(a) Compute , writing the result as a normalised sum of . (b) Phase estimation is to…

PHYSICS · Subjective · 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 The quantum Fourier transform and phase estimation

Problem

(a) Compute F4∣3⟩F_4|3\rangle, writing the result as a normalised sum of ∣0⟩,…,∣3⟩|0\rangle,\dots,|3\rangle. (b) Phase estimation is to resolve an eigenphase to m=6m=6 bits with failure probability below ϵ=0.005\epsilon = 0.005. Using the rule t=m+⌈log⁡2(2+1/(2ϵ))⌉t = m + \lceil\log_2(2+1/(2\epsilon))\rceil, find the number of control qubits tt. (c) The phase gate S=diag(1,i)S=\mathrm{diag}(1,i) has eigenstates ∣0⟩|0\rangle (eigenvalue 11) and ∣1⟩|1\rangle (eigenvalue ii). Phase estimation with t=3t=3 control qubits is run on the input 0.75 ∣0⟩+0.25 ∣1⟩\sqrt{0.75}\,|0\rangle + \sqrt{0.25}\,|1\rangle for this target qubit. List the possible three-bit outputs and their probabilities.

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