Restoring Heisenberg scaling in time via autonomous quantum error correction

Abstract

We establish a sufficient condition under which autonomous quantum error correction (AutoQEC) can effectively restore Heisenberg scaling (HS) in quantum metrology. Specifically, we show that if all Lindblad operators associated with the noise commute with the signal Hamiltonian and a particular constrained linear equation admits a solution, then an ancilla-free AutoQEC scheme with finite $R$ (where $R$ represents the ratio between the engineered dissipation rate for AutoQEC and the noise rate,) can approximately preserve HS with desired small additive error $ε> 0$ over any time interval $0 \leq t \leq T$. We emphasize that the error scales as $ ε= O(κT / R^c) $ where $c$ is a positive integer and $κ$ is the noise rate, indicating that the required $R$ decreases significantly with increasing $c$ to achieve a desired error. Furthermore, we discuss that if the sufficient condition is not satisfied, logical errors may be induced that cannot be efficiently corrected by the canonical AutoQEC framework. Finally, we numerically verify our analytical results by employing the concrete examples of phase estimation under dephasing noise.

Figures (5)

• Figure 1: Schematic illustration of AutoQEC dynamics under different conditions. Each panel depicts the evolution of quantum states across the code space and higher-order error spaces. (a) The engineered dissipation (blue thick arrows) effectively counteracts natural dissipation (red thick arrows), leading to a suppression of the probability of occupying the $0\leq n \leq c$-th order error space, scaling as $P^{[n]}\sim O(1/R^{n})$. Consequently, the decoherence (induced by the natural dissipation) rate is suppressed to $O(\kappa/R^{c})$ at the logical level, also relevant to AutoQEC dynamics in quantum computation. (b) Phase differences between subspaces introduce logical errors, represented by the green thin arrows in the error spaces. As a result, even though the decoherence rate is suppressed to $O(\kappa/R^{c})$, one may not achieve AutoQEC up to order $c$. (c) In addition to natural dissipation, $\hat{U}(wt):=e^{-i\hat{H}wt}$ induces additional errors that transfer states into higher-order error spaces, indicated by $w$-induced red thick arrows. These errors accumulate more rapidly than those caused by natural dissipation, in particular when $w \gg \kappa$.
• Figure 2: QFI as a function of sensing time $t$ when the signal Hamiltonian is given by $\hat{H}=\sum_{i=1}^{3}\hat{Z}_{i}$ in the presence of correlated dephasing noise defined in Eq. \ref{['correlationmatrix']}. With the code defined in Eqs. \ref{['correcodewords1']} and \ref{['correcodewords2']}, AutoQEC of order $c=1$ can be achieved. In the numerics, we consider various values of $R$, with $w=1$, $\kappa=0.1$, and AutoQEC order $c=1$. The black line (labeled "Ideal") and the red line (labeled "Error") represent the QFI of the noiseless case and without AutoQEC case respectively. The inset shows a zoomed-in view of the QFI at smaller times $t$.
• Figure 3: QFI as a function of time when the signal Hamiltonian is given by $\hat{H}=\prod_{i=1}^{5}\hat{Z}_{i}$, in the presence of local dephasing noise. In this case, by exploiting the repetition code defined in Eqs. \ref{['localcodewords1']} and \ref{['localcodewords2']}, AutoQEC order up to $c=2$ can be achieved. For the numerics, we consider different AutoQEC orders $c$, with $R=100$. For $c=1$, we consider $\vert \Phi_{q} \rangle=(\vert \mu_{0} \rangle+\vert \mu_{1} \rangle)/\sqrt{2}~\forall~ 1 \leq q \leq q_{\mathrm{max}}$. Other conventions are identical with those of Fig. \ref{['fig:correlatedepol']}.
• Figure S4: QFI as a function of sensing time $t$, with $R=10^{4}$, $w=1$, $\kappa=0.1$, and AutoQEC order $c=1$. The black line (labeled "Ideal") corresponds to the noiseless case. The blue line (labeled "HNLS: O") represent the QFI when both the Knill-Laflamme condition and HNLS condition are satisfied. We note that the blue line is nearly equal to the black line. The red line (labeled "HNLS: X") shows the QFI when only the Knill-Laflamme condition is satisfied, while HNLS condition is violated. The inset shows a zoomed-in view of the QFI for a smaller range of $t$.
• Figure S5: QFI as a function of sensing time $t$, different values of $R$, with $w=1$, $\kappa=0.1$, and AutoQEC order $c=1$. The black line (labeled "Ideal") and the red line (labeled "Error") represent the QFI of the noiseless case and without AutoQEC case respectively. The green line (labeled "(P1): X") and the pink line (labeled "(P2): X") represent the QFI when (P1) and (P2) are not satisfied respectively. The inset shows a zoomed-in view of the QFI for a smaller range of $t$.

Theorems & Definitions (7)

• Definition 1
• Lemma 1
• Theorem 1
• Theorem 1
• proof
• Corollary 1
• Theorem 2