Robust quantum gates for singlet-triplet spin qubits using composite pulses

We present a comprehensive theoretical treatment of supcode, a method for generating dynamically corrected quantum gate operations, which are immune to random noise in the environment, by using carefully designed sequences of soft pulses. supcode enables dynamical error suppression even when the control field is constrained to be positive and uniaxial, making it particularly suited to counteracting the effects of noise in systems subject to these constraints such as singlet-triplet qubits. We describe and explain in detail how to generate supcode pulse sequences for arbitrary single-qubit gates and provide several explicit examples of sequences that implement commonly used gates, including the single-qubit Clifford gates. We develop sequences for noise-resistant two-qubit gates for two exchange-coupled singlet-triplet qubits by cascading robust single-qubit gates, leading to a 35% reduction in gate time compared to previous works. This cascade approach can be scaled up to produce gates for an arbitrary-length spin qubit array and is thus relevant to scalable quantum computing architectures. To more accurately describe real spin qubit experiments, we show how to design sequences that incorporate additional features and practical constraints such as sample-specific charge noise models and finite pulse rise times. We provide a detailed analysis based on randomized benchmarking to show how supcode gates perform under realistic $1/f^{\alpha }$ noise and find a strong dependence of gate fidelity on the exponent $\alpha$, with best performance for $\alpha >1$. Our supcode sequences can therefore be used to implement robust universal quantum computation while accommodating the fundamental constraints and experimental realities of singlet-triplet qubits.

Figure 1

Parameters for rotations around the $x$ axis for a range of angles, corresponding to the sequence shown in Eq. (16) with $j_2=J=0$. Note that $j_1$ and $j_3$ have been rescaled by a factor of 20 to fall into approximately the same range as the other parameters. In solving for these parameters, we have assumed $g\left(J\right)=J/{\epsilon }_0$.

Figure 2

Parameters for rotations around the axis $\widehat{x}+\widehat{z}$ for a range of angles, corresponding to the sequence shown in Eq. (16) with $j_2=0$ and $J=1$. In solving for these parameters we have assumed $g\left(J\right)=J/{\epsilon }_0$.

Figure 3

Parameters for rotations around the $z$ axis for a range of angles, corresponding to the sequence shown in Eq. (22). The yellow shaded area around $0.6\pi \lesssim \varphi \lesssim 0.9\pi$ indicates a range of $\varphi$ for which the solutions of $j_3$ become negative and thus unphysical. In solving for these parameters we have assumed $g\left(J\right)=J/{\epsilon }_0$.

Figure 4

Parameters for rotations around the $y$ axis for a range of angles, corresponding to the sequence shown in Eq. (31). For $\varphi \lesssim 0.53\pi$, $j_6$ becomes negative. In solving for these parameters, we have assumed $g\left(J\right)=J/{\epsilon }_0$.

Figure 5

Parameters for rotations around the axis $\widehat{x}+\widehat{z}$ vs rotation angle for several $J_{\min }$. In solving for these parameters, we have assumed $g\left(J\right)=(J-J_{\min })/{\epsilon }_0$. The solid lines are for $J_{\min }=0$ and are the same as what is shown in Fig. 2. The dashed lines and dotted lines are for $J_{\min }=0.03$ and 0.06, respectively, as indicated in the figure. As in Fig. 2, the parameters shown correspond to the sequence Eq. (16) with $J=1$, but $j_2=J_{\min }$.

Figure 6

Parameters for rotations around the axis $\widehat{x}+\widehat{z}$ vs rotation angle for $J\left(\epsilon \right)$ as defined in Eq. (34). The parameters in Eq. (34) are $J_{\min }=0.008$, $J_1=67.3$, ${\alpha }_1=0.476$, ${\alpha }_2=0.156$, and $\gamma =0.812$. Here $j_1$ and $j_3$ are rescaled by a certain factor as indicated in the figure. As in Figs. 2 and 5, the parameters shown correspond to the sequence Eq. (16) with $J=1$ and $j_2=0.01>J_{\min }$.

Figure 7

Pulse shape for $R(\widehat{x}+\widehat{z},\pi /2)$ for (a) a square pulse (no rise time, $\tau =0$) and (b) a trapezoidal pulse with finite rise time $\tau =0.1$. In solving for these parameters, we have assumed $g\left(J\right)=J/{\epsilon }_0$. Note that in (a) the pulse sequence ends at around $T_f=31.24$ and in (b) the sequence ends at $T_f=37.22$.

Figure 8

Parameters for rotations around the axis $\widehat{x}+\widehat{z}$ for a range of angles, when a finite pulse rise time $\tau$ as defined in Eqs. (35) and (36) is taken into consideration. The parameters correspond to the sequence shown in Eq. (16) but with a finite pulse rise time $\tau$ incorporated in a similar manner as Fig. 7. Other parameters not shown are $j_2=0$ and $J=1$. In solving for these parameters, we have assumed $g\left(J\right)=J/{\epsilon }_0$.

Figure 9

Quantum circuits for corrected two-qubit gates with the BB1 sequence. Each line represents a quantum dot, linked to its neighboring dots by the exchange interaction. A singlet-triplet qubit then is denoted as $|{\psi }_A\rangle$ and $|{\psi }_B\rangle$ by a pair of lines with corresponding electronic states within $S_z=0$ subspace. (a) The Ising gate [Eq. (37)]. (b) The BB1 sequence [Eq. (45)]. Here ${\varphi }_1$ is defined in Eq. (43) if we assume $g\left(J\right)=J/{\epsilon }_0$. (c) The result after contraction of $z$ rotations and is identical to the sequence shown in Ref. [42] when taking $N=2$ and the plus sign in $\pm$.

Figure 10

Quantum circuits for corrected two-qubit gates with the SK1 sequence. (a) The SK1 sequence [Eq. (49)], with ${\varphi }_1$ defined in Eq. (48) if we assume $g\left(J\right)=J/{\epsilon }_0$. (b) The sequence after contractions are performed as in Fig. 9. Several gates are enclosed by the dashed frame, which are further optimized as explained in the main text [Eq. (50)] and shown in (c).

Figure 11

Implementations of all 24 single-qubit Clifford gates. The thick blue denotes the naive implementation and the fine red line the supcode implementation.

Figure 12

Finite-bandwidth approximation of $1/\omega$ noise via sum of RTS. (a) Thin red lines denote power spectra of individual RTSs, the thick black solid line denotes the sum of RTSs, and the dashed black line is the ideal $1/\omega$. (b) Specific sample of such noise drawn from this distribution.

Figure 13

Fidelity vs number of gates for (a) dc noise and (b) $1/{\omega }^{1.5}$ noise. The red dashed line denotes the naive Clifford implementation and the blue solid line denotes the supcode Clifford implementation. Points are from RB simulation and curves are fits to $(1+e^{-\gamma n})/2$. Note that the exponential decay model does not fully describe the data in (b).

Figure 14

Fidelity decay constant $\gamma$ vs noise amplitude for (a) dc noise and (b) $1/{\omega }^{1.5}$ noise. The red dashed line denotes the naive Clifford implementation and the blue solid line denotes the supcode Clifford implementation. Points are from RB simulation, curves are proportional to ${\delta }^2$ and ${\delta }^4$. For dc noise, as expected, the lowest-order contribution of the noise is canceled by supcode, leaving a residual effect $O\left({\delta }^4\right)$. For the ac noise, the improvement from supcode saturates at approximately tenfold reduction in $\gamma$. Vertical lines indicate the values of $\delta$ used in Fig. 13.

Figure 15

Asymptotic improvement ratio of supcode vs naive pulses, for $1/{\omega }^{\alpha }$ noise. The line is $2\times {76}^{\alpha }$.