MULT20015 Lecture 6 2021 FINAL
MULT20015 Elements of Quantum Computing
Lecture 6
Subject outline
Lecture topics (by week)
1 – Introduction to quantum computing and maths basics
2 – Single qubit representations and logic operations
3 – Two qubit states and logic gates
4 – Multi-qubit states and quantum arithmetic
5 – Basic quantum algorithms
6 – Period finding, cryptography and quantum factoring
7 – Shor’s algorithm, post-quantum crypto, quantum key distribution
8 – Quantum search algorithms
9 – Grover search applications, optimisation problems
10 – Solving optimisation problems on quantum computers
11 – Applications in quantum machine learning
12 – Real quantum computer devices
Assignment schedule:
#1: Hand out in Week 2
#2: Hand out in Week 8
MULT20015 Elements of Quantum Computing
Lecture 6
Week 3
Lecture 5
5.1 Two-qubit systems
5.2 Two qubit example – independent Alice and Bob
5.3 General two qubit states: measurement and operations
Lecture 6
6.1 Two-qubit logic operations
6.2 Entanglement
Practice class 3
Two qubit states and operations
MULT20015 Elements of Quantum Computing
Lecture 6
Lecture 5 recap: binary and decimal in the QUI
| alice&bobi = | alicei ⇥ | bobi
= (
1
p
2
|0i+
1
p
2
|1i)⇥ (
1
p
2
|0i+
1
p
2
|1i)
=
1
2
|0i |0i+
1
2
|0i |1i+
1
2
|1i |0i+
1
2
|1i |1i
=
1
2
|00i+
1
2
|01i+
1
2
|10i+
1
2
|11i
| alice&bobi
=
1
2
|00i+
1
2
|01i+
1
2
|10i+
1
2
|11i
=
1
2
|0i+
1
2
|1i+
1
2
|2i+
1
2
|3i
| alice&bobi = | alicei ⇥ | bobi
= (
1
p
2
|0i+
1
p
2
|1i)⇥ (
1
p
2
|0i+
1
p
2
|1i)
=
1
2
|0i |0i+
1
2
|0i |1i+
1
2
|1i |0i+
1
2
|1i |1i
=
1
2
|00i+
1
2
|01i+
1
2
|10i+
1
2
|11i
binary: amplitude “𝑎!!” = 1/2
deci
mal:
“𝑎!”
= 1/
2
binary amplitude “𝑎””” = 1/2
decimal: “𝑎#” = 1/2
⊗
MULT20015 Elements of Quantum Computing
Lecture 6
Lecture 5 recap: two-qubit systems
| i = a |00i+ b |01i+ c |10i+ d |11i
AAACNXicbZDLSgMxFIYzXmu9jbp0EyyCIJSMCFZBKLhxWcGxhc5QMpm0Dc1cSDJCGeZdfAyfwK1u3bhSt76CmcvCtv4Q+PKfczjJ78WcSYXQu7G0vLK6tl7bqG9ube/smnv7DzJKBKE2iXgkeh6WlLOQ2oopTnuxoDjwOO16k5u83n2kQrIovFfTmLoBHoVsyAhW2hqYV86EqtSJJcvgNcTFDaEMnkKvZCtnUrBV+H7JVjYwG6iJCsFFsCpogEqdgfnl+BFJAhoqwrGUfQvFyk2xUIxwmtWdRNIYkwke0b7GEAdUumnxxwwea8eHw0joEypYuH8nUhxIOQ083RlgNZbztdz8r9ZP1LDlpiyME0VDUi4aJhyqCOaBQZ8JShSfasBEMP1WSMZYYKJ0rDNbPIF1NHku1nwKi2CfNS+b6O680W5VAdXAITgCJ8ACF6ANbkEH2ICAJ/ACXsGb8Wx8GJ/Gd9m6ZFQzB2BGxs8v5lqqfA==AAACNXicbZDLSgMxFIYzXmu9jbp0EyyCIJSMCFZBKLhxWcGxhc5QMpm0Dc1cSDJCGeZdfAyfwK1u3bhSt76CmcvCtv4Q+PKfczjJ78WcSYXQu7G0vLK6tl7bqG9ube/smnv7DzJKBKE2iXgkeh6WlLOQ2oopTnuxoDjwOO16k5u83n2kQrIovFfTmLoBHoVsyAhW2hqYV86EqtSJJcvgNcTFDaEMnkKvZCtnUrBV+H7JVjYwG6iJCsFFsCpogEqdgfnl+BFJAhoqwrGUfQvFyk2xUIxwmtWdRNIYkwke0b7GEAdUumnxxwwea8eHw0joEypYuH8nUhxIOQ083RlgNZbztdz8r9ZP1LDlpiyME0VDUi4aJhyqCOaBQZ8JShSfasBEMP1WSMZYYKJ0rDNbPIF1NHku1nwKi2CfNS+b6O680W5VAdXAITgCJ8ACF6ANbkEH2ICAJ/ACXsGb8Wx8GJ/Gd9m6ZFQzB2BGxs8v5lqqfA==AAACNXicbZDLSgMxFIYzXmu9jbp0EyyCIJSMCFZBKLhxWcGxhc5QMpm0Dc1cSDJCGeZdfAyfwK1u3bhSt76CmcvCtv4Q+PKfczjJ78WcSYXQu7G0vLK6tl7bqG9ube/smnv7DzJKBKE2iXgkeh6WlLOQ2oopTnuxoDjwOO16k5u83n2kQrIovFfTmLoBHoVsyAhW2hqYV86EqtSJJcvgNcTFDaEMnkKvZCtnUrBV+H7JVjYwG6iJCsFFsCpogEqdgfnl+BFJAhoqwrGUfQvFyk2xUIxwmtWdRNIYkwke0b7GEAdUumnxxwwea8eHw0joEypYuH8nUhxIOQ083RlgNZbztdz8r9ZP1LDlpiyME0VDUi4aJhyqCOaBQZ8JShSfasBEMP1WSMZYYKJ0rDNbPIF1NHku1nwKi2CfNS+b6O680W5VAdXAITgCJ8ACF6ANbkEH2ICAJ/ACXsGb8Wx8GJ/Gd9m6ZFQzB2BGxs8v5lqqfA==
| 0i =
a |00i+ b |01i
p
|a|2 + |b|2
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
Measurement/collapse on single qubit, renormalisation:
| 0i =
c |10i+ d |11i
p
|c|2 + |d|2
“0”
on
1st
“1” on 1st
& similar for measurements on qubits #2
Single qubit operations on multi-qubit states, e.g. two-qubits):
We write as: |”($ = 2)⟩ = )!*!+”*”|00⟩ where
*!applies a H-gate to qubit #1 in t=0->1, then
)! applies a Y-gate to qubit #1 in t=1->2
*”applies a H-gate to qubit #2 in t=0->1, then
+” applies a X-gate to qubit #2 in t=1->2
NB. Order for different qubit operators doesn’t matter
i.e. we can write |”($ = 2)⟩ = +”*”)!*!|00⟩
qubit 1
qubit 2
t=0 t=1 t=2
|”($ = 2)⟩
MULT20015 Elements of Quantum Computing
Lecture 6
6.1 Two qubit logic operations
MULT20015
Lecture 6
MULT20015 Elements of Quantum Computing
Lecture 6
Quantum superposition:
Systems can be in indeterminate (multiple) states prior to measurement
Quantum entanglement:
Systems can be linked such that measurement of one part correlates to that of another part
Quantum measurement:
Result of any given measurement a-priori unknown, system “collapses” to an outcome
Recall: Key concepts for quantum computing
We’ve looked at superposition and measurement, now for entanglement
– while it might seem to be a niche interest for physicists and philosophers,
it is actually a crucial aspect of quantum computing…
Entanglement is generated when the qubits interact – e.g. via two-qubit logic operations
MULT20015 Elements of Quantum Computing
Lecture 6
Computing with our quantum register
| alice&bobi = | alicei ⇥ | bobi
= (
1
p
2
|0i+
1
p
2
|1i)⇥ (
1
p
2
|0i+
1
p
2
|1i)
=
1
2
|0i |0i+
1
2
|0i |1i+
1
2
|1i |0i+
1
2
|1i |1i
=
1
2
|00i+
1
2
|01i+
1
2
|10i+
1
2
|11i
| alice&bobi = | alicei ⇥ | bobi
= (
1
p
2
|0i+
1
p
2
|1i)⇥ (
1
p
2
|0i+
1
p
2
|1i)
=
1
2
|0i |0i+
1
2
|0i |1i+
1
2
|1i |0i+
1
2
|1i |1i
=
1
2
|00i+
1
2
|01i+
1
2
|10i+
1
2
|11i
| alice&bobi = | alicei ⇥ | bobi
= (
1
p
2
|0i+
1
p
2
|1i)⇥ (
1
p
2
|0i+
1
p
2
|1i)
=
1
2
|0i |0i+
1
2
|0i |1i+
1
2
|1i |0i+
1
2
|1i |1i
=
1
2
|00i+
1
2
|01i+
1
2
|10i+
1
2
|11i
So far, we considered two qubits each in a superposition, and although we merge the
description into a “quantum binary register” you’ve seen the qubits are still independent
i.e. measurement outcomes for the Alice qubit didn’t depend on those of the Bob qubit
(& vice-versa)
|0ñ or |1ñ
?
nothing happens
to Bob
corresponding quantum register:
numbers only, no computing (yet)
In order to actually compute anything with the numbers in our quantum register,
we need to interact our qubits physically (similar in a sense to classical logic operations).
When we do so, we generate entanglement – during the computation the numbers represented
in the qubit register become quantum connected…in a sense, they know about each other.
Quantum register state:
⊗
MULT20015 Elements of Quantum Computing
Lecture 6
Recall from Lecture 1: Multi qubits – state counting
1 qubit:
|0ñ, |1ñ
2 qubits:
|0ñ, |1ñ |0ñ, |1ñ
Binary combinations
|00ñ, |01ñ, |10ñ, |11ñ
|0ñ, |1ñ
3 qubits:
|0ñ, |1ñ |0ñ, |1ñ
|000ñ, |001ñ, |010ñ, |011ñ
|100ñ, |101ñ, |110ñ, |111ñ
|0ñ, |1ñ
4 qubits:
|0ñ, |1ñ |0ñ, |1ñ |0ñ, |1ñ |0ñ, |1ñ
|0000ñ, |0001ñ, |0010ñ, |0011ñ
|0100ñ, |0101ñ, |0110ñ, |0111ñ
|1000ñ, |1001ñ, |1010ñ, |1011ñ
|1100ñ, |1101ñ, |1110ñ, |1111ñ
binary representation of decimals 0 to 1
binary representation of decimals 0 to 3
binary representation of decimals 0 to 7
binary representation of decimals 0 to 15
MULT20015 Elements of Quantum Computing
Lecture 6
Multiple qubits and quantum processing
Independent quantum superpositions ® superposition over
N-bit binaries |000…0ñ,…, |111…1ñ (and there are 2N of these)
Basic representation of binaries as quantum information:
Quantum computation: qubits interact to create
complex superpositions and entangled states
N qubits …
101100
10
10101110
00101110
11100001
11101010
101100
10
10101110
00101110
11100001
11101010
Not very useful…measurement of qubits collapses to one random N-bit string
11
10
00
01
00
10
00
10
10
11
00
11
binaries time
probability
…
Bubble: Brocken commons.wikimedia.org
MULT20015 Elements of Quantum Computing
Lecture 6
Quantum information processing
|0000ñ
|0001ñ
|0010ñ
|0011ñ
|0100ñ
|1010ñ
|0110ñ
|0111ñ
|1000ñ
|1001ñ
|0101ñ
|1011ñ
|1100ñ
|1101ñ
|1110ñ
|1111ñ
01
01 11
10
10
11
binaries time
probability
|0000ñ
|0001ñ
|0010ñ
|0011ñ
|0100ñ
|0101ñ
|0110ñ
|0111ñ
|1000ñ
|1001ñ|1010ñ
|1011ñ
|1100ñ
|1101ñ
|1110ñ
|1111ñ
11
00
00
10
11
11
• logic gates between qubits perform mathematical operations on binary data
• complex entangled states created ® binary data are quantum “linked”
• quantum interference amplifies probability of desired output (answer)
start
quantum program
finish
Bubble: Brocken commons.wikimedia.org
CNOT
H
Digital quantum computing:
MULT20015 Elements of Quantum Computing
Lecture 6
Two qubit logic gates: CNOT
Control qubit
Target qubit
Two qubit gates can be constructed using an interaction between the two systems.
Most important is the Controlled-NOT (CNOT) gate.
How states transform: CNOT truth table
Rule: The target is flipped
iff the control qubit is “1”
Symbol for “control”
Symbol for binary
addition (bit flip)
a |00i+ b |01i+ c |10i+ d |11i
! a |00i+ b |01i+ d |10i+ c |11i
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
NB. Here, control
qubit is in first
position in the kets
MULT20015 Elements of Quantum Computing
Lecture 6
CNOT in matrix form
As a matrix, the CNOT operation is:
Recall, matrix representation of two qubit states:
CNOT
⟩|𝜓 = 𝑎!! ⟩|00 + 𝑎!” ⟩|01 +𝑎”! ⟩|10 + 𝑎”” ⟩|11 =
𝑎!!
𝑎!”
𝑎”!
𝑎””
⟩|𝜓 = 𝑎!! ⟩|00 + 𝑎!” ⟩|01
+𝑎”! ⟩|10 + 𝑎”” ⟩|11
⟩|𝜓# = 𝑎!! ⟩|00 + 𝑎!” ⟩|01
+𝑎”! ⟩|11 + 𝑎”” ⟩|10
ket form
matrix form
⟩|𝜓# =
1 0
0 1
0 0
0 0
0 0
0 0
0 1
1 0
𝑎!!
𝑎!”
𝑎”!
𝑎””
=
𝑎!!
𝑎!”
𝑎””
𝑎”!
amplitudes
flipped
amplitudes
flipped
⟩|𝜓
CNOT operation: | ⟩𝜓 | ⟩𝜓!
CNOT
MULT20015 Elements of Quantum Computing
Lecture 6
Example: CNOT on superposition
|0i
Before the CNOT, the state is:
After the CNOT, the state is:
↵ |0i+ � |1i
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
| 0i = ↵ |00i+ � |11i
AAACKnicbZBLSgNBEIZ7fMZ31KWbxiAKQpgRwWyEgBuXEYwKmRBqOhXTpOdBd40QhrmFx/AEbvUE7oJbvYc9ySx8/dDw81UVVf0HiZKGXHfizM0vLC4tV1ZW19Y3Nreq2zs3Jk61wLaIVazvAjCoZIRtkqTwLtEIYaDwNhhdFPXbB9RGxtE1jRPshnAfyYEUQBb1qnV/hJT5iZGHOT/nPqhkCHwKXTfnx9wPkErgeXmvWnPr7lT8r/FKU2OlWr3qp9+PRRpiREKBMR3PTaibgSYpFOarfmowATGCe+xYG0GIpptN/5XzA0v6fBBr+yLiU/p9IoPQmHEY2M4QaGh+1wr4X62T0qDRzWSUpISRmC0apIpTzIuQeF9qFKTG1oDQ0t7KxRA0CLJR/tgSaLDRFLl4v1P4a25O6p5b965Oa81GmVCF7bF9dsQ8dsaa7JK1WJsJ9sie2Qt7dZ6cN2fivM9a55xyZpf9kPPxBVjRpqA=AAACKnicbZBLSgNBEIZ7fMZ31KWbxiAKQpgRwWyEgBuXEYwKmRBqOhXTpOdBd40QhrmFx/AEbvUE7oJbvYc9ySx8/dDw81UVVf0HiZKGXHfizM0vLC4tV1ZW19Y3Nreq2zs3Jk61wLaIVazvAjCoZIRtkqTwLtEIYaDwNhhdFPXbB9RGxtE1jRPshnAfyYEUQBb1qnV/hJT5iZGHOT/nPqhkCHwKXTfnx9wPkErgeXmvWnPr7lT8r/FKU2OlWr3qp9+PRRpiREKBMR3PTaibgSYpFOarfmowATGCe+xYG0GIpptN/5XzA0v6fBBr+yLiU/p9IoPQmHEY2M4QaGh+1wr4X62T0qDRzWSUpISRmC0apIpTzIuQeF9qFKTG1oDQ0t7KxRA0CLJR/tgSaLDRFLl4v1P4a25O6p5b965Oa81GmVCF7bF9dsQ8dsaa7JK1WJsJ9sie2Qt7dZ6cN2fivM9a55xyZpf9kPPxBVjRpqA=AAACKnicbZBLSgNBEIZ7fMZ31KWbxiAKQpgRwWyEgBuXEYwKmRBqOhXTpOdBd40QhrmFx/AEbvUE7oJbvYc9ySx8/dDw81UVVf0HiZKGXHfizM0vLC4tV1ZW19Y3Nreq2zs3Jk61wLaIVazvAjCoZIRtkqTwLtEIYaDwNhhdFPXbB9RGxtE1jRPshnAfyYEUQBb1qnV/hJT5iZGHOT/nPqhkCHwKXTfnx9wPkErgeXmvWnPr7lT8r/FKU2OlWr3qp9+PRRpiREKBMR3PTaibgSYpFOarfmowATGCe+xYG0GIpptN/5XzA0v6fBBr+yLiU/p9IoPQmHEY2M4QaGh+1wr4X62T0qDRzWSUpISRmC0apIpTzIuQeF9qFKTG1oDQ0t7KxRA0CLJR/tgSaLDRFLl4v1P4a25O6p5b965Oa81GmVCF7bF9dsQ8dsaa7JK1WJsJ9sie2Qt7dZ6cN2fivM9a55xyZpf9kPPxBVjRpqA=AAACKnicbZBLSgNBEIZ7fMZ31KWbxiAKQpgRwWyEgBuXEYwKmRBqOhXTpOdBd40QhrmFx/AEbvUE7oJbvYc9ySx8/dDw81UVVf0HiZKGXHfizM0vLC4tV1ZW19Y3Nreq2zs3Jk61wLaIVazvAjCoZIRtkqTwLtEIYaDwNhhdFPXbB9RGxtE1jRPshnAfyYEUQBb1qnV/hJT5iZGHOT/nPqhkCHwKXTfnx9wPkErgeXmvWnPr7lT8r/FKU2OlWr3qp9+PRRpiREKBMR3PTaibgSYpFOarfmowATGCe+xYG0GIpptN/5XzA0v6fBBr+yLiU/p9IoPQmHEY2M4QaGh+1wr4X62T0qDRzWSUpISRmC0apIpTzIuQeF9qFKTG1oDQ0t7KxRA0CLJR/tgSaLDRFLl4v1P4a25O6p5b965Oa81GmVCF7bF9dsQ8dsaa7JK1WJsJ9sie2Qt7dZ6cN2fivM9a55xyZpf9kPPxBVjRpqA=
| i | 0i
| i = (↵ |0i+ � |1i)⌦ |0i = ↵ |00i+ � |10i
MULT20015 Elements of Quantum Computing
Lecture 6
Control Phase Gate
Control qubit
Target qubit
Another useful gate is the controlled phase gate (and generalisations using R-gate menu):
As a matrix:
How states transform:
Rule: the phase of the target flipped iff
the control qubit is “1”.
|00i ! |00i
|01i ! |01i
|10i ! |10i
|11i ! �|11i
Fun fact: CZ matrix is “diagonal” so it doesn’t matter which one you think of as control/target.
CZ =
2
66
4
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 �1
3
77
5
a |00i+ b |01i+ c |10i+ d |11i
! a |00i+ b |01i+ c |10i � d |11i
AAACbHicjZHbSsNAEIY38VxP9XBXhMViEcSSiGAvBW+8rGC10IQy2W7bpZsDuxOlhL6m4AN46xN44SYNeKgXDix8/DM/M/wbJFJodJxXy15aXlldW9+obG5t7+xW9/YfdJwqxjsslrHqBqC5FBHvoEDJu4niEAaSPwaTm7z/+MSVFnF0j9OE+yGMIjEUDNBI/WrSoEC9CcfMcWb0jAZzdnNmBbuFPpiz0T2v0qCeEqMxglLx83/851/+frXuNJ2i6CK4JdRJWe1+9c0bxCwNeYRMgtY910nQz0ChYJLPKl6qeQJsAiPeMxhByLWfFcnM6IlRBnQYK/MipIX63ZFBqPU0DMxkCDjWv3u5+Fevl+Kw5WciSlLkEZsvGqaSYkzzmOlAKM5QTg0AU8LcStkYFDA0n/FjS6DARJPn4v5OYREeLpqu03TvLuvXrTKhdVIjx+SUuOSKXJNb0iYdwsgL+bBWrFXr3T60a/bRfNS2Ss8B+VF24xNMsLcyAAACbHicjZHbSsNAEIY38VxP9XBXhMViEcSSiGAvBW+8rGC10IQy2W7bpZsDuxOlhL6m4AN46xN44SYNeKgXDix8/DM/M/wbJFJodJxXy15aXlldW9+obG5t7+xW9/YfdJwqxjsslrHqBqC5FBHvoEDJu4niEAaSPwaTm7z/+MSVFnF0j9OE+yGMIjEUDNBI/WrSoEC9CcfMcWb0jAZzdnNmBbuFPpiz0T2v0qCeEqMxglLx83/851/+frXuNJ2i6CK4JdRJWe1+9c0bxCwNeYRMgtY910nQz0ChYJLPKl6qeQJsAiPeMxhByLWfFcnM6IlRBnQYK/MipIX63ZFBqPU0DMxkCDjWv3u5+Fevl+Kw5WciSlLkEZsvGqaSYkzzmOlAKM5QTg0AU8LcStkYFDA0n/FjS6DARJPn4v5OYREeLpqu03TvLuvXrTKhdVIjx+SUuOSKXJNb0iYdwsgL+bBWrFXr3T60a/bRfNS2Ss8B+VF24xNMsLcyAAACbHicjZHbSsNAEIY38VxP9XBXhMViEcSSiGAvBW+8rGC10IQy2W7bpZsDuxOlhL6m4AN46xN44SYNeKgXDix8/DM/M/wbJFJodJxXy15aXlldW9+obG5t7+xW9/YfdJwqxjsslrHqBqC5FBHvoEDJu4niEAaSPwaTm7z/+MSVFnF0j9OE+yGMIjEUDNBI/WrSoEC9CcfMcWb0jAZzdnNmBbuFPpiz0T2v0qCeEqMxglLx83/851/+frXuNJ2i6CK4JdRJWe1+9c0bxCwNeYRMgtY910nQz0ChYJLPKl6qeQJsAiPeMxhByLWfFcnM6IlRBnQYK/MipIX63ZFBqPU0DMxkCDjWv3u5+Fevl+Kw5WciSlLkEZsvGqaSYkzzmOlAKM5QTg0AU8LcStkYFDA0n/FjS6DARJPn4v5OYREeLpqu03TvLuvXrTKhdVIjx+SUuOSKXJNb0iYdwsgL+bBWrFXr3T60a/bRfNS2Ss8B+VF24xNMsLcyAAACbHicjZHbSsNAEIY38VxP9XBXhMViEcSSiGAvBW+8rGC10IQy2W7bpZsDuxOlhL6m4AN46xN44SYNeKgXDix8/DM/M/wbJFJodJxXy15aXlldW9+obG5t7+xW9/YfdJwqxjsslrHqBqC5FBHvoEDJu4niEAaSPwaTm7z/+MSVFnF0j9OE+yGMIjEUDNBI/WrSoEC9CcfMcWb0jAZzdnNmBbuFPpiz0T2v0qCeEqMxglLx83/851/+frXuNJ2i6CK4JdRJWe1+9c0bxCwNeYRMgtY910nQz0ChYJLPKl6qeQJsAiPeMxhByLWfFcnM6IlRBnQYK/MipIX63ZFBqPU0DMxkCDjWv3u5+Fevl+Kw5WciSlLkEZsvGqaSYkzzmOlAKM5QTg0AU8LcStkYFDA0n/FjS6DARJPn4v5OYREeLpqu03TvLuvXrTKhdVIjx+SUuOSKXJNb0iYdwsgL+bBWrFXr3T60a/bRfNS2Ss8B+VF24xNMsLcy
MULT20015 Elements of Quantum Computing
Lecture 6
SWAP gate
Qubit 1
Qubit 2
A SWAP operation can be implemented using an interaction between the two qubits – the
states of the two qubits are swapped (not the physical qubits).
As a matrix:
How states transform:
Rule: the two qubits are swapped.
NB. Unlike CNOT, SWAP gates do not generate entanglement (but “sqrt SWAP” does!)
SWAP =
2
66
4
1 0 0 0
0 0 1 0
0 1 0 0
0 0 0 1
3
77
5
|00i ! |00i
|01i ! |10i
|10i ! |01i
|11i ! |11i
a |00i+ b |01i+ c |10i+ d |11i
! a |00i+ c |01i+ b |10i+ d |11i
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
MULT20015 Elements of Quantum Computing
Lecture 6
Example: a two-qubit circuit
t=0 t=1 t=2 t=3
Slider at t=1
Slider at t=2
Slider at t=3
Consider a simple two-qubit circuit:
What’s gone on here? – the 11 component
turned red, as did the slider…
MULT20015 Elements of Quantum Computing
Lecture 6
Example: a two-qubit circuit
t=0 t=1 t=2 t=3
State at t=1
State at t=2
State at t=3
Consider a simple two-qubit circuit:
⟩|𝜓(𝑡 = 1) = !
”
⟩|00 + !
”
⟩|01 + !
”
⟩|10 + !
”
⟩|11
CNOT:
⟩|𝜓(𝑡 = 2) = !
”
⟩|00 + !
”
⟩|01 + !
”
⟩|10 + !
”
⟩|11
CZ:
⟩|𝜓(𝑡 = 3) = !
”
⟩|00 + !
”
⟩|01 + !
”
⟩|10 − !
”
⟩|11
a |00i+ b |01i+ c |10i+ d |11i
! a |00i+ b |01i+ c |10i � d |11i
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
-1 <-> 𝜋 phase
But, why has the slider turned red? – entanglement!
CZ:
symmetric -> no change in flip
MULT20015 Elements of Quantum Computing
Lecture 6
6.2 Entanglement
MULT20015
Lecture 6
MULT20015 Elements of Quantum Computing
Lecture 6
Entanglement – the real magic of quantum mechanics
So far, we have focused on qubits in quantum superpositions and subsequent measurement.
Strange and wonderful as that is, superposition is actually the least weird aspect of qubits.
Now we get to the stuff that really bothered Einstein and the like – entanglement!
| alice&bobi = | alicei ⇥ | bobi
= (
1
p
2
|0i+
1
p
2
|1i)⇥ (
1
p
2
|0i+
1
p
2
|1i)
=
1
2
|0i |0i+
1
2
|0i |1i+
1
2
|1i |0i+
1
2
|1i |1i
=
1
2
|00i+
1
2
|01i+
1
2
|10i+
1
2
|11i
| alice&bobi = | alicei ⇥ | bobi
= (
1
p
2
|0i+
1
p
2
|1i)⇥ (
1
p
2
|0i+
1
p
2
|1i)
=
1
2
|0i |0i+
1
2
|0i |1i+
1
2
|1i |0i+
1
2
|1i |1i
=
1
2
|00i+
1
2
|01i+
1
2
|10i+
1
2
|11i
|00ñ
|11ñ
Y
|10ñ
|01ñ
Let’s look at different types of two-qubit states:
| alice&bobi =
✓
1
p
2
|0i+
1
p
2
|1i
◆
⇥
✓
1
p
2
|0i+
1
p
2
|1i
◆
® measurement results are independent
|11ñ
Y
|00ñ
® entangled, measurement results are dependent!
1
p
2
|00i+
1
p
2
|11i
Looks simpler,
but it’s far more
interesting!
Independent superpositions of Alice and Bob
Too “spooky”
to be true
An entangled state of Alice and Bob
“product” | alice&bobi = | alicei ⇥ | bobi
= (
1
p
2
|0i+
1
p
2
|1i)⇥ (
1
p
2
|0i+
1
p
2
|1i)
=
1
2
|0i |0i+
1
2
|0i |1i+
1
2
|1i |0i+
1
2
|1i |1i
=
1
2
|00i+
1
2
|01i+
1
2
|10i+
1
2
|11i
no “product”
analogy
possible!⊗
MULT20015 Elements of Quantum Computing
Lecture 6
Entanglement – why it is strange
Let’s look at that entangled state in more detail – measurement scenarios:
|11ñ
Y
|00ñ
Bob’s measurement
result is completely
determined after
Alice’s qubit
measurement!
i.e. Alice
measurement
somehow determines
Bob’s outcome!
Wow.
1
p
2
|00i+
1
p
2
|11i
Alice measures “0”
(50% of the time)
overall state collapses
® |00ñ
?
1
p
2
|00i+
1
p
2
|11i
50%
𝑎!! = 1/ 2
𝑎”” = 1/ 2
Prob[00] = 𝑎!!
# = ”
#
Prob[11] = 𝑎””
# = ”
#
?
Bob measures “0”
100% of the time
Alice measures “1”
(50% of the time)
overall state collapses
® |11ñ
?
Bob measures “1”
100% of the time
50%
| alice&bobi = | alicei ⇥ | bobi
= (
1
p
2
|0i+
1
p
2
|1i)⇥ (
1
p
2
|0i+
1
p
2
|1i)
=
1
2
|0i |0i+
1
2
|0i |1i+
1
2
|1i |0i+
1
2
|1i |1i
=
1
2
|00i+
1
2
|01i+
1
2
|10i+
1
2
|11i
(NB. this is called
a “Bell State” after
John Bell who figured
out how to really test
all this…)
Looks what happens when Alice measures (first qubit)
MULT20015 Elements of Quantum Computing
Lecture 6
Aside: why did entanglement worry Einstein?
|11ñ
Y
|00ñ Bob’s measurement result is completely determined by
the result of Alice’s qubit measurement!
i.e. Alice measurement determines Bob’s outcome!
1
p
2
|00i+
1
p
2
|11i
| alice&bobi = | alicei ⇥ | bobi
= (
1
p
2
|0i+
1
p
2
|1i)⇥ (
1
p
2
|0i+
1
p
2
|1i)
=
1
2
|0i |0i+
1
2
|0i |1i+
1
2
|1i |0i+
1
2
|1i |1i
=
1
2
|00i+
1
2
|01i+
1
2
|10i+
1
2
|11iEinstein (and others) objected to this because it seemed to imply action or communication
faster than light…
i.e. imagine Alice and Bob qubits are separated to either side of the galaxy…you can see how
it appears that Alice sends a signal to Bob in order for Bob’s measurement to work out.
Important – even though entanglement worried Einstein, you can prove that no actual
communication actually happens between Alice and Bob…it’s more like an “influence”.
We don’t really understand it, but “quantum connections” do occur (Bell showed the way)
More to the point, they are a “resource” that powers quantum computation…
MULT20015 Elements of Quantum Computing
Lecture 6
Back to the example circuit
⟩|𝜓(𝑡 = 3) = !
”
⟩|00 + !
”
⟩|01 + !
”
⟩|10 − !
”
⟩|11
t=0 t=1 t=2 t=3
⟩|𝜓(𝑡 = 1) = !
”
⟩|00 + !
”
⟩|01 + !
”
⟩|10 + !
”
⟩|11 = !
”
⟩|0 + ⟩|1 ⊗ !
”
⟩|0 + ⟩|1
⟩|𝜓(𝑡 = 2) = !
”
⟩|00 + !
”
⟩|01 + !
”
⟩|10 + !
”
⟩|11 = !
”
⟩|0 + ⟩|1 ⊗ !
”
⟩|0 + ⟩|1
Can write as a “product” -> no entanglement
Can’t write as a “product” -> entangled!
Can write as a “product” -> no entanglement
Most will say, yes I can I’ll just use i to get the −
”
)
⟩|11 …but it won’t work…
i.e.
!
”
⟩|0 + ⟩𝑖 |1 ⊗ !
”
⟩|0 + ⟩𝑖 |1 = !
”
⟩|00 + #
”
⟩|01 + #
”
⟩|10 − !
”
⟩|11 See, it’s not the same state…
MULT20015 Elements of Quantum Computing
Lecture 6
Separable states
| i = a|0i+ b|1i |�i = c|0i+ d|1i
A separable state is one which can be written as
|�i = | i ⌦ |�i
All separable states (of two qubits) can be written as:
| i = ac |00i+ ad |01i+ bc |10i+ bd |11i
AAACOnicbVDLSgMxFM34rPVVdekmWARBKBMR7EYpuHFZwT6gM5Q7mbQNzTxIMkIZ+jd+hl/gVjduBRfi1g8wM9OFbb0QOI97uTfHiwVX2rbfrZXVtfWNzdJWeXtnd2+/cnDYVlEiKWvRSESy64FigoespbkWrBtLBoEnWMcb32Z+55FJxaPwQU9i5gYwDPmAU9BG6ldunDHT2IkVx9cYaMZS257icww+LhjJmFdYJLc8vyBk2q9U7ZqdF14GZAaqaFbNfuXT8SOaBCzUVIBSPWLH2k1Bak4Fm5adRLEY6BiGrGdgCAFTbpr/c4pPjeLjQSTNCzXO1b8TKQRKTQLPdAagR2rRy8T/vF6iB3U35WGcaBbSYtEgEVhHOAsN+1wyqsXEAKCSm1sxHYEEqk20c1s8CSaaLBeymMIyaF/UiF0j95fVRn2WUAkdoxN0hgi6Qg10h5qohSh6Qi/oFb1Zz9aH9WV9F60r1mzmCM2V9fMLqUqrow==AAACOnicbVDLSgMxFM34rPVVdekmWARBKBMR7EYpuHFZwT6gM5Q7mbQNzTxIMkIZ+jd+hl/gVjduBRfi1g8wM9OFbb0QOI97uTfHiwVX2rbfrZXVtfWNzdJWeXtnd2+/cnDYVlEiKWvRSESy64FigoespbkWrBtLBoEnWMcb32Z+55FJxaPwQU9i5gYwDPmAU9BG6ldunDHT2IkVx9cYaMZS257icww+LhjJmFdYJLc8vyBk2q9U7ZqdF14GZAaqaFbNfuXT8SOaBCzUVIBSPWLH2k1Bak4Fm5adRLEY6BiGrGdgCAFTbpr/c4pPjeLjQSTNCzXO1b8TKQRKTQLPdAagR2rRy8T/vF6iB3U35WGcaBbSYtEgEVhHOAsN+1wyqsXEAKCSm1sxHYEEqk20c1s8CSaaLBeymMIyaF/UiF0j95fVRn2WUAkdoxN0hgi6Qg10h5qohSh6Qi/oFb1Zz9aH9WV9F60r1mzmCM2V9fMLqUqrow==AAACOnicbVDLSgMxFM34rPVVdekmWARBKBMR7EYpuHFZwT6gM5Q7mbQNzTxIMkIZ+jd+hl/gVjduBRfi1g8wM9OFbb0QOI97uTfHiwVX2rbfrZXVtfWNzdJWeXtnd2+/cnDYVlEiKWvRSESy64FigoespbkWrBtLBoEnWMcb32Z+55FJxaPwQU9i5gYwDPmAU9BG6ldunDHT2IkVx9cYaMZS257icww+LhjJmFdYJLc8vyBk2q9U7ZqdF14GZAaqaFbNfuXT8SOaBCzUVIBSPWLH2k1Bak4Fm5adRLEY6BiGrGdgCAFTbpr/c4pPjeLjQSTNCzXO1b8TKQRKTQLPdAagR2rRy8T/vF6iB3U35WGcaBbSYtEgEVhHOAsN+1wyqsXEAKCSm1sxHYEEqk20c1s8CSaaLBeymMIyaF/UiF0j95fVRn2WUAkdoxN0hgi6Qg10h5qohSh6Qi/oFb1Zz9aH9WV9F60r1mzmCM2V9fMLqUqrow==AAACOnicbVDLSgMxFM34rPVVdekmWARBKBMR7EYpuHFZwT6gM5Q7mbQNzTxIMkIZ+jd+hl/gVjduBRfi1g8wM9OFbb0QOI97uTfHiwVX2rbfrZXVtfWNzdJWeXtnd2+/cnDYVlEiKWvRSESy64FigoespbkWrBtLBoEnWMcb32Z+55FJxaPwQU9i5gYwDPmAU9BG6ldunDHT2IkVx9cYaMZS257icww+LhjJmFdYJLc8vyBk2q9U7ZqdF14GZAaqaFbNfuXT8SOaBCzUVIBSPWLH2k1Bak4Fm5adRLEY6BiGrGdgCAFTbpr/c4pPjeLjQSTNCzXO1b8TKQRKTQLPdAagR2rRy8T/vF6iB3U35WGcaBbSYtEgEVhHOAsN+1wyqsXEAKCSm1sxHYEEqk20c1s8CSaaLBeymMIyaF/UiF0j95fVRn2WUAkdoxN0hgi6Qg10h5qohSh6Qi/oFb1Zz9aH9WV9F60r1mzmCM2V9fMLqUqrow==
MULT20015 Elements of Quantum Computing
Lecture 6
Examples of separable states
Consider the state:
It is separable because:
Consider the state:
It is also separable because:
| i =
|00i+ |01i
p
2
| i = |0i ⌦
|0i+ |1i
p
2
| i =
|0i+ |1i
p
2
⌦
|0i+ |1i
p
2
| i =
|00i+ |01i+ |10i+ |11i
2
MULT20015 Elements of Quantum Computing
Lecture 6
Constructing a Bell state
Refers to maximally entangled 2 qubit states named after the physicist John
Bell (who figured out how to experimentally explore the reality of
entanglement).
Execution:
Question: Is
|00i+ |11i
p
2
separable?
Consider the following circuit in the QUI:
|00i !
|00i+ |10i
p
2
!
|00i+ |11i
p
2H CNOT
MULT20015 Elements of Quantum Computing
Lecture 6
Entanglement
Answer: No! We can never find a, b, c, d, i.e.
A state which is not separable is called an entangled state.
Entanglement is a uniquely quantum mechanical property, with no direct
classical analogue.
|00i+ |11i
p
2
6= (a |0i+ b |1i)⌦ (c |0i+ d |1i)
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
MULT20015 Elements of Quantum Computing
Lecture 6
Entanglement Measure
We would like to have a measure of how much entanglement a state has. Some
states are more entangled than others:
1
p
2
|00i+
1
p
2
|11i
AAACNXicdZDLSgMxFIYz9VbrbdSlm2ARBKFMimDdFdy4rGAv0BlKJs20oZmLyRmhDPMsPoZP4FbXLtypW1/B9LLQVg8Efv7/HM7J5ydSaHCcV6uwsrq2vlHcLG1t7+zu2fsHLR2nivEmi2WsOj7VXIqIN0GA5J1EcRr6krf90dUkb99zpUUc3cI44V5IB5EIBKNgrJ596QaKsozkmavvFGTVPMfuiEPmODk+w/+lhOQ9u+xUnGnhZUHmoozm1ejZH24/ZmnII2CSat0lTgJeRhUIJnleclPNE8pGdMC7RkY05NrLpl/M8Ylx+jiIlXkR4Kn7cyKjodbj0DedIYWhXswm5l9ZN4Wg5mUiSlLgEZstClKJIcYTXrgvFGcgx0ZQpoS5FbMhNVTAUP21xVfUoJlwIYsUlkWrWiFOhdycl+u1OaEiOkLH6BQRdIHq6Bo1UBMx9ICe0DN6sR6tN+vd+py1Fqz5zCH6VdbXN8aOrKI=AAACNXicdZDLSgMxFIYz9VbrbdSlm2ARBKFMimDdFdy4rGAv0BlKJs20oZmLyRmhDPMsPoZP4FbXLtypW1/B9LLQVg8Efv7/HM7J5ydSaHCcV6uwsrq2vlHcLG1t7+zu2fsHLR2nivEmi2WsOj7VXIqIN0GA5J1EcRr6krf90dUkb99zpUUc3cI44V5IB5EIBKNgrJ596QaKsozkmavvFGTVPMfuiEPmODk+w/+lhOQ9u+xUnGnhZUHmoozm1ejZH24/ZmnII2CSat0lTgJeRhUIJnleclPNE8pGdMC7RkY05NrLpl/M8Ylx+jiIlXkR4Kn7cyKjodbj0DedIYWhXswm5l9ZN4Wg5mUiSlLgEZstClKJIcYTXrgvFGcgx0ZQpoS5FbMhNVTAUP21xVfUoJlwIYsUlkWrWiFOhdycl+u1OaEiOkLH6BQRdIHq6Bo1UBMx9ICe0DN6sR6tN+vd+py1Fqz5zCH6VdbXN8aOrKI=AAACNXicdZDLSgMxFIYz9VbrbdSlm2ARBKFMimDdFdy4rGAv0BlKJs20oZmLyRmhDPMsPoZP4FbXLtypW1/B9LLQVg8Efv7/HM7J5ydSaHCcV6uwsrq2vlHcLG1t7+zu2fsHLR2nivEmi2WsOj7VXIqIN0GA5J1EcRr6krf90dUkb99zpUUc3cI44V5IB5EIBKNgrJ596QaKsozkmavvFGTVPMfuiEPmODk+w/+lhOQ9u+xUnGnhZUHmoozm1ejZH24/ZmnII2CSat0lTgJeRhUIJnleclPNE8pGdMC7RkY05NrLpl/M8Ylx+jiIlXkR4Kn7cyKjodbj0DedIYWhXswm5l9ZN4Wg5mUiSlLgEZstClKJIcYTXrgvFGcgx0ZQpoS5FbMhNVTAUP21xVfUoJlwIYsUlkWrWiFOhdycl+u1OaEiOkLH6BQRdIHq6Bo1UBMx9ICe0DN6sR6tN+vd+py1Fqz5zCH6VdbXN8aOrKI=AAACNXicdZDLSgMxFIYz9VbrbdSlm2ARBKFMimDdFdy4rGAv0BlKJs20oZmLyRmhDPMsPoZP4FbXLtypW1/B9LLQVg8Efv7/HM7J5ydSaHCcV6uwsrq2vlHcLG1t7+zu2fsHLR2nivEmi2WsOj7VXIqIN0GA5J1EcRr6krf90dUkb99zpUUc3cI44V5IB5EIBKNgrJ596QaKsozkmavvFGTVPMfuiEPmODk+w/+lhOQ9u+xUnGnhZUHmoozm1ejZH24/ZmnII2CSat0lTgJeRhUIJnleclPNE8pGdMC7RkY05NrLpl/M8Ylx+jiIlXkR4Kn7cyKjodbj0DedIYWhXswm5l9ZN4Wg5mUiSlLgEZstClKJIcYTXrgvFGcgx0ZQpoS5FbMhNVTAUP21xVfUoJlwIYsUlkWrWiFOhdycl+u1OaEiOkLH6BQRdIHq6Bo1UBMx9ICe0DN6sR6tN+vd+py1Fqz5zCH6VdbXN8aOrKI=
p
0.99 |00i+
p
0.01 |11i
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
|00i
AAACAnicbVDLSgNBEJz1GeMr6tHLYhA8hVkRzDHgxWME84BkCbOT3mTI7Owy0yuEJTe/wKt+gTfx6o/4Af6Hs8keTGJBQ1HVTXdXkEhhkNJvZ2Nza3tnt7RX3j84PDqunJy2TZxqDi0ey1h3A2ZACgUtFCihm2hgUSChE0zucr/zBNqIWD3iNAE/YiMlQsEZWqnbnwBmlM4GlSqt0TncdeIVpEoKNAeVn/4w5mkECrlkxvQ8mqCfMY2CS5iV+6mBhPEJG0HPUsUiMH42v3fmXlpl6IaxtqXQnat/JzIWGTONAtsZMRybVS8X//N6KYZ1PxMqSREUXywKU+li7ObPu0OhgaOcWsK4FvZWl4+ZZhxtREtbAs1sNHku3moK66R9XfNozXu4qTbqRUIlck4uyBXxyC1pkHvSJC3CiSQv5JW8Oc/Ou/PhfC5aN5xi5owswfn6BXacmBA=AAACAnicbVDLSgNBEJz1GeMr6tHLYhA8hVkRzDHgxWME84BkCbOT3mTI7Owy0yuEJTe/wKt+gTfx6o/4Af6Hs8keTGJBQ1HVTXdXkEhhkNJvZ2Nza3tnt7RX3j84PDqunJy2TZxqDi0ey1h3A2ZACgUtFCihm2hgUSChE0zucr/zBNqIWD3iNAE/YiMlQsEZWqnbnwBmlM4GlSqt0TncdeIVpEoKNAeVn/4w5mkECrlkxvQ8mqCfMY2CS5iV+6mBhPEJG0HPUsUiMH42v3fmXlpl6IaxtqXQnat/JzIWGTONAtsZMRybVS8X//N6KYZ1PxMqSREUXywKU+li7ObPu0OhgaOcWsK4FvZWl4+ZZhxtREtbAs1sNHku3moK66R9XfNozXu4qTbqRUIlck4uyBXxyC1pkHvSJC3CiSQv5JW8Oc/Ou/PhfC5aN5xi5owswfn6BXacmBA=AAACAnicbVDLSgNBEJz1GeMr6tHLYhA8hVkRzDHgxWME84BkCbOT3mTI7Owy0yuEJTe/wKt+gTfx6o/4Af6Hs8keTGJBQ1HVTXdXkEhhkNJvZ2Nza3tnt7RX3j84PDqunJy2TZxqDi0ey1h3A2ZACgUtFCihm2hgUSChE0zucr/zBNqIWD3iNAE/YiMlQsEZWqnbnwBmlM4GlSqt0TncdeIVpEoKNAeVn/4w5mkECrlkxvQ8mqCfMY2CS5iV+6mBhPEJG0HPUsUiMH42v3fmXlpl6IaxtqXQnat/JzIWGTONAtsZMRybVS8X//N6KYZ1PxMqSREUXywKU+li7ObPu0OhgaOcWsK4FvZWl4+ZZhxtREtbAs1sNHku3moK66R9XfNozXu4qTbqRUIlck4uyBXxyC1pkHvSJC3CiSQv5JW8Oc/Ou/PhfC5aN5xi5owswfn6BXacmBA=AAACAnicbVDLSgNBEJz1GeMr6tHLYhA8hVkRzDHgxWME84BkCbOT3mTI7Owy0yuEJTe/wKt+gTfx6o/4Af6Hs8keTGJBQ1HVTXdXkEhhkNJvZ2Nza3tnt7RX3j84PDqunJy2TZxqDi0ey1h3A2ZACgUtFCihm2hgUSChE0zucr/zBNqIWD3iNAE/YiMlQsEZWqnbnwBmlM4GlSqt0TncdeIVpEoKNAeVn/4w5mkECrlkxvQ8mqCfMY2CS5iV+6mBhPEJG0HPUsUiMH42v3fmXlpl6IaxtqXQnat/JzIWGTONAtsZMRybVS8X//N6KYZ1PxMqSREUXywKU+li7ObPu0OhgaOcWsK4FvZWl4+ZZhxtREtbAs1sNHku3moK66R9XfNozXu4qTbqRUIlck4uyBXxyC1pkHvSJC3CiSQv5JW8Oc/Ou/PhfC5aN5xi5owswfn6BXacmBA=
Not entangled, separable
Entangled, but close to a separable state
Maximally entangled
In the QUI we measure the degree of entanglement using an informatic “entropy”
measure: Entanglement Entropy (EE)
Entanglement is a type of correlation between two systems, say A and B.
To see how much correlation there is between A and B: We can measure
B and ask how many bits of information (as measured by entropy) this can
tell us about the state of A?
MULT20015 Elements of Quantum Computing
Lecture 6
Entanglement Entropy – a recipe (used on QUI)
𝐻 = −𝜆” log)(𝜆”) − 𝜆) log)(𝜆)).
To calculate the entanglement entropy between two qubits, follow these steps…
Consider an arbitrary normalized two-qubit state:
⟩|𝜓 = 𝑎!! ⟩|00 + 𝑎!” ⟩|01 +𝑎”! ⟩|10 + 𝑎”” ⟩|11
1. Construct the matrix from the amplitudes: 𝐴 =
𝑎!! 𝑎!”
𝑎”! 𝑎””
2. Calculate the eigenvalues 𝜆” and 𝜆) of the matrix product 𝐴𝐴*.
3. The entanglement entropy (EE) between the qubits (in number of bits) is given by:
QUI does this for you, EE displayed on red-scale on time-slider
-> generalises to circuits involving more qubits (EE between
qubits above/below the corresponding segment on slider)
NB. Two-qubits: max EE = 1 bit
MULT20015 Elements of Quantum Computing
Lecture 6
Generating Bell states
|11ñ
Y
|00ñ
1
p
2
|00i+
1
p
2
|11i
| alice&bobi = | alicei ⇥ | bobi
= (
1
p
2
|0i+
1
p
2
|1i)⇥ (
1
p
2
|0i+
1
p
2
|1i)
=
1
2
|0i |0i+
1
2
|0i |1i+
1
2
|1i |0i+
1
2
|1i |1i
=
1
2
|00i+
1
2
|01i+
1
2
|10i+
1
2
|11i
John Bell (1964) figured out a protocol to actually measure whether entanglement was
true or not in physical systems (answer is yes to many standard deviations).
The family of simple two-qubit maximally entangled states are named in his honour – Bell States
NB. Two-qubits: max EE = 1 bit
MULT20015 Elements of Quantum Computing
Lecture 6
Dialing in the level of entanglement entropy (EE)
Adjust angle of R-gate rotation (about X + Z axis)
R-gate same as H-gate
-> maximal entanglement
As we reduce
the rotation angle
in R we make the
superposition in
qubit #1 more
unbalanced
before the CNOT
-> slider bar becomes
less red…
(lower entanglement)
EE max
EE med
EE low
MULT20015 Elements of Quantum Computing
Lecture 6
Week 3
Lecture 5
5.1 Two-qubit systems
5.2 Two qubit example – independent Alice and Bob
5.3 General two qubit states: measurement and operations
Lecture 6
6.1 Two-qubit logic operations
6.2 Entanglement
Practice class 3
Two qubit states and operations
MULT20015 Elements of Quantum Computing
Lecture 6
Subject outline
Lecture topics (by week)
1 – Introduction to quantum computing and maths basics
2 – Single qubit representations and logic operations
3 – Two qubit states and logic gates
4 – Multi-qubit states and quantum arithmetic
5 – Basic quantum algorithms
6 – Period finding, cryptography and quantum factoring
7 – Shor’s algorithm, post-quantum crypto, quantum key distribution
8 – Quantum search algorithms
9 – Grover search applications, optimisation problems
10 – Solving optimisation problems on quantum computers
11 – Applications in quantum machine learning
12 – Real quantum computer devices
Assignment schedule:
#1: Hand out in Week 2
#2: Hand out in Week 8