ML Reading Group (Intro to Quantum Computation)

Introduction to Quantum Computing
Antonín Hoskovec
Thursday 10th
May, 2018
Rossum AI

Slides
slideshare.net:
https://bit.ly/2jL2dj2
1

Physical Frameworks
State Observables Time Evolution
Cl. Mechanics q, p f(q, p) Newton equations
Quantum M. |ψ ∈ H s.a. operators Schrödinger eq.
Discrete QM |ψ ∈ Cn s.a. matrices unitary matrices
2

Bit vs. Qubit
bit: b =



1
0
Qubit: |ψ ∈ C2 = span{
1
0
,
0
1
}
|ψ = α
1
0
+ β
0
1
= α |1 + β |0
3

Real world examples
• Spin: |0 - spin down, |1 - spin up
4

Real world examples
Pauli exclusion principle:
4

Real world examples
Pauli exclusion principle:
• Photon polarization: |0 - horizontal polarization, |1 -
vertical polarization
4

Measurements
Physical system in the state given by the vector
|ψ = α |1 + β |0 ,
α, β ∈ C,
|α|2
+ |β|2
= 1,
will be measured to be in the state |1 with probability |α|2
and
in |0 with probability |β|2
. After such measurement it will be in
the state |0 or |1 (further measurements will always give the
same result).
5

How to measure spin
Stern-Gerlach experiment
6

How to measure polarization
Polarization filters
7

Formalism
Dot product
|ψ = α |1 + β |0
|ϕ = γ |1 + δ |0
ψ|ϕ = αγ + βδ (Dirac1939)
Using the bra-ket notation we can write projections onto |ψ :
(|ψ ψ|) |ϕ ≡ |ψ ( ψ|ϕ )
8

Observables
The states that we can measure and corresponding values of
observables: eigenvectors and eigenvalues of self-adjoint
matrices.
For example projection of spin to the z axis:
σz =
1 0
0 −1
has eigenvalues and eigenvectors:
λ1 = 1 :
1
0
= |1 , λ2 = −1 :
0
1
= |0
9

Example of a time evolution
U =
1
√
2
1 1
1 −1
,
U |1 =
1
√
2
1 1
1 −1
1
0
=
1
√
2
1
1
=
1
√
2
|1 +
1
√
2
|0
This could be for example a spin interacting with magnetic
field, phase plates for photons...
10

Multiple qubits
State is a superposition of tensor products:
|ψ1 ⊗ |ψ2 ⊗ |ψ3 . . . ⊗ |ψn ≡ |ψ1 |ψ2 |ψ3 . . . |ψn
Basis of the bigger state space:
span { |0 |0 . . . |0 ,
|1 |0 . . . |0 , |0 |1 . . . |0 , . . . , |0 |0 . . . |1
|1 |1 . . . |0 , . . . , |0 . . . |1 |1
. . .
|1 |1 . . . |1 }
11

Classical Turing machine
Can calculate an arbitrary function of the input bits
b1, b2, b3, . . . , bn → f(b1, b2, b3, . . . , bn),
if we can do
AND, XOR, NOT (NAND)
along with identity,
FANOUT
and auxiliary bits.
13

Single-Qubit operations
Hadamard, Phase, π
8 gates
H =
1
√
2
1 1
1 −1
, S =
1 0
0 i
, T =
1 0
0 exp iπ
4
|q0 H
|q1 S
|q2 T
14

Two-Qubit operations
In the basis:
{|0 |0 , |0 |1 , |1 |0 , |1 |1 } ≡








1
0
0
0





,





0
1
0
0





,





0
0
1
0





,





0
0
0
1








e.g.
CNOT=





1 0 0 0
0 1 0 0
0 0 0 1
0 0 1 0





15

CNOT
CNOT |0 |0 = |0 |0 ,
CNOT |0 |1 = |0 |1 ,
CNOT |1 |0 = |1 |1 ,
CNOT |1 |1 = |1 |0
explicitly e.g.:





1 0 0 0
0 1 0 0
0 0 0 1
0 0 1 0










0
0
1
0





=





0
0
0
1





16

Arbitrary Unitary transformation
Analytically:
CNOT,
1 0
0 1
,
0 1
1 0
,
0 −i
i 0
,
1 0
0 −1
single−qubit operations
With arbitrary precision:
CNOT, H, S, T
18

MBQC
Measurement Based Quantum Computation (One Way)
Measurement on one of the two qubits:
1
√
2
(|0 + |1 )
1
√
2
(|0 + |1 ) →



|0 1√
2
(|0 + |1 ) pro |0 1
|1 1√
2
(|0 + |1 ) pro |1 1
1
√
2
(|0 |0 + |1 |1 ) →



|0 |0 pro |0 1
|1 |1 pro |1 1
21

Graph state
(a) a Greenberger-Horne-Zeilinger (GHZ) state
(b) a 1D cluster state
(c) a 2D cluster state, the latter being a universal resource for
MBQC
Taken from [1]: Measurement-based quantum computation 22

D-Wave
O(a, b, q) =
i
aiqi
i,j
bi,jqiqj
24

Data
• Database with N records.
25

Data
• Lets take log2 N qubits.
25

Data
• Lets take log2 N qubits.
0000 ←→ |0 |0 |0 |0 ≡ |0
0001 ←→ |0 |0 |0 |1 ≡ |1
0010 ←→ |0 |0 |1 |0 ≡ |2
. . .
25

Quantum Oracle)
The searched-for index ω:
Uω |ω = − |ω
Uω |x = |x pro x = ω
26

Starting State, Auxiuliary operation
Starting state:
|s ≡
1
√
N
N−1
x=0
|x ,
Auxiliary operation (Grover diffusion operator):
Us = 2 |s s| − I
27

Algorithm steps
1) Prepare your system in |s
28

Algorithm steps
2) Repeat until you’re satisfied (∼
√
N)
a) Uω
b) Us
28

Algorithm steps
√
N)
a) Uω
b) Us
3) with very high probability the system is in state |ω
28

Algorithm steps
√
N)
a) Uω
b) Us
3) with very high probability the system is in state |ω
Probability of measuring the correct state after n repetitions:
P(ω, n) = sin2
n +
1
2
θ ,
where θ = 2 arcsin 1√
N
, is close to 1 for n ≈ π
√
N
4 .
28

History
1982: Richard Feynman: Simulating physics with computers
29

History
1984: Charles Bennett, Gilles Brassard: Quantum cryptography:
Public key distribution and coin tossing
29

History
1994: Peter Shor: Polynomial-time algorithms for prime
factorization and discrete logarithms on a quantum
computer
factorization in O((log N)2(log log N)(log log log N)) steps
29

History
1994: Peter Shor: Polynomial-time algorithms for prime
factorization and discrete logarithms on a quantum
computer
factorization in O((log N)2(log log N)(log log log N)) steps
1996: Lov Grover: A fast quantum mechanical algorithm for
database search
29

Experiments
2001: IBM, MIT factorization of 15 (NMR, adiabatically)
30

Experiments
2007: 15 refactorized on two different systems
30

Experiments
2011: Quantum teleportation jointly Australia & Japan
30

Experiments
2011: Bristol factorization of 21 (optics)
30

Experiments
2011: China factorizes 143 (NMR)
30

Experiments
2012: First commercial quantum computers (D-Wave, 1QBit...)
30

Experiments
2013: Google Quantum Artificial Intelligence Lab
30

Experiments
2014: The Chinese have also factorized 56153 back in 2011...
30

Experiments
2014: The Chinese have also factorized 56153 back in 2011...
2014: Snowden: NSA is building its own Quantum Computer
30

Experiments
2015: (December) NASA is actively using its Quantum Computer
(D-Wave)
31

Experiments
(D-Wave)
2016: (May)IBM Quantum Experience (first cloud-based QC, 5
qubits)
31

Experiments
(D-Wave)
qubits)
2017: (May) IBM 16 qubit processor
31

Experiments
(D-Wave)
qubits)
2017: (July) Harward and other U.S. research institution: 51
qubits Quantum Simulator
31

Experiments
(D-Wave)
qubits)
2017: (November) IBM 20 qubit commercial processor, and the
first prototype 50 qubit processor
31

Experiments
(D-Wave)
qubits)
2017: (November) IBM 20 qubit commercial processor, and the
first prototype 50 qubit processor
2018: (March) Google Quantum AI Lab 72 qubit processor called
Bristlecone
31

How many qubits do we need?
“If you had 50 or 100 qubits and they really worked well
enough, and were fully error-corrected—you could do
unfathomable calculations that can’t be replicated on any
classical machine, now or ever.”
[Robert Schoelkopf, Yale University]
32

First Computer Game on a Quantum Computer
IBM Quantum Experience:
https://quantumexperience.ng.bluemix.net/
33

H. Briegel, D. Browne, W. Dür, R. Raussendorf, and
M. Van den Nest.
Measurement-based quantum computation.
Nature Physics, 5(1):19–26, 2009.
M. A. Nielsen and I. L. Chuang.
Quantum Computation and Quantum Information: 10th
Anniversary Edition.
Cambridge University Press, New York, NY, USA, 10th
edition, 2011.
34

Questions?
Slides: https://bit.ly/2jL2dj2
34

ML Reading Group (Intro to Quantum Computation)

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