Let be a single-qubit unitary with for some eigenstate and real phase .

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 circuit model

Problem

Let

U

be a single-qubit unitary with

U|\lambda\rangle=e^{i\theta}|\lambda\rangle

for some eigenstate

|\lambda\rangle

and real phase

\theta

. ((a)) Derive the phase kickback identity for

C(U)

acting on

|x\rangle|\lambda\rangle

, for

x\in\{0,1\}

. ((b)) Apply the result to

U=S=\mathrm{diag}(1,i)

with target prepared in the eigenstate

|1\rangle

: find the phase kicked back onto the control when the control is

|1\rangle

. ((c)) The two-qubit unitary

C(S)

has dimension

d=4

and so, by the general two-level decomposition, needs at most

d(d-1)/2=6

two-level factors. Given that

C(S)=\mathrm{diag}(1,1,1,i)

is already diagonal, state and justify the minimum number of two-level unitaries actually needed to realise it exactly.

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