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Equation 4 · Part 12 · How Post-CMOS, Neuromorphic, Photonic, and Quantum AI Compute Actually Works

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C(θ)=⟨ψ0∣V†(x) U†(θ) A U(θ) V(x)∣ψ0⟩C(\theta) = \left\langle \psi_0 \right| V^{\dagger}(x)\, U^{\dagger}(\theta)\, A\, U(\theta)\, V(x) \left|\psi_0\right\rangle
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What this part means

A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.

Its job in the formula

A raised mark can be a power or an index. Its position and the surrounding notation determine which.

The passage around this formula

For a machine-learning task, the near-term architecture almost universally used is the parameterized (or variational) quantum circuit, and its mechanism has four concrete steps. First, classical input data x is encoded into a quantum state by applying a data-dependent unitary V(x) to a fixed initial state — commonly done by rotating each qubit by an angle set from one feature of x, so the input vector is literally written into rotation angles. Second, an “ansatz” unitary U(θ) — a fixed sequence of parameterized single-qubit rotation gates and two-qubit entangling gates, repeated over several layers — is applied, where θ is now a set of continuously adjustable classical numbers rather than…

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Learn the underlying idea

An exponent tells how a base is used in multiplication. In x³, x is the base and 3 is the exponent: x³ = x × x × x.

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Sources cited in the surrounding passage

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