Multi-mode squeezing

for quantum sensing and computing

Jonas S. Neergaard-Nielsen
QPIT/bigQ - Department of Physics, Technical University of Denmark

CVQC, Carlsberg Academy, September 2022

Outline (vises ikke)

hurtig squeezing intro

QPIT :heart: squeezing: Vores squeezere - fotos, performance, type, antal

hurtig project overview

multi-mode squeezing - using just one or two squeezers

show cov.matrix/Wigner efter squeezing, splitting (animation? variabel fase/sqz)

4 modes: distributed phase sensing; 1000s of modes: cluster state for MBQC

intro distributed sensing

experiment

intro cluster state generation

characterised cluster

intro MBQC

gates

Squeezed states

\langle \Delta x^2 \rangle < \frac{\hbar}{2}
\langle \Delta p^2 \rangle < \frac{\hbar}{2}

amplitude squeezed

phase squeezed

\Delta x \Delta p \ge \frac{\hbar}{2}

QPIT 💖️ SQUEEZING

PPKTP

semi-monolithic

PPKTP bow-tie

single- or two-mode squeezing

PPLN waveguide

  • phase sensing
  • Raman spectroscopy
  • entanglement witness
  • integrated squeezers
  • Gaussian Boson Sampling

~8 squeezers in two labs

(many more on the way)

  • QKD
  • teleportation
  • quantum randomness
  • quantum computing

4-mode squeezing

thousands-mode squeezing

POSTERS

Benjamin

Tummas

Renato

Abhinav

Distributed phase sensing

"Cheap" entanglement enhances estimation of a global parameter of spatially separated systems

X Guo, CR Breum, J Borregaard, S Izumi, MV Larsen, T Gehring, M Christandl, JS Neergaard-Nielsen, UL Andersen

Nature Physics 16, 281 (2020)

\phi_\mathrm{avg} = \frac{\phi_1 + \ldots + \phi_M}{M}

Our example

Average of multiple optical phase shifts

Small phase shifts:

Estimate with homodyne detection of phase quadrature

\langle \hat{P}_\mathrm{avg}\rangle \approx \sqrt{2\eta}\alpha \phi_\mathrm{avg}

Task

Estimate some global property of multiple physical systems

\sigma_\mathrm{separable} = \frac{1}{2\sqrt{M}N\sqrt{1+\frac{1}{N}}}
\sigma_\mathrm{entangled} = \frac{1}{2MN\sqrt{1+\frac{1}{MN}}}
\sigma_\mathrm{SQL} = \frac{1}{2\sqrt{MN}}

Heisenberg scaling in probe energy and no. of sites/samples

sensitivity  \(\sigma \equiv 1/SNR\)

- minimum resolvable phase shift

Realistic (lossy) situation:

Heisenberg scaling disappears but sensitivity gain remains

Experiment

\sigma \equiv \frac{1}{SNR} = \frac{\sqrt{\langle\Delta \hat P^2_\mathrm{avg} \rangle}}{|\partial \langle \hat P_\mathrm{avg}\rangle /\partial\phi_\mathrm{avg}|}

Measured sensitivities

Cluster states for CV-MBQC

Universal gate set implemented by homodyne measurements on temporally encoded entangled state of 1000s of modes

(continuous-variable measurement-based quantum computing)

MV Larsen, X Guo, CR Breum, JS Neergaard-Nielsen, UL Andersen

Nature Physics 17, 1018 (2021)

MV Larsen, X Guo, CR Breum, JS Neergaard-Nielsen, UL Andersen

Science 366, 369–372 (2019)

Cluster state

graph state with regular lattice adjacency matrix \(\mathbf{A}\)

{\displaystyle |\psi_\mathbf{A}\rangle=\hat{C}_Z[\mathbf{A}]\ |0\rangle_p^{\otimes m} = \prod_{j=1}^m\prod_{k=1}^m e^{iA_{jk}\hat{x}_j\hat{x}_k}\ |0\rangle_p^{\otimes m}}

In CV:

nodes

edges

Fullfils the nullifier condition: \((\mathbf{\hat{p}} - \mathbf{A}\mathbf{\hat{x}})\ |\psi_\mathbf{A}\rangle=0\)

Finite squeezing → self-loops (non-zero, complex-valued diagonal in \(\mathbf{A}\))

Cluster state generation

  1. EPR states from two single-mode squeezers
  2. Interfere two EPR pairs
  3. Repeat

Use temporal domain: efficient use of resources

EPR adjacency matrix:

\(\mathbf{Z}_\mathrm{EPR} = \begin{pmatrix}i \cosh 2r & -i\sinh 2r \\ -i\sinh 2r & i\cosh 2r\end{pmatrix}\)

BS₂ induces new graph edges

\(\mathbf{Z}_\mathrm{1D}\) is not a true cluster state,
but a phase shift of \(\pi/2\) on every 2nd mode transforms it to \(\mathbf{Z}'_\mathrm{1D}\), a true (approximate) cluster state

Long delay makes nodes separated by \(N=12\) temporal modes adjacent

BS₃ now connects the graph in the second dimension

Nullifiers of the 2D graph state:

\(\hat{n}_k^x=\hat{x}_{k}^A+\hat{x}_{k}^B-\hat{x}_{k+1}^A-\hat{x}_{k+1}^B-\hat{x}_{k+N}^A+\hat{x}_{k+N}^B-\hat{x}_{k+N+1}^A+\hat{x}_{k+N+1}^B\)

\(\hat{n}_k^p=\hat{p}_{k}^A+\hat{p}_{k}^B+\hat{p}_{k+1}^A+\hat{p}_{k+1}^B-\hat{p}_{k+N}^A+\hat{p}_{k+N}^B+\hat{p}_{k+N+1}^A-\hat{p}_{k+N+1}^B\)

 

\(k+1\)

\(k\)

\(k+N+1\)

\(k+N\)

 with variance   \(\langle\Delta\hat{n}_k^{x2}\rangle=4e^{-2r_A},\quad\langle\Delta\hat{n}_k^{p2}\rangle=4e^{-2r_B}\)

The 8 modes making up the nullifiers constitute a unit cell of the graph.

Inseparability of the state

\(k+1\)

\(k\)

\(k+N+1\)

\(k+N\)

van Loock-Furusawa criterion for bipartition of \(S\):

  • Define operators    \(\displaystyle\hat{X} = \sum_{j\in S} h_j\hat{x}_j,\quad \hat{P} = \sum_{j\in S} g_j \hat{p}_j\)
  • If \(\{h_j\},\{g_j\}\) exist such that


    then \(\mathcal{S}_1\) and \(\mathcal{S}_2\) are inseparable

Periodic, so enough to show inseparability of a unit cell

Complete inseparability: any bipartition is inseparable

S
\mathcal{S}_1
\mathcal{S}_2

Task: find suitable \(\hat{X}, \hat{P}\) for all 127 possible bipartitions of the 8-mode unit cell

\(\displaystyle \langle\Delta\hat{X}^2\rangle + \langle\Delta\hat{P}^2\rangle < \Big|\sum_{j\in \mathcal{S}_1} h_j g_j\Big| + \Big| \sum_{j\in\mathcal{S}_2} h_j g_j\Big|\)

We find linear combinations of nullifiers as \(\hat{X}\) and \(\hat{P}\) for all 127 bipartitions of the unit cell

van Loock-Furusawa criterion is fulfilled if all nullifiers are squeezed by >3 dB

\(\displaystyle \langle\Delta\hat{X}^2\rangle + \langle\Delta\hat{P}^2\rangle < \Big|\sum_{j\in \mathcal{S}_1} h_j g_j\Big| + \Big| \sum_{j\in\mathcal{S}_2} h_j g_j\Big|\)

Complete inseperability confirmed for 2 × 15000 modes

To implement gates and circuits on a cluster state, it is reshaped into "wires":

Measuring a node in the computational basis removes it from the graph

Gate implementation

Computational scheme

  • Wires consist of 2-mode EPR states
  • Gates implemented by teleportation
  • Gates determined by homodyne phases
  • Simpler now to consider BS₃ as part of a joint measurement on a 1D cluster

 

Computational scheme

  • Wires consist of 2-mode EPR states
  • Gates implemented by teleportation
  • Gates determined by homodyne phases
  • Simpler now to consider BS₃ as part of a joint measurement on a 1D cluster

 

Computational scheme

\(\hat{U}=(-1)^w\hat{R}\left(\frac{\theta_{+,k}}{2}\right)\hat{S}\left(\tan\frac{\theta_{-,k}}{2}\right)\hat{R}\left(\frac{\theta_{+,k}}{2}\right)\)

\(\theta_{\pm,k}=\pm\theta_{A,k}+\theta_{B,k}\)

Implemented gate:

Computational scheme

\(\hat{U}=(-1)^w\hat{R}\left(\frac{\theta_{+,k}}{2}\right)\hat{S}\left(\tan\frac{\theta_{-,k}}{2}\right)\hat{R}\left(\frac{\theta_{+,k}}{2}\right)\)

\(\theta_{\pm,k}=\pm\theta_{A,k}+\theta_{B,k}\)

Single-mode gates

\(\mathbf{\hat{q}'} = \mathbf{U \hat{q}} + \mathbf{N\hat{p}_i} + \mathbf{Dm} \)

Side effects of the gate teleportation:

Noise from finite squeezing

- achilles heel of CV-MBQC!

Measurement outcome-dependent displacement

- can be treated in post-processing

Both \(\mathbf{U}\) and \(\mathbf{N}\) can be characterised by gate tomography

- use correlations between output and reference input (entangled with input)

\(\hat{U}=(-1)^w\hat{R}\left(\frac{\theta_{+,k}}{2}\right)\hat{S}\left(\tan\frac{\theta_{-,k}}{2}\right)\hat{R}\left(\frac{\theta_{+,k}}{2}\right)\)

\(\theta_{\pm,k}=\pm\theta_{A,k}+\theta_{B,k}\)

Implemented single-mode gates

\textbf{R}_\theta=\begin{pmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{pmatrix}
\begin{pmatrix} \theta_{A,k}\\\theta_{B,k} \end{pmatrix}_R = \frac{1}{2}\begin{pmatrix} \theta-(-1)^w\pi/2\\\theta+(-1)^w\pi/2 \end{pmatrix}

Rotation

\(\hat{U}=(-1)^w\hat{R}\left(\frac{\theta_{+,k}}{2}\right)\hat{S}\left(\tan\frac{\theta_{-,k}}{2}\right)\hat{R}\left(\frac{\theta_{+,k}}{2}\right)\)

\(\theta_{\pm,k}=\pm\theta_{A,k}+\theta_{B,k}\)

Implemented single-mode gates

\textbf{P}_\sigma=\begin{pmatrix} 1 & 0 \\ \sigma & 1 \end{pmatrix}
\begin{pmatrix} \theta_{A,k}\\\theta_{B,k} \end{pmatrix}_P = \begin{pmatrix} 0 \\ \pi/2-\arctan(\sigma/2) \end{pmatrix}

Shear

- actually \(\hat F^j\hat P(\sigma)\) since \(\hat P(\sigma)\) cannot be done in one step

\(\hat{U}=(-1)^w\hat{R}\left(\frac{\theta_{+,k}}{2}\right)\hat{S}\left(\tan\frac{\theta_{-,k}}{2}\right)\hat{R}\left(\frac{\theta_{+,k}}{2}\right)\)

\(\theta_{\pm,k}=\pm\theta_{A,k}+\theta_{B,k}\)

Implemented single-mode gates

\textbf{S}_r= \begin{pmatrix} e^{-r} & 0 \\ 0 & e^r \end{pmatrix}
\begin{pmatrix} \theta_{A,k}\\\theta_{B,k} \end{pmatrix}_S = (-1)^w\arctan e^r\begin{pmatrix} -1 \\ 1 \end{pmatrix}

Squeezing

Two-mode gate

\begin{pmatrix} \pi/4\\-\pi/4\\(-1)^w\pi/4\\-(-1)^w\pi/4\\(-1)^w[\pi/2-\arctan(g/2)]\\0\\(-1)^w\pi/4\\(-1)^w[\pi/4+2\arctan(g/2)]\\(-1)^w[\pi/2-\arctan(g/2)]\\0 \end{pmatrix}

\(\lbrace\hat{R}(\theta),\hat{S}(e^r),\hat{C}_Z(g)\rbrace\)

is a universal multi-mode Gaussian gate set.

source:

Jonathan Home
group, ETH

\(\lbrace\hat{F}=\hat{R}(\pi/2),\hat{P}(1),\hat{C}_Z(g)\rbrace\)

is a multi-mode Clifford gate set for GKP-encoded qubits

A few other recent results

Steering-based randomness certification with squeezed states and homodyne measurements

Marie Ioannou, Bradley Longstaff, MV Larsen, JS Neergaard-Nielsen, UL Andersen, D Cavalcanti, N Brunner, JB Brask, arXiv:2111.06186

Deterministic quantum phase estimation beyond the ideal NOON state limit

Jens AH Nielsen, JS Neergaard-Nielsen, T Gehring, UL Andersen, arXiv:2111.09756

40 km fiber transmission of squeezed light measured with a real local oscillator

Iyad Suleiman, JAH Nielsen, X Guo, N Jain, JS Neergaard-Nielsen,
T Gehring, UL Andersen, arXiv:2101.10177

Casper R. Breum

Xueshi Guo

Casper R. Breum

Xueshi Guo

Mikkel V. Larsen

Mikkel V. Larsen