Eugene Polzik

Akira Furusawa

Ulrik Andersen

Continuous-variable
photonic
quantum computing

Quite different from
qubit encoding

Quite different from
stationary qubit platforms

Measurement-based QC:

Quite different from
gate model of quantum computing

CV and its correspondence with qubits, part 1

CV and its correspondence with qubits, part 2

squeezed light

measurement-based quantum compution

DTU's cluster states

GKP states - towards FTQC

Outline

Qubits (discrete variable) and

CV (continuous variable)
encoding of quantum information

Qubits

CV

Hilbert space

2-dimensional Hilbert space

- atomic levels, superconducting charge,
electronic spin, photon polarisation or path, ...

\(\infty\)-dimensional Hilbert space

- motion of trapped ion, MW resonator field,
single-mode laser field, ...

basis states

\(\ket{0}\) and \(\ket{1}\)

\(\ket{x}\) for any \(x\in\mathbb{R}\)

arbitrary pure states

\[\ket{\psi} = c_0\ket{0} + c_1\ket{1} = \sum_{k=0}^1 c_k\ket{k}\]

\[ \ket{\psi} = \int_{-\infty}^{\infty}dx\ \psi(x) \ket{x}\]

\( c_k = \braket{k|\psi} \)

\( \psi(x) = \braket{x|\psi} \)

conjugate basis

\[ [\hat Z,\hat X] = 2i\hat Y \]

\( [\hat{x},\hat{p}] = i \) (with \(\hbar=1\))

\[ \hat{p} = -i\hbar\frac{\partial}{\partial x} \]

\[ \Delta x \Delta p \ge \frac{1}{2} \]

"phase space"

conjugate basis

\( [\hat{x},\hat{p}] = i \) (with \(\hbar=1\))

\[ [\hat Z,\hat X] = 2i\hat Y \]

basis transformation

\[ \ket{+} = \hat H \ket 0 \quad \ket - = \hat H \ket 1 \]

\( \ket{p=a} = \frac{1}{\sqrt{2\pi}} \int dx\ e^{iax} \ket{x} = \hat F \ket{x=a} \)

Fourier transform = 90° phase space rotation

Hadamard = 1-qubit QFT

\( \hat H = \frac{1}{\sqrt{2}} \begin{pmatrix}1&1\\1&-1\end{pmatrix} \)

\( \hat F = \exp[i\frac{\pi}{4} (\hat x^2 + \hat p^2)] \)

Pauli gate // displacement

\[ \hat X \ket k = \ket{k \oplus 1} \quad \hat Z \ket k = (-1)^k \ket k \]

\( \hat X(s) \ket{a} = \ket{a+s} \quad \hat Z(s) \ket{a} = e^{isa} \ket{a} \)

\( \hat X \) shifts value (position),
\( \hat Z \) shifts phase (momentum)

\(\hat X\) shifts the value, \(\hat Z\) gives a phase

\( \hat X(s) = e^{-is\hat p} \quad \hat Z(s) = e^{is\hat x} \)

Pauli gate // displacement

\[ \hat X \hat Z = -\hat Z \hat X = -i\hat Y\]

\( \hat X(s) \hat Z(t) = e^{-ist} \hat Z(t) \hat X(s) \)
\( = \hat D(s,t) \) (up to phase)

\( \hat X(s) = e^{-is\hat p} \quad \hat Z(s) = e^{is\hat x} \)

Pauli gate // displacement

\( \hat X(s) = e^{-is\hat p} \quad \hat Z(s) = e^{is\hat x} \)

phase gate // shear

\( \hat P(s) = e^{\frac{is}{2}\hat x^2} \)

\[ \hat S \ket k = i^k \ket k \]

\(\hat S = \hat P(\frac{\pi}{2}) = \sqrt{\hat Z} \) gives a \( \pi/2 \) phase

\[ \hat P(s) \ket a = e^{isa^2/2} \ket a \]

the CV phase or shear gate

Quantum harmonic oscillator

quantum harmonic oscillator

\( \hat x = \dfrac{\hat a + \hat a^\dagger}{\sqrt 2} \)

\( \hat p = \dfrac{\hat a - \hat a^\dagger}{i\sqrt 2} \)

Relation between quadratures and raising + lowering operators

monochromatic EM field: harmonic oscillator

\(\hat x\) and \(\hat p\) represent the field quadratures

quantum harmonic oscillator

σx σp ≥ ħ/2

[x,p]=iħ ⇒

quantum harmonic oscillator

quantum noise!

- can be redistributed

σx σp ≥ ħ/2

amplitude squeezing

quantum harmonic oscillator

σx σp ≥ ħ/2

phase squeezing

quantum harmonic oscillator

σx σp ≥ ħ/2

squeezed vacuum

quantum harmonic oscillator

approximate basis states

\( \hat S(r) = \exp[\frac{ir}{2}(\hat x\hat p + \hat p\hat x)] \)

CV basis states \( \ket x \) are unphysical
- infinitely squeezed, i.e. infinite energy

- a fundamental limitation of CV

Use squeezed state as an approximation

squeezed light generation

Same Hamiltonian \( -i\hbar \frac{g}{2} (\hat a^2 - \hat a^{\dagger 2})\) as for SPDC (spontaneous parametric down-conversion) - the workhorse single-photon source

\( \hat S(r) = \exp[\frac{ir}{2}(\hat x\hat p + \hat p\hat x)] = \exp[\frac{r}{2}(\hat a^2 - \hat a^{\dagger 2}] \)

Low gain

High gain

quadrature measurement

\( \hat x, \hat p \) and intermediate quadratures \( \hat x(\theta) = \hat x\cos\theta + \hat p\sin\theta \) can be efficiently measured by homodyne detection

signal field

local oscillator
- strong reference field at phase \(\theta\)

output voltage \(V \propto x(\theta)\)

not "clicks" - data is always there

quadrature measurement

DV vs. CV quantum optics

Main difference between DV and CV photonics [somewhat generalised]:

DV photonic qubits are (often) probabilistic. DV gates are probabilistic.

Squeezing is deterministic. CV gates are deterministic.

DV protocols are typically post-selected - they work (well) when they work. Loss reduces the rate.

CV (Gaussian) protocols always work, but not necessarily well.
Loss reduces the quality.

Lab stuff interlude

QPIT squeezers

QKD
teleportation
phase sensing
displacement sensing
quantum learning

Raman spectroscopy
quantum randomness
Gaussian Boson Sampling
quantum computing
...

Currently ~17 squeezers
of various sorts

QPIT squeezers

Applications:

Bow-tie OPO with PPKTP

ns-scale Pulsed squeezing

LNOI chip squeezing

back to theory...

\( CX\ket{x}_a\ket{y}_b = \ket{x}_a\ket{x\oplus y}_b \)

entangling gates

CNOT / CX

\( CX(g)\ket{x}_a\ket{y}_b = \ket{x}_a\ket{x+gy}_b \)

\( CX = \ket{0}_a\!\bra{0} \otimes \hat I_b + \ket{1}_a\!\bra{1} \otimes \hat X_b \)

\( CX(g) = \exp[-ig\hat x_a \hat p_b] \)

\( CX\ket{x}_a\ket{y}_b = \ket{x}_a\ket{x\oplus y}_b \)

entangling gates

CNOT / CX

\( CX(g)\ket{x}_a\ket{y}_b = \ket{x}_a\ket{x+gy}_b \)

\( CX = \ket{0}_a\!\bra{0} \otimes \hat I_b + \ket{1}_a\!\bra{1} \otimes \hat X_b \)

\( CX(g) = \exp[-ig\hat x_a \hat p_b] \)

\( CZ\ket{x}_a\ket{y}_b = (-1)^{xy}\ket{x}_a\ket{y}_b \)

\( CZ = \ket{0}_a\!\bra{0} \otimes \hat I_b + \ket{1}_a\!\bra{1} \otimes \hat Z_b \)

CZ (controlled-phase)

\( CZ(g)\ket{x}_a\ket{y}_b = e^{igxy}\ket{x}_a\ket{y}_b \)

\( CZ(g) = \exp[-ig\hat x_a \hat x_b] \)

\( CX\ket{x}_a\ket{y}_b = \ket{x}_a\ket{x\oplus y}_b \)

entangling gates

CNOT / CX

\( CX(g)\ket{x}_a\ket{y}_b = \ket{x}_a\ket{x+gy}_b \)

\( CX = \ket{0}_a\!\bra{0} \otimes \hat I_b + \ket{1}_a\!\bra{1} \otimes \hat X_b \)

\( CX(g) = \exp[-ig\hat x_a \hat p_b] \)

\( CZ\ket{x}_a\ket{y}_b = (-1)^{xy}\ket{x}_a\ket{y}_b \)

\( CZ = \ket{0}_a\!\bra{0} \otimes \hat I_b + \ket{1}_a\!\bra{1} \otimes \hat Z_b \)

CZ (controlled-phase)

\( CZ(g)\ket{x}_a\ket{y}_b = e^{igxy}\ket{x}_a\ket{y}_b \)

\( CZ(g) = \exp[-ig\hat x_a \hat x_b] \)

beamsplitter

\( BS(\theta) = \exp[i\theta(\hat x_a \hat p_b - \hat p_a \hat x_b)] \)

Clifford gate set

\(H + S + CNOT\) generate the Clifford group - "easy, non-magical operations"

\(F + P(s) + D(s,t) + BS(\theta)\) generate the Gaussian group

\( \frac{1}{\sqrt 2}(\ket{0}_a\ket{0}_b + \ket{1}_a\ket{1}_b) \)

Bell / EPR states

\( \int dx' \ket{x'}_a \ket{x'}_b \)

perfect bit-value
correlations

perfect \(x\)-value
correlations

\( \frac{1}{\sqrt 2}(\ket{0}_a\ket{0}_b + \ket{1}_a\ket{1}_b) \)

Bell / EPR states

\( \int dx' \ket{x'}_a \ket{x'}_b \)

equivalent (comes from \(\hat H \hat X \hat H = \hat Z\))

realistic / physical
EPR state

perfect bit-value
correlations

perfect \(x\)-value
correlations

ZY Ou, SF Pereira, HJ Kimble, KC Peng,

PRL 68, 3663 (1992)

EPR state generation

N Takei, N Lee, D Moriyama, JS Neergaard-Nielsen, A Furusawa,

PRA 74, 060101 (2006)

EPR state generation

EPR state generation

Bell measurement

a Bell basis measurement essentially inverses the Bell state generation

different input would give different Bell state

teleportation circuit

Bell state

Bell measurement

teleportation circuit

A Furusawa, JL Sørensen, SL Braunstein, CA Fuchs, HJ Kimble, ES Polzik

Science 282, 706 (1998)

Measurement-based
quantum computing

cluster states

a cluster state or graph state is a multiparty entangled state

- the substrate for one-way or measurement-based quantum computing

\(\ket{G}\) can be represented as a graph

stabilized, i.e. \(\hat K_i\ket{G} =\ket G\) by \(\hat K_i = \hat X_i \sum_{j\in N(i)} \hat Z_j\)

CV version has nullifier \(\hat n_i = \hat p_i - \sum_{j\in N(i)} \hat x_j\)

which has variance \(\text{Var}(\hat n_i) \propto e^{-2r} \rightarrow 0 \text{ for } r\rightarrow \infty\)

Single-qubit teleportation

one-way / measurement-based QC

Gate teleportation - depends on

measurement basis

Fowler et al., PRA 86, 032324

Our continuous-variable
MBQC platform

Generating qubit clusters:

Generating CV clusters:

Easier approach:

- requires just squeezed states and linear optics

Continuous variable cluster state

squeezed states

beam splitters

homodyne detection

CV photonic MBQC

Cluster state generation:

EPR states from two single-mode squeezers
Interfere two EPR pairs
Repeat

Use temporal domain: efficient use of resources

CV photonic MBQC

credits:

Mikkel V. Larsen

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.

\(k+1\)

\(k\)

\(k+N+1\)

\(k+N\)

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

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

Cluster state verification

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

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

Science 366, 369 (2019)

W Asavanant, ..., A Furusawa

Science 366, 373 (2019)

Closely related result

One way to implement gates and circuits on our cluster state: Reshape into "wires":

Measuring a node in the computational basis removes it and updates the graph

Gate implementation

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

Gate implementation

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

Gate implementation

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

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

Implemented gate:

Gate implementation

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

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

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

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)

Single-mode gates

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

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

\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

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

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

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

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

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

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

Single-mode gates

\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}