Phase estimation beyond the shot noise limit

Goal

1. Perform Phase estimation for various setup of inputs and outputs, comparing the estimation performance against the Shot Noise limit.
   1. Compare different initial states $\rho$.
   2. Compare different System/Ancillary couplings ( ${\alpha }_{??}$ )
   3. Measure how the uncertainty evolves as the number of trials increases, i.e. estimate the law $\frac{C}{N^{\alpha }}$ as $N\to \infty$.
2. How does optimizing Fisher Information compare with measuring the sensitivity ${\partial }_{{\delta }_S}$?

Potential research plans

Paper URL	Evolution model	Summary (incl. open questions)
https://www.nature.com/articles/s41586-025-09917-9	Matter-wave interferometry (massive particles / nanoparticles)	Demonstrates interferometry with increasingly massive particles, probing how quantum superposition and sensitivity scale with particle size. A central open question is how coherence and sensitivity scale as mass (effective N) grows, especially under decoherence and environmental coupling. (Nature)
https://journals.aps.org/pra/accepted/10.1103/537s-r3t8	Two-channel linear interferometer with squeezed states (Gaussian probes)	Shows Heisenberg scaling ( \sim 1/N ) for multiparameter estimation using realistic squeezed light states. Open direction: how to extend Heisenberg scaling robustness to multi-parameter settings under noise and loss, and whether scaling survives realistic imperfections. (journals.aps.org)
https://www.nature.com/articles/s41534-023-00691-y	Single-photon + orbital angular momentum (OAM) interferometry	Experimental demonstration of sub-shot-noise scaling across a wide range of N (~30k). Open issue: achieving stable Heisenberg scaling across arbitrarily large N without requiring prior information or adaptive strategies. (Nature)
https://www.mdpi.com/2073-8994/15/3/774	General SU(2), SU(1,1), and hybrid interferometers (QCRB framework)	Systematic comparison of scaling laws (shot-noise (1/\sqrt{N}), squeezed improvements, etc.). Open questions concern optimal configurations under realistic constraints (loss, asymmetry, correlations) and whether hybrid schemes can systematically reduce the prefactor/exponent. (MDPI)
https://www.mdpi.com/2304-6732/10/7/749	Nonlinear interferometers with optical parametric amplifiers (OPA)	Studies nonlinear gain-enhanced interferometry approaching Heisenberg scaling. Open problems include trade-offs between gain, losses, and scaling exponent, and whether nonlinear resources can consistently outperform linear schemes in realistic regimes. (MDPI)
https://arxiv.org/abs/2006.08449	Heralded large photon-number entangled states	Uses finite-size entangled photon states (up to (N=8)) to explore scaling. Open question: how to scale to large N while preserving entanglement-enhanced sensitivity in the presence of loss and imperfect detectors. (arXiv)
https://arxiv.org/abs/1012.5787	Interaction-based (nonlinear) quantum metrology	Predicts super-Heisenberg scaling ( \sim N^{-k} ) via k-body interactions. Key open issue: whether such scaling is fundamentally meaningful or survives realistic constraints, since higher-order nonlinearities and noise tend to degrade it. (arXiv)

Alternative plan

1. Do uncertainty quantification via the Stein Variational Gradient Descent method
   1. Condensed Stein Variational Gradient Descent for Uncertainty Quantification of Neural Networks
   2. https://arxiv.org/pdf/1608.04471

Priorities

• Update the calculations to have the standard BS + Phase + BS setup.
• Write the derivation from $H(a^{†}...)$ to $H=J_x$ ( see Jordan-Schwinger representation)
• Understand the optimal time $t$ to increase sensitivity of ${\delta }_S$.
• Verify that the number of particles is equivalent to increasing the number of trials ⇒ Get help? - [ ] This compares the sensitivity improvement from increasing $N_S$ and increasing $N_{trials}$. This also requires being able to estimate ${\delta }_S$ based on the measurement of the number of particles on the left and right well.
• Does the ancilla improve the coefficient of the scaling?
• Does the ancilla change the power law of the scaling? ⇒ This would be the biggest win.
• Lookup: Resource Theory,,
• Formalize the above process of finding the variation of ${\delta }_S$ as $\frac{C}{N^{\alpha }}$ . We want to optimize $C$ and $\alpha$.

Academic Sources

3. Ring lattices
   1. Ring lattices currents (Bose-Hubbard model) with attractive interactions but can be useful https://scipost.org/10.21468/SciPostPhys.12.4.138
   2. One of the 1st experiments on currents on ring currents (GPE and other models)
   3. This and this is what happens to currents states when we put a barrier in the system. For completeness (but no need to go deep on these)
   4. A barrier in a ring lattice avoids crossing in the 1st part, using superposition of current states. A similar phenomenon is seen in Minguzzi’s paper above.
4. AQUIDs
   1. 1st experiments by Boshier
   2. Experiment where they measure rotation, by Boshier
   3. Paper on AQUID (atomtronics SQUID) theory
   4. Measurements and observables of currents in ring lattices (with a barrier AQUID like)
   5. AQUID with attractive interactions
5. Machine Learning
   1. Reinforcement-learning-based matter-wave interferometer in a shaken optical lattice, by Luigi Amico
   2. Rings directly in experimental data: , based on this paper.
6. Bayesian statistics
   1. Quantum theory of phase estimation, by Luca Pezze.
   2. Dirichlet distribution: https://en.wikipedia.org/wiki/Dirichlet_distribution
7. Others
   1. Optimal generators for Quantum Sensing
   2. Phase estimation by pytket

Methodology

1. Describe a process through $H=H_S+H_A+H_{int}$ where
   1. $H_S=-J_S{J}_x\otimes 1+{\delta }_S{J}_z\otimes 1+U_S{J}_z^2\otimes 1$ is the system’s Hamiltonian
   2. $H_A=-J_{A}1\otimes J_x+{\delta }_{A}1\otimes J_z+U_{A}1\otimes J_z^2$ is the ancillary’s Hamiltonian
   3. $H_{int}={\alpha }_{xx}{J}_x\otimes J_x+{\alpha }_{xz}{J}_x\otimes J_z+{\alpha }_{zx}{J}_z\otimes J_x+{\alpha }_{zz}{J}_z\otimes J_z$ represents the system interactions
   4. $\rho =(|0⟩\otimes |\alpha ⟩)(⟨\alpha |\otimes ⟨0|)$ is the initial Quantum state.
      1. Try with a Noon state.
2. Solve ${\rho }_t=e^{-itH}\rho e^{itH}$
3. Get ${\rho }_t^{(S)}={\mathrm{Tr}}_A[{\rho }_t]$ by taking the Partial Trace over the ancillary system.
4. Determine the Measurement probabilities of $J_z$ by calculating $\mathrm{Tr}[{\rho }_t^{\prime }{J}_z]$.
5. Maximize sensitivity by
   1. Option A) finding the $\underset{\theta }{\mathrm{argmin}}\left\{\frac{d}{d\theta }\mathrm{Tr}[{\rho }_t^{\prime }{J}_z]\right\}$, where $\theta$ is any of our coefficients $J_S,U_S,J_A,{\delta }_A,U_A,{\alpha }_{xx},{\alpha }_{xz},{\alpha }_{zx},{\alpha }_{zz},t$.
   2. Option B) Using Bayesian statistics to build the posterior likelihood of $\delta$, and optimize for lower values of $\mathrm{Var}[\delta ]$. The prior can be nicely modeled as the Dirichlet Distribution, which is a generalization of the Beta Distribution.

• Look into writing this with TEBD!

Plots

Informational Gain as a function of $J_S$ and ${\delta }_S$.

Higher resolution informational gain

Higher Resolution Probability Difference System

Higher Resolution Probability Difference Ancilla

High resolution information gain for the Mach Zehnder Interferometer

Growth without ancillary

Growth with ancillary

Feedback

13. Determine the best sensitivity for when the ancillary system is not coupled
14. Plot ${\delta }_S$ as a function of ${\alpha }_i$. If it increases by more than a factor of 2 then it will be a sign of quantum interference improving beyond the shot noise limit.
15. NEXT STEPS:
    1. Learn the Hamiltonian from data based on both observations and time
    2. Build a plot of estimated delta against true delta, as a function of the number of observations
    3. Read the Holland paper https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.3.033279

Constraints

• The ancilla $\Delta$ must be bigger than the system’s $\Delta$
• The ancilla $\gamma$ must be bigger than the system’s $\gamma$
• The ancillary cannot be better than the system while isolated, i.e. $(J_A,U_A,{\delta }_S)$ cannot provide better measurements than $(J_S,U_S,{\delta }_S)$.

Appendix

Assumptions

16. $\hslash =1$: we work in dimensionless units
17. ${\rho }_t=e^{-itH}{\rho }_0{e}^{-itH}$: formal evolution of the system
18. ${\rho }_t^{(S)}={\mathrm{Tr}}_A[{\rho }_t]$ described the Reduced system
19. $H=H_S+H_A+H_{int}$, where
    1. $H_S$ only interacts with ${\rho }^{(S)}$
    2. $H_A$ only interacts with ${\rho }^{(A)}$
    3. $H_{int}$ represents the interactions
20. ${\rho }_0=|0⟩⟨0|\otimes {\rho }^{(A)}$ is our initial state
21. $e^{i\theta \vec{n}\vec{\sigma }}=\cos (\theta )1_2+i\sin (\theta )\vec{n}\vec{\sigma }$, where
    1. $|\vec{n}|=1$
    2. $\vec{\sigma }=[{\sigma }_x,{\sigma }_y,{\sigma }_z]$ are the respective Pauli Matrixes.
22. $J_z|k⟩=\frac{N-2k}{2}|k⟩$, where $|k⟩\in {\mathcal{H}}_A$, and $J_z$ is the Momentum Operator in ${\mathcal{H}}_A$.

On each system, we will often work with the “generic” Hamiltonian $\hat{H}=-J\widehat{J_z}+\frac{U}{2}{\widehat{J_z}}^2+\delta \widehat{J_x}$, where

• $J$ is the tunneling strength
• $\frac{U}{2}$ is the particle interaction
• $\delta$ is the energy shift

Calculating $\mathrm{Tr}[{\rho }_t^{(S)}{\sigma }_z]$

Since the left and right operators are conjugates, I use $(\dots )$ frequently to mean “use the complex conjugate operator as on the left side of ${\rho }_0$ “

$$\begin{array}{rll}{\rho }_t^{(S)}& ={\mathrm{Tr}}_A[{\rho }_t]& \\ & ={\mathrm{Tr}}_A[e^{-itH}{\rho }_0{e}^{itH}]& \\ & ={\sum }_{k=0}^N⟨k|e^{-itH}{\rho }_0{e}^{itH}|k⟩& \\ & ={\sum }_{k=0}^N⟨k|\exp \left[-it(-J_S{\sigma }_x+{\delta }_S{\sigma }_x-J_A{J}_z+U_A{J}_z^2+{\alpha }_x{\sigma }_x{J}_z+{\alpha }_z{\sigma }_z{J}_z)\right]{\rho }_0\exp \left[it(\dots )\right]|k⟩& {\hat{J}}_z|k⟩=\left(\frac{N-2k}{2}\right)|k⟩\\ & ={\sum }_{k=0}^N⟨k|\exp \left[-it(-J_S{\sigma }_x+{\delta }_S{\sigma }_x-J_A\left(\frac{N-2k}{2}\right)+U_A{\left(\frac{N-2k}{2}\right)}^2+{\alpha }_x\left(\frac{N-2k}{2}\right){\sigma }_x+{\alpha }_z\left(\frac{N-2k}{2}\right){\sigma }_z)\right]{\rho }_0\exp \left[it(\dots )\right]|k⟩& \text{Removing constants}\\ & ={\sum }_{k=0}^N⟨k|\exp \left[-it(-J_S{\sigma }_x+{\delta }_S{\sigma }_x+{\alpha }_x\left(\frac{N-2k}{2}\right){\sigma }_x+{\alpha }_z\left(\frac{N-2k}{2}\right){\sigma }_z)\right]{\rho }_0\exp \left[it(\dots )\right]|k⟩& \\ & ={\sum }_{k=0}^N⟨k|\exp \left[-it(({\alpha }_x\left(\frac{N-2k}{2}\right)-J_S){\sigma }_x+({\alpha }_z\left(\frac{N-2k}{2}\right)+{\delta }_S){\sigma }_z)\right]{\rho }_0\exp \left[it(\dots )\right]|k⟩& \{\begin{array}{l}\exp (i\theta \overrightarrow{n}\overrightarrow{\sigma })=\cos (\theta )+i\sin (\theta )\overrightarrow{n}\overrightarrow{\sigma }\\ {\omega }_k:=\sqrt{({\alpha }_x\frac{N-2k}{2}-J_S{)}^2+({\alpha }_z\frac{N-2k}{2}+{\delta }_S{)}^2}\end{array}\\ & ={\sum }_{k=0}^N⟨k|\left[\cos ({\omega }_{k}t)-i\sin ({\omega }_{k}t)\left(\frac{{\alpha }_x\frac{N-2k}{2}-J_S}{{\omega }_k}{\sigma }_x+\frac{{\alpha }_z\frac{N-2k}{2}+{\delta }_S}{{\omega }_k}{\sigma }_z\right)\right]{\rho }_0\left[\dots \right]|k⟩& {\rho }_0=|0⟩⟨0|\otimes {\rho }^{(A)}\\ & ={\sum }_{k=0}^N⟨k|\left[\cos ({\omega }_{k}t)-i\sin ({\omega }_{k}t)\left(\frac{{\alpha }_x\frac{N-2k}{2}-J_S}{{\omega }_k}{\sigma }_x+\frac{{\alpha }_z\frac{N-2k}{2}+{\delta }_S}{{\omega }_k}{\sigma }_z\right)\right]|0⟩⟨0|\otimes {\rho }^{(A)}\left[\dots \right]|k⟩& \text{Splitting }{\mathcal{H}}_S\text{ and }{\mathcal{H}}_A\\ & ={\sum }_{k=0}^N⟨k|{\rho }^{(A)}|k⟩\left[\cos ({\omega }_{k}t)-i\sin ({\omega }_{k}t)\left(\frac{{\alpha }_x\frac{N-2k}{2}-J_S}{{\omega }_k}{\sigma }_x+\frac{{\alpha }_z\frac{N-2k}{2}+{\delta }_S}{{\omega }_k}{\sigma }_z\right)\right]|0⟩⟨0|\left[\dots \right]& \end{array}$$

We care about

$$\begin{array}{rl}\mathrm{Tr}[{\rho }_t^{(S)}{\sigma }_z]& =⟨0|{\rho }_t^{(S)}{\sigma }_z|0⟩+⟨1|{\rho }_t^{(S)}{\sigma }_z|1⟩\\ & =⟨0|{\rho }_t^{(S)}|0⟩-⟨1|{\rho }_t^{(S)}|1⟩\\ & ={\sum }_{k=0}^N⟨k|{\rho }^{(A)}|k⟩\left[{\cos }^2({\omega }_{k}t)+{\sin }^2({\omega }_{k}t){\left(\frac{{\alpha }_x\frac{N-2k}{2}-J_S}{{\omega }_k}\right)}^2-{\sin }^2({\omega }_{k}t){\left(\frac{{\alpha }_z\frac{N-2k}{2}+{\delta }_S}{{\omega }_k}\right)}^2\right]\end{array}$$

We see that the constants $J_a$ and $U_A$ are inconsequential.

Log likelihoods

Given two Dirichlet Distributions, a true distributions with vector $\gamma =\{{\gamma }_k{\}}_k$ and a “wrong” estimated distribution $\alpha =\{{\alpha }_k{\}}_k$, the log likelihood of the observations, with $N$ observations, is given by

$$\begin{array}{rl}\log \left(\frac{\prod {\gamma }_k^{N{\alpha }_k-1}}{\beta (N\alpha )}\right)& =\log \left(\frac{\prod {\gamma }_k^{N{\alpha }_k-1}}{\prod \Gamma (N{\alpha }_k)}\Gamma (N\sum {\alpha }_k)\right)\\ & =\log \left(\prod {\gamma }_k^{N{\alpha }_k-1}\right)-\log \left(\prod \Gamma (N{\alpha }_k)\right)+\log \left(\Gamma (N\sum {\alpha }_k)\right)\\ & =\sum \log \left({\gamma }_k^{N{\alpha }_k-1}\right)-\sum \log \left(\Gamma (N{\alpha }_k)\right)+\log \left(\Gamma (N\sum {\alpha }_k)\right)\\ & =\sum (N{\alpha }_k-1)\log \left({\gamma }_k\right)-\sum \log \left(\Gamma (N{\alpha }_k)\right)+\log \left(\Gamma (N\sum {\alpha }_k)\right)\end{array}$$

This allows us to make use of the loggamma function $\log (\Gamma (\cdot ))$ in calculations, improving numerical stability: https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.loggamma.html.

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Next steps

• Write the section below neatly to organize it
• Check the scaling value ( is it different than 2? )
• Check the pre-factor we get.
• Remake the comparison with the division by 2.
• Ensure that we have the gain $\Vert \delta \theta N^{\alpha }-\frac{C}{2^{\alpha }}\Vert$
• Prepare to present this in 30 seconds / pitch elevator.