Breaking the Quadratic Barrier: The Arithmetic Complexity of the Permanent & Quantum Parallel Repetition

In theoretical computer science, proving circuit lower bounds—demonstrating that a mathematical function cannot be computed with fewer than a certain number of logic gates—is notoriously resistant to progress due to formal barriers (Relativization, Natural Proofs, Algebrization).

1. Superquadratic Circuit Bounds for the Permanent

The fifth chapter in the anthology proves:

• Unrestricted Division-Free Circuits: Lower bound of (\Omega(n^2 \log \log n)) gates.
• Arithmetic Formulas (with division): Lower bound of (\Omega(n^4 / \log n)) variable leaves.

The proof uses reverse-mode automatic differentiation to convert candidate multiplication circuits into gradient generators, establishing that small circuits cannot produce the high-codimension critical loci required by the permanent polynomial.

2. Quantum Parallel Repetition for Entangled Games

In quantum cryptography and Bell nonlocality, parallel repetition determines whether repeating a game decreases the players' winning probability exponentially. For entangled provers, quantum conditioning previously broke classical proofs. The sixth chapter proves general exponential decay for all finite two-player entangled games via postselection-stable sampling.