程序代做 MULT90063 Introduction to Quantum Computing

MULT90063 Introduction to Quantum Computing
Week by week
(1) Introduction to quantum computing
(2) Single qubit representation and operations

(3) Multi-qubit systems
(4) Simple quantum algorithms
(5) Quantum search (Grover’s algorithm)
(6) Quantum factorization (Shor’s algorithm)
(7) Quantum supremacy and noise
(8) Programming real quantum computers (IBM Q)
(9) Quantum error correction (QEC)
(10) QUBO problems and Adiabatic Quantum Computation (AQC)
(11) Variational/hybrid quantum algorithms (QAOA and VQE)
(12) Solving linear equations, QC computing hardware

MULT90063 Introduction to Quantum Computing
5.1 Two qubit systems and operations 5.2 Entanglement
6.1 Dense coding 6.2 Teleportation
Two qubit operations, entanglement, dense coding, teleportation

MULT90063 Introduction to Quantum Computing
Entanglement
We saw how to find the entropy of entanglement for states like these:
p0.99 |00i + p0.01 |11i
p2 |00i + p2 |11i
States have a very particular form:
Not entangled, separable
Entangled, but close to a separable state Maximally entangled
|0> of system A is correlated with |0> of system B.
|1> of system A is correlated with |1> of system B (and so on)
For more general states we need to put them in a similar form first.

MULT90063 Introduction to Quantum Computing

How much entanglement is present in a general state?
| i=a00|00i+a01|01i+a10|10i+a11|11i
Can be hard to tell. It’s not in anything like product form. For that we will use SVD.
Arrange as a matrix:
T a k i n g S i n g u l a r V a l u e D e c o m p o s i t i o n X( S V D ) :
A = i |uiihvi|
A=a00 a01 a10 a11
Allows us to express the state in thisXconvenient form:
This from is known as
| i= i|uii|vii

MULT90063 Introduction to Quantum Computing

| i = Xi |uii|vii
Several terms might have a singular value of 0. The number of non-zero terms is
called the Schmidt rank.
If a state has a Schmidt rank of 1:
| i=|u0i⌦|v0i Then the state is separable, and not entangled.
If a state has a Schmidt rank greater than 1, then the state is entangled. Schmidt rank is a very coarse measure of entanglement. We would like a finer measure.

MULT90063 Introduction to Quantum Computing
Entanglement Entropy
| i = Xi |uii|vii
A more fine-grained measure of entanglement is the entanglement entropy. Form
a probability distribution:
p i = 2i
From which you can calculate the entanglement entropy:
S = Xpi logpi i
This is a measure of entanglement. The higher the entanglement entropy, the more entanglement.

MULT90063 Introduction to Quantum Computing
Entanglement in the QUI
The time scrubber is the vertical bar which moves left and right to show the quantum state at each time step.
The entropy of entanglement is shown in a red colour scale between min and max values possible. Each segment corresponds to the entropy between the system of qubits above and below for that particular bi-partition.
qubit-1 qubit-2 qubit-3 qubit-4
Entanglement entropy between qubit 1 and qubits {2 & 3 & 4} partition
Entanglement entropy between qubits {1 &2} and qubits {3 & 4} partitions
Entanglement entropy between qubit 4 and qubits {1 & 2 & 3} partition

MULT90063 Introduction to Quantum Computing
Lecture 5 Summary
  2 ac 3 Twoqubitgates
a ⌦ c = 6 64 a d 7 75 26 1 0 0 0 37 b d bc CZ=64010075
Tensor Product
bd 0010 0 0 0 1
A state which is not separable is called an entangled state.
26 1 0 0 0 37 CNOT=640 1 0 075
Universal set of gates
26 1 0 0 0 37 SWAP=640 0 1 075
Single Qubit Rotations

MULT90063 Introduction to Quantum Computing
Lecture overview
In this lecture:
6.1 Dense coding 6.2 Teleportation
• Reiffel: 5.3
• Kaye: Ch 5
• Nielsen and Chuang: 1.3.5, 1.3.7, 2.3

MULT90063 Introduction to Quantum Computing
Entanglement and quantum computing
A state which is not separable is entangled. For example: |00i + |11i
In this lecture we will see how entangled states can be critical in various quantum computing tasks and apply these in the Lab to gain experience in how entangled states work.
In particular we will discuss
1. Dense Coding
2. No-cloning theorem
3. Quantum teleportation

MULT90063 Introduction to Quantum Computing
Entanglement as a resource
When asked what practical use electricity was, Faraday reportedly replied:
“Why sir, there is every probability that you will be able to tax it”
Entanglement is similar, a resource useful for many quantum information tasks.

MULT90063 Introduction to Quantum Computing
6.1 Dense coding
MULT90063 Lecture 6

MULT90063 Introduction to Quantum Computing
Dense Coding
Alice would like to send two classical bits to Bob.

01 Wants to send
Alice and Bob can use a quantum NBN, and share some initial entanglement – can they get any advantage?

MULT90063 Introduction to Quantum Computing
Dense Coding
Entanglement makes it possible.

(1) Alice and Bob share an entangled state
(2) Alice flips her qubit one of four ways, based on the state she wants to send
(3) Alice sends her qubit to Bob
(4) Bob measures correlations between the
qubits, to reveal which of the four (ie. two bits) operations Alice applied

MULT90063 Introduction to Quantum Computing
Dense Coding Circuit
Bob state preparation:
Bell state preparation
Alice encoding
Bob’s measurement
|0i + |1i |00i ! p2 |0i
|00i + |11i CNOT ! p2

MULT90063 Introduction to Quantum Computing
Dense Coding Circuit
Bob state preparation
Alice encoding
Bob’s measurement
Alice applies one of four different operations to her qubit, based on the classical information she would like to send.
|0i + |1i |00i ! p2 |0i
|00i + |11i ! p2
|00i + |11i |00i + |11i p2 ! p2
|00i + |11i |01i + |10i p2 ! p2
|00i + |11i |00i |11i
1,0 Z2 p2 ! p2
|00i + |11i |01i |10i X2Z2 p2 ! p2

MULT90063 Introduction to Quantum Computing
Dense Coding Circuit
Bob state preparation
Alice encoding
|00i + |10i p2
|01i + |11i p2
|00i |10i p2
|01i |11i p2
Bob’s measurement
|00i H |01i |10i |11i
|00i + |11i |00i + |11i 0,0 p2 ! p2
|00i + |11i |01i + |10i 0,1 X2 p2 ! p2
|00i + |11i |00i |11i Z2 p2 ! p2
|00i + |11i |01i |10i X2Z2 p2 ! p2

MULT90063 Introduction to Quantum Computing
Dense Coding Circuit
0,1 X p ! p
|00i + |11i |00i + |11i p2 ! p2
2 |00i + |11i |01i + |10i 22
|00i + |11i |00i |11i 1,0 Z2 p2 ! p2
|00i + |11i |01i |10i 1,1 X2Z2 p2 ! p2
|00i + |10i p2
|01i + |11i p2
|00i |10i p2
|01i |11i p2
|0i + |1i = p2
|0i + |1i = p2
|0i |1i = p2
|0i |1i = p2
|0i = |+i |0i |1i = |+i |1i |0i = |i |0i |1i = |i |1i
Bob state preparation
Alice encoding
Bob’s measurement
H |+i = |0i , H |i = |1i
|00i H |01i |10i |11i
Two bits communicated but only one qubit “sent”.
Makes use of pre-existing entanglement.

MULT90063 Introduction to Quantum Computing
6.2 Teleportation
MULT90063 Lecture 6

MULT90063 Introduction to Quantum Computing
A Quantum Computing Bus?
To understand the role entanglement can play in quantum information processing, we will consider how it can be to transmit quantum information around our quantum computer (and potentially between quantum computers)
Wants to send

| i=↵|0i+|1i
Communication around the quantum computer is an important primitive. We could physically move quantum systems, but there is a (potentially) better way: teleportation.
All we require is initial entanglement between A and B and classical communication

MULT90063 Introduction to Quantum Computing
Sending classical information
How would we do this classically? Measure everything about the state, then send that information down (classical) bus and recreate a perfect copy elsewhere.
Image by ChtiTux, Used here under CC-by-SA 1.0 license.
Problem: we can’t do this in quantum mechanics because classical measurement (1) collapses the system, and (2) this clones the system which we can’t do in quantum mechanics.

MULT90063 Introduction to Quantum Computing
No-cloning theorem
Can we make a circuit which clones the input state?
That is, we ask if it is possible to make a unitary transformation s.t.
(↵ |0i + |1i) ⌦ |0i ! (↵ |0i + |1i) ⌦ (↵ |0i + |1i)
(↵ |0i + |1i) ⌦ |0i ! (↵ |0i + |1i) ⌦ (↵ |0i + |1i)
= ↵2 |00i + ↵ |01i + ↵ |10i + 2 |11i
= ↵2 |00i + ↵ |01i + ↵ |10i + 2 |11i
No-cloning theorem: the answer is no.

MULT90063 Introduction to Quantum Computing
Proof of no-cloning theorem
If we had a cloning circuit, we could use it on two arbitrary states, | i and |i U|i|0i=|i|i U| i|0i=| i| i
InnerproductonLHS: h0|h|U†U| i|0i=h| i
h |h |i|i=h |i2
But the only solutions to x2=x are x=0 or x=1. We can only have a circuit clone
states which are orthogonal (x=0 case), not arbitrary states.
There can be no unitary transformation which clones two arbitrary states.
Inner product on RHS:

MULT90063 Introduction to Quantum Computing
Teleportation
|00i + |11i Totalsystem|stiat=e: (↵|0i+|1i) p2
Alice’s state | i Shared entangled state A & B
| i= p2(↵|000i+↵|011i+|100i+|111i)

MULT90063 Introduction to Quantum Computing
Teleportation
|000i + |011i |110i + | p2 + p2
|+i|0i (↵|0i + |1i) 2
+ |+i|1i (↵|1i + |0i) 2
+ |i|0i(↵|0i |1i) 2
+ |i|1i (↵|1i |0i) 2
| i= p2(↵|000i+↵|011i+|100i+|111i)
|000i + |011i |110i + |101i p2 + p2
Rewrite (ex):
(↵ 0 + 1 )

MULT90063 Introduction to Quantum Computing
Teleportation
|000i + |011i |110i + |101i |000i + |011i |110i + |101i |i=↵ p2 + p2 |i=↵ p2 + p2
|+i|0i (↵|0i + |1i) = |0i|0i (↵|0i + |1i) 22
+ |+i|1i (↵|1i + |0i) Hadamard + |0i|1i (↵|1i + |0i) 22
+ |i|0i (↵|0i |1i) + |1i|0i (↵|0i |1i) 22
+ |i|1i (↵|1i |0i) + |1i|1i (↵|1i |0i) 22

MULT90063 Introduction to Quantum Computing
Teleportation
|000i + |011i |110i + |101i i=↵ p2 + p2
|0i|0i (↵|0i + |1i) 2
+ |0i|1i (↵|1i + |0i) 2
+ |1i|0i (↵|0i |1i) 2
+ |1i|1i (↵|1i |0i) 2
Alice measures her two qubits.
Bob’s qubit collapses to one of the four possibilities.
Alice now tells Bob her outcomes (double lines indicate classical communication).
Bob will perform simple corrections shown.

MULT90063 Introduction to Quantum Computing
Teleportation
|000i + |011i |110i + |101i =↵ p2 + p2
|0i|0i (↵|0i + |1i) 2
+ |0i|1i (↵|1i + |0i) 2
+ |1i|0i(↵|0i|1i) 2
+ |1i|1i(↵|1i|0i) 2
Alice measures
Bob’s qubit
↵|0i + |1i
↵|1i+|0i ↵|0i+|1i
1,0 ↵|0i|1i
1,1 ↵|1i|0i ↵|0i|1i
↵ |0i + |1i ↵ |0i + |1i

MULT90063 Introduction to Quantum Computing
Teleportation
| i=↵|0i+|1i
Alice measures
0, 0 0, 1 1, 0 1, 1
| i=↵|0i+|1i
i.e. after correction Bob has successfully reconstructed
Alice’s original state.
↵|0i + |1i ! ↵|0i + |1i X(↵|1i + |0i) ! ↵|0i + |1i Z(↵|0i |1i) ! ↵|0i + |1i ZX(↵|1i |0i) ! ↵|0i + |1i
Bob’s qubit
↵|0i + |1i ↵|1i + |0i ↵|0i |1i ↵|1i |0i

MULT90063 Introduction to Quantum Computing
Teleportation
Send a qubit with only classical communication?
Initially shared entanglement makes it possible.
(1)Alicehasaqubit| i=↵|0i+|1i (2) Alice and Bob share an entangled state
(3) Alice measures correlations between her
qubit and half of the entangled state
(4) Alice sends the results of the
measurements to Bob
(5) Bob uses them to reconstruct the
original state in his qubit
| i=↵|0i+|1i ULT90063 Introduction to Quantum Computing
Week by week
(1) Introduction to quantum computing
(2) Single qubit representation and operations
(3) Multi-qubit systems
(4) Simple quantum algorithms
(5) Quantum search (Grover’s algorithm)
(6) Quantum factorization (Shor’s algorithm)
(7) Quantum supremacy and noise
(8) Programming real quantum computers (IBM Q)
(9) Quantum error correction (QEC)
(10) QUBO problems and Adiabatic Quantum Computation (AQC)
(11) Variational/hybrid quantum algorithms (QAOA and VQE)
(12) Solving linear equations, QC computing hardware

MULT90063 Introduction to Quantum Computing
5.1 Two qubit systems and operations 5.2 Entanglement
6.1 Dense coding 6.2 Teleportation
Two qubit operations, entanglement, dense coding, teleportation

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