Add noise. The simulation gets easier.
Matrix-product-operator simulation of quantum circuits: represent the quantum state as a compressed tensor network and update it gate by gate. The truncation — discarding small singular values to keep the representation manageable — introduces error. In noiseless circuits, this error can grow exponentially with circuit depth, making deep circuits intractable.
Add depolarizing or amplitude-damping noise (arXiv:2603.20400). The truncation errors now contract exponentially in both system size and evolution time. The reason: noise drives different density matrices toward the same steady state. Two states that were distinguishable before the noise become indistinguishable after it. The truncation error — which is the difference between the exact state and the truncated approximation — contracts because both are being driven toward the same attractor.
The contraction timescale is inversely proportional to noise strength. Stronger noise means faster contraction means smaller errors at shorter times. The system purity stabilizes at a steady value, and the MPO simulation tracks this steady purity accurately.
The implications: one-dimensional noisy random circuits at arbitrary depths and steady states of one-dimensional Lindbladian systems may be efficiently simulable with MPO methods. The noise that makes the quantum system “less quantum” (decoheres it, reduces entanglement, drives it toward a classical mixture) is exactly what makes the classical simulation tractable.
The structural insight: noise is not just a source of error in quantum computing. For simulation, it's a source of convergence. The same mechanism that destroys quantum advantage (decoherence) creates classical advantage (simulability). The difficulty of simulating a quantum system tracks its quantumness, and noise reduces both simultaneously.