The Quantum Compressibility Threshold

We prove a structural dichotomy for open quantum systems. A single dimensionless parameter \(\rho_Q = e^{|\lambda_1|/v_{LR} - 1}\), formed from the Lindblad spectral gap \(|\lambda_1|\) and the Lieb-Robinson velocity \(v_{LR}\), sharply separates two regimes:

- Supercritical (\(\rho_Q > 1\), equivalently \(|\lambda_1| > v_{LR}\)): the \(k\)-th cumulant truncation error is \(O((er)^k)\) with \(r = e^{-|\lambda_1|/v_{LR}}\), and \(k^* = O(\log(1/\varepsilon) / \log \rho_Q)\) orders suffice independently of \(n\). - Subcritical (\(\rho_Q < 1\), equivalently \(|\lambda_1| < v_{LR}\)): the cumulant series diverges; \(N_{99\%}\) grows as \(4^n\).

We show that useful quantum computation requires \(\rho_Q < 1\): a platform with \(\rho_Q > 1\) has dissipation dominating entanglement generation and cannot sustain the correlations needed for computational advantage. The incompressibility of the subcritical regime is therefore not a technological limitation but a structural consequence of quantum mechanics — a spectral reformulation of the no-cloning principle.

The negative result is complemented by a positive one: while global dynamics are incompressible, the spectral gap \(|\lambda_1|\) is locally recoverable from a constant-size (\(64 \times 64\)) cluster Lindblad, independent of \(n\). This enables one-shot spectral error mitigation at \(O(n)\) cost using a single circuit execution.

We verify five platforms (IBM Eagle, IBM Heron, Quantinuum H2, QuEra Aquila, Google Willow) to be deeply subcritical (\(\rho_Q \approx 0.37\) for all), with rigorous Lieb-Robinson bounds and propagated uncertainty \(\delta\rho_Q < 10^{-6}\). While \(\rho_Q\) is universal, the operational budget \(B_Q = v_{LR}/(n \cdot |\lambda_1|)\) — the maximum useful circuit depth — varies by an order of magnitude (555 to 5,437 for \(n = 10\)), providing a hardware-differentiating metric. We unify decoherence and gate errors into a single effective useful depth law: \(B_{\text{eff}}^{-1} = B_Q^{-1} + c \cdot \varepsilon_{2q}\), and show that all current hardware is control-limited (gate errors dominate), with the coherence wall 19–269\(\times\) away from current gate fidelities. We derive an algorithmic advantage frontier: the single condition \(r_{\text{eff}} \cdot D_{\text{req}}(n) < \ln(1/F^*)\) determines which algorithms can run on which hardware; at \(n = 50\), only low-depth algorithms (QAOA, random circuit sampling) are feasible on any platform, while VQE and Hamiltonian simulation hit scale-out walls at \(n = 12\)–\(23\). The one-shot spectral correction shifts this frontier by \(\ln(\kappa_{\max})\), recovering 2–3 algorithms per platform; the ratio \(F_{\text{obs}} / F_{\text{pred}}(B_{\text{eff}})\) directly quantifies how much existing experiments rely on error mitigation (1.3\(\times\) for Quantinuum to \(> 10^{15}\times\) for IBM). Cru