Phase estimation's controlled-power ladder reads a phase to bits of precision using…

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

Quantum Information and Technology, Ch Quantum metrology and sensing

Problem

Phase estimation's controlled-power ladder reads a phase

\varphi

t

bits of precision using

t

control qubits, the

j

th of which controls

U^{2^j}

on the target register, for

j=0,1,\ldots,t-1

; this ladder is the engine inside both order finding (used by Shor's algorithm) and quantum counting (used inside amplitude estimation). Resolving the same

t

-bit phase ambiguity with GHZ states, as in a GHZ magnetometer, uses a ladder of GHZ states of sizes

2^0,2^1,\ldots,2^{t-1}

, the state of size

2^j

giving a parity fringe

\cos(2^j\varphi)

read out by a chain of CNOTs, one Hadamard and one measurement, mirroring the powers of

U

. ((a)) For

t=6

bits of precision, state the number of controlled-power gates in the phase-estimation ladder, and the total number of applications of

U

they contain if each controlled-

U^{2^j}

is built by applying controlled-

U

2^j

times. ((b)) For the same

t=6

bits, state the total number of atoms consumed if the GHZ ladder uses one fresh copy of each sized state, used once. ((c)) For the same

t=6

bits, state the size of the largest GHZ state the ladder needs, and the power of

U

it mirrors. ((d)) Using the result that under independent dephasing no entangled state of

N

atoms beats independent atoms by more than a factor

\sqrt e

, explain in two or three sentences why adding rungs to the GHZ ladder stops improving precision, whereas adding control qubits to an ideal phase-estimation ladder does not, although both ladders spend the same total evolution.

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