Second law of quantum complexity

For a generic chaotic finite-dimensional quantum system evolving unitarily from a low-complexity state, let $C(t)$ denote its quantum circuit complexity and let $t_{\mathrm{eq}}$ be the time at which ordinary thermal equilibrium is reached. The conjecture is that, for $t_{\mathrm{eq}}\leq t\leq t_{\mathrm{sat}}$ and apart from fluctuations, $C(t)$ continues to increase, eventually reaching a plateau $C_{\max}$ at a saturation time $t_{\mathrm{sat}}$ of order exponential in the equilibrium entropy, after which $C(t)$ remains close to $C_{\max}$ until extraordinarily late recurrence times: $C(t_2)\gtrsim C(t_1)$ whenever $t_{\mathrm{eq}}\leq t_1\leq t_2\leq t_{\mathrm{sat}}$ , and $C(t)\approx C_{\max}$ for $t\gtrsim t_{\mathrm{sat}}$ before recurrences.

References

Primary source

arXiv

Additional references

`It from Bit': is there a second law of quantum complexity? — arXiv

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A new manuscript claims to derive the proposed law, but no independent or formal confirmation exists.

Proposed by Adam Brown and Leonard Susskind in 2018, the conjecture says that quantum circuit complexity keeps increasing after ordinary thermal equilibrium, eventually reaching a complexity plateau. The original work explicitly treated this as conjectural rather than proved.

Known results

Brown and Susskind, 2017: argued for complexity growth, saturation after times of order $e^K$ , and recurrences on timescales of order $\exp[e^K]$ .
Brandão and coauthors: obtained approximate growth results for random-interaction models, but with simplifications that do not exactly match the black-hole setting.
Iliesiu, Mezei, and Sárosi, 2021: found quantum effects that could eventually arrest black-hole interior growth; this does not prove the general conjecture.

August 24, 2026 claimed derivation

A manuscript titled It from Bit: is there a second law of quantum complexity? claims that the law follows from an information-quantization principle and supplies near- and far-equilibrium asymptotics. The claim is theoretical and has no reported formal proof or independent verification.

Current status (as of August 2026): The conjecture remains unverified; the August 2026 manuscript claims a derivation, but its correctness and the proposed information-quantization principle remain open.

Sources

No solutions have been posted yet.