Quantum computing and information processing

Plethora of different kinds of computing devices and computing models besides ordinary CPUs

• Abacus (counting frame)
• GPUs
• TPUs
• DNA computing
• Processors in Minecraft

and quantum computers.

Quantum computing works on various hardware

Although the hardware varies the abstract model of quantum computing is quite standard (circuit):

• IQM (circuit)
• IBM (circuit, superconductor)
• Google (circuit, superconductor)
• IonQ (circuit, trapped ion)
• Honeywell with Quantinuum (circuit, trapped ion)
• Rigetti (circuit, integrated)
• Microsoft (circuit, topological?)
• Xanadu (circuit, photonic)
• D-wave (annealing & circuit coming)

Why does it work?

The abstract foundation of quantum computing is based on quantum mechanics.

See section Introduction to quantum mechanics in book Quantum Computation and Quantum Information 10th Anniversary Edition by Nielsen and Chuang.

Approaching from machine learning perspective

You have developed a machine learning model to classify cats and dogs.

Model outputs that the image has 80% prob of represeting cat and 20% prob of representing a dog.

Approaching from machine learning perspective

If you modify the model, you want:

the prob of cat + the prob of dog = 100%

Making things more abstract:

Instead of cats and dogs, we have vectors.

Instead of probabilites, we have complex numbers which implicitly encode the probabilities.

Single qubit systems

$|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$

where $\alpha,\beta \in \Complex$ are amplitudes and

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

The states $|0\rangle = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$ and $|1\rangle = \begin{bmatrix} 0 \\ 1 \end{bmatrix}$ are called computational basis.

Multiple qubits

\[\begin{aligned} |\psi\rangle &= (a_0|0\rangle + b_0|1\rangle) \otimes (a_1|0\rangle + b_1|1\rangle) \\ & = a_0a_1|00\rangle + a_0b_1|01\rangle + b_0a_1|10\rangle + b_0b_1|11\rangle. \end{aligned} \]

We see that $n$ qubit system has $2^n$ amplitudes (i.e. the coefficients).

Evolution of quantum systems

Initial system of qubits (usually $|00\ldots0\rangle$)

Apply gates the qubits. Gates are modelled as unitary matrixes. The idea is that the sum of the squared norms of complex amplitudes is preserved.

Measure the system. The system ends up to some classical state. The sequence of classical bits is the result of the quantum computation.

Two big advantages of quantum computing

Superposition

One qubit can be in multiple states simultaneously.

For example, the most common method to put a qubit to a superposition is to apply Hadamar-gate

$H|0\rangle = \frac{1}{\sqrt{2}}\begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}\begin{bmatrix} 1 \\ 0 \end{bmatrix} = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 \\ 1 \end{bmatrix} = \frac{|0\rangle + |1\rangle}{\sqrt{2}}$.

Entanglement

Entanglement roughly means that two or more qubits are connected and they can interact in ways such that the state of each qubit cannot be described independently of the state of the other qubits.

For example, the most common method to entangle two qubits is to use CNOT gate:

CNOT$(|x\rangle \otimes |y\rangle) = |x\rangle \otimes |x + y \mod 2\rangle$.

Some challenges in quantum information processing

that are related to "data"

No-cloning theorem

Theoretically provable theorem that quantum states cannot be copied or cloned.

See Box 12.1: The no-cloning theorem in book Quantum Computation and Quantum Information.

Measuring destroys states

Measuring a qubit collapses it to either 0 or 1. After the measurement we cannot modify the measured qubit anymore.

Error

Quantum computing is never performed in absolutely isoleted system and the environment creates decoherence and other quantum noise.

We can tackle the error by developing quantum error correction.