Correlated noise enhancement of coherence and fidelity in coupled qubits

Figures & data

Figure 1. Sketch of the 2-site model with correlated noise interactions. Here, each site is taken as a 2-state qubit and is coupled to its neighbour via J (c.f. Equation (13(14) $|{\psi }_{\pm }〉=\frac{1}{\sqrt{2}}(|01〉\pm |10〉)$(14) )) and to its local environment via ${\overrightarrow{E}}_i\cdot {\overrightarrow{\sigma }}_i$ as per Equation (15(16) $S\left(\omega \right)\propto {|\frac{1}{\mathrm{i\hslash }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{\mathrm{i\omega t}}〈\mu (t\left)\right[\mu \left(0\right),\rho (-\mathrm{\infty })\left]\right]〉|}^2$(16) ). However, the local environments are correlated via Σ.

Figure 2. Linear response absorption spectra and associated line-width for a pair of qubits subject to transverse (a,b) and longitudinal (c,d) noise terms with various degrees of correlation.

Figure 3. Purity(a,b) and fidelity (c,d) of composite $\mathrm{SU}\left(2\right)\otimes \mathrm{SU}\left(2\right)$ qubit pair vs time for systems prepared in a maximally entangled (Bell) state corresponding to (a,c)$|{\mathrm{\Phi }}^{+}〉$ or (b,d) ${\mathrm{\Psi }}^{+}$.

Table A1. Redfield tensor elements for $\mathrm{SU}\left(2\right)$ qubit with cross-correlation between ${\hat{\sigma }}^x$ and ${\hat{\sigma }}^z$ noise terms.

i	j	k	l	Secular	${\mathcal{R}}_{\mathrm{ij};\mathrm{kl}}$	Ornstein-Uhlenbeck
1	1	1	1	Secular	$J_{\mathrm{xx}}(-ϵ)+J_{\mathrm{xx}}\left(ϵ\right)$	$\frac{2s_x(s_x+2\xi s_{\mathrm{xz}})}{{\gamma }_x^2+{ϵ}^2}$
1	1	1	2	Non-Secular	$-J_{\mathrm{xz}}\left(0\right)-J_{\mathrm{zx}}\left(0\right)$	$-\frac{2(s_x(s_{\mathrm{xz}}+\xi s_z)+\xi s_{\mathrm{xz}}^2)}{{\gamma }_x{\gamma }_z}$
1	1	2	1	Non-Secular	$J_{\mathrm{xz}}(-ϵ)-J_{\mathrm{zx}}(-ϵ)-2J_{\mathrm{zx}}\left(0\right)$	$-\frac{2({\gamma }_x^2(\mathrm{iϵ}{\gamma }_z+{\gamma }_z^2+{ϵ}^2)-\mathrm{iϵ}{\gamma }_x{\gamma }_z^2+{ϵ}^2({\gamma }_z^2+{ϵ}^2))(s_x(s_{\mathrm{xz}}+\xi s_z)+\xi s_{\mathrm{xz}}^2)}{{\gamma }_x{\gamma }_z({\gamma }_x^2+{ϵ}^2)({\gamma }_z^2+{ϵ}^2)}$
1	1	2	2	Secular	$-J_{\mathrm{xx}}(-ϵ)-J_{\mathrm{xx}}\left(ϵ\right)$	$-\frac{2s_x(s_x+2\xi s_{\mathrm{xz}})}{{\gamma }_x^2+{ϵ}^2}$
1	2	1	1	Non-Secular	$-2J_{\mathrm{xz}}(-ϵ)-J_{\mathrm{xz}}\left(0\right)+J_{\mathrm{zx}}\left(0\right)$	$-\frac{2(s_x(s_{\mathrm{xz}}+\xi s_z)+\xi s_{\mathrm{xz}}^2)}{(ϵ+i{\gamma }_x)(ϵ-i{\gamma }_z)}$
1	2	1	2	Secular	$2\left(J_{\mathrm{xx}}\right(ϵ)+2J_{\mathrm{zz}}(0\left)\right)$	$\frac{4s_{\mathrm{xz}}^2({\gamma }_x^2+{ϵ}^2)+4\xi s_{\mathrm{xz}}(2s_z({\gamma }_x^2+{ϵ}^2)+s_x{\gamma }_z^2)+2s_x^2{\gamma }_z^2}{{\gamma }_z^2({\gamma }_x^2+{ϵ}^2)}$
1	2	2	1	Non-Secular	$-2J_{\mathrm{xx}}(-ϵ)$	$-\frac{2s_x(s_x+2\xi s_{\mathrm{xz}})}{{\gamma }_x^2+{ϵ}^2}$
1	2	2	2	Non-Secular	$J_{\mathrm{xz}}(-ϵ)+J_{\mathrm{zx}}(-ϵ)$	$\frac{2({\gamma }_x{\gamma }_z+{ϵ}^2)(s_x(s_{\mathrm{xz}}+\xi s_z)+\xi s_{\mathrm{xz}}^2)}{({\gamma }_x^2+{ϵ}^2)({\gamma }_z^2+{ϵ}^2)}$
2	1	1	1	Non-Secular	$-J_{\mathrm{xz}}\left(ϵ\right)-J_{\mathrm{zx}}\left(ϵ\right)$	$-\frac{2({\gamma }_x{\gamma }_z+{ϵ}^2)(s_x(s_{\mathrm{xz}}+\xi s_z)+\xi s_{\mathrm{xz}}^2)}{({\gamma }_x^2+{ϵ}^2)({\gamma }_z^2+{ϵ}^2)}$
2	1	1	2	Non-Secular	$-2J_{\mathrm{xx}}\left(ϵ\right)$	$-\frac{2s_x(s_x+2\xi s_{\mathrm{xz}})}{{\gamma }_x^2+{ϵ}^2}$
2	1	2	1	Secular	$2\left(J_{\mathrm{xx}}\right(-ϵ)+2J_{\mathrm{zz}}(0\left)\right)$	$\frac{4s_{\mathrm{xz}}^2({\gamma }_x^2+{ϵ}^2)+4\xi s_{\mathrm{xz}}(2s_z({\gamma }_x^2+{ϵ}^2)+s_x{\gamma }_z^2)+2s_x^2{\gamma }_z^2}{{\gamma }_z^2({\gamma }_x^2+{ϵ}^2)}$
2	1	2	2	Non-Secular	$2J_{\mathrm{xz}}\left(ϵ\right)+J_{\mathrm{xz}}\left(0\right)-J_{\mathrm{zx}}\left(0\right)$	$\frac{2(s_x(s_{\mathrm{xz}}+\xi s_z)+\xi s_{\mathrm{xz}}^2)}{(ϵ-i{\gamma }_x)(ϵ+i{\gamma }_z)}$
2	2	1	1	Secular	$-J_{\mathrm{xx}}(-ϵ)-J_{\mathrm{xx}}\left(ϵ\right)$	$-\frac{2s_x(s_x+2\xi s_{\mathrm{xz}})}{{\gamma }_x^2+{ϵ}^2}$
2	2	1	2	Non-Secular	$-J_{\mathrm{xz}}\left(ϵ\right)+J_{\mathrm{zx}}\left(ϵ\right)+2J_{\mathrm{zx}}\left(0\right)$	$\frac{2({\gamma }_x^2(-\mathrm{iϵ}{\gamma }_z+{\gamma }_z^2+{ϵ}^2)+\mathrm{iϵ}{\gamma }_x{\gamma }_z^2+{ϵ}^2({\gamma }_z^2+{ϵ}^2))(s_x(s_{\mathrm{xz}}+\xi s_z)+\xi s_{\mathrm{xz}}^2)}{{\gamma }_x{\gamma }_z({\gamma }_x^2+{ϵ}^2)({\gamma }_z^2+{ϵ}^2)}$
2	2	2	1	Non-Secular	$J_{\mathrm{xz}}\left(0\right)+J_{\mathrm{zx}}\left(0\right)$	$\frac{2(s_x(s_{\mathrm{xz}}+\xi s_z)+\xi s_{\mathrm{xz}}^2)}{{\gamma }_x{\gamma }_z}$
2	2	2	2	Secular	$J_{\mathrm{xx}}(-ϵ)+J_{\mathrm{xx}}\left(ϵ\right)$	$\frac{2s_x(s_x+2\xi s_{\mathrm{xz}})}{{\gamma }_x^2+{ϵ}^2}$

Notes: The second column indicates whether the term survives under the secular approximation, which separates the evolution of the population and the coherence terms. The last column gives the tensor element within the correlated Ornstein-Uhlenbeck model.

Data availability

Data supporting the findings of this study are available from the corresponding author on a reasonable request.