Downloading many-body continuous variable entanglement to qubits
Zhihua Han, Kero Lau
Simon Fraser University
December 6, 2024
Outline
- Background and Motivation
- Qubit and CV quantum computation
- Cluster states
- Entanglement Download Protocol
- The Displaced GKP basis
- Understanding noise
- Equivalent Circuit Model
- Finite squeezing and thermalization
- Suppressing noise
- Multiple-qubits-per site
- Weak conditional displacement
- Physical Implementations
Imagine I have some qubits:
|+\rangle
Qubit cluster state
and now I entangle the edges with the (qubit) CZ gate.
|+\rangle
Qubit cluster state
\hat{C}_Z = \begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 &-1
\end{bmatrix}
\hat{C}_Z
For this talk, I will define a entanglement gate as a two-qubit/qumode gate.
and now I entangle the edges with the (qubit) CZ gate.
|+\rangle
Qubit cluster state
\hat{C}_Z
In the circuit model, it looks like this:
|+\rangle
Qubit cluster state
Adjacency matrix
\mathbf{A} = \begin{bmatrix} 0 & 1 & 0 \\1 & 0 & 1\\ 0 & 1 & 0\end{bmatrix}
\(A_{ij} = 1 \) if \(i\) and \(j\) have an edge
\begin{aligned}
&= |G\rangle
\end{aligned}
The quantum state specified by \(G\) is called a qubit cluster state.
\begin{aligned}
&= |G\rangle\\
&= \hat{\mathbf{C}}_{Z}(\mathbf{A}) |+\rangle^{\otimes N}
\end{aligned}
\hat{C}_Z
|+\rangle
Qubit cluster state
Bold CZ means apply according to the graph structure.
Qubit cluster state
+
Single qubit measurements
Fault tolerant universal quantum computation
(Raussendorf 2001)
Why we need qubit cluster state
|+\rangle
First entangle 1 and 2, then entangle 2 and 3
"Bottom up approach"
How to make qubit cluster states?
|+\rangle
Two qubit gates are typically the noisiest gate in hardware
"Bottom up approach"
How to make qubit cluster states?
Superconducting qubits: 51
Photonic qubits: 14
Trapped ion: 32
How to make qubit cluster states?
Qubit cluster state
+
Single qubit measurements
Fault tolerant universal quantum computation
- "Want to replace circuit model because two qubit gates have too much noise"
- "Need better two qubit gates to replace the circuit model"
How to make qubit cluster states?
Qubit cluster state
Why we need qubit cluster state
Goal: Make many body entanglement in physical qubits
How do we make scalable qubit cluster states?
"Downloading entanglement from a CV cluster state"
Continuous variable (CV) cluster state
Physical qubits
\hat{S}(r)|\mathrm{vac}\rangle
\hat{C}_Z^\mathrm{CV}
Entanglement Download Protocol
How do we make scalable qubit cluster states?
Entanglement Download Protocol
Continuous variable (CV) cluster state
Qubit cluster state
\hat{S}(r)|\mathrm{vac}\rangle
\hat{C}_Z^\mathrm{CV}
Entanglement Download Protocol
"Top down approach"
Now if I have some qumodes:
\hat{S}(r)|\mathrm{vac}\rangle
CV cluster state
Squeezed vacuum state
and entangle them with CV CZ gate:
\hat{S}(r)|\mathrm{vac}\rangle
\hat{C}_Z^\mathrm{CV} = e^{i\hat{q}_1 \hat{q}_2}
CV cluster state
We say it is a CV cluster state.
\hat{S}(r)|\mathrm{vac}\rangle
\hat{C}_Z^\mathrm{CV} = e^{i\hat{q}_1 \hat{q}_2}
\begin{aligned}
&= |G\rangle^\mathrm{CV}
\end{aligned}
\begin{aligned}
&= |G\rangle^\mathrm{CV}\\
&= \hat{\mathbf{C}}_{Z}^\mathrm{CV}(\mathbf{A}) (\hat{S}(r)|\mathrm{vac}\rangle )^{\otimes N}
\end{aligned}
CV cluster state
\(r\) is the squeezing of the squeezed state.
\hat{S}(r)|\mathrm{vac}\rangle
When \(r \to \infty\), the CV cluster state is an ideal CV cluster state.
\(p\)
\hat{S}(r)|\mathrm{vac}\rangle
Finite vs Ideal CV cluster state
\(r\) is the squeezing of the squeezed state.
\hat{S}(r)|\mathrm{vac}\rangle
\(p\)
|0\rangle_p
Finite vs Ideal CV cluster state
When \(r \to \infty\), the CV cluster state is an ideal CV cluster state.
10000 modes! 1D, (Furusawa 2013)
How to make CV cluster state
1 million modes, 1D, (Furusawa 2016)
How to make CV cluster state
5x1240 modes, 2D, (Furusawa 2019)
24x1250, 2D (Andersen 2019)
How to make CV cluster state
Can generate CV cluster state with engineered dissipation
Ideal CV cluster state
|G\rangle^\mathrm{CV} = \hat{\mathbf{C}}_{Z}^\mathrm{CV}(\mathbf{A}) |0\rangle_p^{\otimes N}
|0\rangle_p
\hat{C}_Z^\mathrm{CV} = e^{i\hat{q}_1 \hat{q}_2}
Qubit cluster state
\begin{aligned}
|G\rangle= \hat{\mathbf{C}}_{Z}(\mathbf{A}) |+\rangle^{\otimes N}
\end{aligned}
|+\rangle
\hat{C}_Z = \text{diag}(1, 1, 1, -1)
CV vs qubit cluster state
Continuous variable (CV) cluster state
Physical qubits
|0\rangle_p
\hat{C}_Z^\mathrm{CV}
How to perform entanglement transfer?
\hat{C}_Z^\mathrm{CV} = e^{i\hat{q}_1 \hat{q}_2}
\hat{C}_Z = \text{diag}(1, 1, 1, -1)
Entanglement download protocol
How to perform entanglement transfer?
Continuous variable (CV) cluster state
Qubit cluster state
|0\rangle_p
\hat{C}_Z
\hat{C}_Z^\mathrm{CV} = e^{i\hat{q}_1 \hat{q}_2}
\hat{C}_Z = \text{diag}(1, 1, 1, -1)
Entanglement download protocol
We need:
- A CV cluster state*
- \(\hat{q}\) quadrature homodyne detection
- Conditional displacement gate \( \hat{C}_D\)
\hat{C}_D = |0\rangle \langle 0| \hat I + |1\rangle \langle 1| \hat{D}_q({\sqrt{\pi}})
How to perform entanglement transfer?
Entanglement download protocol
1. Initialize all qubits to \(|+\rangle\).
(These are physical qubits)
|+\rangle
Entanglement Download Protocol
Step 2: Prepare a CV cluster state.
|0\rangle_p
\hat{C}_Z^\mathrm{CV}
Entanglement Download Protocol
Step 3. Apply conditional displacement to each pair:
\(\hat{C}_D = |0\rangle \langle 0| \hat I + |1\rangle \langle 1| \hat{D}_q({\sqrt{\pi}})\)
|0\rangle_p
\hat{C}_Z^\mathrm{CV}
Entanglement Download Protocol
Step 3. Apply conditional displacement to each pair:
\(\hat{C}_D = |0\rangle \langle 0| \hat I + |1\rangle \langle 1| \hat{D}_q({\sqrt{\pi}})\)
\hat{C}_D
Entanglement Download Protocol
Step 4. Measure \(q\) quadrature.
Entanglement Download Protocol
1. Initialize all qubits to \(|+\rangle\).
|+\rangle
Entanglement Download Protocol
|+\rangle
Entanglement Download Protocol
|0\rangle_p
\hat{C}_Z^\mathrm{CV}
2. Prepare a CV cluster state.
Entanglement Download Protocol
|+\rangle
|0\rangle_p
Step 3. Apply conditional displacement to each pair:
\(\hat{C}_D = |0\rangle \langle 0| \hat I + |1\rangle \langle 1| \hat{D}_q({\sqrt{\pi}})\)
Entanglement Download Protocol
|+\rangle
|0\rangle_p
\hat{C}_D
Step 3. Apply conditional displacement to each pair:
\(\hat{C}_D = |0\rangle \langle 0| \hat I + |1\rangle \langle 1| \hat{D}_q({\sqrt{\pi}})\)
Entanglement Download Protocol
Step 4. Measure the \(q\) quadrature and correct phases.
|+\rangle
|0\rangle_p
You now have a qubit cluster state!
But why does it work?
Entanglement Download Protocol
We show there is a hidden qubit cluster state inside a CV cluster state!
|0\rangle_p
\hat{C}_Z^\mathrm{CV}
\(q\)
\(q\)
\(q\)
\(q\)
CV cluster state
Qubit cluster inside CV cluster
Logical qubit cluster state!
|0^\mathrm{GKP}\rangle = \sum_{n=-\infty}^{\infty} |2n\sqrt{\pi}\rangle_q
Gottesman-Kitaev-Preskill (GKP state)
\(q\)
\(|2n\sqrt{\pi}\rangle\)
\(0\)
\(2\sqrt \pi\)
\(4\sqrt \pi\)
\(6\sqrt \pi\)
GKP Background
|1^\mathrm{GKP}\rangle = \sum_{n=-\infty}^{\infty} |(2n+1)\sqrt{\pi}\rangle_q
\(q\)
\(|(2n+1)\sqrt{\pi}\rangle\)
\(0\)
\(2\sqrt \pi\)
\(4\sqrt \pi\)
\(6\sqrt \pi\)
\(\sqrt \pi\)
\(3\sqrt \pi\)
\(5\sqrt \pi\)
\hat{X}^{\mathrm{GKP}} = e^{-i \sqrt \pi \hat p} = \hat{D}_q(\sqrt{\pi})
GKP Background
|1^\mathrm{GKP}\rangle = \sum_{n=-\infty}^{\infty} |(2n+1)\sqrt{\pi}\rangle_q
\(q\)
\(|(2n+1)\sqrt{\pi}\rangle\)
\(0\)
\(2\sqrt \pi\)
\(4\sqrt \pi\)
\(6\sqrt \pi\)
\(\sqrt \pi\)
\(3\sqrt \pi\)
\(5\sqrt \pi\)
\(|2n\sqrt{\pi}\rangle\)
|0^\mathrm{GKP}\rangle = \sum_{n=-\infty}^{\infty} |2n\sqrt{\pi}\rangle_q
GKP Background
|+^\mathrm{GKP}\rangle = \sum_{n=-\infty}^{\infty} |n\sqrt{\pi}\rangle_q = \frac{1}{\sqrt{2}}(|0^\mathrm{GKP}\rangle + |1^\mathrm{GKP}\rangle)
\(|2n\sqrt{\pi}\rangle\)
\(q\)
\(|(2n+1)\sqrt{\pi}\rangle\)
\(0\)
\(2\sqrt \pi\)
\(4\sqrt \pi\)
\(6\sqrt \pi\)
\(\sqrt \pi\)
\(3\sqrt \pi\)
\(5\sqrt \pi\)
GKP Background
\(p\)
|0\rangle_p
Node of ideal CV cluster
\(q\)
|0\rangle_p
Node of ideal CV cluster
\(q\)
|0\rangle_p
|+^\mathrm{GKP}\rangle
GKP state
Node of ideal CV cluster
\(q\)
|0\rangle_p
\(\mu_q\)
\mu_q \in [-\frac{\sqrt{\pi}}{2}, -\frac{\sqrt{\pi}}{2})
\hat{D}_q(\mu_q)|+^\mathrm{GKP}\rangle
Displaced GKP state
|+^\mathrm{GKP}_{\mu_q, 0} \rangle \equiv \hat{D}_q(\mu_q)|+^\mathrm{GKP}\rangle
(Glancy 2006)
Displaced GKP
\(q\)
|0\rangle_p
|+^\mathrm{GKP}_{\mu_q, 0} \rangle \equiv \hat{D}_q(\mu_q)\hat{D}_p(0)|+^\mathrm{GKP}\rangle
So if we integrate over \(\mu_q\), we form an ideal \(|0\rangle_p\) state with \(\mu_p = 0\).
Node of ideal CV cluster is superposition of displaced GKP
Displaced GKP Basis
(Glancy 2006)
\(\mu_q\) displacement is color coded
\(q\)
|0\rangle_p
+
+
+
+
= \int d\mu_q
Node of ideal CV cluster is superposition of displaced GKP
=
Displaced GKP Basis
CV Basis
A single zero momentum state is a superposition of displaced GKP states.
\(q\)
|0\rangle_p
= \int d\mu_q
Node of ideal CV cluster is superposition of displaced GKP
=
Displaced GKP Basis
CV Basis
A single zero momentum state is a superposition of displaced GKP states.
\(q\)
= \int d\bm{\mu_q }
Node of ideal CV cluster is superposition of displaced GKP
Multiple zero momentum states is multiple integrals of displaced GKP states.
\(q\)
\(q\)
\(q\)
\pmb{\mu_q} = \begin{bmatrix}
\mu_{q_1} \\
\vdots\\
\mu_{q_N}\\
\end{bmatrix}
\ket{+^{\mathrm{GKP}}_{\pmb{\mu_q, 0}}}^{\otimes N}
\pmb{\mu_q} = \begin{bmatrix}
\mu_{q_1} \\
\vdots\\
\mu_{q_N}\\
\end{bmatrix}
To keep track of displaced GKP states, we define the pair \((\bm{\mu_q}, \bm{\mu_p})\), where
Node of ideal CV cluster is superposition of displaced GKP
\pmb{\mu_p} = \begin{bmatrix}
\mu_{p_1} \\
\vdots\\
\mu_{p_N}\\
\end{bmatrix}
Ideal zero momentum eigenstates: \(\bm{\mu_p} = 0\)
\(q\)
|0\rangle_p
= \int d\mu_q
Node of ideal CV cluster is superposition of displaced GKP
=
Displaced GKP Basis
CV Basis
\(q\)
Node of ideal CV cluster is superposition of displaced GKP
The next step as per the protocol is applying CV CZ.
\int d\bm{\mu_q}\ket{+^{\mathrm{GKP}}_{\pmb{\mu_q, 0}}}^{\otimes N}
Let's translate these gates into the displaced GKP basis , which will reveal the mystery.
Edge of ideal CV cluster is logical CZ
= \frac{1}{2}(|00\rangle_\mathrm{GKP} + |01\rangle_\mathrm{GKP} + |10\rangle_\mathrm{GKP} -|11\rangle_\mathrm{GKP})
\hat{C}_Z^\mathrm{CV} \ket{++}_{\mathrm{GKP}}
Logical qubit CZ gate on GKP states!
= \sum_{n_1, n_2}e^{i \pi n_1 n_2} |n_1\sqrt{\pi}\rangle_q |n_2\sqrt{\pi}\rangle_q
\equiv \hat{C}_Z^\mathrm{GKP} \ket{++}_{\mathrm{GKP}}
\hat{C}_Z^\mathrm{CV}
CV CZ applied to a (non-displaced) GKP state
Edge of ideal CV cluster is logical CZ
The generalization is much more involved, but it can be written in a single line.
\hat{\mathbf{C}}_{Z}^\mathrm{CV}(\mathbf{A}) \ket{+_{\bm{\mu_q}, \bm{0}}}^{\otimes N}_{\mathrm{GKP}} = \hat{\mathbf{C}}_{Z}^\mathrm{GKP}(\mathbf{A}) \ket{+_{\bm{\mu_q}, \mathbf{A}\bm{\mu_q}}}^{\otimes N}_{\mathrm{GKP}}
(Global phase omitted)
\(q\)
|0\rangle_p
Edges of ideal CV cluster
\(q\)
|0\rangle_p
\hat{C}_Z^\mathrm{CV}
=
\(q\)
|0\rangle_p
Edges of ideal CV cluster
+
+
+
+
+
+
+
+
\(q\)
|0\rangle_p
\hat{C}_Z^\mathrm{CV}
=
=
\(q\)
|0\rangle_p
+
+
+
+
+
+
+
+
\(q\)
|0\rangle_p
\hat{C}_Z^\mathrm{CV}
= \iint d \mu_{q_1} d \mu_{q_2}
\hat{C}_Z^\mathrm{CV}
\hat{C}_Z^\mathrm{CV}
\hat{C}_Z^\mathrm{CV}
Edges of ideal CV cluster
\hat{C}_Z^\mathrm{CV}
Displaced GKP Basis
CV Basis
\(q\)
|0\rangle_p
+
+
+
+
+
+
+
+
\(q\)
|0\rangle_p
\hat{C}_Z^\mathrm{CV}
\hat{C}_Z^\mathrm{CV}
\hat{C}_Z^\mathrm{CV}
\hat{C}_Z^\mathrm{CV}
Edges of ideal CV cluster
= \iint d \mu_{q_1} d \mu_{q_2}
\hat{C}_Z^\mathrm{CV}
\hat{C}_Z^\mathrm{CV}
Edge of ideal CV cluster is logical CZ
\hat{\mathbf{C}}_{Z}^\mathrm{CV}(\mathbf{A}) \ket{+_{\bm{\mu_q}, \bm{0}}}^{\otimes N}_{\mathrm{GKP}} = \hat{\mathbf{C}}_{Z}^\mathrm{GKP}(\mathbf{A}) \ket{+_{\bm{\mu_q}, \mathbf{A}\bm{\mu_q}}}^{\otimes N}_{\mathrm{GKP}}
(Global phase omitted)
A CV CZ gate is a logical CZ gate, but it changes the displacement in \(\bm{\mu_p}\).
An ideal CV cluster state: \(\bm{\mu_p} = \mathbf{A} \bm{\mu_q}\).
Edge of ideal CV cluster is logical CZ
\hat{C}_D(\sqrt{\pi})|+\rangle|L_{\mu_q, \mu_p}^{\mathrm{GKP}}\rangle = |0\rangle|L_{\mu_q, \mu_p}^{\mathrm{GKP}}\rangle + e^{-i\mu_p\sqrt{\pi}}|1\rangle|(L+1)_{\mu_q, \mu_p}^{\mathrm{GKP}} \rangle
Next is conditional displacement, which acts as a logical X gate + a phase shift on the qubit.
From the last slide, an ideal CV cluster state has \(\bm{\mu_p} = \mathbf{A} \bm{\mu_q}\).
Edge of ideal CV cluster is logical CZ
Next is conditional displacement, which acts as a logical X gate + a phase shift on the qubit.
\hat{C}_D(\sqrt{\pi})|+\rangle|L_{\mu_q, \mu_p}^{\mathrm{GKP}}\rangle = |0\rangle|L_{\mu_q, \mu_p}^{\mathrm{GKP}}\rangle + e^{-i\mu_p\sqrt{\pi}}|1\rangle|(L+1)_{\mu_q, \mu_p}^{\mathrm{GKP}} \rangle
=
\(q\)
|0\rangle_p
\(\mu_q\)
\mu_q \in [-\frac{\sqrt{\pi}}{2}, -\frac{\sqrt{\pi}}{2})
q = (2n+L)\sqrt\pi + \mu_q
Finally, if we measure the \(q\) quadrature of the displaced GKP state, we can infer the \(\mu_q\) displacement and the logical value \(L\)!
\(q\)
\(|(2n+1)\sqrt{\pi}\rangle\)
\(0\)
\(2\sqrt \pi\)
\(4\sqrt \pi\)
\(6\sqrt \pi\)
\(\sqrt \pi\)
\(3\sqrt \pi\)
\(5\sqrt \pi\)
\(|2n\sqrt{\pi}\rangle\)
qubit
qubit
qubit
One bit teleportation
For qubits, the one-bit teleportation circuit is
qubit
qubit
qubit
One bit teleportation
One bit teleportation
We can add in a rotation by \(\theta\) without changing the end result.
One bit teleportation
Hybrid one-bit teleportation
Normally, we can't do much since we don't know \(\mu_p\)...
One bit teleportation
If this state was part of an ideal CV cluster state, \(\bm{\mu_p} = \mathbf{A} \bm{\mu_q}\).
This means for an ideal CV cluster state we can download whatever logical information is in the displaced GKP state, i.e. the entanglement!
The final thing is to match the post-processing.
If we use the identity \(\hat{X}^L\ket{+} = \ket{+}\) and:
Our protocol is equivalent to hybrid quantum teleportation.
|0\rangle_p
\hat{C}_Z^\mathrm{CV}
Displaced GKP cluster inside a CV cluster
|0\rangle_p
\hat{C}_Z^\mathrm{CV}
= \int \ldots
+
+
Displaced GKP cluster inside a CV cluster
|0\rangle_p
\hat{C}_Z^\mathrm{CV}
\hat{C}_Z^{\mathrm{GKP}}
Displaced GKP cluster inside a CV cluster
= \int \ldots
\ket{+^{\mathrm{GKP}}_{\bm{\mu_q}, \mathbf{A}\bm{\mu_q}}}^{\otimes N}
\hat{\mathbf{C}}_{Z}^\mathrm{GKP}(\mathbf{A})
Interpret the GKP cluster as a qubit cluster.
We perform qubit-qubit quantum teleportation.
Displaced GKP cluster to qubit cluster
\hat{C}_Z^{\mathrm{GKP}} = \text{diag}(1, 1, 1, -1)
Interpret the GKP cluster as a qubit cluster.
We perform qubit-qubit quantum teleportation.
\hat{C}_Z = \text{diag}(1, 1, 1, -1)
\ket+
Displaced GKP cluster to qubit cluster
|0\rangle_\mathrm{GKP} = \sum_{n=-\infty}^{\infty} |2n\sqrt{\pi}\rangle_q
Gottesman-Kitaev-Preskill (GKP state)
\(q\)
\(|2n\sqrt{\pi}\rangle\)
\(0\)
\(2\sqrt \pi\)
\(4\sqrt \pi\)
\(6\sqrt \pi\)
GKP-qubit one bit teleportation: X gate
|1\rangle_\mathrm{GKP} = \sum_{n=-\infty}^{\infty} |(2n+1)\sqrt{\pi}\rangle_q
\(q\)
\(|(2n+1)\sqrt{\pi}\rangle\)
\(0\)
\(2\sqrt \pi\)
\(4\sqrt \pi\)
\(6\sqrt \pi\)
\(\sqrt \pi\)
\(3\sqrt \pi\)
\(5\sqrt \pi\)
\hat{X}^{\mathrm{GKP}} = e^{-i \sqrt \pi \hat p} = \hat{D}_q(\sqrt{\pi})
GKP-qubit one bit teleportation: X gate
|0\rangle_\mathrm{GKP} = \sum_{n=-\infty}^{\infty} |2n\sqrt{\pi}\rangle_q
Gottesman-Kitaev-Preskill (GKP state)
\(q\)
\(|2n\sqrt{\pi}\rangle\)
\(0\)
\(2\sqrt \pi\)
\(4\sqrt \pi\)
\(6\sqrt \pi\)
GKP-qubit one bit teleportation: \(\mu_q, \mu_p\)
|1\rangle_\mathrm{GKP} = \sum_{n=-\infty}^{\infty} |(2n+1)\sqrt{\pi}\rangle_q
\(q\)
\(0\)
\(2\sqrt \pi\)
\(4\sqrt \pi\)
\(6\sqrt \pi\)
\(\sqrt \pi\)
\(3\sqrt \pi\)
\(5\sqrt \pi\)
\(|2n\sqrt{\pi}+\mu_q\rangle\)
\(\mu_q\): Rotational X gate
\(\mu_p\): Rotational Z gate
\(\mu_q, \mu_p\) as rotational X, Z
= \int \ldots
q = (2n+L)\sqrt\pi + \mu_q
|0\rangle_p
\hat{C}_Z^\mathrm{CV}
+
+
Homodyne detection collapses the GKP cluster
However, we can have mode preparation noise.
\(q\)
\(p\)
\hat{S}(r)|\mathrm{vac}\rangle
We need to prepare the zero momentum eigenstate, which requires infinite squeezing and is physically impossible.
However, we can have mode preparation noise.
We need to prepare the zero momentum eigenstate, which requires infinite squeezing and is physically impossible.
\(q\)
\(p\)
\hat{S}(r)|\mathrm{vac}\rangle
Finite squeezing noise.
Another possibility is that the mode is not in the ground state before squeezing.
However, we can have mode preparation noise.
Another possibility is that the mode is not in the ground state before squeezing.
\(q\)
\(p\)
\rho(\bar{n})
A thermal state is a statistical mixture of these coherent states, so effectively the variance is larger.
However, we can have mode preparation noise.
Another possibility is that the mode is not in the ground state before squeezing.
\(q\)
\(p\)
\hat{S}(r)\rho(\bar{n})\hat{S}^\dagger(r)
A thermal state is a statistical mixture of these coherent states, so effectively the variance is larger.
And a general qumode will have \(\mu_p \neq 0\) in the displaced GKP basis:
\ket{\psi} = \int_{-\sqrt{\pi}/2}^{\sqrt{\pi}/2} \int_{-\sqrt{\pi}/2}^{\sqrt{\pi}/2} \, d\mu_q d\mu_p \, a_0(\mu_q, \mu_p) \ket{0_{\mu_q, \mu_p}^\mathrm{GKP}}+ \,a_1(\mu_q, \mu_p) \ket{1_{\mu_q, \mu_p}^\mathrm{GKP}}
\ket{0}_p = \frac{1}{\sqrt{\pi}}\int_{-\sqrt{\pi}/2}^{\sqrt{\pi}/2} d\mu_q \ket{+_{\mu_q, 0}^\mathrm{GKP}}.
Infinite squeezing is not physically possible:
😵💫
Displaced GKP basis cannot describe mixed states!
However, it turns out that CV CZ can be converted to a qubit CZ gate at the operator level, before we even specify what the state is:
\bra{\bm{q}} \hat{\mathbf{C}}_D \hat{\mathbf{C}}_Z^\mathrm{CV}(\mathbf{A}) = \hat{\mathbf{R}}_Z^\dagger(\bm{\phi}) \hat{\mathbf{C}}_Z(\mathbf{A}) \bra{\bm{q}} \hat{\mathbf{C}}_D
CV CZ
Qubit CZ
Phase shifts
Equivalent circuit model
=
\ket{\Psi(q)}
Mode preparation error = single qubit preparation error
- We can now handle all the mode preparation errors, including thermalization errors and finite squeezing!
Finite squeezing:
\(p\)
\hat{S}(r)\ket{\mathrm{vac}}
Loss: Finite Squeezing
\(q\)
\hat{S}(r)\ket{\mathrm{vac}}
Finite squeezing:
Loss: Finite Squeezing
\(q\)
\hat{D}_q(\sqrt{\pi})\hat{S}(r)\ket{\mathrm{vac}} \ket{1}^{\mathrm{Qubit}}
After conditional displacement:
\hat{S}(r)\ket{\mathrm{vac}} \ket{0}^{\mathrm{Qubit}}
Loss: Finite Squeezing
\(q\)
\hat{D}_q(\sqrt{\pi})|0, \sigma_p \rangle_p \ket{1}^{\mathrm{Qubit}}
Now, the probability of measuring \(q\):
|0, \sigma_p \rangle_p \ket{0}^{\mathrm{Qubit}}
Loss: Finite Squeezing
\(q\)
After measuring \(q\):
Loss: Finite Squeezing
\(q\)
After measuring \(q\):
Loss: Finite Squeezing
\(q\)
After measuring \(q\):
\ket{0} + \, \, \ket 1
Amplitude imbalance error
The qubit is:
Loss: Finite Squeezing
\ket{0} + \, \, \ket 1
Amplitude imbalance error
The qubit is:
We can correct the qubit by performing weak measurement POVMs \(M_0, M_1\), which commute with qubit CZ.
Failure:
\ket{1}
\ket{0} + \, \, \ket 1
Success:
Loss: Finite Squeezing
\ket{0} + \, \, \ket 1
Amplitude imbalance error
The qubit is:
We can correct the qubit by performing weak measurement POVMs \(M_0, M_1\).
Failure: \(p\)
\ket{1}
\ket{0} + \, \, \ket 1
Success: \(1-p\)
\hat M_0
\hat M_1
Loss: Finite Squeezing
Finitely squeezed CV cluster state
Loss: Finite Squeezing
Loss: Finite Squeezing
Amplitude imbalance error!
After weak measurement:
Loss: Finite Squeezing
Failure: \(p\)
Can convert initial squeezing error to deletion error!
Loss: Finite Squeezing
How much squeezing do we need?
Loss: Finite Squeezing
Failure: \(p\)
(Stace 2009)
(Barrett and Stace 2010)
11.9 dB
5.4 dB
Loss: Finite Squeezing
5.4 dB: Loss-tolerant Quantum computation
Loss: Finite Squeezing
5.4 dB: Loss-tolerant Quantum computation
Loss: Finite Squeezing
5.4 dB: Robust quantum memory
Loss: Finite Squeezing
11.9 dB: Fault-tolerant quantum computation
Loss: Finite Squeezing
11.9 dB: Fault-tolerant quantum computation
Loss: Finite Squeezing
Can we suppress the deletion probability?
Loss: Finite Squeezing
Loss: Finite Squeezing
After weak measurement, qubits 1 and 3 are no longer entangled.
Loss: Finite Squeezing
Site 1
Site 2
Site 3
Loss: Finite Squeezing
Loss: Finite Squeezing
After weak measurement:
Loss: Finite Squeezing
Dual rail encoding (n=2)
Loss: Finite Squeezing
In order to break entanglement between site 1 and 3 both qubits has to be deleted.
Dual rail encoding (n=2)
Loss: Finite Squeezing
Deletion probability of a site: \(p^n\)
What happens to the qubit if you send in a squeezed thermal state?
\(q\)
\(p\)
\rho^{\mathrm{thermal}}
= \rho
\(q\)
\(p\)
|0, \sigma_p \rangle_p
\hat{D}_p(p_0)
Squeezed thermal state
Loss: Channel and detector loss
What happens to the qubit if you send in a squeezed thermal state?
\(q\)
\(p\)
\rho^{\mathrm{thermal}}
= \rho
\(q\)
\(p\)
\hat{S}(r)|\mathrm{vac}\rangle
\hat{D}_p(p_0)
Mixture of displaced squeezed states
Squeezed thermal state
=
Loss: Channel and detector loss
What happens to the qubit if you send in a squeezed thermal state?
Loss: Channel and detector loss
\(q\)
\(p\)
\hat{S}(r)|\mathrm{vac}\rangle
\hat{D}_p(p_0)
\hat{S}(r)|\mathrm{vac}\rangle
\hat{D}_p(p_0)
Displaced squeezed state
What happens to the qubit if you send in a squeezed thermal state?
Loss: Channel and detector loss
\(q\)
\(p\)
\hat{S}(r)|\mathrm{vac}\rangle
\hat{D}_p(p_0)
\hat{D}_p(p_0)
\ket{\Psi(q)}
Amplitude imbalance
Displaced squeezed state
What happens to the qubit if you send in a squeezed thermal state?
Loss: Channel and detector loss
\(q\)
\(p\)
\hat{S}(r)|\mathrm{vac}\rangle
\hat{D}_p(p_0)
\hat{R}_Z(-p_0\sqrt{\pi})
\ket{\Psi(q)}
Amplitude imbalance + Phase shift
What happens to the qubit if you send in a squeezed thermal state?
Loss: Channel and detector loss
\(q\)
\(p\)
\hat{S}(r)|\mathrm{vac}\rangle
\hat{D}_p(p_0)
\hat{R}_Z(-p_0\sqrt{\pi})
\ket{\Psi(q)}
Amplitude imbalance + Phase shift
Infinite squeezing:
Same result as displaced GKP basis
What happens to the qubit if you send in a squeezed thermal state?
Loss: Channel and detector loss
\hat{R}_Z(-p_0\sqrt{\pi})
\ket{\Psi(q)}
\int_{-\infty}^{\infty} dp_0 \cdot f(p_0, \sigma_{p_0}^2)
What happens to the qubit if you send in a squeezed thermal state?
Loss: Channel and detector loss
(1-s)\ket{\Psi(q)}\bra{\Psi(q)}+ s\hat{Z} \ket{\Psi(q)}\bra{\Psi(q)} \hat{Z}
The qubit becomes mixed and dephases
In experiment, conditional displacement interaction can be weak, or reduced by a factor of \(\sqrt{R}\).
Weak conditional displacement
\hat C_D(\sqrt{\pi/R}) = \ket 0 \bra 0 \otimes I + \ket 1 \bra 1 \otimes \hat D_q (\sqrt {\pi / R})
We now introduce a scheme to address this.
Suppose \(\hat{C}_D\) is \(R = 3\) times weaker
Weak conditional displacement
Weak conditional displacement
Suppose \(\hat{C}_D\) is \(R = 3\) times weaker
Weak conditional displacement
Suppose \(\hat{C}_D\) is \(R = 3\) times weaker
Weak conditional displacement
Suppose \(\hat{C}_D\) is \(R = 3\) times weaker
Only one round of weak measurement correction.
Suppose \(\hat{C}_D\) is 3 times weaker
Weak conditional displacement
The noise introduced by preparing \(R\) fresh CV cluster states and performing conditional displacement \(R\) times at strength \(\sqrt{\pi/R}\) has the same noise as preparing 1 CV cluster state and 1 round of CD at \(\sqrt \pi\).
Weak conditional displacement
Noise occurs at the middle of the circuit!
Optical loss
Detection inefficiency
Mode preparation error
Optical loss:
Optical loss:
- Absorbed by the material
- Scattering loss
- Reflection loss
- Bending / misalignment / manufacturing defects
- Loss of signal / energy / intensity / power / photons
Intensity is reduced
Optical loss:
\ket{\mathrm{vac}}
Optical loss is modelled as a beam splitter:
\hat{a}_1 \to \hat{a}_1 \sqrt{\eta} + \hat{a}_2 \sqrt{1 - \eta}.
(\(\eta \to 1\) is no optical loss)
Bob can amplify the signal (either digitally or using an optical parametric amplifier)
Amplifier
Bob can amplify the signal (either digitally or using an optical parametric amplifier)
Amplifier
Bob can amplify the signal (either digitally or using an optical parametric amplifier)
Amplifier
This (phase insensitive) amplification process is a transformation on phase space, which increases the variance of the Gaussian state.
\(q\)
\(p\)
Amplitude
\(q\)
\(p\)
Amplitude
This (phase insensitive) amplification process is a transformation on phase space, which increases the variance of the Gaussian state.
This (phase insensitive) amplification process is a transformation on phase space, which increases the variance of the Gaussian state.
Optical parametric amplification using
Spontaneous parametric down conversion
\(q\)
\(p\)
Digital amplification
\hat{a}_1 \to \frac{1}{\sqrt{\eta}}\hat{a}_1- \sqrt{\frac{1 - \eta}{\eta}} \hat{a}_3^\dagger
Real life photodetectors do not perfectly absorb photons:
Amplifier
- Absorption inefficiency
- Shot noise
- Loss on the electronics level e.g. temperature
Detector inefficiency is modelled as optical loss + amplification before a perfect detection.
\hat{a}_1 \to \hat{a}_1 + \sqrt{\frac{1 - \eta}{\eta}}(\hat{a}_2 + \hat{a}^\dagger_3)
added noise
\hat{a}_1 \to \hat{a}_1 + \sqrt{\frac{1 - \eta}{\eta}}(\hat{a}_2 + \hat{a}^\dagger_3)
added noise
\(q\)
\(p\)
Detector inefficiency is modelled as optical loss + amplification before a perfect detection.
\hat{a}_1 \to \hat{a}_1 + \sqrt{\frac{1 - \eta}{\eta}}(\hat{a}_2 + \hat{a}^\dagger_3)
added noise
\(q\)
\(p\)
Detector inefficiency is modelled as optical loss + amplification before a perfect detection.
Returning to the entanglement download protocol, channel loss and detector inefficiency complicate the protocol.
\hat{a}_1 \to \hat{a}_1 \sqrt{\eta} + \hat{a}_2 \sqrt{1 - \eta}.
Channel loss
\hat{a}_1 \to \hat{a}_1 + \sqrt{\frac{1 - \eta}{\eta}}(\hat{a}_2 + \hat{a}^\dagger_3)
Detector inefficiency
Beginning with detector inefficiency, we expand the Gaussian random noise in terms of a circuit.
Beginning with detector inefficiency, we expand the Gaussian random noise in terms of a circuit.
After tracing out mode 3, we have only Gaussian random noise in the \(q\) quadrature, and all loss is lumped together.
The CV cluster state is entangled before optical loss:
Thus, we cannot have independent errors, and the errors must be correlated.
However, we can add a beam splitter before entangling all modes to ensure the modes are uncorrelated after loss.
The properties of the input modes, beam splitter, and squeezing depend on the adjacency matrix \(\mathbf{A}\).
Implementation details:
- A scalable CV cluster state,
- A qubit platform,
- Conditional displacement gate,
- Measurement of the \(q\) quadrature.
Three possible implementations:
- Cavity QED
- Circuit QED
- Free electron qubits
Possible implementations: Cavity QED
CV cluster: Furusawa protocol
|\alpha\rangle
Cavity QED system with Jaynes-Cumming interaction
(Dhara, Jiang, and Guha 2024)
Possible implementations: Cavity QED
|\alpha\rangle
Controlled Rotation gate
(Dhara, Jiang, and Guha 2024)
Possible implementations: Cavity QED
Cat state coupled to qubit
Free photons generated from Furusawa protocol
\eta \to 1
(Dhara, Jiang, and Guha 2024)
Possible implementations: Cavity QED
\eta \to 1
(Dhara, Jiang, and Guha 2024)
Possible implementations: Cavity QED
Free photons generated from Furusawa protocol
\eta \to 1
(Dhara, Jiang, and Guha 2024)
Possible implementations: Cavity QED
Free photons generated from Furusawa protocol
Apply X gate to qubit
Reroute entangled pulse back to cavity
(Dhara, Jiang, and Guha 2024)
Possible implementations: Cavity QED
Free photons generated from Furusawa protocol
Apply another X gate
Disentangle with
controlled phase
Net result: Conditional displacement
(Dhara, Jiang, and Guha 2024)
Possible implementations: Cavity QED
(Dhara, Jiang, and Guha 2024)
Possible implementations: Cavity QED
Free photons generated from Furusawa protocol
(Dhara, Jiang, and Guha 2024)
Homodyne detection
Possible implementations: Cavity QED
Homodyne detection
Homodyne detection in optics
Coherent state
as a reference mode
State we are interested in measuring
I_1
I_2
|\alpha\rangle
Homodyne detection
Homodyne detection in optics
Coherent state
as a reference
mode
State we are interested in measuring
I_1
I_2
I_1 - I_2 \propto |I_1 I_2| \cos(\phi_1 - \phi_2)
|\alpha\rangle
Possible implementations: Free electron qubits
CV cluster: Furusawa protocol
Possible implementations: Free electron qubits
Free electron qubits
(Reinhardt 2021)
CV cluster: Furusawa protocol
\ket{0} = \sum_n \ket{2n\hbar \omega}
\ket{1} = \sum_n \ket{(2n+1)\hbar \omega}
Possible implementations: Free electron qubits
Free electron qubits
(Reinhardt 2021)
\ket{0} = \sum_n \ket{2n\hbar \omega}
\ket{1} = \sum_n \ket{(2n+1)\hbar \omega}
photon-induced nearfield electron
microscopy (PINEM)
Possible implementations: Free electron qubits
Free electron qubits
(Reinhardt 2021)
\ket{0} = \sum_n \ket{2n\hbar \omega}
\ket{1} = \sum_n \ket{(2n+1)\hbar \omega}
Possible implementations: Free electron qubits
Electron-photon scattering can be used as a CD gate
(Baranes 2023)
Possible implementations: Free electron qubits
(Baranes 2023)
Possible implementations: Free electron qubits
S = \exp(g\hat{b} \hat{a}^\dagger - g^* \hat{b}^\dagger \hat{a})
Free electron-photon scattering (Dahan 2021):
\(\hat b^\dagger = \hat b = \sigma_x \) are Fermionic operators
PINEM: when the mode \(\hat a\) is a coherent state
General mode \(\hat a\) implements a conditional displacement
(Baranes 2023)
Possible implementations: Free electron qubits
Prepare free electrons as qubits using PINEM and coherent laser pulses
Prepare CV cluster state using the Furusawa protocol
S = \exp(g\hat{b} \hat{a}^\dagger - g^* \hat{b}^\dagger \hat{a})
Electron-photon coupling:
CD gate
Homodyne detection
Use free electrons as qubits or couple to other qubits
Possible implementations: Superconducting qubits
CV cluster: Frequency comb in microwave resonator
95 mode CV cluster state
(Hernández 2024)
Possible implementations: Superconducting qubits
CV cluster: Frequency comb in microwave resonator
Josephson Parametric Amplifier
Vacuum fluctuations
CV cluster state
Time varying flux to modulate the signal
Possible implementations: Superconducting qubits
Transmon qubit
CV cluster state: Frequency comb in microwave resonator
Jaynes-Cummings interaction with dispersive coupling
\(H_I = \chi \sigma_z \hat{a}^\dagger \hat{a} \)
Conditional rotation
Possible implementations: Superconducting qubits
Transmon
Echoed conditional displacement gate
(A. Eickbusch 2018)
Possible implementations: Superconducting qubits
Transmon
How to measure \(q\) quadrature?
Possible implementations: Superconducting qubits
Transmon
1. Qubitdyne detection
(Strandberg 2023)
Ancilla qubit for measuring
Jaynes-Cumming interaction:
\(H_I = \hat{a} \sigma_+ + \hat{a}^\dagger \sigma_-\)
Possible implementations: Superconducting qubits
Transmon
1. Qubitdyne detection
(Strandberg 2023)
Jaynes-Cumming interaction:
\(H_I = \hat{a} \sigma_+ + \hat{a}^\dagger \sigma_-\)
Ancilla qubit for measuring
Possible implementations: Superconducting qubits
Transmon
2. Quantum Phase Estimation (Terhal and Weigand 2016)
ECD gate + dispersive coupling
\(H_I = \sigma_z (\alpha \hat{a}^\dagger - \alpha^* \hat{a})\)
Ancilla qubit for measuring
Conditional displacement \(U = (\hat{X}^{\mathrm{GKP}})^2\)
Possible implementations: Superconducting qubits
Transmon
Ancilla oscillator for measuring
Optomechanical coupling:
\(H_I = g\hat{b}^\dagger\hat{b} (\hat a + \hat a^\dagger)\)
\hat{q} \propto (\hat{a} + \hat{a}^\dagger )
3. Photon-pressure coupling (Terhal and Weigand 2020)
Waveguide
| Circuit QED | Free electron qubits | Cavity QED | |
|---|---|---|---|
| CV cluster state | Frequency comb in cavity | Furusawa protocol | Furusawa protocol |
| Conditional displacement | Echoed conditional displacement gate (ECD gate) | Electron-photon coupling | Mediated interaction with coherent state |
| q quadrature measurement | 1. Quantum phase estimation 2. Qubitdyne detection 3. Photon-pressure coupling |
Homodyne detection | Homodyne detection |
| Qubit | Transmon | Free electrons | Atom in cavity |
Possible implementations: Summary
Downloading many-body continuous variable entanglement to qubits
- We can make many body entanglement in qubits!
- Entanglement transfer from CV cluster state to qubit cluster state is possible
- Quality of the qubit cluster state depends on initial CV cluster
- Weak measurement / qubit deletion protocol can reduce requirements
- 6dB squeezing for robust quantum memory
- 12dB squeezing for fault tolerant quantum computing
- No generation of GKP states is needed
- arXiV in progress
- This research was sponsored by the QuantumBC CREATE Program.
Zhihua Han: zhi_han@sfu.ca
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bonus slides
Entanglement Download
By Zhi Han
