God really plays dice
Introduction to Quantum Computing

strathweb.com • @filipw@mathstodon.xyz • Filip W

Quantum Computing

Nature isn't classical, dammit, and if you want to make a simulation of nature, you'd better make it quantum mechanical.

Quantum computer

a device capable of performing computation using the principles of quantum information theory

Quantum strangeness

❓ Superposition
Quantum objects be in multiple different "superposed" states
🧶 Entanglement
Stronger than classical correlations, nonlocality

📡 Wave-particle duality
Quantum objects behave both as waves and point-like objects
🎲 Randomness
Irreducible role of probabilities

Quantum Computing milestones

📝 David Deutsch, 1985
first universal quantum model for computation
💯 Peter Shor, 1994
Shor's algorithm for integer factorization

🖥 Chuang et al. 1998
first experimental 2-qubit quantum hardware
☁️ IBM, 2016
first quantum computing cloud platform

Hybrid computing

Quantum hype

💥 NISQ era
noisy intermediate-scale quantum computers
⚠️ only QPUs
there are only quantum processors - no storage or memory, and they are severely limited in input and output
🔬 specialized use cases
applicability will likely be limited to specialized uses cases where quantum algorithms can provide dramatic speedup

Chemistry

subatomic modeling, molecular simulations, drug discovery

Materials science

problem optimization, energy research, simulation and discovery

Security

secure communication, cryptoanalysis

Machine Learning

quantum classifiers, hybrid quantum/classical training

Financial Services

risk analysis, complex simulations, portfolio optimization

Q# and Azure Quantum

Quantum Dev Kit

💻 Q# and Python support
Use a quantum-specific or a popular language
📦 Familiar to .NET developers
VS Code and VS support, dotnet CLI, Nuget, MSBuild
📚 Rich family of libraries
Math, machine learning, chemistry
🔬 Standalone & Jupyter support
Simulate locally, develop in IDE or Jupyter Notebooks

Azure Quantum

💻 Q#, Qiskit, Cirq support
Cloud platform for QIR
⚛️ Quantum Hardware
Run programs on variety of real quantum hardware
🤖 Simulators
Simulate jobs on vendor-specific simulators
💸 Free credits
At the moments $500 free credits to try things

Azure Quantum Execution

                            // queue job
az quantum job submit --target-id {target}

// check job status
az quantum job show --job-id {id}

// fetch job output
az quantum job output --job-id {id}

Result     Frequency
---------  -----------  ----------------------
[0,1,0,0]  0.60000000   |████████████        |
[0,0,1,0]  0.20000000   |████                |
[1,0,1,0]  0.20000000   |████                |

Azure Quantum Execution

Basic Theory

In classical computing a bit can have only two discrete values

0 or 1

In quantum information theory an information unit is called a qubit.

The qubit is described by a two dimensional state vector, where $\alpha$ and $\beta$ are complex numbers.

$\begin{bmatrix} \alpha \cr \beta \end{bmatrix}$

$$|\alpha|^2 + |\beta|^2 = 1$$

To describe the state of a multi-qubit system, we use a tensor product of the individual states.

$$ \begin{bmatrix}\alpha_1 \\ \beta_1 \end{bmatrix} \otimes \begin{bmatrix}\alpha_2 \\ \beta_2 \end{bmatrix} = \begin{bmatrix}\alpha_1\alpha_2 \\ \alpha_1\beta_2 \\ \beta_1\alpha_2 \\ \beta_1\beta_2 \end{bmatrix} $$

A quantum state vector for $n$ qubits has

$2^n$ dimensions

which makes quantum simulations on classical devices almost impossible

To simulate a molecule of penicillin we need

$10^{86}$ bits

which exceeds the amount of atoms in the universe

The same simulation can in principle be done on a

286 qubit

error-free quantum computer.

Similarly to a bit, a qubit has two distinguishable states.

$$ \begin{bmatrix} 1 \cr 0 \end{bmatrix} = \ket{0} = 0 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \begin{bmatrix} 0 \cr 1 \end{bmatrix} = \ket{1} = 1 $$

Together, we call them the $\mathbf{Z}$ basis

Qubit state can be visualized using a Bloch sphere, where $\ket{0}$ and $\ket{1}$ are vectors starting at the center and pointing to "north pole" and "south pole".

All other states are a linear combination of $\ket{0}$ and $\ket{1}$. This is a superposition.

$$\ket{\psi} = \alpha\ket{0} + \beta\ket{1}$$

A superposition state is

not directly observable

Only the basis states can be observed in the measurement process

Upon observation, superposition state collapses to one of the observable states (e.g. $\ket{0}$ or $\ket{1}$).

outcome is probabilistic

We refer to $\alpha$ and $\beta$ as probability amplitudes. They can be converted to probabilities using the Born rule.

$$P(0) = P(\ket{0}) = |\alpha|^2$$ $$P(1) = P(\ket{1}) = |\beta|^2$$

$$|\alpha|^2 + |\beta|^2 = 1$$

Program evolution

type		input		execution		output
Classical		010	$\rightarrow$	110	$\rightarrow$	001
Quantum		$\ket{1}\ket{0}\ket{0}$	$\rightarrow$	$\ket{\psi}$	$\rightarrow$	$\ket{0}\ket{1}\ket{0}$

I am convinced that God does not play dice

Circuits

Classical circuits

Classical computation models are expressed by logic gates (e.g. NOT, OR, XOR, AND) composed into circuits.

Quantum circuits

Quantum computation circuits are composed of reversible linear operators. They are also referred to as gates.

$\mathbf{H}$ Gate

Hadamard gate is used to create a uniform superposition from a state $\ket{0}$ or $\ket{1}$.

$$ \mathbf{H}=\frac{1}{\sqrt{2}}\begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}$$

$$\mathbf{H}\ket{0} = \frac{1}{\sqrt{2}}\begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}\begin{bmatrix} 1 \\ 0 \end{bmatrix} = \frac{1}{\sqrt{2}}\ket{0} + \frac{1}{\sqrt{2}}\ket{1}$$

$$\mathbf{H}\ket{1} = \frac{1}{\sqrt{2}}\begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}\begin{bmatrix} 0 \\ 1 \end{bmatrix} = \frac{1}{\sqrt{2}}\ket{0} - \frac{1}{\sqrt{2}}\ket{1}$$

Upon measurement, there is an equal 50% probability of a collapse to $\ket{0}$ or a $\ket{1}$.

$$\mathbf{H} \times \mathbf{H} = \begin{bmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \end{bmatrix} \times \begin{bmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = I $$

Applying $H$ a second time, without a measurement, will return the qubit to its earlier state due to interference. This results in the randomness disappearing again.

The strength of quantum computers is that they are able to take algorithmic advantage of superposition. This is called

quantum parallelism

...quantum parallelism allows quantum computers to evaluate a function $f(x)$ for many different values of x simultaneously.

Deutsch's problem - given an unknown function

$f : \{0,1\} \rightarrow \{0,1\}$

determine if the function is constant or balanced

$f_0$ is constant
$f_1$ is constant
$f_2$ is balanced
$f_3$ is balanced

$f_0(0) = f_0(1) = 0$
$f_1(0) = f_1(1) = 1$
$f_2(0) = 0 \;\;\; f_2(1) = 1$
$f_3(0) = 1 \;\;\; f_3(1) = 0$

Classically, two function evaluations are needed.

After the first execution, we still cannot say if the function is balanced or constant.

On a quantum computer, thanks to interference, the answer can be obtained in just one call.

The linear algebra for the circuit produces (for the measured qubit)

$$\ket{\psi} = \begin{cases} \; \ket{0} \;\;\;\; \text{when f(0) = f(1)} \\ \; \ket{1} \;\;\;\; \text{when f(0)} \neq \text{f(1)} \\ \end{cases} $$

This solves the problem in a single shot.

Entanglement

$\mathbf{CNOT}$ Gate

Applies the NOT logic on one qubit, based on the state of another qubit. It can create entanglement.

$$ \mathbf{CNOT}=\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{bmatrix}$$

$\mathbf{CNOT}$ flips the target qubit if the control qubit is in state $\ket{1}$

$$\mathbf{CNOT}(\ket{0}\ket{0}) = \ket{0}\ket{0}$$

$$\mathbf{CNOT}(\ket{0}\ket{1}) = \ket{0}\ket{1}$$

$$\mathbf{CNOT}(\ket{1}\ket{0}) = \ket{1}\ket{1}$$

$$\mathbf{CNOT}(\ket{1}\ket{1}) = \ket{1}\ket{0}$$

But what if the control qubit was in a

superposition?

Then we have an entangled state.

$$\mathbf{CNOT} \biggl(\mathbf{H}\ket{0}\ket{0}\biggr) = \frac{1}{\sqrt{2}}\ket{0}\ket{0} + \frac{1}{\sqrt{2}}\ket{1}\ket{1}$$

Upon measurement of either qubit, there is an equal 50% probability of a state collapse to $\ket{00}$ or $\ket{11}$.

Entangled states

cannot be decomposed

into a tensor product of the individual quantum states.

A single quantum state can span multiple quantum objects which are

spatially separated

Spooky action at a distance

We call this

non-locality

It is based on the Bell's theorem from 1964.

Exploring entanglement in algorithms is a critical factor contributing to computational advantage of quantum computing.

Thank you

📝 filipw.github.io/azurebootcamp-2023
💻 github.com/filipw/2023-azurebootcamp-demos
📚 My Quantum Computing book
📸 Photos credit: CERN

God really plays dice Introduction to Quantum Computing

Quantum Computing

Quantum computer

Quantum strangeness

Quantum Computing milestones

Hybrid computing

Quantum hype

Chemistry

Materials science

Security

Machine Learning

Financial Services

Q# and Azure Quantum

Quantum Dev Kit

Azure Quantum

Azure Quantum Execution

Azure Quantum Execution

Basic Theory

0 or 1

$\begin{bmatrix} \alpha \cr \beta \end{bmatrix}$

$2^n$ dimensions

$10^{86}$ bits

286 qubit

$$\ket{\psi} = \alpha\ket{0} + \beta\ket{1}$$

not directly observable

outcome is probabilistic

Program evolution

Circuits

Classical circuits

Quantum circuits

$\mathbf{H}$ Gate

quantum parallelism

$f : \{0,1\} \rightarrow \{0,1\}$

Entanglement

$\mathbf{CNOT}$ Gate

superposition?

cannot be decomposed

spatially separated

non-locality

Thank you

God really plays dice
Introduction to Quantum Computing