Trains in this tab

The interference
they left on the table.

The people who built quantum machine learning and then pivoted to thermodynamic chips moved the sampling workload to classical physics — because a thermodynamic sampler draws from a non-negative Gibbs distribution P(x) ∝ e−E(x). What that hardware physically cannot do is interference: a complex amplitude ψ(x) whose wrong answers destructively cancel. That is the genuinely-quantum resource — and it runs on classical hardware via a state vector.

Below, a Quantum Circuit Born Machine — Pθ(x) = |ψθ(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 field's central limit, 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 contract: reproducible f64.

Born machine · learns by interference

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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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decision boundary + data
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Metered · the energy price of interference

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no browser meter — illustrative rate
signed phasor receipt · —
 
 
 
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The phasor bridge · same interference, two substrates
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The honest limit · barren plateaus
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The surviving moat · learning from quantum data
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   worst-case fidelity error vs channel copies — log–log 
 
 
Bell (quantum memory) vs entanglement-free
 
 
 
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Why this is the real unlock — and where it deploys
 

A thermodynamic sampler is sign-problem-free by construction — non-negative weights that only
            add probability — which is exactly why it cannot reproduce interference (the sign structure of a
            wavefunction). The Born-machine expressivity separation is unconditional: a minimal quantum
            model is strictly more expressive than any classical energy-based model of comparable size, and the
            resource is precisely this negative/complex quasi-probability. That is what the pivot left behind.

 

The honest catch — 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
Σω cω eiωx; complex-phasor binding is
interference (phase addition); and Fourier-holographic representations (FHRR / HDC) are the same
            algebra in polynomial space. qFHRR — the integer-native phasor substrate — is the cheapest
            deployable realization of exactly what a Born machine computes. Same interference, two substrates:
            the state vector here, the phasor fabric for scale. Reference engine + spec in the open-standards
            repo, Apache-2.0.