Introduction to the principles of Quantum Computing and its applications to Machine Learning. No previous knowledge of Quantum Mechanics is required.

1. Quantum Computing and
Machine Learning
Nicolò Musmeci – Jan 2021

2. Executive Summary
1. Quantum computers: principles
• Example of a quantum algorithm:
Shor’s factorisation algorithm
• Debunking Quantum computing
2. Quantum Machine Learning
3. Physical realisations
• Technical difficulties
• Examples of hardware
• Timeline of Quantum Computing

3. INPUT OUTPUT
A Q
0 1
1 0
INPUT OUTPUT
A B Q
0 0 0
0 1 0
1 0 0
1 1 1
Classic computers: bits and logic gates

4. 𝑎𝑢
2: probability of measuring spin up
𝑎𝑑
2
: probability of measuring spin down
Measurement (accessing the information)
Transformation (quantum gates)
𝐻11 𝐻12
𝐻21 𝐻22
𝑎𝑢
𝑎𝑑
=
𝑎𝑢𝐻11 + 𝑎𝑑𝐻12
𝑎𝑢𝐻21 + 𝑎𝑑𝐻22
𝜓 = (𝑎𝑢𝐻11 + 𝑎𝑑𝐻12) + (𝑎𝑢𝐻21 + 𝑎𝑑𝐻22)
Quantum computers: qubits and quantum gates
Superposition
Amplitudes
Electron with
spin up
Electron with
spin down

5. INPUT OUTPUT
A Q
0 1
1 0
0 1
1 0
0 1
1 0
1
3
+
2
3
1 0
2
3
+
1
3
1 0
0 1
1 0
×
1
3
2
3
=
2
3
1
3
INPUT OUTPUT INPUT OUTPUT
Classical logic gate Quantum logic gate
Example: 1-bit logic gate vs 1-qubit quantum gate

6. =
1
3
+
2
3
0
1
=
3
4
+
1
4
0
1
=
1
2
+
1
2
1 0 0 1
Entangled state
From 1-qubit to n-qubit: the magic of Entanglement
2 independent 1-qubits don’t
make a 2-qubit
1
2
: probability of measuring
1
2
: probability of measuring
The two 1-qubits are correlated
in their evolution: 2-qubit
2-qubit

7. From 1-qubit to n-qubit: the magic of Entanglement
1 1
𝑎1 … + 𝑎2 … + … + 𝑎2𝑛 …
1 1 0 1 1
0 1
By using Entanglement we can create n-qubits with “n” arbitrary large: the amount of information grows exponentially!
“Giving a complete description of all the correlations among just a few hundred qubits may require more bits
than the number of atoms in the visible universe” – John Preskill
𝑛

8. Example: Shor’s algorithm
Factorisation problem: 𝑁 = 𝑝 ∙ 𝑞 (𝑝, 𝑞 ∶ 𝑝𝑟𝑖𝑚𝑒 𝑛𝑢𝑚𝑏𝑒𝑟𝑠) Given 𝑁, what are 𝑝 and 𝑞 ?
Peter Shor’s idea:
1. Let’s transform the question into a problem for which
using qubits is an advantage
2. If the sequence is represented as a vector, its period can be found through matrix multiplication
(Fourier transformation):
This approach to factorisation is not computationally convenient for classical computers.
However, with quantum computers matrix operations are super fast!
For any number 𝑥 , the sequence
𝑥 𝑚𝑜𝑑 𝑁, 𝑥2
𝑚𝑜𝑑 𝑁, 𝑥3
𝑚𝑜𝑑 𝑁, … repeats itself; if
you can calculate the period of this sequence you
can then derive 𝑝 and 𝑞
𝑎, b, c, 𝑎, b, c, 𝑎, b, c × = 𝑎, 0, 0, 𝑎, 0, 0, 𝑎, 0, 0
Period = 3 Calculate 𝑝 and 𝑞

9. Example: Shor’s algorithm
Peter Shor’s idea:
3. Create a qubit large enough to represent the sequence
4. Apply the Fourier transformation to the qubit to reveal the sequence’s period (Quantum-Fourier gate):
1 1
𝑥 𝑚𝑜𝑑 𝑁 … + (𝑥2
𝑚𝑜𝑑 𝑁) … + (𝑥3
𝑚𝑜𝑑 𝑁) … + …
1 1 0 1 1
0 1
𝑥 𝑚𝑜𝑑 𝑁, 𝑥2
𝑚𝑜𝑑 𝑁, 𝑥3
𝑚𝑜𝑑 𝑁, … ×
Sequence’s
period
Calculate 𝑝 and 𝑞

10. 𝑥 𝑚𝑜𝑑 𝑁, 𝑥2
𝑚𝑜𝑑 𝑁, 𝑥3
𝑚𝑜𝑑 𝑁, … ×
Sequence’s
period
Calculate 𝑝 and 𝑞
Time complexity: ~ 𝑝𝑜𝑙𝑦(log 𝑁)
Best non-quantum algorithm (general number field sieve)
Time complexity: ~ 𝑒𝑝𝑜𝑙𝑦(log 𝑁)
Exponential speed up!
Example: Shor’s algorithm

11. Debunking Quantum Computing
Quantum computers cannot solve
problems that are unsolvable by
classic computers
Quantum computing is not a fancy
version of parallel computing!
Not all classic algorithms can have a
quantum speedup! E.g. NP-hard
problems are not
Universal
Turing machine
Alan Turing
In principle a classic computer can
simulate a quantum computer (although
we don’t know how to do it)
For some problems
maybe QC can provide
fast approximate
solutions
We only have limited access to the qubit’s information

12. Quantum Machine Learning
Matrix Inversion (HHL algorithm)
Recommendation systems
Search problems (Grover’s algorithm)
Applications: LQ regression/SVM/Gaussian
processes/Bayesian Deep Learning
Applications: DB queries, k-medians,
k-nearest neighbours algorithms
~ 𝑝𝑜𝑙𝑦(log 𝑁) , exponential speedup
~ 𝑁 , quadratic speedup
~ 𝑝𝑜𝑙𝑦(log m ∙ n) , exponential speedup
Sampling high-dimensional distributions

13. 1
2
+
1
2
1 0 0 1
1
3
+
2
3
0
1
3
4
+
1
4
0
1
Why is Quantum Computing hard to realise?
Quantum decoherence
Noise from the
external environment
2-qubit
1-qubit
1-qubit
Solution:
Quantum error correction
(but it requires even more qubits!)

14. Physical realisations of Quantum Computers
Superconducting
circuits
Trapped ion
quantum computer
Quantum dot
Optical quantum
computer
Spin of electrons
Energy level of pairs
of electrons
Energy level of
charged atoms
Polarisation and other
characteristics of light
Qubit:

15. Timeline for Quantum Computing
NISQ era
(Noisy Intermediate-Scale Quantum):
• 50-100 qubits (Intermediate-Scale)
• No quantum error-correction (Noisy)
September 2019: Google claims to have reached quantum
supremacy with an array of 54 qubits, which were used to perform a
series of operations in 200 seconds that would take a supercomputer
about 10,000 years to complete

16. Questions?
nik.musmeci87@gmail.com