Finite-round quantum error correction on symmetric quantum sensors

TL;DR

It is proved that in the limit of a large number of qubits, the quantum field sensing protocol has a precision that approaches the Heisenberg limit despite a linear rate of deletion errors.

Abstract

The Heisenberg limit provides a quadratic improvement over the standard quantum limit, and is the maximum quantum advantage that quantum sensors could provide over classical methods. This limit remains elusive, however, because of the inevitable presence of noise decohering quantum sensors. Namely, if infinite rounds of quantum error correction corrects any part of a quantum sensor's signal, a no-go result purports that the standard quantum limit scaling can not be exceeded using Markovian quantum error correction. We side-step this no-go result and prove that in the limit of a large number of qubits, our quantum field sensing protocol has a precision that approaches the Heisenberg limit despite a linear rate of deletion errors. This is achieved by using an optimally determined, finite number of rounds of quantum error correction married with adaptive, non-Markovian signal recovery procedures. Our protocol is based on permutation invariant quantum error correction codes, which can be designed to admit correction for a tunable number of bit-flip, phase-flip, and deletion errors. We discuss near-term implementations using quantum control assisted by coupling the spins to a common bosonic mode.

Figures (5)

• Figure 1: Signal accumulation in Protocol 1: $r$ rounds of signal accumulation, each over a time $\tau t_0$, followed by QEC. If the projection is on the codespace $\mathcal{C}^{t,\sigma}$, the subsequent unitary does nothing. Otherwise, if the projection is on the orthogonal space $\mathcal{Q}^{t,\sigma}$, the subsequent unitary maps $\mathcal{Q}^{t,\sigma}$ back to the codespace. Here the dimensionless time is $\tau = r^{-1}$, and $t_0$ is a dimension full time unit. Here, the size of $r$ is determined by the gnu code parameter $g$ and a positive exponent $\delta$ that shows how $r/g$ grows with $N$.
• Figure 2: Amplitude rebalancing in Protocol 1: Rounds of amplitude rebalancing until the amplitudes are balanced. Each timestep takes a time of $t_0 / N^{1+\delta},$ and so essentially no deletions occur in this stage. Successful rebalancing occurs with projection onto $S^0_\pm$ with probability at least 4/5. If deletions do happen, we calculate the number of deletions and the resultant random shift in Dicke weight, and calculate the resultant change in the distortion ratio.
• Figure 3: We can approach the HL exponentially fast in $k$, using $k$ iterations of Protocol 1. At the $k$th iteration, we estimate $b$ with standard deviation ${{\bar{B}}}_k = \Theta(N^{- {{\bar{b}}}_k} )$. The SQL corresponds to estimating $b$ with ${{\bar{B}}}_k = \Theta(N^{-1/2})$, which is achieveable using classical techniques. The HL corresponds to estimating $b$ with ${{\bar{B}}}_k = \Theta(N^{-1})$, which is the best possible precision using quantum techniques. Note that $\delta$ determines how close we can approach the Heisenberg scaling. The smaller the $\delta$, the larger the value of $N$ needs to be before we begin to see the illustrated advantages.
• Figure 4: FI and QFI of shifted gnu codes on 1000 qubits in the absence of noise. Here, ($u=1$). The QFI for shifted gnu codes is $g^2n$. The FI is obtained by measuring the evolved shifted gnu states in the code's logical plus-minus basis. The plot shows how the ratio of the FI to the QFI depends on the true value of $\theta$. For reference, the 1000-qubit GHZ state corresponds to $g=1000,n=1$.
• Figure 5: Quantum sensing with the shifted gnu code to infer the strength of a field $E$ which acts uniformly on the spins. We depict the amplitudes of the logical basis states in the Dicke basis. A unitary $U_{\theta}=e^{-i\theta \hat{J}^z}$ rotates the logical basis to a new logical frame, which satisfies the same error correction criteria. Here, we estimate $\theta$.

Theorems & Definitions (45)

• Lemma 1
• proof
• Theorem 2
• Theorem 3: SLD for pure symmetric states
• proof : Proof of Theorem \ref{['thm:SLD-pure-state']}
• Lemma 4: Impact of deletions
• Lemma 5
• proof
• Lemma 6: Impact of AD errors
• Lemma 7