Perceptron theory · deeper into the quantum · two coupled cubi
The perceptron in two cubi
One cubi could only turn its own state. Two cubi can do something no classical pair can: become entangled — a single joint state where neither cubi has an identity of its own. Here the feature map isn't a rotation I bolt on; it's a real two-cubi coupling, a gate that physically links them, so the cross-correlations are computed by the hardware. We build it honestly — and then tell you the truth about what it does and doesn't buy, which is not what most quantum-ML demos will admit.
two cubi = a 4-state joint system · the bond = a real entangling gate (CNOT / ZZ) · feature map = the coupling itself · readout = the quantum kernel
✓ STRONG
Real entanglement & quantum kernels. Two-cubi gates make genuine Bell pairs (concurrence 1, CHSH 2.83 — measured in labs for decades); the quantum-kernel feature map runs on hardware (Havlíček et al., IBM, 2019).
◐ MIDDLING
Quantum-kernel classifiers. Demonstrated — but kernel estimation is noisy, and many entangling feature maps you can actually build turn out to be classically simulable after all.
◔ FRONTIER
A proven advantage on real data. Whether entanglement gives a genuine machine-learning edge over classical methods is open and contested — the honest seam between physics and hype.
I · Two cubi, and the bond that has no classical picture
Start both cubi in |0⟩. Put the first into superposition (an H gate), then apply a CNOT — the second cubi flips if the first is |1⟩. Now the pair is (|00⟩+|11⟩)/√2: both 00 and both 11, never 01 or 10. Measure one and you instantly know the other — yet neither had a definite value. That linkage is entanglement, and the meter is the concurrence: 0 for two separate cubi, 1 for a perfect Bell pair.
cubi A and cubi B · the bond brightens with entanglement · below: the 4 joint amplitudes |00⟩|01⟩|10⟩|11⟩
0.00
concurrence (entanglement)
|00⟩
joint state
A separable pair (concurrence 0) is just two arrows you can describe one at a time. The instant CNOT links them, the description of "A alone" stops existing — only the pair has a state. That's the resource the whole rest of this page runs on.
II · The feature map is the coupling
To classify, encode the inputs into the pair: a rotation Rz(x₁) on A, Rz(x₂) on B, and then the two-cubi ZZ coupling that depends on x₁·x₂ — the gate that does the entangling. The data is now a point in an entangled 4-dimensional state |Φ(x)⟩, and similarity between two inputs is the quantum kernelK(x, x′) = |⟨Φ(x′)|Φ(x)⟩|² Toggle the coupling and watch the concurrence — and the kernel — change. The cross-term is real hardware, not arithmetic.
x₁ x₂
|Φ(x)⟩ — the 4 joint amplitudes after the feature map · the kernel to a fixed reference point shown as the bar
—
concurrence of |Φ(x)⟩
—
kernel K(x, x_ref)
encode= Rz(x₁), Rz(x₂)couple= ZZ(x₁·x₂) — entanglescompare= the quantum kernel
With the coupling ON the state is entangled (concurrence > 0) and the kernel mixes both inputs. OFF, it factorizes into two independent cubi. This is exactly the knob the next module interrogates.
III · It learns XOR — and here's what nobody tells you
Run a kernel classifier on the four XOR points. With the entangling coupling, it nails XOR 4/4. Now switch the coupling OFF — and it still gets 4/4. Both work. That's the verified, deflationary truth: for two inputs, you do not need entanglement to solve XOR; a product map already carries the cross-term.
press classify
So what does entanglement actually buy?
Not toy problems. What it buys is correlations a classical model literally cannot fake — the entangled pair violates the Bell/CHSH bound (S = 2.83 > 2, verified), which no separable description can reach. That un-fakeable structure is the only honest basis for a quantum advantage: feature maps too entangled to simulate on a classical computer. Whether that translates into a real edge on real machine-learning data — once you pay for noise, state preparation, and readout — is unproven and actively argued. We built the real entangler; we won't sell you the magic.
the resource= entanglement (concurrence, CHSH)the toy= doesn't need it (XOR works either way)the bet= classically-hard feature mapsthe verdict= advantage unproven → FRONTIER