Overtone

A variational quantum circuit that encodes classical data is, exactly and provably, a truncated Fourier series in that data. The frequencies it can reach are fixed by the eigenvalues of the data-encoding gates; the coefficients are set by everything else in the circuit.

That is Schuld, Sweke and Meyer, Physical Review A 103, 032430. The theorem is not new. Pointing it at a reinforcement-learning policy while the policy trains, and watching what it forbids, is.

Above, an agent has one encoding layer. Its reachable frequencies are lambda × {-1, 0, 1}, and the environment rewards frequency three. So it cannot score until lambda reaches three, and you can watch the peak slide until it does. It is a string being tuned, and that is not a metaphor: it is what is happening.

The environment is a Fourier coefficient extractor

SpectralControl-k is a contextual bandit. A state is drawn uniformly, there are two actions, and the reward is r(s, a) = (2a - 1) cos(k s). Its expected return works out to J = E[cos(ks) (2 pi(1|s) - 1)], which picks out precisely the frequency-k Fourier coefficient of the policy and ignores everything else.

So a policy that cannot reach frequency k scores exactly zero. Not worse, not near chance. Zero, for every parameter setting, forever. Set the layer count below the environment frequency in the panel below and the return curve will sit flat on the axis while the agent trains against it.

Where the softmax policy escapes its own circuit

A RAW-PQC reads the Born rule directly, so pi(1|s) = (1 + <Z>)/2 is a trig polynomial of bounded degree and its spectrum stops dead at the ceiling. A SOFTMAX-PQC applies a sigmoid to that same band-limited quantity, and a sigmoid of a trig polynomial is not band-limited. Switch the policy above and bars appear past the rule.

They appear at odd harmonics of whichever frequency the policy has concentrated on, not at odd multiples of the layer count. Those coincide when the agent has locked onto the environment, which is why the looser phrasing looks right; at a layer count too low to concentrate, the leakage is broadband instead. Even harmonics are suppressed a few hundred fold, which is the signature of an odd nonlinearity rather than of noise.

The consequence worth stating: the ceiling drawn behind the return curve bounds a strictly band-limited policy. It does not bound a softmax one, and a softmax agent will climb straight past it.

Barren plateaus, as a slider

Sample random parameter vectors, take the gradient, and look at its variance as the register grows. With a global observable the variance collapses exponentially and training becomes hopeless at scale. With a local one, while the circuit stays shallow, it does not.

The fitted exponent is printed beside the curves, because a number is more convincing than a slope. Set the depth policy to linear and the local observable stops escaping — the result is shallow and local, not local.

What the algebra knows before you train

Take the generators of the circuit's gates and close them under nested commutators. The set grows, alarmingly, and then stops. Its dimension is one integer, and that integer predicts how flat the loss landscape will be, how many directions the circuit can move in, and how many parameters it takes before the landscape stops having spurious minima — before a single gradient step.

None of this needs a matrix. A Pauli string is two bitsets; two of them anticommute when a symplectic form is odd, and their commutator is an XOR. The closure of a hundred-qubit circuit fits in a browser tab.

The right-hand panel is the discipline that makes it a prediction rather than a description. The line is computed from the algebra alone. The points are measured afterwards, from circuits with random parameters. The line is drawn first, every time.

Worlds, and the one number that separates them

Write a word on the lattice — periodic, Fibonacci, Thue–Morse, Rudin–Shapiro, or noise — and let each letter choose which coin the amplitude field meets at that site. The field's spread obeys sigma(t) ~ t^beta, and that exponent moves across its whole range as the word changes: one for ballistic, a half for diffusive, and no exponent at all when the field stops spreading and simply saturates.

The three aperiodic words are ordered by the nature of their spectrum. Fibonacci is pure point and spreads; Thue–Morse is singular continuous and carries a localized component alongside a spreading one; Rudin–Shapiro is absolutely continuous — the closest a deterministic word gets to noise — and it is the one that localizes. Part IV's own table attaches those spectra the other way round.

The race is the part that surprised us. Three of the four agents play the same coin at every site, and an agent whose coin does not depend on the local letter cannot read the world at all: its trace on Fibonacci and on the clean lattice is not similar, it is identical. Only the periodic world repays conditioning on the letter. On the aperiodic ones the best strategy found is to ignore the substrate entirely.

Beside it, Wave Function Collapse — which borrows every word here as a metaphor. Its cells are in "superposition", its lowest-entropy cell is "observed", and its constraints propagate. None of it is quantum. Its entropy is a count of the solver's remaining options; the entropy on the Lab panel is a property of the state. That contrast is the panel.

One eigendecomposition, three fields

Mahadevan's proto-value functions are basis functions learned from the topology of a state space rather than from its rewards: in the discrete case, the eigenvectors of the graph Laplacian. A classical random walk on the same maze evolves under e^(-Lt); a continuous-time quantum walk under e^(-iLt). Both are diagonal in that one basis, so the eigenvectors of this maze are simultaneously the RL basis functions, the eigenmodes of the quantum walk, and the graph Fourier transform of the corridors.

Part V writes the toggle as e^(-Lt) against e^(-iAt). Those are two different operators, and they agree up to an unobservable phase only on a regular graph. A maze is never regular — a dead end has degree one and a crossing four — so both sides here use L, and the claim that one character changed is then literally true.

The spread readout is there because "the quantum walk spreads faster" is false without a time attached. It leads early, is passed near t = 4, and is behind again by t = 8: unitary on a finite graph, so it never settles and the modes come back into phase. Diffusion, having only decaying modes, only ever spreads.

Below, the same maze as a control problem. Todorov's restriction — the controller may only reshape the passive dynamics, and pays a KL cost for it — makes the Bellman equation linear in z = exp(-v). So the desirability field appears without training, and two solved tasks add: move the sliders and a correct policy for a third task that nobody solved appears as a weighted sum. The error against solving it from scratch is printed beside it. The linearity is bought with that restriction, which belongs on the panel next to the result.