Correction to: npj Quantum Information https://doi.org/10.1038/s41534-021-00449-4, published online 16 July 2021

The original PDF version of this Article contained typesetting errors in Eqs. (9) and (12) and incorrectly read as follows:

Equation (9)

$$\hat U_{{{{\mathrm{CPHASE}}}}} = \left( {100001000010000 - 1} \right),\hat U_{{{{\mathrm{CNOT}}}}} = \left( {1000010000010010} \right).(9)$$

Equation (12)

$$\begin{array}{l}\widetilde U_{{\mathop{{{\rm{int}}}}} }\left( {t_0} \right) = \hat U_{{\mathop{{{\rm{int}}}}} }\left( {t_0} \right)\hat U_\delta \left( {t_0} \right) = \left( {e^{ - {{{\mathrm{i}}}}V_1t_0}0000e^{ - {{{\mathrm{i}}}}V_2t_0}0000e^{ - {{{\mathrm{i}}}}V_3t_0}0000e^{ - {{{\mathrm{i}}}}V_4t_0}} \right)\\ \left( {e^{ - {{{\mathrm{i}}}}\mathop {\int}\limits_0^{t_0} {\delta V_1\left( {t\prime } \right)dt\prime } }0000e^{ - {{{\mathrm{i}}}}\mathop {\int}\limits_0^{t_0} {\delta V_2} \left( {t\prime } \right)dt\prime }0000e^{ - {{{\mathrm{i}}}}\mathop {\int}\limits_0^{t_0} {\delta V_3} \left( {t\prime } \right)dt\prime }0000e^{ - {{{\mathrm{i}}}}\mathop {\int}\limits_0^{t_0} {\delta V_4\left( {t\prime } \right)dt\prime } }} \right).\end{array}$$

The correct form of Eq. (9) is:

$$\widehat U_{{{{\mathrm{CPHASE}}}}} = \left( {\begin{array}{*{20}{l}} 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & { - 1} \hfill \end{array}} \right),\; \widehat U_{{{{\mathrm{CNOT}}}}} = \left( {\begin{array}{*{20}{l}} 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 1 \hfill \\ 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill \end{array}} \right).$$
(1)

The correct form of Eq. (12) is:

$$\begin{array}{c}\widetilde U_{{\mathop{{{\rm{int}}}}} }\left( {t_0} \right) = \hat U_{{\mathop{{{\rm{int}}}}} }\left( {t_0} \right)\hat U_\delta \left( {t_0} \right)\\ = \left( {\begin{array}{*{20}{l}} {{\mathrm{e}}^{ - {{{\mathrm{i}}}}V_1t_0}} \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & {{\mathrm{e}}^{ - {{{\mathrm{i}}}}V_2t_0}} \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & {{\mathrm{e}}^{ - {{{\mathrm{i}}}}V_3t_0}} \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & {{\mathrm{e}}^{ - {{{\mathrm{i}}}}V_4t_0}} \hfill \end{array}} \right)\\ \left( {\begin{array}{*{20}{l}} {{\mathrm{e}}^{ - {{{\mathrm{i}}}}\mathop {\int}\limits_0^{t_0} {\delta V_1\left( {t^\prime } \right)} dt^\prime }} \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & {{\mathrm{e}}^{ - {{{\mathrm{i}}}}\mathop {\int}\limits_0^{t_0} {\delta V_2\left( {t^\prime } \right)} dt^\prime }} \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & {{\mathrm{e}}^{ - {{{\mathrm{i}}}}\mathop {\int}\limits_0^{t_0} {\delta V_3\left( {t^\prime } \right)} dt^\prime }} \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & {{\mathrm{e}}^{ - {{{\mathrm{i}}}}\mathop {\int}\limits_0^{t_0} {\delta V_4\left( {t^\prime } \right)} dt^\prime }} \hfill \end{array}} \right).\end{array}$$
(2)

In the in-line equation right below Eq. (12), the tilde on the second U was rendered across “U”, instead of above it. It incorrectly read:

$${\mathcal{F}}=\frac{1}{{d}^{2}}| \,\text{Tr}\,({U}^{\dagger }{\tilde{U}}){|}^{2}$$

The correct form of this equation is:

$${{{\mathcal{F}}}} = \frac{1}{{d^2}}\left| {{{{\mathrm{Tr}}}}\left( U^{\dagger} {\tilde{U}} \right)} \right|^2,$$

The equation label “(12)” was also incorrectly omitted.

This has now been corrected in the PDF version of the Article. The HTML version is correct and remains unchanged.