A superconducting phase qubit implements a fixed single-qubit gate in the…

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 superconducting phase qubit implements a fixed single-qubit gate

U=\mathrm{diag}(1,e^{2\pi i\times 3/8})

in the computational basis. The state

|u\rangle=|1\rangle

is an eigenstate of

U

U|u\rangle=e^{2\pi i\varphi}|u\rangle

with

\varphi=3/8

. Phase estimation is run with a control register of

t=3

qubits (qubit

1

the most significant), each prepared in

|0\rangle

then a Hadamard, followed by the controlled gates

C(U^{2^{0}})

C(U^{2^{1}})

C(U^{2^{2}})

(controlled respectively by qubit

3

, qubit

2

, qubit

1

), and finally the inverse quantum Fourier transform on the three control qubits, which are then measured in the computational basis. All gates act exactly as specified and there is no noise. ((a)) Using the product form of the quantum Fourier transform, state how many Hadamard gates, controlled-rotation gates and swap gates the (inverse) QFT circuit on

t=3

qubits uses. ((b)) Derive, by phase kickback, the state of the three control qubits immediately after the three controlled-

U^{2^j}

gates and before the inverse QFT. ((c)) Show that the inverse QFT returns one computational basis state with probability

1

, and identify it as a

3

-bit binary string.

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