The sibling of The Comb. Each cell of the paper is now a qubit, and a sheet no longer holds one groove per cell. It holds one complex amplitude for every whole pattern of the cells: 2ⁿ of them. Each comb is a row of two-qubit gates. Every other comb sits one cell further round, so the comb turns as it goes.
Every comb is unitary. Total groove energy Σ|a|² stays exactly 1, and there is no ReLU. Any cut can be uncut by running the comb backwards.
An unknown quantum state cannot be cloned, so a real device can't keep every sheet. It has one sheet, cut in place. Only a classical simulator can keep them all.
Reading the sheet returns one pattern, drawn with probability |a|². n qubits give n bits per reading, however many amplitudes were inside.
Each cell's groove shows the chance that qubit reads 1. A solid dark groove means the qubit can be described on its own. A faded one, with a dashed blue halo round its socket, is entangled with the rest of the ring, so its own cell can't hold its description. The outer halo shows all 2ⁿ amplitudes of the newest sheet: length is |a|, colour is phase.
Each history contributes one small arrow. Most of them cancel. The path wanders, spirals and folds back, and ends exactly on the amplitude the kept sheets give. That cancellation is interference, done by arithmetic.
Patterns are laid out left to right by number. The phase strip under the bars is what the shots can't see: every colour there is information inside the sheet that no single reading returns.
One amplitude ⟨y|U|x⟩, three ways. The numbers follow the lathe settings.
| Schedule | Paper held at once | Work | What it costs you |
|---|---|---|---|
| Schrödinger: keep the sheets | – | – | Memory doubles with every qubit. Simulators stall around 50 qubits for this reason. |
| Feynman: sum the histories, depth first | – | – | Memory is tiny and work doubles with every qubit in every layer. This schedule is the proof that quantum computers can be simulated in polynomial space (BQP ⊆ PSPACE): the Regrow button of The Comb, with complex teeth. |
| The physical comb | – | – | Small in space and small in time. The bill moves to the readout: one pattern of n bits per run, drawn at random. |
Reversibility replaces erasure here. A classical comb that crushes values has to keep them or regrow them. A quantum comb never crushes, so anything it computed on the side has to be uncomputed by running backwards (Bennett 1989). That's the reversible cousin of Williams' regrowing tree.