Quantum Computing 量子计算代写

Ph219C/CS219C Due: Thursday 5 May 2022 2.1 A quantum version of Fano’s inequality a) In a d-dimensional system, suppose a density operator ρ approxi- mates the pure state |ψ⟩ with fidelity F =⟨ψ|ρ|ψ⟩=1−ε. (1) H(ρ) ≤ H2(ε) + ε log2(d − 1). (2) Hint: Recall that if a complete orthogonal measurement per- formed on the […]

Ph219C/CS219C Due: Thursday 14 April 2022 1.1 Positivity of quantum relative entropy a) Show that ln x ≤ x−1 for all positive real x, with equality iff x = 1. b) The (classical) relative entropy of a probability distribution {p(x)} relative to {q(x)} is defined as D(p∥q)≡􏰄p(x)(logp(x)−logq(x)) . (1) x with equality iff the probability

P1 = √1 (|0⟩+i|1⟩)· √1 (⟨0|−i⟨1|) and P2 = √1 (|0⟩−i|1⟩)· √1 (⟨0|+i⟨1|). 2222 Check that the above matrices are projection operators. Check that the above matrices can be used for organizing a von Neumann measurement. Find the classical and quantum outputs of the measurements of a qubit in 1. state |0⟩, 2. state |1⟩.

Homework 5 1. For the unitary transformation U from Problem 5 of HW 4, find all the vectors |us⟩, s = 0,…,r−1, and check that they are eigenvectors of U. 2. Compute explicitly 􏰀r−1 |us⟩. 3. Find the joint quantum state of all t + L qubits right before the measurement blocks in the quantum

QCC 2022, Home Work 3 1. A circuite for the Quantum Discrete Fourier Transform (QDFT) with N = 4 (n = 2 qubits) is shown below. Let |v1⟩ = |10⟩ and |v2⟩ = |11⟩ be used at the input of this circuite and |w1⟩ and |w2⟩ be the states at the output of the circuite.