Combining dynamical decoupling with fault-tolerant quantum computation

We study how dynamical decoupling (DD) pulse sequences can improve the reliability of quantum computers. We prove upper bounds on the accuracy of DD-protected quantum gates and derive sufficient conditions for DD-protected gates to outperform unprotected gates. Under suitable conditions, fault-tolerant quantum circuits constructed from DD-protected gates can tolerate stronger noise and have a lower overhead cost than fault-tolerant circuits constructed from unprotected gates. Our accuracy estimates depend on the dynamics of the bath that couples to the quantum computer and can be expressed either in terms of the operator norm of the bath’s Hamiltonian or in terms of the power spectrum of bath correlations; we explain in particular how the performance of recursively generated concatenated pulse sequences can be analyzed from either viewpoint. Our results apply to Hamiltonian noise models with limited spatial correlations.

Figure 1

Parameters characterizing a sequence of uniformly spaced rectangular pulses. $\delta$ is the pulse width, and ${\tau }_0-\delta$ is the interval between the end of one pulse and the beginning of the following pulse.

Figure 2

Illustration of the local-bath assumption. Solid (blue) lines are system qubits, and dashed (black) lines are bath subsystems. Each open rectangle is a quantum gate, and its associated solid (light gray) rectangle represents the joint evolution of system qubits and bath subsystems that are coupled by the noise Hamiltonian. The solid rectangles do not overlap, indicating that when two gates act in parallel on distinct sets of system qubits, the associated bath subsystems are also distinct.

Figure 3

Schematic representation of $H_M(t)$ for time-symmetric DD pulse sequences. The time axis is bent in half, with time flowing counterclockwise from the upper right to the lower left, so that times aligned on the upper and lower branches are related by time symmetry. (a) Two pulses (marked as black boxes) in a time-symmetric memory sequence with $\mathrm{H̃}(t_{\mathrm{DD}}-t)=\mathrm{H̃}(t)$; the pulse on the bottom branch is the time-reversed version of the pulse on the top branch. (b) Appending the gate pulse (box $G$) to the memory sequence spoils the time symmetry; the black pulses on the upper and lower branches are no longer aligned. (c) Appending fictitious time evolution governed by $-H_B$ during $t\in [0,\delta ]$ (box $B$) restores the time symmetry of the memory sequence for $t\in [\delta ,T-\delta ]$, where $T=t_0+\delta$.

Figure 4

Two scenarios where DD-protected gates outperform unprotected gates. (a) Quantum computing is scalable with DD-protected gates, but not with unprotected gates. (b) Quantum computing is scalable with either DD-protected gates or with unprotected gates, but DD reduces the overhead cost of fault tolerance.

Figure 5

Plot of ${\eta }_{\mathrm{DD}}/\eta$ versus $\epsilon {\tau }_0$ for the universal decoupling sequence [Eq. (69)] and for the time-symmetric sequence [Eq. (74)], assuming zero-width pulses. The noise strength for the DD-protected gate is weaker than the noise strength for the unprotected gate for $\epsilon {\tau }_0<0.0711$ in the case of the universal decoupling sequence and for $\epsilon {\tau }_0<0.0403$ in the case of the time-symmetric sequence. For $\epsilon {\tau }_0$ sufficiently small, using the time-symmetric sequence reduces the noise strength further than the universal decoupling sequence.

Figure 6

Plot of $L_{\mathrm{DD}}^{*}/L_{\mathrm{un}}^{*}$ [Eq. (62)] versus $\epsilon {\tau }_0$ for the universal decoupling sequence and the time-symmetric sequence in the limit $\delta /{\tau }_0\to 0$. We have assumed ${\eta }_0/\eta =2$ and have used the value $a=\log {}_2(291)\approx 8.18$ appropriate for the fault-tolerant gadget constructions in[31].

Figure 7

Plot of $\Vert {\Omega }_3\Vert {}_{\mathrm{Bound}}/\Vert {\Omega }_3\Vert {}_{\mathrm{Actual}}$ versus $\beta {\tau }_0$ for the time-symmetric DD sequence Eq. (72). The noise Hamiltonian is $H=H_B+H_{\mathit{SB}}$ ($H_S^0=0$), where $H_B=(\beta /2)\sum _i{\sigma }_i^z$ and $H_{\mathit{SB}}=(J/4)\sum _{\alpha =x,y,z}{\sigma }_S^{\alpha }\otimes (\sum _{i=1}^n{\sigma }_i^{\alpha })$; the index $i$ labels the bath spins. Here $\Vert {\Omega }_3\Vert {}_{\mathrm{Bound}}$ is computed using Eq. (50) and Table (where $\delta =0$ and $T=8{\tau }_0$), while $\Vert {\Omega }_3\Vert {}_{\mathrm{Actual}}$ is computed by evaluating the integral in Eq. (28) exactly. The kinks in the plots arise because the operator norm can have a discontinuous first derivative when eigenvalues cross.

Figure 8

Comparison of effective noise strengths ${\eta }_{\mathrm{DD}}$ and ${\eta }_{\mathrm{EDD}}$ for the universal decoupling sequence given in Eq. (65) (for different pulse widths) and the Eulerian decoupling sequence given in Eq. (78), respectively. The universal decoupling sequence is always worse than no decoupling ($\eta =J{\tau }_0$) for $\delta /{\tau }_0⩾1/4$, and Eulerian decoupling is worse than no decoupling for $\epsilon {\tau }_0⩾0.0239$. The Eulerian sequence is always better than universal DD for $\delta /{\tau }_0⩾0.1983$. For smaller values of $\delta /{\tau }_0$, as the pulse width increases, the Eulerian sequence outperforms the universal sequence for small values of $\epsilon {\tau }_0$. However, because of its longer length, the Eulerian sequence offers no advantage over the universal sequence or no decoupling when $\epsilon {\tau }_0$ is too large.

Figure 9

Illustration of the parameter regions in which no DD, the universal decoupling sequence (DD) [Eq. (65)], or the Eulerian decoupling sequence (EDD) [Eq. (78)] emerges as the best strategy. Different regions indicated correspond to the following inequalities: (1) ${\eta }_{\mathrm{EDD}}<{\eta }_{\mathrm{noDD}}<{\eta }_{\mathrm{DD}}$, (2) ${\eta }_{\mathrm{EDD}}<{\eta }_{\mathrm{DD}}<{\eta }_{\mathrm{noDD}}$, (3) ${\eta }_{\mathrm{DD}}<{\eta }_{\mathrm{EDD}}<{\eta }_{\mathrm{noDD}}$, (4) ${\eta }_{\mathrm{DD}}<{\eta }_{\mathrm{noDD}}<{\eta }_{\mathrm{EDD}}$, (5) ${\eta }_{\mathrm{noDD}}<{\eta }_{\mathrm{DD}}<{\eta }_{\mathrm{EDD}}$, (6) ${\eta }_{\mathrm{noDD}}<{\eta }_{\mathrm{EDD}}<{\eta }_{\mathrm{DD}}$. The noise strengths are given by ${\eta }_{\mathrm{noDD}}=J{\tau }_0$ and Eqs. (69) and (82).

Figure 10

Upper bounds on the effective noise strengths achieved by the concatenated universal DD pulse sequence [blue dashed line, based on Eq. (147) with $R=4$] and by the concatenated time-symmetric DD pulse sequence [red solid line, based on Eq. (149) with $R=8$], as a function of $\mathrm{c̄}\epsilon {\tau }_0$, where $\mathrm{c̄}$ is defined in the text.