A description of basic quantum computing opertions

1. Quantum Computing
Len Bass

2. © Len Bass 2019 2
In the beginning of the
computer age
• Programmers worked in machine
language
• Required a good knowledge of the
architecture of the underlying hardware
• Limited in terms of size and complexity of
programs.

3. © Len Bass 2019 3
In the beginning of the
quantum computer age
• Programmers work in quantum machine
language
• Requires a good knowledge of the
architecture of the underlying hardware
• Limited in terms of size and complexity of
programs.

4. © Len Bass 2019 4
This talk
• Will focus on the logic of the quantum
hardware
• Qubits
• And how you program for this hardware
• Gates
• Operators
• Will sketch some common algorithms

5. © Len Bass 2019 5
What is a bit in a classical
computer?
• A bit has a value of either 0 or 1.
• There is no ambiguity
• Reading a classical bit does not affect
its value.

6. © Len Bass 2019 6
What is a bit (qubit) in a
quantum computer?
• A qubit has the value 0 with some
probability
• It has the value 1 with 1-probability(0)
• i.e you can think of a qubit has having
both values simultaneously

7. © Len Bass 2019 7
A qubit is fundamentally
different from a classical bit
• Measuring (reading) a qubit has the
following effects
• It returns a 0 or a 1. The value returned is
based on probabilities of the values.
• It replaces the original values in the qubit
with the returned value (0 or 1)

8. © Len Bass 2019 8
Copying qubits not possible
• A copy would result in two instances of the
same value
• A copy is a read followed by a store
• A read destroys the value and results in a
0 or a 1
• Therefore – cannot copy qubits
• It is possible to move the value in one qubit to
another qubit at the cost of destroying the
original value

9. © Len Bass 2019 9
Probabilities
• A measurement of a qubit will return 0 with a
probability of α and 1 with a probability of b =
1- α
• I.e. if α = 40% then 4 out of 10 measurements
will return 0 and 6 will return 1 (roughly).

10. © Len Bass 2019 10
Superposition
• The ability of a qubit to hold values with
different probabilities is called
“superposition”

11. © Len Bass 2019 11
Third value for qubit
specification
• A qubit also has a “phase”.
• A phase is a value between 0 and 2π
• The phase provides an additional
handle for manipulating qubits but does
not enter into probabilities for reading 0
or 1.

12. © Len Bass 2019 12
Notations
• Two notations:
• Ket (Dirac) notation |0> for 0 and |1> for 1
• Matrix notation
|0> =
1
0
|1> =
0
1

13. © Len Bass 2019 13
Arbitrary qubit
• An arbitrary qubit g is
• In ket notation |g> = α|0> + b |1>
• In matrix notation g =
α
b
• The phase is captured by making α and b
be complex numbers. We won’t go into
phases.

14. © Len Bass 2019 14
How do you manipulate qubit?
• A qubit is manipulated using an
operator O
• g → O → d
• An operator is realized as a gate.

15. © Len Bass 2019 15
Single qubit operators
• Single qubit operators include
• NOT
• Hanamard
• Rotate

16. © Len Bass 2019 16
Representing qubit operators
• A qubit operator can be represented as
• a gate in a circuit for ket notation or
• a matrix for matrix notation

17. © Len Bass 2019 17
NOT
• NOT reverses α and b
Matrix form Circuit form
0 1
1 0
0
1
1
0 = |0> |1>NOT

18. © Len Bass 2019 18
Hanamard
• Hanamard places the qubit into
superposition with probabilities of 50%
for both α and b
1
2
1 1
1 −1
1
0
=
1
2
1
1
Matrix form Circuit form
|0> H
|0>+|1>
2

19. © Len Bass 2019 19
Rotate
• Rotate changes the phase by π. It does
not affect the probabilities.
Matrix form Circuit form
= |0> |0>Z1 0
0 −1
1
0
1
0

20. © Len Bass 2019 20
Gates can be cascaded
• What is the output of this?
|0> H NOT

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Two qubit notation
• |00> =
• |01> =
1
0
0
0
0
1
0
0
• |10> =
• |11> =
0
0
1
0
0
0
0
1

22. © Len Bass 2019 22
Measuring two qubits
|gh> = a|00> + b|01> + c|10> + d|11>
• Measuring the two qubits will yield two
classical bits (0 0, 0 1, 1 0, 1 1) with
probabilities a, b, c, d
• It will also collapse the two qubits.
Suppose, b is chosen as the value, then
the measured values is 0 1 and g is
collapsed to 0 and h is collapsed to 1.

23. © Len Bass 2019 23
Entanglement
• Suppose |gh> = a|00> + d|11>
• i.e. b and c are 0. (The probabilities for the
middle two terms of the definition.)
• Now measure gh It will be 0 0 or 1 1 with
probabilities a and d.
• Both qubits will have the same value after the
measurement.
• The two qubits are “entangled”

24. © Len Bass 2019 24
Weirdness of entanglement
• A subsequent measurement of h can be
at a different time from a subsequent
measurement of g.
• h can be in a different location than g.
• They will always have the same value.
• Physicists have verified this
phenomenon over kilometers.

25. © Len Bass 2019 25
Exploiting entanglement
• We will use entanglement to
communicate information over
distances
• First we show how to generate
entangled qubits
• This requires the Controlled Not
operation

26. © Len Bass 2019 26
Operation on two qubits
• Controlled Not – CNOT
• The first qubit acts as a control. If it is 0
then there is no change to the second
qubit. If it is 1 then the second qubit is
flipped.

27. © Len Bass 2019 27
CNOT
|0>
|0> CNOT
|0>
|0>
Circuit form
1
0
0
0
1
0
0
0
=
1 0 0 0
0 1 0 0
0 0 0 1
0 0 1 0
Matrix form

28. © Len Bass 2019 28
Bell Pair
• Entangling two qubits can be done by
creating a Bell Pair
|0> H
|00>+|11>
2===
=
|0> CNOT

29. © Len Bass 2019 29
Quantum teleportation
Three qubits involved: A, B, 
• A and  are in one location. B in
another.
•  Is teleported to B and is destroyed in
the process.
• Although a qubit cannot be copied, its
contents can be moved at the cost of
destroying the original qubit.

30. © Len Bass 2019 30
Steps in teleportation - 1
1. Entangle qubits A and B.
2. Prepare the payload. The payload
qubit  will have the state to be
teleported.

31. © Len Bass 2019 31
Steps in teleportation - 2
3. Propagate the payload. The
propagation involves two classical bits
that are transferred to the location of B.
The propagation also involves
measuring A and  This will destroy
the state of both of these qubits.
4. Recreate the state of  in B.

32. © Len Bass 2019 32
Why is this interesting?
• It is not possible to copy qubits but we can
transfer the state of a qubit to a different
location at the cost of destroying the state of
the original qubit.
• This will be th the basis of quantum based
communication protocols.
• NIST is currently considering creation of a
httpq protocol to ultimately replace https

33. © Len Bass 2019 33
Other quantum algorithms
• Other algorithms exist that are not
currently realizable
• Notably:
• Grover’s – breaks password hash
• Shor’s – breaks RSA encryption
• HHL – matrix inversion – used in machine
learning
• …

34. © Len Bass 2019 34
Grover’s Algorithm
• Computes inverse of a function – in particular a hash
function
• Uses superposition to identify inverse of a value
• Problem is the identified value has same probability
of being measured as all of the other values
• Phase manipulation and amplitude amplification are
used to increase probability of measuring inverse
value.

35. © Len Bass 2019 35
Shor’s algorithm
• Uses number theory results to break
RSA
• Quantum used to find period of an
exponential factor.
• Results are not directly readable but
must be inferred

36. © Len Bass 2019 36
HHL
• Inverts large matrices.
• Used in machine learning
• Uses Amplitude amplification as in
Grover’s

37. © Len Bass 2019 37
Current state of quantum
computers
• Google has announced they have
achieved “quantum supremacy”
• This means they have demonstrated a
problem that can be solved
exponentially faster on a quantum
computer than on a classical computer.
(the problem is not interesting)

38. © Len Bass 2019 38
Future state
• Suppose Moore’s Law holds (exponential
growth in computing power over time)
• Then quantum computers will become real in
the 5-10 year time frame.
• Quantum computers will NOT replace
classical computers.
• Quantum computers will be used for
problems involving combinatorics.

39. © Len Bass 2019 39
Summary
• Qubits are the basic computation unit of
a quantum computer
• Qubits can be in superposition
• Qubits can be entangled.
• Algorithms exist for some problems
intractable on classic computers but, as
yet, none of them are realizable.