logo Use CA10RAM to get 10%* Discount.
Order Nowlogo
(5/5)

Measurement and distinguishability in different bases

INSTRUCTIONS TO CANDIDATES
ANSWER ALL QUESTIONS

1. Bra-Ket Gymnastics. (15 Points)

(a) Let ψ and ϕ be normalized states. Prove that ψ ϕ 2 1. (5 Points)

(b) Suppose we are given an orthonormal basis set i : i = 1, . . . , n . Write the identity matrix in terms of these basis vectors. (5 Points)

(c) Let A = ϕ ψ and B = ϕ′ ψ′ . Write down Tr(AB) in terms of inner products. (Here Tr(X) is the trace of the matrix X.) (5 Points)

2. Measurement and distinguishability in different bases (40 Points)

(a) Show that no measurement can distinguish ϕ and eiθ ϕ for any θ R. Here “distinguishing” means that the statistical distribution of the measurement outcome is different. (10 Points)

(b) Suppose  we are given |+⟩  = √1  (|0⟩ + |1⟩) and  |−⟩  = √1  (|0⟩ − |1⟩).  Can we distinguish these two

states using a measurement in the 0 , 1 basis? Explain you reasoning. (10 Points)

(c) Suppose we are given two states, ψ and ϕ , which are orthogonal to each other. Explain why there exists a measurement basis that can perfectly distinguish the two states. That is, there is a measure- ment basis in which the statistical distribution of the measurement outcome of ψ and ϕ are perfectly distinguishable from each other, with zero error. (10 Points)

(d) Suppose we are given two states, ψ and ϕ , which have a nonzero overlap, i.e., ψ ϕ 2 > 0. Explain why there is no measurement that can perfectly distinguish the two states. (10 Points)

3. Partial Measurements (20 Points)

(a) Consider a quantum state over two qubits

1|ψ⟩ = √2 (|00⟩ + |11⟩).

Suppose we meausure the first qubit in the {|+⟩, |−⟩} basis,  where |±⟩ =

 √1  (|0⟩ ± |1⟩).  Write  down  the 

post-measurement state (over the second qubit) and their respective probabilities. (5 Points)

(b) Using partial measurement, we can teleport a qubit. This means that we can send an unknown quantum state from one location to another. Here I will explain the procedure, while omitting the last step. Figure out this last step.

1. Start with the state |ψ⟩|+⟩, where |ψ⟩ is an arbitrary single-qubit state and |+⟩ = √1  (|0⟩ + |1⟩).

2. Apply the CZ gate to this state. The action of the CZ gate on the basis states is defined as

CZ|00⟩ = |00⟩, CZ|01⟩ = |01⟩, CZ|10⟩ = |10⟩, and CZ|11⟩ = −|11⟩.

 

3. Measure the first qubit.

4. Depending on the measurement outcome, apply some unitary operator to the second qubit.

5. The end result: The state of the second qubit is |ψ⟩.

The question is this: What operator should be applied in the fourth step and under what condition? (15 Points)

 

4. Pauli Matrices (15 Points)

(a) Prove that all single-qubit Pauli operators square to one. (5 Points)

(b) Write the eigenvalues of the single-qubit Pauli operators. (Hint: You can use the answer in (a).) (5 Points)

(c) Write all possible eigenvalues of multi-qubit Pauli operators. (5 Points)

 

5. Clifford Hierarchy (10 Points)

(a) The full Clifford group can be generated by H, S, and CNOT . Explain why that is true. (5 Points)

(b) Another useful gate in quantum computing is CCZ gate, defined as CCZ x y z = x y z only if x = y = z = 1 and CCZ x y z = x y z otherwise. (Here x, y, z 0, 1 .) Where does this gate belong in the Clifford hierarchy? (5 Points)

 

6. (Optional Challenge Problem) Find a way to implement Toffoli using Clifford + T . Conversely, find a way to implement Clifford + T using Toffoli + H. (20 Points)

(5/5)
Attachments:

Related Questions

. Introgramming & Unix Fall 2018, CRN 44882, Oakland University Homework Assignment 6 - Using Arrays and Functions in C

DescriptionIn this final assignment, the students will demonstrate their ability to apply two ma

. The standard path finding involves finding the (shortest) path from an origin to a destination, typically on a map. This is an

Path finding involves finding a path from A to B. Typically we want the path to have certain properties,such as being the shortest or to avoid going t

. Develop a program to emulate a purchase transaction at a retail store. This program will have two classes, a LineItem class and a Transaction class. The LineItem class will represent an individual

Develop a program to emulate a purchase transaction at a retail store. Thisprogram will have two classes, a LineItem class and a Transaction class. Th

. SeaPort Project series For this set of projects for the course, we wish to simulate some of the aspects of a number of Sea Ports. Here are the classes and their instance variables we wish to define:

1 Project 1 Introduction - the SeaPort Project series For this set of projects for the course, we wish to simulate some of the aspects of a number of

. Project 2 Introduction - the SeaPort Project series For this set of projects for the course, we wish to simulate some of the aspects of a number of Sea Ports. Here are the classes and their instance variables we wish to define:

1 Project 2 Introduction - the SeaPort Project series For this set of projects for the course, we wish to simulate some of the aspects of a number of

Ask This Question To Be Solved By Our ExpertsGet A+ Grade Solution Guaranteed

expert
Atharva PatilComputer science

992 Answers

Hire Me
expert
Chrisantus MakokhaComputer science

652 Answers

Hire Me
expert
AyooluwaEducation

895 Answers

Hire Me
expert
RIZWANAMathematics

843 Answers

Hire Me

Get Free Quote!

443 Experts Online