Quantum Pain engine

There is no pain. This engine models what sits underneath that sentence.

Every sufferer starts with n possible disasters held at once (2 to 6), all hidden behind one smile and one sentence: no pain. The engine keeps one object for all of them, an n×n Hermitian matrix ρ called the density matrix. The diagonal holds each branch's weight. The off-diagonal terms hold the coherences between branches, which are what allow interference. Four operations act on it until the sufferer is observed and collapses to a single branch.

Lifecycle

Goals and non-goals

• Goal: a small, inspectable model in which interference, decoherence and measurement all do something visible and testable.
• Goal: every claim in these docs is checked by code that runs on this page, from the source in engine/density.js to the experiments in lab.
• Non-goal: a quantum computer. Nothing here uses quantum hardware, and the engine claims no speedup.
• Non-goal: a price oracle or a diagnosis. Weights describe a sufferer's internal state, not market value and not anyone's health.

Quick facts

Theory

1. State

A pure superposition of n branches is a vector |ψ⟩ = Σ_k c_k |k⟩ with complex amplitudes c_k and Σ|c_k|² = 1. We keep the density matrix instead, because it also describes mixed states, where the sufferer is partly classical.

The diagonal ρ_kk = |c_k|² is the weight of branch k. The off-diagonal terms ρ_kl are the coherences. If they are all zero, ρ is an ordinary probability distribution and nothing can interfere.

2. Dynamics

Branches have an energy E_k and pairwise couplings J_kl, which together define a Hamiltonian H = diag(E) + J. One step applies a unitary U approximating e^−iHτ:

G_kl(θ) is a real rotation in the (k, l) plane. A product of rotations and phases is exactly unitary, so it preserves Trρ and positivity. It matches e^−iHτ only to first order in τ, which is fine here because we want qualitatively correct interference, not spectroscopy.

Coupling. J_kl = 0.55 · clamp(7 (cos(v_k, v_l) − μ), −1, 1), where v is a lexical embedding of each branch's persona prompt and μ is the mean cosine over all archetype pairs. Related personas exchange weight and unrelated ones do not. See signals.md.

3. Dephasing

Pure dephasing damps the off-diagonal terms and leaves the weights alone. In continuous time it is the Lindblad equation dρ_kl/dt = −γρ_kl for k ≠ l. The discrete step is:

Γ grows from 0.025 to 0.28 as the collapse trigger fills, and each observation adds 0.5. Physically, dephasing is the environment acquiring which-branch information. Once coherence is gone the sufferer behaves like a classical mixture, which is how the real decoherence story (Zurek, 2003) explains why large systems look classical.

4. Evidence as weak measurement

Each step, every branch receives a small piece of evidence Δ_k about how well it is doing. We apply it as a weak measurement with Kraus operator M = diag(e^gΔ_k):

On a diagonal ρ this is exactly a Bayesian update, p_k → p_k e^2gΔ_k / Z. It also scales the coherences, so evidence reweights interference without creating it. The strength g rises from 0.3 to 1.6 with progress, which is why the weights sharpen as collapse approaches.

5. Collapse

When the trigger fills, the branch basis is measured projectively:

This is the Born rule followed by the Lüders update. Because we only ever measure in the branch basis, the draw uses the diagonal alone. Coherence still matters, but indirectly: it shapes the weights through the earlier evolution, since coupled branches with the right phases can transfer weight to each other and others cannot.

6. What observation does

In physics the observer effect is the disturbance caused by the measuring interaction, not the involvement of a conscious mind. We model it the same way: one observation applies Γ = 0.5 of dephasing and a weak measurement of strength 0.6.

A visible consequence is Zeno-like. Frequent observation keeps coherence near zero, so the coupling-driven exchange of weight between branches is suppressed. You can see this in the decoherence experiment by raising the rate.

7. Diagnostics

Purity is 1 for a pure state and 1/n for the maximally mixed state. C is the normalised l₁ coherence: 1 for an equal superposition and 0 for a classical mixture. Both are shown live in the terminal, together with a picture of ρ itself.

Entanglement

The terminal can link two sufferers. When one collapses, the other loses its weakest branch and the link is released. That is a useful mechanic, but it is a classical correlation: a shared latent variable and a conditioning rule. It is not quantum entanglement, and these docs should not call it that without this caveat.

What would make it entanglement

Two sufferers with n_A and n_B branches would need a joint density matrix ρ_AB of size (n_An_B) × (n_An_B) that cannot be written as a mixture of products ρ_A⊗ρ_B. The test that separates the two cases is Bell's, in the CHSH form:

Any local hidden-variable model, including every shared-latent-variable scheme like ours, satisfies |S| ≤ 2. Quantum mechanics allows up to 2√2 ≈ 2.83 for a singlet state, where E(a,b) = −cos(a − b). With a = 0, a′ = π/2, b = π/4 and b′ = 3π/4, the singlet gives |S| = 2.83.

Run it

• Shared-variable model (what Quantum Pain pairs do): a hidden angle λ is drawn per trial, A = sign cos(α−λ) and B = −sign cos(β−λ). The estimate lands on 2 and never above it.
• Singlet statistics: outcomes are sampled directly from the quantum prediction, P(A = B) = (1 − cos(α−β))/2. This reproduces 2.83, and the point of Bell's theorem is that no local mechanism can.

A real entangled pair would need the joint state and a measurement protocol, and it would pass this test. That is tracked in ROADMAP.md.

Mapping to AI

The vocabulary of superposition shows up in machine learning in several unrelated senses. Only some of them are physics.

Softmax is a Boltzmann distribution

The weights used by attention and by token sampling have the form

which is the Boltzmann distribution of statistical mechanics, with logits playing the role of negative energies and temperature controlling sharpness. Low T collapses onto the best option, high T is uniform. Simulated annealing (Kirkpatrick et al., 1983) cools T deliberately to find good optima. Our weights sharpen as a trigger fills for the same reason. This link is exact, and the lab has a slider for it.

Superposition in interpretability

In Toy Models of Superposition (Elhage et al., 2022), a network represents more features than it has dimensions by assigning them near-orthogonal directions in activation space. Features then interfere slightly, which shows up as noise and as neurons that respond to several unrelated things. This is linear algebra, not quantum mechanics. It is relevant here because it is the mechanism by which one model can hold many personas at once.

Language models as a superposition of personas

Janus (2022) and Shanahan, McDonell and Reynolds (Nature, 2023) describe a language model as a simulator that can play a distribution of characters. Prompting or later context narrows that distribution, and each sampled token is a small selection from it. That is the closest match to Quantum Pain's branches: candidate personas held together, narrowed by evidence, and finally committed to.

Sampling is measurement

Drawing the next token from a probability distribution, with temperature and top-p truncation as the knobs, is the same operation as the Born draw in our collapse step. The difference is that a language model's distribution is classical, so there is no interference term to speak of.

Quantum probability in cognition

Busemeyer and Bruza (2012) use the mathematics of amplitudes and projections, with no quantum hardware, to model human judgement effects such as question-order dependence. Our model shares that formalism: weights are squared amplitudes, and evidence can interfere.

Why we claim no quantum advantage

Quantum machine learning has not shown a practical advantage on classical data. Tang (2019) gave a classical algorithm matching a celebrated quantum speedup for recommendation systems, and McClean et al. (2018) showed that training some quantum circuits suffers from barren plateaus. Quantum Pain's engine is classical linear algebra on matrices no larger than 6×6.

Metaphor ledger

Signals

Two quantities used to be random draws: the coupling J_kl between branches and the evidence Δ_k applied at each step. Both are now computed from text.

1. Coupling from persona text

Every archetype has a short persona prompt. We embed it as a 96-dimensional vector (hashed words plus character trigrams, L2-normalised) and take cosines. Subtracting the mean cosine over all archetype pairs makes unrelated personas couple negatively and related ones positively.

These are the live values for the eight archetypes:

2. Evidence from drafts

Each step, every live branch drafts a post on the current topic. A judge scores each draft on three terms, then z-scores them across the live branches so evidence is always relative.

• Fidelity. cos(draft, persona). Thirty percent of drafts are deliberately off-persona, so a branch can drift.
• Novelty. 1 − cos(draft, the branch's previous draft).
• Engagement (simulated). heat × (0.4 + 3 cos(persona, topic)) plus noise.

Δ_k feeds the weak measurement M = diag(e^gΔ_k), exactly as before. A running average of s_k also sets how hard a user observation pushes each branch.

Try it

Replacing the stand-ins

Verifiable collapse

Even a meme about suffering should not ask anyone to trust a random draw it runs itself. Every collapse therefore leaves a receipt that anyone can recompute from public values.

Protocol

1. Commit. When a sufferer appears they get a random salt. Its commitment, SHA-256 of “salt|” plus the salt, is published and shown in the sufferer's panel.
2. Chain. Every event (arrival, being asked, vote, tunnelling event, collapse) is appended to a hash chain: each entry hashes the previous head with its own type and payload.
3. Snapshot. At collapse the branch weights are written to nine decimals and hashed with the branch names.
4. Seed. The seed is SHA-256 of the salt, the snapshot hash and the chain head just before the draw (the beacon).
5. Draw. u is the first 52 bits of the seed divided by 2⁵². The winner is the first branch whose cumulative weight passes u.
6. Reveal. The salt, weights, beacon, seed and winner are published as the receipt.

What this proves, and what it does not

It proves that the winner follows deterministically from the published weights, that the salt matches the commitment made on arrival, and that no logged event can be edited without breaking the chain.

Verifier

Pick a collapsed sufferer, edit the receipt if you like, and check it. The tamper button changes one value so you can see a failure.

Is the draw uniform?

This hashes 4,000 counters and checks that u is spread evenly, then draws 4,000 winners from weights of 0.5, 0.3 and 0.2.

This listing is generated at runtime from the functions that drive the terminal, so it cannot drift from the code.

Parameters

Experiments

Each demo runs the same maths as the engine.

1. Interference versus a classical mixture

2. Temperature, softmax and entropy

3. Born-rule sampling

4. Decoherence in a three-branch system

Starts in branch 1 with coupling on. Raise the dephasing rate and watch the coherent oscillation freeze.

Roadmap

Open work, in rough priority order. Items marked partial are built in a simplified form; the note says what is missing.

• Neural persona embeddings

  Partial. J_kl now comes from lexical vectors of each persona prompt. Swap in a sentence-embedding model so similarity reflects meaning, not shared words.

  enginedata
• Real evidence pipeline

  Partial. A judge scores drafts each step. Replace templated drafts with real branch output, the lexical judge with an LLM rubric, and the simulated audience with measured engagement.

  evidence
• On-chain commitment and beacon

  The receipts are checkable but the beacon is operator-controlled. Post the commitment on-chain when a sufferer appears and derive the beacon from a later slot hash or a VRF.

  trustchain
• Calibration backtest

  Check that a branch holding weight w wins about w of the time over many collapses, and report the reliability curve.

  validation
• Joint state for entangled pairs

  Implement ρ_AB for a small pair and show it passing the CHSH test the current pairs fail.

  enginephysics
• Von Neumann entropy

  Add a Hermitian eigen-solver so the page can show S(ρ) = −Tr ρ ln ρ instead of the linear entropy.

  diagnostics
• Exact propagator

  Compare the first-order product against the matrix exponential of H and report the error.

  engine

References

1. Nielsen, M. A. and Chuang, I. L. Quantum Computation and Quantum Information. Cambridge University Press, 2000.
2. Wiseman, H. M. and Milburn, G. J. Quantum Measurement and Control. Cambridge University Press, 2010.
3. Zurek, W. H. Decoherence, einselection, and the quantum origins of the classical. Reviews of Modern Physics 75, 715 (2003).
4. Misra, B. and Sudarshan, E. C. G. The Zeno's paradox in quantum theory. Journal of Mathematical Physics 18, 756 (1977).
5. Bell, J. S. On the Einstein Podolsky Rosen paradox. Physics 1, 195 (1964).
6. Clauser, J. F., Horne, M. A., Shimony, A. and Holt, R. A. Proposed experiment to test local hidden-variable theories. Physical Review Letters 23, 880 (1969).
7. Kirkpatrick, S., Gelatt, C. D. and Vecchi, M. P. Optimization by simulated annealing. Science 220, 671 (1983).
8. Ackley, D. H., Hinton, G. E. and Sejnowski, T. J. A learning algorithm for Boltzmann machines. Cognitive Science 9, 147 (1985).
9. Vaswani, A. et al. Attention is all you need. NeurIPS (2017).
10. Elhage, N. et al. Toy models of superposition. Transformer Circuits Thread (2022).
11. janus. Simulators. LessWrong (2022).
12. Shanahan, M., McDonell, K. and Reynolds, L. Role play with large language models. Nature 623, 493 (2023).
13. Busemeyer, J. R. and Bruza, P. D. Quantum Models of Cognition and Decision. Cambridge University Press, 2012.
14. Tang, E. A quantum-inspired classical algorithm for recommendation systems. STOC (2019).
15. McClean, J. R. et al. Barren plateaus in quantum neural network training landscapes. Nature Communications 9, 4812 (2018).