Enhancing distributed sensing with imperfect error correction

Entanglement has shown promise in enhancing information processing tasks in a sensor network via distributed quantum sensing protocols. As noise is ubiquitous in sensor networks, error correction schemes based on Gottesman-Kitaev-Preskill (GKP) states are required to enhance the performance, as shown in [Q. Zhuang, J. Preskill, and L. Jiang, New J. Phys. 22, 022001 (2020)] assuming homogeneous noise among sensors and perfect GKP states. Here, we extend the analyses of performance enhancement to finite-squeezed GKP states in a heterogeneous noise model. To begin with, we study different concatenation schemes of GKP-two-mode-squeezing codes. While traditional sequential concatenation schemes in previous works do improve the suppression of noise, we propose a balanced concatenation scheme that outperforms the sequential schemes in the presence of finite GKP squeezing. We then apply these results to two specific tasks in distributed quantum sensing (parameter estimation and hypothesis testing) to understand the trade-off between imperfect squeezing and performance. For the first task, we consider an energy-constrained scenario and provide an optimal way to distribute the energy of the finite-squeezed GKP states among the sensors. For the later task, we show that the error probability can still be drastically lowered via concatenation of realistic finite-squeezed GKP codes.

Figure 1

(a) CV-QEC scheme to protect an entangled probe state from the (heterogeneous) noise accumulated during the distribution of sensor probes. The state $\widehat{\rho }$ is the approximate entangled probe state after QEC. One can use the approximate probe state $\widehat{\rho }$ (b) as an input to a DQS protocol [14, 15] or (c) as an input for SLAEN [23].

Figure 2

Different schemes for concatenation codes. Panels (a,c) are circuit diagram and simplified schematic of a sequential scheme. Panels (b,d) are circuit diagram and simplified schematic of a balanced scheme. For simplicity, we did not show the measurement of quadratures' module $\sqrt{2\pi }$, which needs another GKP ancilla to be performed. Panel (e) is the decoding circuit, where we use the second noisy GKP state. “$\widehat{\mathrm{TS}}$” represents two-mode squeezing and “${\widehat{\mathrm{TS}}}^{†}$” represents the inverse two-mode squeezing. $\mathcal{N}$ represents the additive white Gaussian noise (AWGN).

Figure 3

Performance of sequential type (red line), and balanced type (blue line), with (a) initial $\sigma =0.15$ and mean photon number in ancillary modes $\overline{n}=200$ [the GKP squeezing is defined as $s_{\mathrm{GKP}}=-10{log}_{10}\left(2{\sigma }_{\mathrm{GKP}}^2\right)$, corresponding to $26\text{dB}$], and (b) initial $\sigma =0.25$ and $\overline{n}=50$ (corresponding to $20\text{dB}$). Dashed lines are the lower bound of each concatenation codes.

Figure 4

Performance of concatenation codes of different initial $\sigma$, with constant mean photon number in ancilla modes $\overline{n}=400$ (${\sigma }_{\mathrm{GKP}}=0.025$) (a) sequential type and (b) balanced type. Red lines with circles present $\sigma =0.1$, orange lines with squares present $\sigma =0.12$, green lines with diamonds present $\sigma =0.15$, blue lines with up triangles present $\sigma =0.17$, and purple lines with down triangles present $\sigma =0.2$.

Figure 5

Schematic of a distributed quantum sensing protocol with GKP-TMS error correction in the distribution step. The label “$\widehat{\mathrm{TS}}$” represents two-mode squeezing, while “${\widehat{\mathrm{TS}}}^{†}$” represents the inverse two-mode squeezing; “homo” represents homodyne detection.

Figure 6

Contour plot of the two sensor modes, with (a) identical noise 0.05 and weight $w_1=w_2=0.5$ and (b) asymmetric noise $0.2,0.1$, with weight $w_1=0.3,w_2=0.7$. The $x$ axis is the photon numbers of mode 2, $y$ axis is the photon numbers of mode 1. Contours are the ratio of output ${\sigma }_{\mathrm{L}}^2$ with and without error correction.

Figure 7

Error probability of different schemes, assuming $N_s=20$ and $M=10$. Black dashed lines with filled circles represent the separable state scheme. Red lines with squares represent no-error correction. Green lines with diamonds represent a one-layer finite-squeezed GKP error correction code. Orange lines with down triangles represent the lower bound of sequential concatenation code. Blue lines with up triangles represent the lower bound of balanced concatenation code. Black dashed lines with empty circles represent ideal distributed sensing with no loss. Mean photon number in GKP states $\overline{n}=400$, (a) $\eta \in [0.99,1],\overline{\eta }=0.992$ and (b) $\eta \in [0.9,1],\overline{\eta }=0.96$.

Figure 8

(a) Asymptotic expression for heterogeneous noise. The $x$ axis is ${\sigma }_1$, and $y$ axis is the output logical noise ${\sigma }_{\mathrm{L}}$. In the procedure, we keep ${\sigma }_1\times {\sigma }_2={\overline{\sigma }}^2$ constant and vary ${\sigma }_1$. (b,c) Numerical expression and asymptotic expression result for different $\sigma$. The $x$ axis is the noise of finite-squeezed GKP state ${\sigma }_{\mathrm{GKP}}$, the $y$ axis is the output noise ${\sigma }_{\mathrm{L}}$.