As a consequence, our algorithms for uniform noise can be thought of as working explicitly in the regime $d=\mathcal{O}\mathbf{(}{\gamma }^{-1}\log (n/ϵ)\mathbf{)}$. After this depth, one can approximate any expectation value with its value in the maximally mixed state [13].

In particular, threshold theorems typically assume two ingredients [9, 10, 11, 12]: midcircuit measurement and feedforward operations, and the ability to insert “fresh qubits” unaffected by idle errors. In effect, both noise models allow midcircuit measurement and feedforward operations, since one can implement the feedforward operations via controlled quantum gates. (For a fixed input state, this can be made fault tolerant by encoding the state of the “measured” qubit in a classical error correcting code at the layer at which it is “measured.”) Crucially however, the gate-based noise model allows the insertion of fresh qubits, while the uniform noise model does not.

See Ref. [31], where this is referred to as “orthogonality.” This requirement is satisfied in random circuits, where averaged properties of operator time evolution are governed by an incoherent classical Markov process, but not more generally.

At this halting step, we also group together all weights above the threshold in order to achieve a tighter bound on the error. See Appendix pp3 for full details.

We note that the maximum nontrivial depth for quantum circuits with random input states, $d=\mathcal{O}\mathbf{(}{\gamma }^{-1}\log (1/ϵ)\mathbf{)}$, is slightly shorter than the maximum depth for quantum circuits with a fixed input state, $d=\mathcal{O}\mathbf{(}{\gamma }^{-1}\log (n/ϵ)\mathbf{)}$ [13]. See Appendix pp3 for further details.

A similar but less precise connection was made in Ref. [109].

For example, for the entanglement entropy and OTOC, one can consider a deep Clifford circuit. This generates high entanglement despite being classically simulable via the Gottesman-Knill theorem. For the magic, one can consider a 1D circuit with many $T$ gates but low circuit depth, or a Gaussian fermionic state. Both have an extensive number of $T$ gates, while remaining simulable via matrix product state and match gate algorithms, respectively. For the operator entanglement entropy (and all aforementioned metrics as well), one can consider a generic highly ergodic many-body system. Time-evolved operators in such systems typically feature a linear growth in operator entanglement entropy over time, despite the operators’ expectation values often being efficiently simulated via Pauli truncation methods [46, 49, 50, 51, 54, 55, 56, 57, 58, 59].

If the individual coefficients can be computed in time $\tau$ instead of time $\mathcal{O}(1)$, then the time to read in all coefficients will be $\mathcal{O}(\tau ·D_{\ell })$ instead of $\mathcal{O}(D_{\ell })$. Our final run-time will be unchanged as long as $\tau =\mathcal{O}(d·\ell !)$, and will be quasipolynomial in $n$ as long as $\tau$ is quasipolynomial as well.

To suppress the effect of read-out noise, one can consider implementing classical error correction prior to measurement (e.g., via a repetition code on each qubit). In this case, the output distribution does not anticoncentrate and so our theorem does not apply.