An optical platform for measurement-based quantum computing

Jonas S. Neergaard-Nielsen

bigQ - Center for Macroscopic Quantum States

Technical University of Denmark (DTU)

Main credits: Mikkel Vilsbøll Larsen - now at Xanadu

Outline

Continuous-variable Measurement-based Quantum Computing
Cluster state generation
Gate implementation
Scheme for fault-tolerant computing

CV-MBQC

Crazy Cryogenics + individual qubit control

= pretty hard to scale up...

F. Arute et al., Nature 574, 505 (2019)

53 qubit sampling

Photonic quantum computing advantages:

mostly room temperature (except detectors)
potentially better scalability
direct network-compatibility

Challenges:

photonic qubits don't like to stick around
photonic qubits don't like to interact

→ use measurement-based quantum computing
instead of standard gate-based approach

Y. Wu et al., Phys. Rev. Lett. 127, 180501 (2021)

H-S. Zhong et al., Phys. Rev. Lett. 127, 180502 (2021)

56 qubit sampling

144 mode, 113 photons Gaussian Boson Sampling

Standard GATE MODEL of quantum computation

Measurement-based quantum computing (MBQC)

Photonics is ideally suited for this!

Coherent, high-fidelity qubit interactions

Creation of massively entangled cluster state
- the "computational substrate"

challenge:

Continuous-variable MBQC

squeezed states

beam splitters

homodyne detection

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 \(\mathbf{A}\))

Our work is inspired by

theory from Nicolas Menicucci's group
experiments from Akira Furusawa's group

Wallace & Gromit: The Wrong Trousers (1993)

Menicucci et al., PRL 104, 250503 (2010)

Uses temporal encoding:

resources are re-used over and over
computation is performed in parallel with generation of the cluster state

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

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.

Experiment in more detail

λ = 1550 nm: squeezed light generation in free space, the rest in fibre

3 cavity locks + 7 phase locks (chopped)

50 m fibre = 247 ns delay

600 m fibre

Laser is cw; homodyne power spectra reveal detailed experimental parameters

We define "pulses" as temporal modes optimised for high squeezing

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

\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

Science 366, 369–372 (2019)

Science 366, 373–376 (2019)

Very similar result from Tokyo University:

Gate implementation

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

With our \(N=12\) cylindrical graph,
we can project out 6 wires

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

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

Carefully calibrated AWG (arbitrary waveform generator) + EOMs (phase shifters) implement fully programmable phase settings

frames of 228 modes - allows for full characterisation of 8 single-mode gates

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

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

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}