An optical platform for measurement-based quantum computing

Jonas S. Neergaard-Nielsen

bigQ - Center for Macroscopic Quantum States

Technical University of Denmark (DTU)

CV-MBQC

Cluster state: graph state with regular lattice adjacency matrix $\mathbf{A}$ .

|\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}

{\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

S. Yokoyama et al., Nat. Photonics 7, 982 (2013)

Generate 1D cluster state following Yokoyama et al.:

Produce an x-squeezed and a p-squeezed state
Generate EPR state by combining them on a 50:50 BS
Delay one mode by a predefined delay time $\tau$
Entangle neighbouring EPR states by interfering on another 50:50 BS

Then, generate cylindrical cluster state (~2D) by "coiling up" the 1D string and stitching together

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

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.

But how can we be sure the state is entangled (more precisely: inseparable)?

$k+1$

$k$

$k+N+1$

$k+N$

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}_1

\mathcal{S}_2

\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

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

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

$\theta_{\pm,k}=\pm\theta_{A,k}+\theta_{B,k}$

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
(W. Asavanant et al., PR Applied 16, 034005 (2021))

- 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}

\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}

\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}

\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}

\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}

\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}

\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}

\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}

1 . 1

An optical platform for measurement-based quantum computing Jonas S. Neergaard-Nielsen bigQ - Center for Macroscopic Quantum States Technical University of Denmark (DTU)

An optical platform for measurement-based quantum computing

Outline

CV-MBQC

Standard GATE MODEL of quantum computation

Measurement-based quantum computing (MBQC)

Measurement-based quantum computing (MBQC)

challenge:

Continuous-variable MBQC

Cluster state generation

Experiment in more detail

Gate implementation

Computational scheme

Computational scheme

Computational scheme

Computational scheme

Single-mode gates

Implemented single-mode gates

Implemented single-mode gates

Implemented single-mode gates

Two-mode gate

Small circuit

Scheme for fault-tolerance