High-fidelity gates in quantum dot spin qubits

Significance

This paper addresses a critical issue in the development of a practical quantum computer using semiconducting quantum dots: the achievement of high-fidelity quantum gates in the presence of environmental noise. The paper shows how to maximize the fidelity, which is the key figure of merit, for several different implementations of quantum gates in semiconducting quantum dot qubits. The paper also shows how to optimize the fidelity over the various control parameters, and that the different implementations display an unexpected commonality in how the fidelity depends on these parameters. The optimum fidelity for a given implementation is determined by experimental constraints on the control parameters, which are different for different qubit designs.

Abstract

Several logical qubits and quantum gates have been proposed for semiconductor quantum dots controlled by voltages applied to top gates. The different schemes can be difficult to compare meaningfully. Here we develop a theoretical framework to evaluate disparate qubit-gating schemes on an equal footing. We apply the procedure to two types of double-dot qubits: the singlet–triplet and the semiconducting quantum dot hybrid qubit. We investigate three quantum gates that flip the qubit state: a DC pulsed gate, an AC gate based on logical qubit resonance, and a gate-like process known as stimulated Raman adiabatic passage. These gates are all mediated by an exchange interaction that is controlled experimentally using the interdot tunnel coupling g and the detuning ϵ, which sets the energy difference between the dots. Our procedure has two steps. First, we optimize the gate fidelity (f) for fixed g as a function of the other control parameters; this yields an that is universal for different types of gates. Next, we identify physical constraints on the control parameters; this yields an upper bound that is specific to the qubit-gate combination. We show that similar gate fidelities should be attainable for singlet-triplet qubits in isotopically purified Si, and for hybrid qubits in natural Si. Considerably lower fidelities are obtained for GaAs devices, due to the fluctuating magnetic fields ΔB produced by nuclear spins.

The fundamental building block of a quantum information processor is a two-state quantum system, or qubit. Solid-state qubits based on electrons confined in top-gated quantum dots in semiconductor heterostructures (1) are promising, due to the promise of manipulability and the overall maturity of semiconductor technology. In a charge qubit, the information is stored in the location of an electron in a double quantum dot. Because charge qubits are subject to strong Coulomb interactions, they can be manipulated quickly, at gigahertz frequencies, using control electronics (2–5); however, they also couple strongly to environmental noise sources, such as thermally activated charges on materials defects, leading to short, subnanosecond decoherence times (6). Spin qubits, which couple more weakly to environmental noise, have much longer coherence times (1, 7–15). However, because magnetic couplings are weak, gate operations between spin qubits are slow. For this reason, in most gating protocols, spin qubits adopt a charge character briefly during gate operations. Successful gate operations generally entail a tradeoff: charge-like for faster gates vs. spin-like for better coherence.

Several types of logical qubits have been designed to enable electrically controlled manipulation and measurement of qubits encoded in spin degrees of freedom formed of two or more electrons in two (7, 9, 16) or three (11) coupled dots. These logical qubits share experimental control knobs; however, their spin-charge characteristics vary widely, yielding variations in gating speeds, dephasing rates, and gating protocols.

When characterizing quantum gates, instead of considering the gating time and decoherence time separately, it is important to consider the gate fidelity, a measure of the fraction of the wave function that is in the targeted state, which depends on the ratio of the gating time to the decoherence time. Here we argue that achieving meaningful comparisons between logical qubits and gating schemes is greatly facilitated by first optimizing the specific gate operations, taking into account the different dephasing rates of the spin and charge sectors. We compute and optimize gate fidelities for different qubits and gating protocols using a master equation approach.

We consider two types of logical qubits in a double quantum dot: a singlet–triplet (ST) qubit formed with two electrons, one in each dot (7, 8), and a quantum dot hybrid qubit formed with three electrons, two in one dot and one in the other (9, 17). Logical qubit states for the ST qubit are and , where and are spin-up or -down states, and L and R refer to the left or right dots. Logical qubit states of the quantum dot hybrid qubit are and , where and are singlet (S) and triplet (T) states. Energy differences between the qubit states drive z-rotations around the Bloch sphere. For ST qubits, the energy splitting is caused by a magnetic field difference on the two sides of the double dot. occurs naturally in GaAs and natural Si, and may be enhanced by nuclear polarization (18, 19), or with micromagnets (20) or striplines (10). Typical values of are in the range T. Hybrid qubits do not require local magnetic fields; the qubit energy splitting is dominated by the ST energy splitting of the two-electron dot. is typically of order 0.1 meV (21), yielding much faster z-rotations than in ST qubits.

Although z-rotations can never be extinguished in ST or hybrid qubits, the rotation axis may be varied in the x–z plane by adjusting the tunnel coupling between the two sides of the double dot. As Fig. 1 indicates, the main experimental parameters are g and the detuning ϵ, which characterizes the energy difference between different charge configurations [(1,1) vs. (0,2) for the ST and (2,1) vs. (1,2) for the hybrid qubit]. We use analytical and numerical calculations to find the relationship between ϵ and g that maximizes the fidelity of x- and z-rotations. Physical limits on ϵ and g for a given qubit scheme then determine the maximum achievable fidelity.

Fig. 1.

Physics of x- and z-rotations of ST and hybrid qubits in a double quantum dot. Transitions between qubit states yield x-rotations, whereas z-rotations change the relative phase of the qubit states. (A) Schematics of processes underlying x-rotations for ST and hybrid qubits, which are implemented by changing the detuning to turn on an exchange interaction by mixing in an excited state with different charge character. (Left and Right) Qubit states. (Center) Intermediate states. Increasing the strength of the exchange interaction increases the gate speed but also increases dephasing from charge fluctuations. Reducing the gate speed increases the exposure to spin dephasing. In this paper, we determine the optimal gate speeds that maximize the gate fidelity. (B and C) The three lowest energy levels for ST and hybrid qubits, including the two qubit states and the excited charge state, as a function of the detuning, ϵ. The energy splitting of the qubit states drives z-rotations, and is typically much smaller for ST qubits than for hybrid qubits (i.e., ). For a given value of the tunnel coupling g, the exchange interaction J is large when is small. Large J causes faster gate speed but also faster decoherence from charge noise. For ST qubits, optimum fidelity is obtained when . For hybrid qubits, AC gates are optimized with , whereas DC pulsed gates are optimized when ϵ is set to the energy level anticrossings (24). (D) LQR is performed by oscillating the tunnel coupling [and therefore ] at the primary resonant frequency , or at the secondary harmonic . (The secondary period is identified in the figure.) (E) STIRAP (25) is implemented at the resonant frequencies, and , using overlapping Gaussian envelopes known as Stokes and pump pulses.

We consider three different schemes for performing x-rotations: (i) DC pulsed gates (8), in which the detuning is changed suddenly between different values; (ii) logical qubit resonance (LQR), an AC resonant technique analogous to electron spin resonance (ESR) for single spins (22), and (iii) stimulated Raman adiabatic passage (STIRAP) (23), another AC resonant technique in which each qubit state is coupled to an auxiliary excited state. Given a tunnel coupling g, pulse-gating and LQR are optimized over the detuning ϵ, whereas STIRAP is optimized over the duration of the pulses used. Remarkably, we find that the g-dependence of the optimal fidelity is very similar for all three gating schemes. However, physical constraints that differ between the gating schemes limit the achievable fidelity .

The paper is organized as follows. The following section provides relevant details concerning ST and hybrid qubits and their decoherence rates. We describe the physical mechanisms for implementing x-rotations (transitions between qubit states) and z-rotations (changes in the phase difference between the qubit states). We discuss the “slow” or “pure” spin-dephasing rate γ, arising from dephasing of the qubit states themselves, and the “fast” charge-dephasing rate Γ, involving the intermediate state (26, 27). We then present the calculations and results for qubit fidelities, based on the master equations presented in Materials and Methods (see SI Text for further details). Figs. 2 and 3 show the key results, plots of optimized fidelities as a function of the tunnel coupling g. In Discussion, we describe the physical constraints that determine the upper bounds on for each type of qubit, gate operation, and materials system (Si vs. GaAs).

Fig. 2.

Fidelities for LQR and STIRAP^† x-rotations in ST and hybrid qubits for isotopically pure ²⁸Si (see legend at top left) using material parameters given in the main text. We assume a magnetic field difference of , from an external micromagnet. Data points are numerically optimized fidelities vs. g₀. Solid lines are analytical estimates for (Eq. 2). The corresponding gate speeds increase with g₀. (Lower, Left Inset) At fixed g₀, the optimized value is found by minimizing the infidelity 1 − f as a function of ϵ. Data points are numerical results for , and solid lines are analytical estimates (SI Text). The numerically obtained agree with the analytical estimate (Eq. 1). (Upper, Right Inset) A cartoon showing the tunnel barrier and detuning in a double dot, which are both controlled electrostatically.

Fig. 3.

Optimized fidelities for pulsed gate x-rotations in ST and hybrid qubits, for isotopically pure ²⁸Si. Numerical results for the infidelity vs. tunnel coupling g₀ for an ST qubit with (magenta squares) and (red circles). The solid black line is an upper bound on the fidelity, obtained when . The analytical form is the same as for LQR gates in ST qubits at the secondary resonance (solid black curve in Fig. 2). The numerical results deviate from this limiting behavior most significantly at small g₀, in the regime where and the rotation axis points away from . The horizontal lines describe the infidelity of z-rotations for the same . (Inset) Numerical results for the infidelity of a pulse-gated hybrid qubit, using the five-step pulse sequence described in ref. 24 (green triangles). As a comparison, the solid line is the analytical estimate for an LQR gate in a hybrid qubit at the secondary resonance (solid blue curve in Fig. 2). Pulsed gates in hybrid qubits have relatively poor fidelity because ϵ cannot be optimized.

Logical Qubits, Gates, and Decoherence Mechanisms

Fig. 1 shows gating schemes and energy levels for ST and hybrid quantum dot qubits. The horizontal energy levels in the portion of Fig. 1 B and C correspond to the logical qubit states. Only states that can be reached by spin-conserving processes are shown. A third state with a different charge configuration that plays a prominent role during gating is shown for both ST and hybrid qubits ( or , respectively). At (or near) the detuning value , states with different charge configurations are energetically degenerate [(1,1) and (0,2) states for the ST qubit and (2,1) and (1,2) states for the hybrid qubit]. We focus on the regime .

Implementations of x-Rotations.

The implementations of x-rotations for ST and hybrid qubits discussed here involve the exchange interaction, which is mediated by the excited state . Fig. 1A demonstrates the exchange process for ST and hybrid qubits. Decreasing increases the occupancy of , which enhances the speed of x-rotations, but also increases the coupling to external charge noise (28). For both ST and hybrid qubits, the rate Γ of charge dephasing between and the qubit states is much faster than the rate γ of pure dephasing between the qubit states, so changing ϵ strongly affects the gate fidelity. Charge noise couples to both ϵ and g, yielding distinct dephasing mechanisms (29). However, as shown below, the highest fidelities are obtained when , in the region where fluctuations in g are dominant, because the qubit energy levels have very nearly the same dependence on detuning.* Therefore, we consider only g-noise here.

DC Pulsed Gates.

ST qubit experiments typically keep the tunnel coupling fixed and use ϵ to tune the exchange coupling (8, 13), as indicated in Fig. 1. z-rotations are obtained when , whereas x-rotations are obtained when . In pulsed-gating protocols, ϵ is switched between these two positions quickly, so that the quantum state does not evolve significantly during the switching time.

In a hybrid qubit, the energy splitting between the qubit states is much larger than the tunnel couplings . The energy level diagram then has two distinct anticrossings, as indicated by vertical dotted lines in Fig. 1C. The pulse-gating scheme proposed in ref. 24 to implement an arbitrary rotation on the Bloch sphere has five steps, three of which are at anticrossings. Below, we show that this requirement leads to serious constraints on gate fidelities using current technology.

Logical Qubit Resonance.

In conventional ESR (30), a DC magnetic field applied along induces a Zeeman splitting, . A small AC transverse magnetic field applied along at the resonant frequency induces transitions between states with different values of spin component S_z. In the analogous LQR scheme (Fig. 1D), the qubit energy splitting plays the role of the Zeeman energy, whereas an oscillating exchange interaction plays the role of the transverse field. As for pulsed gates, ϵ is increased from a value << 0 to a value closer to zero, where an increase in tunnel coupling g creates a significant exchange interaction. ϵ is then held constant, whereas is modulated. Here, we assume that g oscillates between zero and a positive value 2g₀, as indicated in Fig. 1D. The amplitude of the AC component of determines the speed of the x-rotation.

LQR differs from conventional ESR in two main ways. First, because the oscillating exchange interaction J has a nonzero DC component (J_DC), the precession axis tilts slightly away from . Within the rotating wave approximation discussed below (SI Text), this leads to infidelity in the LQR gate because the effective B-field has a shifted magnitude . (This error is accounted for in all of the calculations shown here.) Second, the primary resonance occurs at half the Larmor frequency ; this is because the tunnel coupling, , generates two different AC components. For example, yields a primary component, , and a secondary component, (SI Text). The numerical results reported here all correspond to the secondary resonance, because it yields slightly higher fidelities.

Stimulated Raman Adiabatic Passage.

The STIRAP protocol (23) generates x-rotations on the Bloch sphere by inducing transitions between the qubit states and . A simple STIRAP protocol is shown in Fig. 1E. The tunneling processes and are controlled independently by oscillating at the resonant frequencies and . Again, we assume the tunnel coupling is nonnegative, with a DC component g₀, and an AC amplitude 2g₀. Counterintuitively, an adiabatic pulse sequence with followed by produces a rotation from to that never populates and therefore never experiences charge dephasing. Realistic pulse sequences have a finite duration however; this yields a small population of , and therefore dephasing. Similar to pulsed gates and LQR, we anticipate there will be an optimal gate speed that maximizes the process fidelity. We note that the standard STIRAP protocol shown in Fig. 1E is not a true qubit gate.^† True gates can be achieved by using longer, STIRAP-like pulses (31), which must be optimized over many more parameters. We only study the standard, two-pulse sequence here, to focus on the fundamental physics limiting the fidelity of the protocol.

Decoherence Mechanisms.

The key physics incorporated in our calculations is that the qubit states, which have a spin character, often have a much lower dephasing rate than the states accessed during a gate operation, which typically have a substantial charge character (26). To understand the achievable fidelities of real devices, our calculations use experimentally realistic numbers, which we list here.

Charge qubit experiments indicate that the fast charge noise-dephasing rate Γ is very similar in Si and in GaAs (2, 4, 5). Here, we adopt the value . The much slower pure dephasing rate γ depends significantly on the material host and the type of qubit. For ST qubits, pure dephasing is caused by the slow diffusion of nuclear spins. We adopt the values for 99.99% isotopically purified ²⁸Si, 4.5 MHz for natural Si, and 0.14 GHz for GaAs, which are obtained as quadrature sums of contributions from the nuclear hyperfine coupling (32) and the electron–phonon coupling (27, 33). For hybrid qubits, we use for 99.99% isotopically purified ²⁸Si, 4.6 MHz for natural Si, and 5.9 GHz for GaAs, with the main contributions to dephasing coming from charge noise and optical phonons (26, 27).

Though application of echo sequences can be used to greatly increase the coherence times of quiescent qubits (34) and of z-rotations, it is nontrivial to correct for low-frequency noise during a gate sequence. Though several correction schemes have been proposed (35, 36), and noise suppression schemes have been implemented (37), the required pulse sequences are rather complicated. Here, we only study short sequences, so the dephasing rates in our calculations must include the low-frequency noise.

Calculations and Results

We now present our results for the optimized fidelity of single-qubit gate operations in the presence of both fast and slow dephasing mechanisms. We first focus on the fidelity of x-rotations. As described in Materials and Methods, we solve a master equation for the density matrix ρ. For both ST and hybrid qubits, the coherent evolution is governed by a three-state Hamiltonian, H, involving the two logical qubit states, and , and the excited charge state, , with a fast dephasing rate Γ between the excited state and each of the qubit states, and a slow dephasing rate γ between the qubit states. The pulsed, LQR, and STIRAP protocols are implemented by modulating the detuning and the tunnel coupling . Dephasing is introduced through a Markovian phenomenological term D (38) that incorporates dephasing associated with charging transitions in a double quantum dot (28).

We present the fidelities of different gating schemes for the specific gate operation of a π-rotation about the x-axis from the initial state [initial density matrix ] to the final state [target density matrix ], for a gate that is implemented in a time τ. Our fidelity measure is the distance between the actual and ideal density matrices for a π-rotation (39), which is the calculated value of (SI Text). For pulsed gates, we consider a one-step pulse sequence for ST qubits (8, 13), and a five-step sequence for hybrid qubits (24). For the AC gates, we solve the master equation within the rotating wave approximation (RWA) (25).

We first optimize the LQR gates. For a fixed value of g₀, the value of ϵ at which the fidelity f is maximized, , is found. (Fig. 2, Lower, Left Inset shows the infidelity 1 − f, which exhibits a minimum.) For small (large detuning), the gate speed is slow and the fidelity is limited by the pure dephasing rate γ. For large , the gate speed is fast and the fidelity is limited by the charge-noise dephasing rate Γ. The optimum fidelity, which is achieved at the cross-over between the two regimes, is determined numerically. SI Text presents the derivation of analytical estimates for the fidelity as a function of ϵ (Fig. 2, Lower, Left Inset) and of the optimal detuning and fidelity for LQR gates driven at the secondary resonance (solid lines in the main panel of Fig. 2):

and

Numerical results are also shown. Results for the fidelity of LQR at the primary resonance can be obtained by replacing in Eqs. 1 and 2, yielding a lower fidelity.

Pulsed gates in ST qubits are optimized similarly to LQR gates, yielding similar results. Fig. 3 shows numerically optimized fidelities for two different interdot magnetic field differences, . In the low-field regime , the rotation axis points nearly along . When , we can obtain analytical estimates for the optimized detuning and fidelity, obtaining the same results as Eqs. 1 and 2, with and (SI Text). In Fig. 3, we see that the numerically optimized fidelities approach this limiting behavior for large g₀ or small . For smaller g₀, the fidelity is suppressed by a combination of dephasing effects, and a misalignment of the rotation axis from . A three-step pulse sequence that corrects the rotation angle (40) yields only small improvements in the fidelity (SI Text).

Pulsed gates in hybrid qubits differ significantly from the other gating schemes because the optimal value of ϵ does not depend on g₀. To understand this, we note that a general, pulsed-gate rotation sequence for a hybrid qubit requires five steps (24), with three of these steps occurring at anticrossings. Dephasing errors are minimized by maximizing the transition speed, i.e., by tuning ϵ directly to the level anticrossings in Fig. 1C; this yields the results shown in Fig. 3 Inset. The inset also shows the optimal fidelity for an LQR gate in a hybrid qubit; LQR typically achieves a much higher fidelity.

We next present results for optimized fidelities of the STIRAP protocol for ST and hybrid qubits.^† The pulse shape determines the gate speed of STIRAP (Fig. 1E). For a given value of 2g₀, the pulse shape parameters t_width and t_delay are optimized for maximum fidelity. There are no simple analytical methods for treating the STIRAP protocol, so the optimal fidelities are obtained numerically, yielding the results shown in Fig. 2. Remarkably, we find that the optimal fidelities for STIRAP and LQR gates exhibit the same dependence on g₀, differing only by a small factor (SI Text). The various gate speeds are indicated by the calibration bars at the top of Fig. 2.

The analysis of the fidelity of z-rotations is considerably simpler than the analysis of x-rotations presented above. For both ST and hybrid qubits, the fidelity of a z-rotation should be understood simply as a competition between the pure dephasing rate γ, and the gate speed, where the latter is determined by the energy splitting between the qubit states.

Discussion

The previous section presents relations between the control parameters that yield optimized gate fidelities . In principle, the fidelity can be made arbitrarily close to 1 by increasing g₀. In practice, physical constraints on the experimental control parameters bound the fidelity. We now list the constraints that bound the fidelity of x-rotations.

Level Spacing.

The anticrossings in the qubit energy level diagram should be well separated; otherwise, transitions may occur between three levels rather than two. The g₀ should therefore be smaller than the spacing between single-particle levels in the dots. Recently, this condition was found to be satisfied in an electrostatically defined SiGe double dot for which (41). We therefore assume a bound of GHz for systems of this type. For hybrid qubits, g₀ should also be smaller than the qubit energy splitting, or . When this constraint is not satisfied, the infidelity rises, as on the right-hand side of Fig. 3 Inset.

RWA.

Resonant gating requires that many fast, resonant oscillations fit inside a single pulse envelope; this is the basis for the RWA, and it yields the following constraints (SI Text): for the secondary resonance of LQR, and for STIRAP. When , the LQR requirement further simplifies to . For ST qubits, the constraint is quite strict and yields relatively low fidelities. Our numerical calculations suggest that in this situation the fidelity can be improved by deviating from Eq. 1. For the STIRAP scheme, there is no analogous relation between and g₀, and the RWA is far less restrictive.

Adiabaticity.

Pulsed gating methods require instantaneous pulses. However, rise times in real experiments are finite. For pulsed gates, the evolution is effectively instantaneous when the time dependence of the energy difference at an anticrossing satisfies (42). Using experimental measurements and numerical calculations, and assuming a realistic rise time of ∼100 ps for currently available pulse generators (4, 5), we deduce a bound of GHz for pulsed gates.

Misorientation.

If z-rotations cannot be turned off, the fidelity of an x-rotation will be limited by the misorientation of the rotation axis; this is always true for simple pulsed gates in hybrid and ST qubits because of the energy splitting between the qubit states. For ST qubits, the problem is mitigated by reducing the magnetic field difference (e.g., by modifying the micromagnets) or by increasing g₀ (and therefore the x component of the rotation). In the latter case, the fidelity can be improved by increasing both ϵ and the charging energy, as described below. Alternatively, a three-step pulse sequence can be used to correct the misorientation (40). For hybrid qubits, the problem is more severe, and a single-step sequence is untenable (24). For LQR gates, the misorientation of the rotation axis occurs because the tunnel coupling has a DC component. If all of the DC components of the rotation axis are known, a three-step sequence could also be used to correct the misorientation in LQR.

Charging Energy.

When g₀ satisfies Eq. 1, constraints on ϵ translate into constraints on g₀. In the far-detuned regime, for the dot occupation to remain constant, must satisfy , where U is the charging energy. When , numerical optimization indicates that we may improve the fidelity slightly by deviating from Eq. 1.

The results in Table 1 were obtained by numerically maximizing the fidelity. The reported values of were obtained by using the most restrictive of the constraints described above. In Table 1, we list the dominant constraints, and corresponding modifications that could enhance the fidelity. Generally, we observe different constraints for different types of gates. The STIRAP scheme appears particularly promising because optimization does not involve Eq. 1. (Hence, the charging energy constraint does not apply.) Additional work is needed to clarify this scheme, however.^†

Table 1.

Numerically maximized gate fidelities

Qubit	z-gate	x: Pulsed	LQR	STIRAP*
ST, natural Si†	97‡	99.6§	98§,¶	99.8║
ST, purified 28Si†	99.4‡	99.6§	99.3§,¶	99.9║
ST, GaAs†	66‡	98§	92§,¶	98║
Hybrid, natural Si†	99.995‡	83**	99§	99.6║
Hybrid, purified 28Si†	99.999‡	83**	99.4§	99.8║
Hybrid, GaAs†	94‡	83**	89§	96║

^*The standard, Stokes-pump STIRAP protocol yields but not , and is therefore not an x-rotation.

^†For the ST qubits we use , 0.3, and 10 mT, respectively; for the hybrid qubits we use .

^‡Constrained by the z-gate speed. Improve by increasing or .

^§Constrained by the charging energy and/or the misorientation of the rotation axis. Improve by decreasing or increasing U.

^¶Constrained by the RWA. Improve by increasing , , or U.

^║Constrained by . Improve by increasing .

^**Constrained because anticrossings must be distinct. Improve by increasing .

Next, we consider z-rotations. Because g₀ (and hence J) can be turned off completely and the pure dephasing rate γ is fixed, the gate fidelity can only optimized by maximizing the gate speed. For a π-rotation, the gate period is and the fidelity is , where for ST qubits and for hybrid qubits. Here, we use for isotopically pure ²⁸Si (0.01% ²⁹Si), 3.0 neV for natural Si, and 92 neV for GaAs (32). Our results for are presented in Table 1.

The simulations reported here are for simple gating schemes, including a one-step pulsed gate sequence for ST qubits, a five-step pulsed gate sequence for hybrid qubits, and a simple STIRAP scheme, which does not provide a true gate. More sophisticated pulse sequences have also been proposed. A three-step pulsed gate sequence was proposed to correct for the misorientation of the x-rotation axis in ST qubits (40). For pulsed gates, we find that this sequence does improve the fidelity over a small range of control parameters (SI Text). However, the procedure incorporates an intermediate z-rotation step, so the final fidelity is bounded by the fidelity of the z-rotation. Similar considerations should apply to LQR gates in ST qubits, although we do not study that problem here. For hybrid qubits, the misorientation effect is quite weak for LQR, and corrective pulses have little effect.

For z-rotations of ST qubits, because the noise spectrum of the nuclear spins is dominated by low frequencies (43), pulse sequences similar to spin echoes (34) can improve the fidelity. For hybrid qubits, the noise spectrum of optical phonons (27) and charge fluctuators (44) has weight at higher frequencies, so echo-type pulse sequences may be less effective. In principle, the fidelity of spin echoes can always be improved by increasing the sophistication of the pulse sequence (45); in practice, however, they are constrained by pulse imperfections and by dephasing that occurs during the x-rotations in the sequence.

In conclusion, we have presented a method for optimizing the fidelity of gate operations of logical ST and hybrid qubits in the presence of both spin and charge dephasing, and we have identified upper bounds on the fidelity for simple gating schemes. We obtain the following general results. The fidelity of z-rotations in hybrid qubits in Si is high because their gate speeds are much greater than their rate of pure dephasing. The fidelity of z-rotations in ST qubits in dots without external field gradients is low but can be improved greatly using spin echo methods. Therefore, the limits on overall performance are those of the x-rotations.

For x-rotations in ²⁸Si, the maximum achievable fidelities of ST qubits (pulsed and LQR gates) and hybrid qubits (LQR gates) are similar, with . STIRAP gates appear quite promising, although further work is required to clarify this scheme. Hybrid qubits are probably not viable in GaAs due to fast pure dephasing (27). For ST qubits, the maximum fidelities for x-rotations are considerably larger in Si than in GaAs. There are two reasons for this: (i) the large, intrinsic in GaAs causes a misorientation of the x-rotation axis, and (ii) the large in GaAs makes it difficult to implement corrective protocols involving z-rotations (40). Hence, the nuclear spins that complicate the implementation of z-rotations in GaAs also ultimately constrain the x-rotations.

Materials and Methods

The dynamical evolution of the logical qubit density matrix ρ is governed by the master equation (38)

where H is the Hamiltonian describing coherent evolution, and D describes the fast (Γ) and slow (γ) dephasing processes. Though D is phenomenological, its form can be justified in a bosonic environment, assuming Markovian dynamics, as has been argued for the double quantum dot system (28). The analytical and numerical methods used for solving Eq. 3 are described in SI Text.

To treat the ST and hybrid qubits on equal footing, we define our logical qubit basis states and in the far-detuned limit, . (Note that for ST qubits, this basis choice differs from that of ref. 8.) We also consider a third, excited charge state that is tunnel-coupled to the logical qubit states for both the ST and hybrid qubit systems. In the ordered basis, we have

In principle, the tunnel couplings g₁ and g₂ are independently tunable (9); here we take them to be equal, with .