Scrooge ensemble approximation for shallow random quantum circuits

Establish that, for every constant depth d>d^* above a critical depth in two-dimensional brickwork shallow random quantum circuits, the post-measurement ensemble on region A obtained by measuring regions B and C, conditioned on the gates in the backward light cone of AB and the outcome on B, has low-order moments approximated by the Scrooge ensemble (or the stabilizer Scrooge ensemble for random Clifford circuits) associated with its background state, with exponentially small error in the size of C and the distance between A and C, and that the background state has min-entropy proportional to the boundary size of A.

Background

The paper studies conditional dependence created when measurements are performed on shallow random quantum circuits. Although the unmeasured output state has only short-range correlations because of the circuit lightcone, measuring a complementary region can induce long-range entanglement and conditional correlations. The authors propose that these post-measurement states possess a universal statistical description despite retaining the local lightcone structure of the circuit.

The conjecture concerns a tripartition of a two-dimensional brickwork circuit into regions A, B, and C, with B shielding A from C. For depth above a constant critical value d*, the ensemble of states on A generated by varying the gates and measurement outcomes in C is conjectured to agree, in fixed-order moments, with the Scrooge ensemble determined by the averaged or background state on A. The corresponding stabilizer Scrooge ensemble is proposed for random Clifford circuits. The claimed approximation would provide a universal account of conditional correlations and underlies the paper’s consequences for classical simulation under weak depolarizing noise.

The supplementary analysis supports the claim through statistical-mechanical mappings and replica calculations, but explicitly notes that the replica continuation required to obtain the desired moments is not rigorous. Thus the formal Scrooge hypothesis remains an unresolved conjecture rather than a theorem.

References

For a tripartition $ABC$ of the qubits, we consider the ensemble of post-measurement states on $A$ that is conditioned on a specific measurement outcome on $B$ and ranges over all possible measurement outcomes on $C$. For circuit depths exceeding a constant critical value $d*$, we conjecture that this ensemble is well approximated by a certain generalization of the Haar ensemble, called the Scrooge ensemble~[Jozsa et al., \href{https://doi.org/10.1103/PhysRevA.49.668}{Phys. Rev. A 49, 668 (1994)}]; we also provide supporting numerical and analytical evidence.

— Conditional dependence and Scrooge ensembles in shallow random quantum circuits  (2608.12255 - Liu et al., 12 Aug 2026) in Introduction and formal statement in “The Scrooge hypothesis” (Hypothesis, Eq. (scroogeeps)); supplementary material, Section “The Scrooge ensemble and shallow 2D Haar-random quantum circuits”