Deutsch-Jozsa Circuit in 3D

Hidden function

Qubits n 3

Watch the pulse cross the circuit: H on every wire, the oracle box (one query), then H again on the n data wires. The bar landscape behind the circuit rises with the real computed amplitudes once it reaches the end.

Run

N = 2ⁿ states

Quantum queries

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Classical worst case

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P(measure 0…0)

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Verdict

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How it works

The wires are the n data qubits (top) plus the ancilla (bottom, prepared in |−⟩). The pulse travelling along them is not decorative — its arrival at the final H gate is exactly when the amplitude landscape behind the circuit is recomputed from the real Walsh-Hadamard transform below, using whichever f you built on the left.

|0⟩ⁿ|1⟩ --H⊗(n+1)--> uniform superposition
        --U_f-->        aₓ → (−1)^f(x) aₓ    (one oracle query)
        --H⊗n-->        a_y = (1/√N) Σₓ (−1)^(x·y) aₓ
a₀…₀ = (1/N) Σₓ (−1)^f(x)  →  ±1 (P=1) if f constant, 0 (P=0) if f balanced

Qubits n — register size; N=2ⁿ bars in the landscape. Classical worst case is 2ⁿ⁻¹+1 queries; quantum stays at 1 regardless of n.
Oracle box — the single query. It never measures or branches on x — it only imprints a phase, which is why one pass suffices.
Bar landscape — height is the real |amplitude|² for each of the 2ⁿ possible measurement outcomes y; the y=0…0 bar is highlighted, since that is the only outcome the algorithm needs to read.

Real-world relevance: the same phase-kickback-then-interfere structure shown here — encode a query result as a phase, then use a global transform to convert phase differences back into readable amplitude differences — is reused, with variations, by Simon's algorithm, Shor's algorithm and quantum phase estimation.

Qubit wire H gate Oracle U_f y = 0…0 bar

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