<aside> 🧭

Landing page for this research thread. Start here: short overview, then jump to any section below. Status: Exploring · Opened 2026-10-25.

</aside>

<aside> 🧭

Guiding connections (the Director's framing) • The connections being made between infinity and the quantum, the mind and consciousness. • Also focusing on the biology of the brain. • Quantum neural networks and the way neurons work in our own brains. • Interested in the physiology of the brain and neural network abstractions with Bayesian statistics. • Tying into quantum mechanics and infinity arguments with Rudy Rucker. • The idea of capturing the concepts. • Having all of that as a thread of understanding we are trying to distill into a simple picture of what's going on. Shares this guiding thread with: Infinity thread

</aside>

Overview

We are opening a thread at the philosophy + science intersection: what logic looks like when the underlying physics is quantum, what it means for a quantum computer to work "in superposition", and whether proofs could be checked — or even filtered — by a quantum process. The short honest answer so far: a quantum computer cannot simply keep only the valid proofs. Measurement is probabilistic, so wrong answers must be suppressed by interference (as in Grover's algorithm), which gives at most a quadratic speed-up for unstructured search. "Keep only the good outcomes" is called postselection, and Aaronson showed PostBQP = PP: very powerful, but not something nature gives for free. What is well developed is quantum proof verification in complexity theory (QMA, QIP = PSPACE, MIP* = RE, Mahadev's classical verification) and formally verified quantum programs in Coq.

Key concepts

• Quantum logic (Birkhoff & von Neumann 1936): propositions as closed subspaces of Hilbert space; the lattice is orthomodular, not distributive.
• Proof as a physical process (Deutsch, Ekert & Lupacchini 2000): our knowledge of mathematical truth depends on the physics of the machines that compute it.
• Superposition and interference: a quantum computer evolves many basis states at once, but you only read one measurement outcome; advantage comes from arranging interference.
• Amplitude amplification (Grover 1996; Brassard–Høyer–Mosca–Tapp): repeatedly boosts the amplitude of marked ("correct") answers; ~√N queries for search.
• Optimality limit (Bennett–Bernstein–Brassard–Vazirani 1997): no quantum algorithm beats ~√N for black-box search, so no generic exponential shortcut to NP.
• Postselection (Aaronson 2005): PostBQP = PP — "only valid proofs survive" is equivalent to an unphysical resource.
• QMA (Kitaev): quantum analogue of NP — a quantum proof state checked by an efficient quantum verifier; local Hamiltonian is QMA-complete.
• QIP = PSPACE (Jain–Ji–Upadhyay–Watrous 2010); MIP = RE* (Ji–Natarajan–Vidick–Wright–Yuen 2020).
• Classical verification of quantum computation (Mahadev 2018).
• Formally verified quantum programs: QWIRE, SQIR/VOQC, CoqQ (Coq).
• Streaming intuition: a quantum algorithm acts on a superposition of many possible states at once. Each step rotates the state so the correct answer's amplitude grows, like a wave building toward a peak. Concrete example: Grover's algorithm peaks after about (π/4)√N iterations; going past that lowers the success probability, because it is a rotation, not endless growth.
• States, collapse and measurement: measuring gives a random outcome with probability equal to |amplitude|², and the superposition collapses to that one outcome. For a physical qubit, the essential requirement is two reliably distinguishable (orthogonal) states.
• Director's curiosity: the most efficient valid path? Could collapse surface not just a valid path but the most efficient valid one among many? Amplitude amplification can be adapted to optimization if each possibility can be scored — e.g. Dürr & Høyer, "A Quantum Algorithm for Finding the Minimum" (1996, arXiv quant-ph/9607014). But it still gives only a quadratic speed-up, and computing or comparing the score may be as hard as the search itself. Open question.
• Error correction measures the errors, not the answer: repeated checks catch and fix mistakes without collapsing the protected superposition. Below threshold, errors shrink exponentially as the code grows, which is what makes long Grover-style runs physically possible.