In Refs. [43, 44] this ensemble is called the “Gaussian Adjusted Projected (GAP)” ensemble.

The term Scrooge ensemble was coined in Ref. [49]. It is most likely a reference to the fictional character Ebenezer Scrooge from Charles Dickens’s 1843 novel A Christmas Carol, as this character was famous for being stingy with his money.

The relative error is a strictly stronger notion of approximation: A relative error $ϵ$ implies an additive error $ϵ$, but the converse is not true.

In quantum many-body systems, $\Vert \sigma {\Vert }_2=\sqrt{\mathrm{Tr}({\sigma }^2)}$ is often exponentially small in the system size (or, equivalently, inverse power in $D$). Therefore, we expect the low-purity condition to be valid up to exponentially high moments $k$.

For brevity, we suppress constants in the definition of $\exp (k)$, writing $\exp (k)=e^{c_1{k}^{c_2}}$ for some constants $c_1$, $c_2>0$.

This is consistent with the result of recent work by Ghosh et al. [63], though there they prove this for the weaker additive error and here we prove this for the stronger relative error.

This can be derived by a counting argument: Measuring subsystem $B$ of a stabilizer state $|\mathrm{\Psi }{⟩}_{AB}$ in a stabilizer basis will only generate at most $2^{\ell }$ distinct stabilizer states on $A$, where $\ell \le N_A$ is the number of ebits shared between $A$ and $B$. This is insufficient to form approximate 2-designs, which require about $4^{N_A}$ states.

Note that in order to guarantee that the projected ensemble approximates a 2-design, the additive error must satisfy $ϵ\ll 1/(D_A{D}_B)$ since $N$-qubit stabilizer states form approximate 4-designs with additive error $ϵ=\mathrm{\Theta }(2^{-N})$ [87, 88], yet their projected ensembles only form a 1-design.

If $H$ has time-reversal symmetry, ${\xi }_j$ is a real-valued Gaussian variable.

These are certain properties of the state or evolution needed for quantum information processing to achieve quantum advantage over classical information processing.

This can be rigorously quantified by so-called relative entropy of coherence $C(|{\mathrm{\Psi }}_0⟩)=-{\sum }_z|c_z{|}^2\ln (|c_z{|}^2)$, which is just the Shannon entropy of the populations in the computational basis, if the state $|{\mathrm{\Psi }}_0⟩$ is pure. Low coherence here means $C=\alpha N$ for some $\alpha$ less than the model-dependent critical value ${\alpha }^{*}$.

These are closely related to instantaneous quantum polynomial (IQP) circuits [119, 120] studied in quantum complexity theory.

More generally, we may also apply an arbitrary unitary $U_A$ on $A$, without affecting the discussion.