-level systems and followed by single-qudit measurements can be efficiently simulated classically with high accuracy. In contrast, we prove that such strong simulations are hard for 3-local circuits. Using sampling methods we further show that all commuting circuits composed of exponentiated Pauli operators

e^{i\theta P}

can be simulated efficiently classically when followed by single-qubit measurements. Finally, we show that commuting circuits can efficiently simulate certain non-commutative processes, related in particular to constant-depth quantum circuits. This gives evidence that the power of commuting circuits goes beyond classical computation.

DOI: 10.26421/qic13.1-2-5

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d

e^{i\theta P}

n

-qubit circuits whose action commutes with the qubit-permuting representation of the symmetric group

S_n

-can be simulated in polynomial time. However, existing approaches scale as

O(n^7)

, and can rapidly become prohibitively expensive. In this work, we introduce a practical algorithm for simulating

S_n

-equivariant circuits under the assumption that the gate generators are at most

k

-local, with

k\in O(1)

. The resulting method runs in

O(n^{ω+1})

time for constant depth, where

ω

is the matrix multiplication exponent, significantly lowering the polynomial degree compared to existing techniques. Finally, we numerically validate this scaling by simulating the dynamical evolution of the Lipkin-Meshkov-Glick model, and show that for

n=512

spins, a standard laptop can compute the concurrence of the evolved state in under two minutes.