Trains in this tab

Machine learning
by interference.

Train a small circuit to reproduce a distribution, and measure the point where training stops working.

A sampler over a non-negative Gibbs distribution P(x) ∝ e^−E(x) can only add probability. Interference is different: a complex amplitude ψ(x) whose wrong answers destructively cancel. That is the quantum resource this page works with — and at this size it runs on classical hardware, via a state vector.

Below, a Quantum Circuit Born Machine — P_θ(x) = |ψ_θ(x)|² — trained on a laptop by the exact parameter-shift gradient to match a target only interference can hit. Then the barren plateau: the central limit on training these models, made measurable — the reason you don't just scale up circuits, and why the deployable form of this is shallow, structured, and phasor-based. Determinism: float (f64), exact on this build — not a cross-build byte guarantee.

Born machine · learns by interference

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target

KL loss over training

target (outline) vs learned (fill)

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Deploy on the phasor substrate · the full model on qFHRR

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exact (outline) vs qFHRR 4-bit (fill)

TV distance from exact vs phase bits

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Supervised · the quantum kernel as a phasor classifier

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dataset

decision boundary + data

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The receipt · exact work, and an energy slot

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energy: not metered in this tab — the receipt is sealed unmetered

signed phasor receipt · energy not metered · —

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The phasor bridge · same interference, two substrates

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The limit · barren plateaus

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Learning from quantum data · a sample-complexity separation

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system size worst-case fidelity error vs channel copies — log–log

Bell (quantum memory) vs entanglement-free

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What interference adds — and where it deploys

A model with non-negative weights is sign-problem-free by construction — its weights only add probability — so it has no way to reproduce interference (the sign structure of a wavefunction). A Born machine's resource is exactly this complex amplitude: contributions that can cancel, which is what lets it place the exact zeros the parity target needs.

The barren plateau above is why the deployable form isn't "scale up a parameterized circuit" (that regime is both untrainable and, where it is trainable, classically simulable). It is the shallow, structured, phasor regime. And here the whole family collapses into one object: a quantum feature map is a truncated Fourier series of phasors Σ_ω c_ω e^iωx; complex-phasor binding is interference (phase addition); and Fourier-holographic representations (FHRR / HDC) are the same algebra in polynomial space. qFHRR, the same phasor algebra in integers, computes the interference a Born machine computes: one idea, in the state vector here and in integer phasors.

Where it lives. The engine is reference code in the open-standards repo, Apache-2.0. Determinism: float (f64), exact on this build — not a cross-build byte guarantee. Energy: nothing is metered in the browser, so the receipt on this page is sealed unmetered and its energy is shown as not metered.

Machine learning by interference.