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This is the Past Examination Paper of Mathematical Tripos which includes Quantum Computation, Planetary System Dynamics, Physical Cosmology, Perturbation and Stability Methods etc. Key important points are: Quantum Computation, Standard Gate Notation, Identity Transformation, Computational Basis Measurement, Quantum Oracle Problem, Hamming Distance, Shift Operation, Unitary Operation, Single Query to OracleQuantum Computation, Standard Gate Notation, Identity Transformation, Computational Basis
Typology: Exams
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Friday, 3 June, 2011 1:30 pm to 3:30 pm
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 transformation throughout.
NM
Computational basis measurement
Part III, Paper 49
This question is about the quantum Fourier transform and periodicity determina- tion.
In this question you may assume the following statements (S1) and (S2) about QF T , the quantum Fourier transform mod N : (S1): if N = Ar where A and r are integers and 0 6 x 0 < r then
k=
|x 0 + krใ =
r
rโโ 1
l=
ฯx^0 lA^ |lAใ where ฯ = e^2 ฯi/N^.
(S2): QF T may be implemented in poly(log N ) time.
(a) Let ZN denote the integers mod N. Let f : ZN โ ZN be a periodic function with period r and with the property that f is one-to-one within each period. Suppose we are given one instance of the quantum state
|f ใ =
x=
|xใ |f (x)ใ.
Using (S1) and (S2), describe an efficient procedure that may be used to determine the period r with probability O(1/ log log N ). (You may also assume that the number of integers less than K that are coprime to K is O(K/ log log K)).
(b) Consider the function f : Z 12 โ Z 10 defined by f (x) = 3x^ mod 10.
(i) Suppose we are given the state |f ใ = โ^112
x=0 |xใ |f^ (x)ใ^ and we measure the second register. What are the possible measurement values y and their probabilities? (ii) Suppose the measurement result was y = 3. Find the resulting state |ฯใ of the first register after the measurement. (iii) Suppose we measure the state QF T |ฯใ (with |ฯใ from (ii)). What is the probability of each outcome 0 6 c 6 11?
Part III, Paper 49
This question is about measurement-based quantum computation. Given a graph, the corresponding graph state is obtained by preparing a qubit in the state |+ใ = โ^12 (| 0 ใ + | 1 ใ) for every node in the graph, then applying a CZ gate between every pair of qubits linked by an edge. Consider the graph state |ฯ 2 ร 2 ใ corresponding to the graph
(a) Show that if one of the qubits is measured in the computational basis with result r, the remaining qubits will be left in the state (Zr^ โ I โ Zr) |ฯ 3 ใ, where |ฯ 3 ใ is the graph state corresponding to the graph
(b) Next, the first qubit of the state (Zr^ โ I โ Zr) |ฯ 3 ใ is measured in the basis {|v 0 (ฮธ)ใ , |v 1 (ฮธ)ใ}, where |vs(ฮธ)ใ = โ^12
| 0 ใ + (โ1)seiฮธ^ | 1 ใ
, and result s is obtained. Show that the remaining qubits are left in the state
CZ
X(r+s)U (ฮธ) โ Zr
where U (ฮธ) = | 0 ใ ใv 0 (ฮธ)| + | 1 ใ ใv 1 (ฮธ)|.
(c) Using your previous answers, explain how you could simulate the results of the circuit |+ใ U (ฮธ) โข NM
โ k 1
|+ใ โข NM
โ k 2 using single-qubit measurements on |ฯ 2 ร 2 ใ and classical processing of the results.
(d) Suppose that you prepare a qubit in the state | 0 ใ for every node in a graph, and apply the same two-qubit gate V along every edge. Find a V , and appropriate single-qubit measurements, which would allow you to perform universal quantum computation using such states.
Part III, Paper 49 [TURN OVER
Part III, Paper 49
