Fault-Tolerant Quantum Dynamical Decoupling

Dynamical decoupling pulse sequences have been used to extend coherence times in quantum systems ever since the discovery of the spin-echo effect. Here we introduce a method of recursively concatenated dynamical decoupling pulses, designed to overcome both decoherence and operational errors. This is important for coherent control of quantum systems such as quantum computers. For bounded-strength, non-Markovian environments, such as for the spin-bath that arises in electron- and nuclear-spin based solid-state quantum computer proposals, we show that it is strictly advantageous to use concatenated pulses, as opposed to standard periodic dynamical decoupling pulse sequences. Namely, the concatenated scheme is both fault tolerant and superpolynomially more efficient, at equal cost. We derive a condition on the pulse noise level below which concatenation is guaranteed to reduce decoherence.

Figure 1

Left: Performance of CDD (solid line, $n=4$) vs PDD (dot-dashed line) as a function of random jitter fraction $|w_P|≔\Vert W_P\Vert /\Vert H_P\Vert$, with pulse-width $\delta ={10}^{-5}T$, coupling strength $j=.2/T$ ($T$ is the total evolution time), and number of bath spins $K=2$, averaged over 90 jitter realizations. For comparison, the horizontal dashed line corresponds to free evolution. Right: CDD as a function of random jitter fraction $|w_P|$ and concatenation level $n$. The vertical axis denotes $l={log}_{10}(1-\mathrm{\text{purity}})$ here and in Figs. 2, 3.

Figure 2

CDD (left) and PDD (right), as a function of system-bath coupling $j$; pulse width $\delta ={10}^{-4}T$, number of bath spins $K=5$, and without jitter ($W_P=0$). Note the $l$-axis scale difference between CDD and PDD.

Figure 3

Left: CDD performance as a function of concatenation level and systematic jitter. The pulse-width $\delta ={10}^{-4}T$, number of bath spins $K=5$, $jT=15.0$, averaged over 7–80 realizations (more realizations for higher $n$). Right: CDD (solid line) vs PDD (dot-dashed line) as a function of systematic jitter for $n=5$, $\delta ={10}^{-5}T$, $K=5$, $j{\tau }_0=3.0$, averaged over 14–80 realizations. The dashed line is pulse-free evolution. CDD performance is unaffected up to $\sim 20\%$ jitter level.

Figure 4

Projections involved in DD. (a) Perfect cancellation in first-order Magnus case. (b) Extra rotation induced by higher-order Magnus terms, and the effect of concatenation.