Fast pulse sequences for dynamically corrected gates in singlet-triplet qubits

We present a set of experimentally feasible pulse sequences that implement any single-qubit gate on a singlet-triplet spin qubit and demonstrate that these new sequences are up to three times faster than existing sequences in the literature. We show that these sequences can be extended to incorporate built-in dynamical error correction, yielding gates that are robust to both charge and magnetic field noise and up to twice as fast as previous dynamically corrected gate schemes. We present a thorough comparison of the performance of our new sequences with that of several existing ones using randomized benchmarking, considering both quasistatic and $1/f^{\alpha }$ noise models. We provide our results both as a function of evolution time and as a function of the number of gates, which respectively yield both an effective coherence time and an estimate of the number of gates that can be performed within this coherence time. We determine which set of pulse sequences gives the best results for a wide range of noise strengths and power spectra. Overall, we find that the traditional, slower sequences perform best when there is no field noise or when the noise contains significant high-frequency components; otherwise, our new, fast sequences exhibit the best performance.

Figure 1

Plots of $\alpha$ (left) and $\chi$ (right) as functions of $\theta$ for the Ramon sequence, Eq. (7), for positive values of $\varphi$. Note that we add $2\pi$ to the (negative) value of $\alpha$ obtained from Eq. (8); this results in an overall minus sign on top of the rotation.

Figure 2

Plots of $\alpha$ (left) and $\chi$ (right) as functions of $\theta$ for the Ramon sequence, Eq. (7), for negative values of $\varphi$. Unlike the case of positive values of $\varphi$, we do not need to add $2\pi$ to the value of $\alpha$ obtained from Eq. (9), and thus we perform a proper $z$ rotation.

Figure 3

Plots of $\alpha$ (left) and $\chi$ (right) for the modified Ramon sequence, Eq. (17) with ${\theta }^{\prime }=\pi /4$ (top row), $\pi /8$ (middle row), and $\pi /16$ (bottom row) and for positive values of $\varphi$.

Figure 4

Similar to Fig. 3, but for negative values of $\varphi$.

Figure 5

Plots of the time required to execute our $\theta -2\theta -\theta$ sequence, Eq. (13) (solid lines), and the Ramon sequence, Eq. (7) (dashed lines), as a function of $\theta$ for positive values of $\varphi$.

Figure 6

Plots of the time required to execute our $\theta -2\theta -\theta$ sequence, Eq. (13) (solid lines), and the Ramon sequence, Eq. (7) (dashed lines), as a function of $\theta$ for negative values of $\varphi$.

Figure 7

(Top left) Plot of the time required to execute the UCO-I Hadamard-$z$-Hadamard sequence, Eq. (57), for all possible values of $\theta$, namely, $\pi /4<\theta <\pi /2$. The remaining plots compare the times required to execute this sequence (solid lines) with the time required to execute the UCO-II modified Ramon sequence, Eq. (58) (dashed lines), for ${\theta }^{\prime }=\frac{\pi }{4}$ (top right), $\frac{\pi }{8}$ (bottom left), and $\frac{\pi }{16}$ (bottom right). All plots shown here are for positive values of $\varphi$.

Figure 8

Similar to Fig. 7, but for negative values of $\varphi$.

Figure 9

Plots of $t_{\text{total}}$ for the five-pulse sequence (solid lines) and for the $z-x-z$ sequence (dashed lines), given by Eqs. (63) and (64), respectively, for positive $\varphi$ (left) and negative $\varphi$ (right).

Figure 10

Plots of average fidelity extracted from randomized benchmarking simulations as a function of time for our uncorrected sequence sets (UCUO, UCO-I, UCO-II) in the presence of quasistatic noise. The left column [(a), (c), and (e)] represents a quantum dot operated with barrier control [18, 19], while the right column [(b), (d), and (f)[ uses tilt control. These two methods of control correspond to charge noise strengths of ${\sigma }_J=0.00426J$ and $0.0563J$, respectively [19]. The solid curves are data, and the dashed curves are fits to Eq. (73). The top row (a, b) represents a system with no field noise (e.g., isotopically purified Si) and magnetic field gradient $h_0=23\text{MHz}$, the middle row [(c) and (d)] a system with ${\sigma }_h=11.5\text{MHz}$ and $h_0=40\text{MHz}$, and the bottom row [(e) and (f)] a system with ${\sigma }_h=23\text{MHz}$ and $h_0=40\text{MHz}$. For the UCO-II pulse sequences, we use the naive versions of the corresponding CO-II pulse sequences, with the same parameters. The values of ${\sigma }_J$ that our parameters correspond to are, assuming $\frac{1}{30}h\le J\le 30h, 3.27\text{kHz}$ to $2.94\text{MHz}$ (a), $5.68\text{kHz}$ to $5.11\text{MHz}$ [(c) and (e)], $43.1\text{kHz}$ to $38.8\text{MHz}$ (b), and $75.1\text{kHz}$ to $67.6\text{MHz}$ [(d) and (f)].

Figure 11

Plots of average fidelity extracted from randomized benchmarking simulations as a function of time for our corrected sequence sets (CUO, CO-II) in the presence of quasistatic noise. The left column [(a), (c), and (e)] represents a quantum dot operated with barrier control [18, 19], while the right column [(b), (d), and (f)] uses tilt control. These two methods of control correspond to charge noise strengths of ${\sigma }_J=0.00426J$ and $0.0563J$, respectively [19]. The solid curves are data, and the dashed curves are fits to Eq. (73). The top row [(a) and (b)] represents a system with no field noise (e.g., isotopically purified Si) and magnetic field gradient $h_0=23\text{MHz}$, the middle row [(c) and (d)] a system with ${\sigma }_h=11.5\text{MHz}$ and $h_0=40\text{MHz}$, and the bottom row [(e) and (f)] a system with ${\sigma }_h=23\text{MHz}$ and $h_0=40\text{MHz}$. The values of ${\sigma }_J$ that our parameters correspond to are, assuming $\frac{1}{30}h\le J\le 30h, 3.27\text{kHz}$ to $2.94\text{MHz}$ (a), $5.68\text{kHz}$ to $5.11\text{MHz}$ [(c) and (e)], $43.1\text{kHz}$ to $38.8\text{MHz}$ (b), and $75.1\text{kHz}$ to $67.6\text{MHz}$ [(d) and (f)].

Figure 12

Plots of average fidelity extracted from randomized benchmarking simulations as a function of number of gates for our uncorrected sequence sets (UCUO, UCO-I, UCO-II) in the presence of quasistatic noise. The left column [(a), (c), and (e)] represents a quantum dot operated with barrier control [18, 19], while the right column [(b), (d), and (f)] uses tilt control. These two methods of control correspond to charge noise strengths of ${\sigma }_J=0.00426J$ and $0.0563J$, respectively [19]. The solid curves are data, and the dashed curves are fits to Eq. (74). The top row [(a) and (b)] represents a system with no field noise (e.g., isotopically purified Si) and magnetic field gradient $h_0=23\text{MHz}$, the middle row [(c) and (d)] a system with ${\sigma }_h=11.5\text{MHz}$ and $h_0=40\text{MHz}$, and the bottom row [(e) and (f)] a system with ${\sigma }_h=23\text{MHz}$ and $h_0=40\text{MHz}$. For the UCO-II pulse sequences, we use the naive versions of the corresponding CO-II pulse sequences, with the same parameters. The values of ${\sigma }_J$ that our parameters correspond to are, assuming $\frac{1}{30}h\le J\le 30h, 3.27\text{kHz}$ to $2.94\text{MHz}$ (a), $5.68\text{kHz}$ to $5.11\text{MHz}$ [(c) and (e)], $43.1\text{kHz}$ to $38.8\text{MHz}$ (b), and $75.1\text{kHz}$ to $67.6\text{MHz}$ [(d) and (f)].

Figure 13

Plots of average fidelity extracted from randomized benchmarking simulations as a function of number of gates for our corrected sequence sets (CUO, CO-II) in the presence of quasistatic noise. The left column [(a), (c), and (e)] represents a quantum dot operated with barrier control [18, 19], while the right column [(b), (d), and (f)] uses tilt control. These two methods of control correspond to charge noise strengths of ${\sigma }_J=0.00426J$ and $0.0563J$, respectively [19]. The solid curves are data, and the dashed curves are fits to Eq. (74). The top row [(a) and (b)] represents a system with no field noise (e.g., isotopically purified Si) and magnetic field gradient $h_0=23\text{MHz}$, the middle row [(c) and (d)] a system with ${\sigma }_h=11.5\text{MHz}$ and $h_0=40\text{MHz}$, and the bottom row [(e) and (f)] a system with ${\sigma }_h=23\text{MHz}$ and $h_0=40\text{MHz}$. The values of ${\sigma }_J$ that our parameters correspond to are, assuming $\frac{1}{30}h\le J\le 30h, 3.27\text{kHz}$ to $2.94\text{MHz}$ (a), $5.68\text{kHz}$ to $5.11\text{MHz}$ [(c) and (e)], $43.1\text{kHz}$ to $38.8\text{MHz}$ (b), and $75.1\text{kHz}$ to $67.6\text{MHz}$ [(d) and (f)].

Figure 14

Plots of average fidelity extracted from randomized benchmarking simulations as a function of time for our uncorrected sequence sets (UCUO, UCO-I, UCO-II) in the presence of $1/f^{\alpha }$ noise. The solid curves are data, and the dashed curves are fits to Eq. (73). The left column [(a) and (c)] represents a quantum dot operated with barrier control [18, 19], while the right column [(b) and (d)] uses tilt control. The top row [(a) and (b)] represents a system with no field noise (e.g., isotopically purified Si) and magnetic field gradient $h_0=23\text{MHz}$, and the bottom row [(c) and (d)] a system with field noise and $h_0=40\text{MHz}$. For the UCO-II pulse sequences, we use the naive versions of the corresponding CO-II pulse sequences, with the same parameters.

Figure 15

Plots of average fidelity extracted from randomized benchmarking simulations as a function of number of gates for our uncorrected sequence sets (UCUO, UCO-I, UCO-II) in the presence of $1/f^{\alpha }$ noise. The solid curves are data, and the dashed curves are fits to Eq. (73). The left column [(a) and (c)] represents a quantum dot operated with barrier control [18, 19], while the right column [(b) and (d)] uses tilt control. The top row (a, b) represents a system with no field noise (e.g., isotopically purified Si) and magnetic field gradient $h_0=23\text{MHz}$, and the bottom row [(c) and (d)] a system with field noise and $h_0=40\text{MHz}$. For the UCO-II pulse sequences, we use the naive versions of the corresponding CO-II pulse sequences, with the same parameters.

Figure 16

Plots of average fidelity extracted from randomized benchmarking simulations as a function of time for our corrected sequence sets (CUO, CO-II) in the presence of $1/f^{\alpha }$ noise. The left column [(a) and (c)] represents a quantum dot operated with barrier control [18, 19], while the right column [(b) and (d)] uses tilt control. The top row (a, b) represents a system with no field noise (e.g., isotopically purified Si) and magnetic field gradient $h_0=23\text{MHz}$, and the bottom row [(c) and (d)] a system with field noise and $h_0=40\text{MHz}$.

Figure 17

Plots of average fidelity extracted from randomized benchmarking simulations as a function of number of gates for our corrected sequence sets (CUO, CO-II) in the presence of $1/f^{\alpha }$ noise. The left column [(a) and (c)] represents a quantum dot operated with barrier control [18, 19], while the right column [(b) and (d)] uses tilt control. The top row [(a) and (b)] represents a system with no field noise (e.g., isotopically purified Si) and magnetic field gradient $h_0=23\text{MHz}$, and the bottom row [(c) and (d)] a system with field noise and $h_0=40\text{MHz}$.

Figure 18

Plots of $T_2$ extracted from randomized benchmarking simulations in the presence of $1/f^{\alpha }$ noise for (a) the uncorrected sequence sets and (b) the corrected sets with no field noise as functions of ${\alpha }_J=\alpha$.

Figure 19

Plots of $n_0$ extracted from randomized benchmarking simulations in the presence of $1/f^{\alpha }$ noise for (a) the uncorrected sequence sets and (b) the corrected sets with no field noise as functions of ${\alpha }_J=\alpha$.

Figure 20

Plots of $T_2$ extracted from randomized benchmarking simulations in the presence of $1/f^{\alpha }$ noise for (a) the uncorrected sequence sets and (b) the corrected sets with ${\alpha }_h=2.6$ and as functions of ${\alpha }_J=\alpha$.

Figure 21

Plots of $n_0$ extracted from randomized benchmarking simulations in the presence of $1/f^{\alpha }$ noise for (a) the uncorrected sequence sets and (b) the corrected sets with ${\alpha }_h=2.6$ and as functions of ${\alpha }_J=\alpha$.

Figure 22

Plots of $T_2$ extracted from randomized benchmarking simulations in the presence of $1/f^{\alpha }$ noise for (a) the uncorrected sequence sets and (b) the corrected sets as functions of ${\alpha }_h={\alpha }_J=\alpha$.

Figure 23

Plots of $n_0$ extracted from randomized benchmarking simulations in the presence of $1/f^{\alpha }$ noise for (a) the uncorrected sequence sets and (b) the corrected sets as functions of ${\alpha }_h={\alpha }_J=\alpha$.

Figure 24

Plots of $\alpha$ (left) and $\chi$ (right) for the “generalized Ramon sequence” with ${\theta }^{\prime }=\frac{\pi }{4}$ (top row), $\frac{\pi }{8}$ (middle row), and $\frac{\pi }{16}$ (bottom row) and for positive values of $\varphi$.

Figure 25

Plots of times required to execute the $\theta -2\theta -\theta$ sequence, Eq. (13) (solid lines), and the “generalized Ramon sequence”, Eq. (A1) (dashed lines), with ${\theta }^{\prime }=\frac{\pi }{4}$ (top left), $\frac{\pi }{8}$ (top right), and $\frac{\pi }{16}$ (bottom) and for positive values of $\varphi$.

Figure 26

As Fig. 24, but for negative values of $\varphi$.

Figure 27

As Fig. 25, but for negative values of $\varphi$.

Figure 28

(Top left) Plot of the time required to execute the sequence, Eq. (B1), for all possible values of $\theta$, namely, $\pi /4<\theta <\pi /2$. The remaining plots compare the times required to execute the sequence, Eq. (B1) (solid lines), with the time required to execute the “modified Ramon sequence,” Eq. (17) (dashed lines), for ${\theta }^{\prime }=\frac{\pi }{4}$ (top right), $\frac{\pi }{8}$ (bottom left), and $\frac{\pi }{16}$ (bottom right). All plots shown here are for positive values of $\varphi$.

Figure 29

As Fig. 28, but for negative values of $\varphi$.