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:
\alpha | 0 \rangle + \beta |1 \rangle \rightarrow \alpha | 00...0 \rangle + \beta |11...1 \rangle
GHZ states:
\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:
|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
\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
| W^4_0 \rangle \propto | 0 \rangle + | 1 \rangle + |2 \rangle + | 3 \rangle
| W^4_1 \rangle \propto | 0 \rangle + \omega_4| 1 \rangle + \omega_4^2|2 \rangle + \omega_4^3| 3 \rangle
| W^4_2 \rangle \propto | 0 \rangle + \omega_4^2| 1 \rangle + \omega_4^4|2 \rangle + \omega_4^6| 3 \rangle
| 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
e^{i\theta_0}
e^{i\theta_1}
e^{i\theta_2}
0
1
2
\mathcal{E}^{\vec\theta} :
\phantom{x}^{\theta_0}
\phantom{x}^{\theta_1}
\phantom{x}^{\theta_2}
Random variables with
distribution
p(\theta_i)
| W \rangle
| W^{\vec\theta} \rangle
=
\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}}
...
...
\color{red}\mathcal{E}^{\vec\theta}
| W \rangle \langle W |
=
\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}
...
...
=
\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}
| W^{\vec\theta} \rangle \langle W^{\vec\theta} |
\int p(\vec\theta)| W^{\vec\theta} \rangle \langle W^{\vec\theta} | d\theta
= {\color{yellow} f(p) } | W \rangle \langle W | + (1 - {\color{yellow} f(p) }) \Delta(|W \rangle \langle W |)
Dephasing channel
\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}
= {\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
\color{yellow}f(\theta)
by noticing that the integral
\int p(\theta) e^{-i\theta z} d\theta = \Phi_p(z)
where
\Phi_p(z)
is the characteristic function of the distribution
p(\theta)
\rho_{W} = |\Phi_p(1)|^2 | W \rangle \langle W | + (1 - | \Phi_p(1) |^2) \Delta(|W \rangle \langle W |)
P_{H} =\eta + (1 -\eta) \left\{ e^{-\delta^2} + \frac{2}{N}(1 - e^{-\delta^2}) \right\}
With a loss rate
\eta
, the heralding probability will be
If
\theta
is normally distributed with variance
\delta^2
For a logical input
| L \rangle = \alpha |0 \rangle + \beta | 1 \rangle
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
\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