A quantum simulator estimates an energy gap of a two-level system by phase estimation on

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 The quantum Fourier transform and phase estimation

Problem

A quantum simulator estimates an energy gap

E

of a two-level system by phase estimation on

U = e^{-iHt_0/\hbar}

, where a single controlled application of

U

evolves the system for a base time

t_0 = 10\,\mathrm{fs}

. ((a)) Using

\delta t \approx \hbar/\delta E

, estimate the total controlled-evolution time needed to resolve the energy to

\delta E = 10\,\mu\mathrm{eV}

, whatever the circuit used to achieve it. ((b)) The standard phase-estimation circuit with

t

control qubits applies

U

a total of

2^t - 1

times. Find the smallest

t

for which

2^t - 1

applications of

U

, each of duration

t_0

, provide at least the total evolution time found in (a), and state the resulting total evolution time.

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