Error filtration using W-state encoding
Madhav Krishnan V
Collaborators: Peter Rohde, Austin Lund
Correct it!
Detect error
Filtration
Throw it away!
Filter
- Error correction - redundancy
Eg:
α∣0⟩+β∣1⟩→α∣00...0⟩+β∣11...1⟩
\alpha | 0 \rangle + \beta |1 \rangle \rightarrow \alpha | 00...0 \rangle + \beta |11...1 \rangle
GHZ states:
21(∣00...0⟩+∣11...1⟩)
\frac{1}{\sqrt{2}} ( | 00...0 \rangle + |11...1 \rangle )
- Error correction
- Cryptography
- Communication
- Not robust against loss.
W-states
- Superposition of single excitation
Eg:
∣W13⟩=31(∣100⟩+∣010⟩+∣001⟩)
|W^3_1 \rangle = \frac{1}{\sqrt{3}} (|1 0 0 \rangle + | 0 1 0 \rangle + | 0 0 1 \rangle )
- Single photon passive linear optics system
- Atomic Ensembles
- High coherence times
Tri(∣WN⟩⟨WN∣)=NN−1∣WN−1⟩⟨WN−1∣+N1∣vac⟩⟨vac∣
\text{Tr}_i(|W^N \rangle \langle W^N |) = \frac{N-1}{N} | W^{N-1} \rangle \langle W^{N-1} | + \frac{1}{N} | vac \rangle \langle vac |
|W_i^N \rangle = \hat\mathrm{QFT_N} | i \rangle
∣W04⟩∝∣0⟩+∣1⟩+∣2⟩+∣3⟩
| W^4_0 \rangle \propto | 0 \rangle + | 1 \rangle + |2 \rangle + | 3 \rangle
∣W14⟩∝∣0⟩+ω4∣1⟩+ω42∣2⟩+ω43∣3⟩
| W^4_1 \rangle \propto | 0 \rangle + \omega_4| 1 \rangle + \omega_4^2|2 \rangle + \omega_4^3| 3 \rangle
∣W24⟩∝∣0⟩+ω42∣1⟩+ω44∣2⟩+ω46∣3⟩
| W^4_2 \rangle \propto | 0 \rangle + \omega_4^2| 1 \rangle + \omega_4^4|2 \rangle + \omega_4^6| 3 \rangle
∣W34⟩∝∣0⟩+ω44∣1⟩+ω46∣2⟩+ω49∣3⟩
| W^4_3 \rangle \propto | 0 \rangle + \omega_4^4| 1 \rangle + \omega_4^6|2 \rangle + \omega_4^9| 3 \rangle
Protocol
Random phase error on each mode
eiθ0
e^{i\theta_0}
eiθ1
e^{i\theta_1}
eiθ2
e^{i\theta_2}
0
0
1
1
2
2
Eθ:
\mathcal{E}^{\vec\theta} :
xθ0
\phantom{x}^{\theta_0}
xθ1
\phantom{x}^{\theta_1}
xθ2
\phantom{x}^{\theta_2}
Random variables with
distribution
p(θi)
p(\theta_i)
∣W⟩
| W \rangle
∣Wθ⟩
| W^{\vec\theta} \rangle
=∣c0∣2c1c0∗c2c0∗xc0c1∗∣c1∣2c2c1∗c0c2∗c1c2∗∣c2∣2.........CB
=
\begin{pmatrix}
|c_0|^2 & c_0c_1^* &c_0c_2^*&... \\
c_1c_0^* &|c_1|^2 &c_1c_2^* &... \\
c_2c_0^* &c_2c_1^* & |c_2|^2 &... \\
\phantom{x}
\end{pmatrix}_{\text{CB}}
...
...
...
...
Eθ
\color{red}\mathcal{E}^{\vec\theta}
∣W⟩⟨W∣
| W \rangle \langle W |
=∣c0∣2c1c0∗ei(θ1−θ0)c2c0∗ei(θ2−θ0)xc0c1∗ei(θ0−θ1)∣c1∣2c2c1∗ei(θ2−θ1)c0c2∗ei(θ0−θ2)c1c2∗ei(θ1−θ2)∣c2∣2.........
=
\begin{pmatrix}
|c_0|^2 & c_0c_1^* \color{red}e^{i(\theta_0 - \theta_1)} &c_0c_2^* \color{red}e^{i(\theta_0 - \theta_2)}&... \\
c_1c_0^* \color{red}e^{i(\theta_1 - \theta_0)} &|c_1|^2 &c_1c_2^* \color{red}e^{i(\theta_1 - \theta_2)} &... \\
c_2c_0^* \color{red}e^{i(\theta_2 - \theta_0)} &c_2c_1^* \color{red}e^{i(\theta_2 - \theta_1)} & |c_2|^2 &... \\
\phantom{x}
\end{pmatrix}
...
...
...
...
=∣c0∣2c1c0∗f(p)c2c0∗f(p)xc0c1∗f(p)∣c1∣2c2c1∗f(p)c0c2∗f(p)c1c2∗f(p)∣c2∣2.........
=
\begin{pmatrix}
|c_0|^2 & c_0c_1^* \color{yellow}f(p) &c_0c_2^* \color{yellow}f(p)&... \\
c_1c_0^* \color{yellow}f(p)&|c_1|^2 &c_1c_2^* \color{yellow}f(p) &... \\
c_2c_0^* \color{yellow}f(p)&c_2c_1^*\color{yellow}f(p) & |c_2|^2 &... \\
\phantom{x}
\end{pmatrix}
...
...
...
...
\color{yellow}\overline\mathcal{E}^{\vec\theta}
∣Wθ⟩⟨Wθ∣
| W^{\vec\theta} \rangle \langle W^{\vec\theta} |
∫p(θ)∣Wθ⟩⟨Wθ∣dθ
\int p(\vec\theta)| W^{\vec\theta} \rangle \langle W^{\vec\theta} | d\theta
=f(p)∣W⟩⟨W∣+(1−f(p))Δ(∣W⟩⟨W∣)
= {\color{yellow} f(p) } | W \rangle \langle W | + (1 - {\color{yellow} f(p) }) \Delta(|W \rangle \langle W |)
Dephasing channel
ρW=∣c0∣2c1c0∗f(p)c2c0∗f(p)xc0c1∗f(p)∣c1∣2c2c1∗f(p)c0c2∗f(p)c1c2∗f(p)∣c2∣2.........
\rho_{W} = \begin{pmatrix}
|c_0|^2 & c_0c_1^* \color{yellow}f(p) &c_0c_2^* \color{yellow}f(p)&... \\
c_1c_0^* \color{yellow}f(p)&|c_1|^2 &c_1c_2^* \color{yellow}f(p) &... \\
c_2c_0^* \color{yellow}f(p)&c_2c_1^*\color{yellow}f(p) & |c_2|^2 &... \\
\phantom{x}
\end{pmatrix}
=f(p)∣c0∣2c1c0∗c2c0∗xc0c1∗∣c1∣2c2c1∗c0c2∗c1c2∗∣c2∣2.........+(1−f(p))∣c0∣200x0∣c1∣2000∣c2∣2.........
= {\color{yellow} f(p)}
\begin{pmatrix}
|c_0|^2 & c_0c_1^* &c_0c_2^* &... \\
c_1c_0^* &|c_1|^2 &c_1c_2^* &... \\
c_2c_0^* &c_2c_1^* & |c_2|^2 &... \\
\phantom{x}
\end{pmatrix}
+ (1-{\color{yellow} f(p)})
\begin{pmatrix}
|c_0|^2 & 0 &0 &... \\
0 &|c_1|^2 &0 &... \\
0 &0 & |c_2|^2 &... \\
\phantom{x}
\end{pmatrix}
...
...
...
...
...
...
...
...
...
...
...
...
We can determine the function
f(θ)
\color{yellow}f(\theta)
by noticing that the integral
∫p(θ)e−iθzdθ=Φp(z)
\int p(\theta) e^{-i\theta z} d\theta = \Phi_p(z)
where
Φp(z)
\Phi_p(z)
is the characteristic function of the distribution
p(θ)
p(\theta)
ρW=∣Φp(1)∣2∣W⟩⟨W∣+(1−∣Φp(1)∣2)Δ(∣W⟩⟨W∣)
\rho_{W} = |\Phi_p(1)|^2 | W \rangle \langle W | + (1 - | \Phi_p(1) |^2) \Delta(|W \rangle \langle W |)
PH=η+(1−η){e−δ2+N2(1−e−δ2)}
P_{H} =\eta + (1 -\eta) \left\{ e^{-\delta^2} + \frac{2}{N}(1 - e^{-\delta^2}) \right\}
With a loss rate
η
\eta
, the heralding probability will be
If
θ
\theta
is normally distributed with variance
δ2
\delta^2
For a logical input
∣L⟩=α∣0⟩+β∣1⟩
| L \rangle = \alpha |0 \rangle + \beta | 1 \rangle
FH=(1−η)×e−δ2+N2(1−e−δ2)e−δ2+(1−e−δ2)N1+2∣α∣2∣β∣2
F_H = (1 - \eta) \times \frac{e^{-\delta^2} + (1 - e^{-\delta^2})\frac{1 + 2|\alpha|^2|\beta|^2}{N} }{e^{-\delta^2} + \frac{2}{N}(1 - e^{-\delta^2}) }
, the heralded fidelity is
η=0.5,∣α∣=21
\eta = 0.5, |\alpha| = \frac{1}{\sqrt{2}}
Ongoing / future work
- Qudit generalization
- Using failure mode information
- Compare with other encoding
- Show that W-encoding is ideal for this scheme
Thank you
Protocol
Error filtration using W-state encoding Madhav Krishnan V Collaborators: Peter Rohde, Austin Lund