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This is the Exam Paper of Mathematical Tripos which includes Solitons and Instantons, Nonlinear Schr¨Odinger Equation, Stationary Soliton Solutions, Differential Equation, Effective Lagrangian, Harmonic Potential, Derrick Scaling Argument, Abelian Higgs Model etc. Key important points are: Quantum Computation, Standard Gate Notation, Computational Basis Measurement, Unitary Oracle, Addition Modulo, Two-Qubit Unitary Operation, Oracle Implements, Particular Input, Grover’s Search Algorithm
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Friday, 5 June, 2009 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
NM
Computational basis measurement
Part III, Paper 51
2 Consider an oracle encoding a function f : { 0 , 1 }n^ → { 0 , 1 }, that gives a non-zero output only for a particular input a:
f (x) =
1 if x = a 0 if x 6 = a.
We want to find a using Grover’s search algorithm. However, we are also given the promise that a lies in a small subset A of all possible bit strings, a ∈ A ⊆ { 0 , 1 }n, where A contains |A| elements. Define the state |ψA〉 as the uniform superposition of all states in A:
|ψA〉 =
x∈A
|x〉.
(a) Consider the 2-dimensional subspace V spanned by |a〉 and |ψA〉. Find a vector |ω〉 ∈ V such that |a〉 and |ω〉 form an orthonormal basis, and
|ψA〉 = sin(θ) |a〉 + cos(θ) |ω〉
for 0 6 θ 6 π 2. Find θ as a function of |A|.
(b) The Grover operation is G = −V|ψA〉V|a〉, where
V|ψA〉 = I − 2 |ψA〉 〈ψA| , V|a〉 = I − 2 |a〉 〈a|.
Show that when applied to states in V, G acts as
G = cos(2θ) (|a〉 〈a| + |ω〉 〈ω|) + sin(2θ) (|a〉 〈ω| − |ω〉 〈a|).
(c) It follows that Gk^ |ψA〉 = sin ((2k + 1) θ) |a〉 + cos ((2k + 1) θ) |ω〉. Show that the minimal integer k such that a computational basis measurement on Gk^ |ψA〉 yields a with probability > cos^2 (θ) satisfies
k < π
(d) Consider the case in which |A| = 2m^ for an integer m = O(log 2 n). Given that we can efficiently (i.e. with a circuit of size poly(n)) implement the unitary operation
Ux,y = I − |x〉 〈x| − |y〉 〈y| + |x〉 〈y| + |y〉 〈x| ,
where |x〉 and |y〉 are computational basis states of n + m qubits, show that we can efficiently prepare |ψA〉.
Part III, Paper 51
3 Consider a family of single qubit phase measurements, parameterised by θ, that are characterised by the measurement operators
M 0 (θ) = |v 0 (θ)〉 〈v 0 (θ)| , M 1 (θ) = |v 1 (θ)〉 〈v 1 (θ)| ,
where |v 0 (θ)〉 =
| 0 〉 + eiθ^ | 1 〉
, |v 1 (θ)〉 =
| 0 〉 − eiθ^ | 1 〉
A phase measurement with angle θ that gives result k is denoted vk(θ):=; <.
(a) Consider the single qubit unitary
U (θ) = | 0 〉 〈v 0 (θ)| + | 1 〉 〈v 1 (θ)|.
Prove the relations U (θ)Z = XU (θ), and U (θ)X = e−iθZU (−θ).
(b) Consider the following circuit, incorporating a CZ gate:
|ψ〉 • vk(θ):=;<
| 0 〉 (^) H • |ψ′〉
Show that |ψ′〉 = XkU (θ) |ψ〉 and that the results k = 0 and k = 1 are equiprobable.
Consider the procedure below, which consists of preparing a four qubit graph state (the dashed section) then performing a sequence of measurements on it. The lowest qubit is measured in the computational basis, then δ ∈ { 0 , 1 } is added to the outcome (modulo 2) to give the result r.
| 0 〉 (^) H • vk(α):=;<
| 0 〉 (^) H • • vl(β):=;<
| 0 〉 (^) H • • vm(γ):=;<
| 0 〉 (^) H • NM
(⊕ δ)
_ _ _ _ _ _ _ _
_ _ _ _ _ _ _ _
(c) Show that for an appropriate (adaptive) choice of α, β, γ and δ, the result r will perfectly simulate the measurement result in the circuit
| 0 〉 (^) H U (θ) (^) H X U (φ) NM
(d) Show that when φ is an integer multiple of π 2 , it is possible to simulate the circuit above by making all measurements on the graph state simultaneously (i.e. non- adaptively).
Part III, Paper 51 [TURN OVER
