Qubit noise deconvolution

Mangini, Stefano; Maccone, Lorenzo; Macchiavello, Chiara

doi:10.1140/epjqt/s40507-022-00151-0

EPJ Quantum Technology

Table 1 Table summarizing the results of the present analysis, consisting of some of the most common single-qubit noisy channels $\mathcal{N}$, along with their inverse noise maps $\mathcal{N}^{-1}$, defined as the map such that $(\mathcal{N}^{-1}\circ \mathcal{N})(\rho ) = \rho$ ∀ρ. Clearly, all noise channels are CPTP maps, while the inverse channels are not, yet they admit an operator-sum representation. All the noise maps except for amplitude damping and 2-Kraus channel have trivial adjoint channels, so one must pay attention in using the adjoint channel inside the deconvolution formula (19)

	Noise channel $\mathcal{N}(\rho )$	Inverse map $\mathcal{N}^{-1}(O)$
Bit-Flip	$(1-p)\rho + p\sigma _{x}\rho \sigma _{x}$	${\frac{1-p}{1-2p}O - \frac{p}{1-2p}\sigma _{x}O \sigma _{x}}$
Phase-Flip (or dephasing)	$(1-p)\rho + p\sigma _{z}\rho \sigma _{z}$	${\frac{1-p}{1-2p}O - \frac{p}{1-2p}\sigma _{z}O \sigma _{z}}$
Bit-Phase-Flip	$(1-p)\rho + p\sigma _{y}\rho \sigma _{y}$	${\frac{1-p}{1-2p}O - \frac{p}{1-2p}\sigma _{y}O \sigma _{y}}$
Depolarizing	$(1 - p) ρ + p \frac{1}{2}$	$\frac{1}{1 - p} (O - \frac{p}{2} Tr [O] 1)$
General Pauli Channel	$p_{0} \rho + p_{x} \sigma _{x}\rho \sigma _{x}+ p_{y} \sigma _{y}\rho \sigma _{y}+ p_{z} \sigma _{z}\rho \sigma _{z}$	$\beta _{0} O + \beta _{1} \sigma _{x}O \sigma _{x}+ \beta _{2} \sigma _{y}O \sigma _{y}+ \beta _{3} \sigma _{z}O \sigma _{z}$ (see Eq. (41) for the coefficients)
Amplitude Damping	$V_{0}\rho V_{0} + V_{1} \rho V_{1}^{\dagger}$	$K_{0} O K_{0} - K_{1} O K_{1}^{\dagger}$
	$V_{0} = \|0 \rangle \langle 0\|+ \sqrt{1-\gamma} \|1 \rangle \langle 1\|$	$K_{0} = \|0 \rangle \langle 0\|+ \sqrt{\frac{1}{1-\gamma}} \|1 \rangle \langle 1\|$
	$V_{1} = \sqrt{\gamma} \|{0}\rangle \langle {1}\|$	$K_{1} = \sqrt{\frac{\gamma}{1-\gamma}} \|{0}\rangle \langle {1}\|$
2-Kraus Channel	$A_{0}\rho A_{0} + A_{1} \rho A_{1}^{\dagger}$	$B_{1} O B_{1}^{\dagger }- B_{2} O B_{2}^{\dagger}$
	$A_{0} = \cos \alpha \|0 \rangle \langle 0\|+ \cos \beta \|1 \rangle \langle 1\|$	$B_{0} = \frac{\sqrt{2}\cos \beta}{\sqrt{\cos 2\alpha +\cos 2\beta}} \|0 \rangle \langle 0\|+ \frac{\sqrt{2}\cos \alpha}{\sqrt{\cos 2\alpha +\cos 2\beta}} \|1 \rangle \langle 1\|$
	$A_{1} = \sin \beta \|{0}\rangle \langle {1}\|+\sin \alpha \|{1}\rangle \langle {0}\|$	$B_{1} = \frac{\sqrt{2}\sin \beta}{\sqrt{\cos 2\alpha +\cos 2\beta}} \|{0}\rangle \langle {1}\|+\frac{\sqrt{2}\sin \alpha}{\sqrt{\cos 2\alpha +\cos 2\beta}} \|{1}\rangle \langle {0}\|$

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	Noise channel \(\mathcal{N}(\rho )\)	Inverse map \(\mathcal{N}^{-1}(O)\)
Bit-Flip	\((1-p)\rho + p\sigma _{x}\rho \sigma _{x}\)	\({\frac{1-p}{1-2p}O - \frac{p}{1-2p}\sigma _{x}O \sigma _{x}}\)
Phase-Flip (or dephasing)	\((1-p)\rho + p\sigma _{z}\rho \sigma _{z}\)	\({\frac{1-p}{1-2p}O - \frac{p}{1-2p}\sigma _{z}O \sigma _{z}}\)
Bit-Phase-Flip	\((1-p)\rho + p\sigma _{y}\rho \sigma _{y}\)	\({\frac{1-p}{1-2p}O - \frac{p}{1-2p}\sigma _{y}O \sigma _{y}}\)
Depolarizing	$(1 - p) ρ + p \frac{1}{2}$	$\frac{1}{1 - p} (O - \frac{p}{2} Tr [O] 1)$
General Pauli Channel	\(p_{0} \rho + p_{x} \sigma _{x}\rho \sigma _{x}+ p_{y} \sigma _{y}\rho \sigma _{y}+ p_{z} \sigma _{z}\rho \sigma _{z}\)	\(\beta _{0} O + \beta _{1} \sigma _{x}O \sigma _{x}+ \beta _{2} \sigma _{y}O \sigma _{y}+ \beta _{3} \sigma _{z}O \sigma _{z}\) (see Eq. (41) for the coefficients)
Amplitude Damping	\(V_{0}\rho V_{0} + V_{1} \rho V_{1}^{\dagger}\)	\(K_{0} O K_{0} - K_{1} O K_{1}^{\dagger}\)
	\(V_{0} = \|0 \rangle \langle 0\|+ \sqrt{1-\gamma} \|1 \rangle \langle 1\|\)	\(K_{0} = \|0 \rangle \langle 0\|+ \sqrt{\frac{1}{1-\gamma}} \|1 \rangle \langle 1\|\)
	\(V_{1} = \sqrt{\gamma} \|{0}\rangle \langle {1}\|\)	\(K_{1} = \sqrt{\frac{\gamma}{1-\gamma}} \|{0}\rangle \langle {1}\|\)
2-Kraus Channel	\(A_{0}\rho A_{0} + A_{1} \rho A_{1}^{\dagger}\)	\(B_{1} O B_{1}^{\dagger }- B_{2} O B_{2}^{\dagger}\)
	\(A_{0} = \cos \alpha \|0 \rangle \langle 0\|+ \cos \beta \|1 \rangle \langle 1\|\)	\(B_{0} = \frac{\sqrt{2}\cos \beta}{\sqrt{\cos 2\alpha +\cos 2\beta}} \|0 \rangle \langle 0\|+ \frac{\sqrt{2}\cos \alpha}{\sqrt{\cos 2\alpha +\cos 2\beta}} \|1 \rangle \langle 1\|\)
	\(A_{1} = \sin \beta \|{0}\rangle \langle {1}\|+\sin \alpha \|{1}\rangle \langle {0}\|\)	\(B_{1} = \frac{\sqrt{2}\sin \beta}{\sqrt{\cos 2\alpha +\cos 2\beta}} \|{0}\rangle \langle {1}\|+\frac{\sqrt{2}\sin \alpha}{\sqrt{\cos 2\alpha +\cos 2\beta}} \|{1}\rangle \langle {0}\|\)