Let be the NOT function, , (a balanced function)

PHYSICS · Multi-part · Fermi University Physics · Quantum Information & Technology · The quantum circuit model and quantum algorithms · The quantum circuit model and quantum algorithms

Quantum Information and Technology, Ch Oracles, quantum parallelism and the first algorithms

Problem

Let f:{0,1}→{0,1}f:\{0,1\}\to\{0,1\} be the NOT function, f(0)=1f(0)=1, f(1)=0f(1)=0 (a balanced function). Using the Cleve-Ekert-Macchiavello-Mosca circuit for Deutsch's problem, prepare ∣0⟩∣1⟩|0\rangle|1\rangle, apply H⊗HH\otimes H, apply the bit oracle Uf:∣x⟩∣y⟩↦∣x⟩∣y⊕f(x)⟩U_f:|x\rangle|y\rangle\mapsto|x\rangle|y\oplus f(x)\rangle, apply HH to the top qubit, and measure it. ((a)) Write the state of the two qubits after each of the three gates. ((b)) Show that the measured value of the top qubit equals f(0)⊕f(1)=1f(0)\oplus f(1)=1 with certainty. Then explain why this output alone cannot distinguish the NOT function from the other balanced one-bit function.

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