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The questions and instructions for paper 49 of the mathematical tripos part iii exam focused on quantum computation. It includes details on the use of gate notation, grover's search algorithm, graph states, and error correction schemes.
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Friday, 4 June, 2010 9:00 am to 11:00 am
Attempt no more than THREE questions. There are FOUR questions in total. The questions carry equal weight.
Cover sheet None Treasury Tag Script paper
You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.
The following standard gate notation is used in this paper. Note that I denotes the identity operator throughout.
Toffoli = (| 00 〉〈 00 | + | 01 〉〈 01 | + | 10 〉〈 10 |) ⊗ I + | 11 〉〈 11 | ⊗ X
NM
Computational basis measurement
Part III, Paper 49
A graph state is obtained by preparing a qubit in the |+〉 state for each node in a graph, where |±〉 = √^12 (| 0 〉 ± | 1 〉), and then applying a CZ gate along each edge of the graph. Any quantum computation can be carried out by performing a sequence of single-qubit measurements on an appropriate graph state.
Of particular interest are single-qubit phase measurements, characterised by projec- tors {|v 0 (θ)〉 〈v 0 (θ)| , |v 1 (θ)〉 〈v 1 (θ)|} for θ ∈ [0, 2 π] which satisfy
( |vr(θ)〉 〈vr(θ)| ⊗ I
CZ |ψ〉 |+〉 =
I ⊗ XrU (θ)
|vr(θ)〉 |ψ〉
for all |ψ〉, where U (θ) = |+〉 〈 0 | + e−iθ^ |−〉 〈 1 | and r ∈ { 0 , 1 }.
(a) Prove the relation U (θ)X = e−iθ^ ZU (−θ).
(b) Show how to simulate the quantum circuit
|+〉 U (α) U (β) NM
→ k
by a sequence of single qubit measurements on an appropriate graph state, and deterministic classical processing of the results. Your procedure should generate a bit k with the same probability distribution as the circuit.
(c) Prove that
U (β) U (α) |+〉 = e−i^
α 2
cos
( (^) α 2
|+〉 + i sin
( (^) α 2
e−iβ^ |−〉
(d) Suppose that Alice has the first two qubits of the three qubit graph state (I ⊗ CZ )(CZ ⊗ I) |+〉 |+〉 |+〉, and Bob has the third qubit. Alice is also given two angles α, β ∈ [0, 2 π]. Give a protocol in which Alice sends only two classical bits to Bob, after which Bob is left with a qubit in the state U (β) U (α) |+〉, up to an irrelevant global phase factor.
Part III, Paper 49
Consider the circuit below showing a unitary coding and error correction scheme, where the error E is also unitary.
Encoding Error Detection Recovery Decoding
|ψ〉 • •
E
We say that the circuit protects against the error E if for any |ψ〉 the output state is the tensor product of |ψ〉 for the top qubit with any state for the remaining four qubits.
(a) Briefly explain the structure of the circuit, and give the output state when E = X ⊗ I ⊗ I, corresponding to a bit-flip error on the top qubit.
(b) Find the reduced density operator ρ describing the state of the top output qubit when the error is E = H ⊗ I ⊗ I and |ψ〉 = α | 0 〉 + β | 1 〉.
(c) Show that if the circuit protects against errors E 1 and E 2 , it will also protect against an error given by E = αE 1 + βE 2 , where α and β are any complex numbers such that E is unitary.
(d) How would you change the circuit to protect against the four error cases
E = I ⊗ I ⊗ I , E = Z ⊗ I ⊗ I , E = I ⊗ Z ⊗ I , E = I ⊗ I ⊗ Z.
Part III, Paper 49 [TURN OVER
