Quantum Computing, Encode, Classical Bits & Polynomial-Size Quantum Circuit - IT Assignment Help

Assignment Task -                 
 

Encode Two Classical Bits1.  Show how to encode two classical bits into one qubit such that anyone's bit can be recovered correctly with a probability greater than 85%. 
2. Let f : {0, 1}n → {0, 1}m, g : {0, 1}m → {0, 1}n be such that g(f(x)) = x for all x ∈ {0, 1}n. Assume that f, g can be computed by polynomial-size (classical) circuits.

Q-Show that there exists a polynomial-size quantum circuit that maps |x, 0i to |f(x), 0i. Would you expect this to be possible without the assumption that g has a polynomial-size circuit? 
3. ( For x, y ∈ {0, 1}2n such that x 6= y, and for b ∈ {0, 1} denote by |ψx,y, bi the 2n-qubits state |ψx,y, bi = √12(|xi + (−1)b|yi). Suppose Alice and Bob share this state, i.e., Alice holds the first n qubits and Bob holds the last n qubits. Alice and Bob do not know which state they are sharing (i.e, they do not know x, y, and b). Each of them can send one classical message to their mutual friend Charlie, who happens to know x and y, and wishes to learn b. Suggest a protocol for Alice and Bob to help Charlie achieve his goal. 
4. All groups considered in this exercise are finite and abelian. 
(a)For a group G denote by Gb the set of its characters. The point-wise multiplication χ · µ of χ and µ is defined by (χ · µ)(x) = χ(x) · µ(x). Show that Gb is a group with respect to point-wise multiplication. 
(b) For cyclic groups G, H show that G\× H = {χ · µ : χ ∈ G, µ b ∈ Hb}. 
(c) Let H be a subgroup of G, and define 
H⊥ = {χ : χ ∈ Gb and ∀x ∈ H χ(x) = 1}. 
Show that H⊥ is a subgroup of Gb. 
(d)  Consider the group G` = Z2 × Z4 × Z8 × . . . × Z2` with the group action 
(ai)`i=1,(bi)`i=1 →ai + bi (mod Z2i ) `i=1.  
Give an efficient quantum algorithm to solve the following problem: 
• Input: An oracle for f: G` → M. 
• Promise: There exists a subgroup HG` such that f(x) = f(y) if and only if x − y ∈ H. • Output: H (a generating set). 
You may assume there exists an efficient algorithm that given a linear system of modular equations over the integers, mod n, samples a uniformly chosen solution for it (if one exists). 
5. (a) Alice and Bob play the following game. Alice holds x ∈ {0, 1}2n. Bob holds a matching 
(i1, j1), . . . ,(in, jn) ∈ ([2n] × [2n])n. (Meaning that ∪nk=1{ik, jk} = [2n].) Alice wants to send an  
O(log n)-qubit state |ψi to Bob such that Bob can use it to output like ⊕xjkfor a k of his choice, with a success probability of 0.9. Show such a protocol for Alice and Bob, or prove that no such protocol exists. 

5 – Concluding Assignment 2 (b) Alice and Bob play the following game. Alice holds x ∈ {0, 1}2n. Bob holds a matching (i1, j1), . . . ,(in, jn) ∈ ([2n] × [2n])n. (Meaning that ∪nk=1{ik, jk} = [2n].) Alice wants to send an  O(log n)-qubit state |ψi to Bob such that Bob can use it to output (k, xik ⊕ xjk) for a uniformly chosen k (without erring). Show such a protocol for Alice and Bob, or prove that no such protocol exists. 
6. Alice holds an n-qubit state ρ and wishes to send it to Bob by sending a single quantum message, without revealing too much information to Eve who might be eavesdropping. Unfortunately, Alice and Bob share only n + O(log n + log( 1ε)) uniformly random bits, for some ε > 0. 
(a)  For any δ > 0, show that there exists a set S ⊆ {0, 1}2n of size O(nδ2 ) such that for any nonzero t ∈ {0, 1}2n it holds that     1|S|X x∈S 
(Hint: Chernoff bound.) (−1)ht,xi    ≤ δ. 

(b)  Show that if σ is an m-qubit state and T r(σ2) ≤1+ε2 
2m then D(σ, I2m ) ≤ ε. 
(c)  Describe an explicit protocol for Alice and Bob with the following guarantee. Suppose Alice chooses to use the protocol to either send ρ or I2n, according to a coin flip. Then, Eve should not be able to tell which of the states was sent with a probability greater than 12 + ε. You may use the set from (6a). 
7.  Let ρ be an n-qubit state and let B = {|vii}2n to B results in a distribution p = (pi)2n 
i=1 be an orthonormal basis. Measuring ρ according to i=1. Show (from basic principles) that S(ρ) ≤ H(p). 
8. Alice wants to commit on a bit b to Bob. Let |ψ00i = |0i, |ψ01i = |1i, |ψ10i = |+i, |ψ11i = |−i. Consider the following protocol (for honest players): 
• Commit: Alice chooses a random r ∈ {0, 1} and sends |ψbri to Bob. 
• Reveal: Alice reveals b and r to Bob. Bob measures his state according to the appropriate basis (|ψb0i, |ψb1i) and accepts if and only if he got |ψbri. 
(a)  Show explicitly that the protocol is wrong. (That is, a concrete way for the parties to violate binding or concealing.) 
(b) Prove that the protocol fulfills one of the requirements (binding or concealing). 
9.  Alice and Bob live in an n-restaurant world. Each one of them has a list of restaurants he likes (n-bit string). They would like to find out whether there exists a restaurant which both of them like. Give a bounded-error quantum protocol in which they exchange O(√n log n) qubits.

This IT Assignment has been solved by our IT Experts at My Uni Paper. Our Assignment Writing Experts are efficient to provide a fresh solution to this question. We are serving more than 10000+Students in Australia, UK & US by helping them to score HD in their academics. Our Experts are well trained to follow all marking rubrics & referencing style.

Be it a used or new solution, the quality of the work submitted by our assignment Experts remains unhampered. You may continue to expect the same or even better quality with the used and new assignment solution files respectively. There’s one thing to be noticed that you could choose one between the two and acquire an HD either way. You could choose a new assignment solution file to get yourself an exclusive, plagiarism (with free Turnitin file), expert quality assignment or order an old solution file that was considered worthy of the highest distinction.