A two-level system has energies and

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 two-level system has energies

E_0=0

and

E_1>0

, both known to lie in a window

0\le E<E_{\max}

for some upper bound

E_{\max}

, but

E_1

itself is unknown. Phase estimation is to be run on

U=e^{-iHt_0/\hbar}

, whose eigenphases are

\varphi=-Et_0/2\pi\hbar\bmod1

. (a) Explain why, to avoid confusing

E_1

's reading with that of some other energy in the window once the phase is reduced modulo

1

, the base evolution time

t_0

must be chosen so that

E_{\max}t_0/2\pi\hbar\le1

, that is

t_0\le2\pi\hbar/E_{\max}

, and say what goes wrong if

t_0

is chosen any larger. (b) Suppose

t_0

is chosen, consistent with part (a), so that

E_1t_0/2\pi\hbar=3/8

exactly. Phase estimation is run with

t=3

control qubits on a trial state with overlap

|c_0|^2=0.8

on the ground state

|0\rangle

(and the remaining amplitude on

|1\rangle

). List every possible three-bit reading and its probability. (c) One run returns the reading

101

. Show that this corresponds to

\varphi=5/8

, and, using only the fact that

0\le E_1t_0/2\pi\hbar<1

from part (a), unwrap this reading to recover

E_1t_0/2\pi\hbar=3/8

and hence

E_1

in terms of

t_0

and

\hbar

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