2πwη/g 1 g 2 , or while constraint (b) becomes, using Eq. (20) with ∆ j = p j η, W j = 2Ω j = q j η, and ∆ = ∆ − 1 = ∆ − 2 = wη and incorporating constraint (c), We also note that q 1 , q 2 , p 1 , p 2 , w, and m must all be integers.
There are multiple sets of parameter values that satisfy these constraints, and we choose one set (given in Table II) for the gate fidelity analysis described in this work. To show that this approach effectively restores the ideal iSWAP evolution in the absence of the dispersive shift terms, we choose n = 0 and compare the ideal state at time τ m for Λ r = 0 with the numerical solution to the time-dependent Schrödinger equation for the density matrix in the interaction picture, ρ I = −i [V I , ρ I ] , for V I given by Eq. (9), the initial state ρ I (0) = |ψ i ψ i | = |eg, 0 eg, 0| , and the chosen set of parameters. For the numerical calculations, we work in the photon subspace with n = 0, 1, 2. The ideal final state is given by For the comparison, we calculate the fidelity where ρ (0) I (τ m ) denotes the numerical solution in the absence of decay, i.e., for γ j = κ = 0 in the master equation of Eq. (25). For the chosen parameters, we find F 0 ≈ 0.998, confirming that setting χ 1 = χ 2 effectively eliminates the modification to the dynamics induced by the dispersive shift terms Λ r and restores the evolution expected for the iSWAP gate.
We now apply this approach to resonance condition 9, which from Table I  eff P + QH (n) eff Q. As described in the main text, ideal evolution within the n-photon subspace can be expressed as U m = e −iJ τmΣ x , where Σ x ≡ |ee gg| + |gg ee|. The ideal evolution in this case occurs nontrivially only within the two-dimensional subspace associated with P in Eq. (D4), and we find P H (n) eff P = n + where Σ z ≡ |ee ee| − |gg gg| . If we now assume that the initial state |ψ i lies within the subspace associated with P, evolution for Λ n = 0 remains within this space such that U m |ψ i = P U m P |ψ i . Here, we consider the initial state |ψ i = |ee . For the double-excitation gate U iDE described in the main text, which is equivalent to U m at time τ m = −π/2J [see Eq. (23)] the ideal final state is |ψ f = U iDE |ψ i = U iDE |ee = i |gg . Again, we set n = 0 for simplicity [with straightforward generalization of expressions to any n via (χ 1 ± χ 2 ) /2 → n + 1 2 (χ 1 ± χ 2 )]. In terms ofû ≡ the modified action of the gate in the presence of Λ r is given by F s now oscillates with the modified frequency Ω instead of J in the ideal case. We can again recover the ideal dynamics and obtain F s = 1 for the gate time τ m = −π/2J by choosing parameters such that χ 1 = −χ 2 , which yields P H (n) eff P = V qq and P U m P = P U m P. Thus, the constraints that must be satisfied by the parameters for resonance condition 9 (in addition to those listed in Table I) are (a) τ m = −π/2J , (b) χ 1 = −χ 2 , and (c) w ≡ p 1 − q 1 = p 2 + q 2 from the resonance condition. From constraint (a) and using ∆ = wη, we find τ m ≡ 2πm/η = −π/2J = −2πwη/g 1 g 2 , which gives On the other hand, constraint (b) with constraint (c) incorporated becomes [again using Eq. (20) with ∆ j = p j η, W j = 2Ω j = q j η, and ∆ = ∆ − 1 = ∆ − 2 = wη], As before, q 1 , q 2 , p 1 , p 2 , w, and m must all be integers. There are once again multiple sets of parameter values that satisfy these constraints, and we choose one set (given in Table II) for the analysis described in this work. We now show that the ideal evolution described by U iDE in the absence of the dispersive shift terms is restored for χ 1 = −χ 2 and the chosen parameters. Working in the photon subspace with n = 0, 1, 2 and numerically solvinġ ρ I = −i [V I , ρ I ] for resonance condition 9 with the initial state ρ I (0) = |ψ i ψ i | = |ee, 0 ee, 0| , we compare the γ j = κ = 0 solution ρ (0) I (τ m ) at time τ m with the ideal (Λ r = 0) final state via the fidelity in Eq. (D12). We find F 0 ≈ 0.998, showing that setting χ 1 = −χ 2 effectively eliminates the modification to the dynamics induced by the dispersive shift terms and restores the expected gate evolution generated by V qq for resonance condition 9. The small error 1 − F 0 found for both resonance conditions 7 and 9 also serves as a measure of the validity of the model that we develop in this work. By choosing parameters that satisfy additional constraints, a similar approach can be used to eliminate dispersive shift dynamics for general initial states |ψ i with arbitrary photon number n, as well as for other resonance conditions and the corresponding interactions and twoqubit gates listed in Table I. For a general initial state |ψ i = (c ee |ee, n + c eg |eg, n +c ge |ge, n + c gg |gg, n ) , both the subspaces associated with P and Q must be taken into account. For resonance condition 7, we find P U m P |ψ i = P e −iΛnτm P = e −i(n+ 1 2 )(χ1+χ2)τm |ee, n ee, n| + e i(n+ 1 2 )(χ1+χ2)τm |gg, n gg, n| (D20) and P U m P |ψ i = P |ψ i , while for resonance condition 9, QU m Q |ψ i = Qe −iΛnτm Q = e −i(n+ 1 2 )(χ1−χ2)τm |eg, n eg, n| + e i(n+ 1 2 )(χ1−χ2)τm |ge, n ge, n| (D21) and QU m Q |ψ i = Q |ψ i . From the forms of Eqs. (D20) and (D21), we see that setting χ j = 2πk j with k j an integer for j = 1, 2, along with k 1 = k 2 (k 1 = −k 2 ) for resonance condition 7 (9) to satisfy χ 1 = χ 2 (χ 1 = −χ 2 ) eliminates the dynamical phases due to Λ r and restores the ideal evolution due to V qq within a subspace of fixed n, or for an integer average photon number a † a . We now consider the other resonance conditions in Table I. Resonance condition 1 is given by ∆ ≡ ∆ 1 = ∆ 2 and (assuming the associated constraints listed in the table are satisfied by W 1 and W 2 ) yields V qq = −2J σ z 1 σ z 2 . This interaction is equivalent to that which generates a Mølmer-Sørensen gate in the original qubit basis [95] and can be used to construct a controlled-phase gate in the dressed qubit basis considered here. Using U m = e −iVqqτm = e i2J τmσ z 1 σ z 2 , a sequence for a controlled-phase gate is given by where ϕ ≡ 8J τ m , which shows that U m for ∆ ≡ ∆ 1 = ∆ 2 is equivalent to a controlled-phase gate up to single-qubit rotations. For ϕ = (2l + 1) π or τ m = (2l + 1) π/8J with l an integer, U ϕ represents a controlled-Z gate U CZ .
To determine conditions for eliminating the dispersive shift dynamics in the case of resonance condition 1, we can compare the action of U m and U m = e −i(Vqq+Λn)τm = e −iΛnτm U m , both of which now act in the full two-qubit space, on a general state |ψ i in this space [Eq. (D19)] via (D23) In this case, we can in principle recover the ideal dynamics and obtain F s = 1 for any state |ψ i by choosing parameters that satisfy χ j = 2πk j with k j an integer for j = 1, 2, and that also satisfy τ m = −ϕ/8J , p 1 = p 2 from the resonance condition, and q 1 = ±q 2 from Table I to obtain a controlled-phase gate U ϕ according to the construction in Eq. (D22). For U CZ , τ m = (2l + 1) π/8J , and we can proceed to identify potential parameter sets by choosing, e.g., l = 0 and setting τ m ≡ 2πm/η = −π/8J in analogy to the analysis described above for resonance conditions 7 and 9.
The remaining physically distinct case in Table I is that of resonance condition 4, given by ∆ ≡ ∆ + 1 = ∆ 2 and corresponding to V qq = J σ x 1 σ z 2 when W 1 and W 2 satisfy the listed constraints. This interaction can be used to construct a controlled-NOT (CNOT) gate [45]. We note that V qq , and therefore U m and U m , are block-diagonal with the associated subspace projectors P = |ee ee| + |ge ge| and Q = |eg eg|+|gg gg| . If we choose |ψ i = |eg such that P U m P |ψ i = P U m P |ψ i = 0, we find from a calculation similar to that in Eqs. (D5) and (D8), withΩ ≡ χ 1 + 4J 2 . In this case, recovering the dynamics for Λ r = 0 requires χ 1 = 0, which is not possible for δ j = 0 and g 1 , 2Ω 1 = 0 [see Eq. (20)]. Similar results hold for other initial states |ψ i . While F s = 1 is not achievable exactly for resonance condition 4 when δ j = 0, choosing appropriate parameters to minimize χ j in principle allows F s to be made arbitrarily close to unity, enabling the dynamics for Λ r = 0 and thus the gate generated by V qq = J σ x 1 σ z 2 to be well approximated. The dispersive shift dynamics can also be eliminated in this case for δ j = 0, as we describe below. The other resonance conditions in Table I lead to effective interactions V qq that are either identical in form to those considered above or have the roles of qubits 1 and 2 reversed.
Finally, we consider the conditions under which the one-qubit contribution to H eff (τ ) , including the dispersive shift terms Λ, can be made to vanish for each qubit individually. In this case, H eff (τ ) = V qq (τ ) consists solely of qubit-qubit interaction terms. Setting Λ = 0 in Eq. (19) yields the condition δ 2 j + 2∆ j δ j + W 2 j = 0, or equivalently δ 2 j + 2p j ηδ j + q 2 j η 2 = 0. We note that for resonant driving, described by δ j = 0, the one-qubit contribution Λ = 0 unless W j = 0, which is not physically meaningful (see Appendix B). Thus, provided the drive frequency is not on resonance with the qubit frequency ω d j = ω j , the dispersive shift terms can be eliminated individually for each qubit by an appropriate choice of parameters.