Phase Measurement Beyond the Standard Quantum Limit Using a Quantum Neuromorphic Platform

Phase measurement constitutes a key task in many fields of science, both in the classical and quantum regime. The higher precision of such measurement offers significant advances, and can also be utilized to achieve finer estimates for quantities such as distance, the gravitational constant, electromagnetic field amplitude, etc. Here we theoretically model the use of a quantum network, composed of a randomly coupled set of two-level systems, as a processing device for phase measurement. An incoming resource state carrying the phase information interacts with the quantum network, whose emission is trained to produce a desired output signal. We demonstrate phase-precision scaling following the standard quantum limit and Heisenberg limit. This can be achieved using quantum resource states such as NOON states or other entangled states, however, we also find that classically correlated mixtures of states are alone sufficient, provided that they exhibit quantum coherence. Our proposed setup does not require conditional measurements, and is compatible with many different types of coupling between the quantum network and the phase-encoding state, hence making it attractive to a wide range of possible physical implementations.

Figure 1

Setup for high-precision phase measurement. A quantum resource state $|{\psi }_N⟩$ picks up a phase, after which it interacts with a quantum network consisting of randomly coupled two-level nodes. The phase information is embedded in the QN and retrieved through its emission (estimated mean values after a finite number of repetitions), which is processed via a trained output layer. With this method, we show the phase-precision limit following the standard quantum limit and Heisenberg limit.

Figure 2

Exemplary output signals showing superresolution. Different NOON states $|{\psi }_N⟩$ are used as input for a quantum network of size 4. The corresponding solid curves indicate the ideal signals $I_{\mathrm{ideal}}$. In all cases, the training and testing size is 10 and 50, respectively, with $\xi =0.01$.

Figure 3

Performance of phase-estimation task. (a)–(c) Estimated phase versus ideal phase for different measurement error $\xi$, indicated on the bottom right of each panel. A QN with four nodes is used in these simulations. (d) SD at different $\varphi$ for different network sizes: (2, 3, 4) indicated by (squares, triangles, circles), respectively, and $\xi ={10}^{-3}$. (e) $\mathrm{\Delta }{\varphi }_1$ at $\pi /2$ indicating the scaling with QN size. Each SD is averaged over 50 sets of realizations of the random network parameters. In all considered cases, for each realization of the network parameters, $|{\psi }_1⟩$ is used with ten training and 100 testing sets.

Figure 4

Phase precision showing the standard quantum limit and Heisenberg limit. (a) Phase-estimation errors $\mathrm{\Delta }{\overline{\varphi }}_N$ and the SD at highest output slope $\mathrm{\Delta }{\varphi }_N$, plotted against the measurement error $\xi$ ($\propto 1/\sqrt{M}$). We use $N$ to denote the use of $N$-NOON state. (b) The ratio of phase-estimation errors, see Eq. (10). Empty triangles are for ${\overline{\eta }}_{12}$, empty squares for ${\overline{\eta }}_{13}$, and empty diamonds for ${\overline{\eta }}_{14}$. Also the corresponding filled ones: ${\eta }_{12}$ (triangles), ${\eta }_{13}$ (squares), and ${\eta }_{14}$ (diamonds). We report that $\mathrm{\Delta }{\overline{\varphi }}_N,\mathrm{\Delta }{\varphi }_N\propto 1/\sqrt{M}$—demonstrating the SQL, and $\mathrm{\Delta }{\overline{\varphi }}_N,\mathrm{\Delta }{\varphi }_N\propto 1/N$—demonstrating the HL. Panels (c),(d) are the phase-estimation errors and their ratios, respectively, using the classically correlated state ${\rho }_N$ in Eq. (11). (e) Estimation errors using ${\rho }_N$ dephased in the Fock basis, modeled by multiplying the off-diagonal elements of ${\rho }_N$ with a positive coefficient $p\le 1$. The inset shows the amount of coherence in the state ${\rho }_N$ with respect to $p$. In all cases, each data point represents an error evaluation of a trained output layer with ten training and 100 testing sets, which is averaged over 50 different realizations of network parameters. (f) Quantum Fisher information of dephased ${\rho }_N$ against $p$.

Figure 5

SD ratio ${\eta }_{1N}$ at different times for quantum network with two (a) and four (b) nodes. Notation as in Fig. 4. At some times, the ratio ${\eta }_{1N}$ exceeds not only the SQL ($\sqrt{N}$) but also the HL ($N$). Although the ratio, at some times, for QN-$2$ is higher, the QN-$4$ produces lower phase-estimation errors because its performance for single-excitation NOON state (the reference for the ratio) is much better, see Fig. 3.

Figure 6

(a) SD of highest slope $\mathrm{\Delta }{\varphi }_1$ (filled symbols) and $\mathrm{\Delta }{\varphi }_2$ (empty symbols) in the presence of energy decay (circles), dephasing (triangles), and depolarizing (squares) noise. The axis $\gamma /\hslash \mathrm{\Omega }$ refers to the ratio of the strength of respective noise to the energy unit of the quantum network parameters. The simulations are conducted averaging over 50 realizations of network parameters, with ten training and 100 testing sets. A QN with size $4$ is used, and $\tau =6/\mathrm{\Omega }$. (b) The ratio ${\eta }_{12}$ of the SDs in (a) for the case of energy decay (circles), dephasing (triangles), and depolarizing (squares) noise. Quantum advantage is present for all these noise strengths.

Figure 7

Comparison of different input-network coupling mechanisms and resource states. (a) Energy-preserving coupling (Jaynes-Cummings or Josephson). (b) Ultrastrong coherent interactions (note different vertical scale). (c) Cascading of input into QN nodes. (d) Energy-preserving coupling with maximally entangled states as the input. In all the panels, SDs $\mathrm{\Delta }{\varphi }_1$ and $\mathrm{\Delta }{\varphi }_2$ are plotted for different evolution times. The right axis indicates the ratio ${\eta }_{12}=\mathrm{\Delta }{\varphi }_1/\mathrm{\Delta }{\varphi }_2$. All the considered cases are capable of beating the SQL threshold (red dash-dotted lines).

Figure 8

Comparison of the phase-measurement errors using a quantum neuromorphic platform to the ultimate quantum Cramér-Rao bound. With respect to ${10}^3$ realizations of the quantum network parameters, the average and minimum phase-measurement errors are denoted by the black circles and blue triangles, respectively. The errors are plotted against (a) different NOON states $|{\psi }_N⟩$ using QN of size $\mathcal{Q}=2$; (b) QN sizes using $|{\psi }_1⟩$; (c) different mixed states ${\rho }_N$ using QN of size 2; and (d) different QN sizes using ${\rho }_1$. In all panels, the quantum Cramér-Rao bound is indicated by the black dashed-dotted curve or line, where the number of repetitions is $M={10}^4$. The testing is performed at the preferred highest output slope.

Figure 9

Phase-estimation error versus ridge parameter (a) and number of training sets (b). The arrow indicates the chosen $N_{\mathrm{train}}$ in this paper.

Figure 10

Minimizing the SD with shifted output signal. The same training set and testing set are used for different target output $\theta =0$ (a) and $\pi /4$ (b). The corresponding phase-estimation tasks in the range indicated by the green dashed boxes are plotted in (c),(d), respectively. The minimum error shifts from $\theta =0$ (a) to $\pi /4$ (b).