What is a quantum algorithm?

\hat H \mapsto \hat \mathcal H_\ell = \hat \mathcal U_\ell^\dagger \hat H \hat\mathcal U_\ell

\partial_\ell \hat \mathcal H_\ell = [[A(\hat \mathcal H_\ell),B(\hat \mathcal H_\ell)], \hat \mathcal H_\ell]

0

C

What is a universal quantum computer?

It's an experimental setup which includes a quantum system and that setup allows to manipulate its quantum state.

It's an experimental setup which includes few-level quantum systems and that setup allows to manipulate the quantum state using gates.

These gates form a universal gate set which means that if you apply sufficiently many you will be able to reach any desired quantum state.

What is our quantum computer?

\hat H \mapsto \hat \mathcal H_\ell = \hat \mathcal U_\ell^\dagger \hat H \hat\mathcal U_\ell

\partial_\ell \hat \mathcal H_\ell = [[A(\hat \mathcal H_\ell),B(\hat \mathcal H_\ell)], \hat \mathcal H_\ell]

What is a universal quantum computer?

|\psi\rangle = a | 0 \rangle + b | 1\rangle

|\psi\rangle = a | 0 \rangle + b | 1\rangle

|\psi\rangle = a | 00 \rangle + b | 10\rangle+c | 01 \rangle + b | 11\rangle

|\psi\rangle = a | 000 \rangle + b | 100\rangle+c | 010 \rangle + d | 100\rangle+ e | 011 \rangle + f | 101\rangle+g | 110 \rangle +h | 111\rangle

|\psi\rangle = a | 0000 \rangle + b | 1000\rangle+c | 0100 \rangle + d | 1000\rangle+ e | 0011 \rangle + f | 0101\rangle+g | 1010 \rangle +h | 1100\rangle + \ldots

1 qubit

2 qubits

3 qubits

4 qubits

And you get the idea lah

In the circuit model we will apply gates. Today we will see some special examples.

What are quantum gates?

|\psi\rangle = a | 0 \rangle + b | 1\rangle

|\psi\rangle = a | 0 \rangle + b | 1\rangle

|+\rangle = \frac 1 {\sqrt 2} (| 0 \rangle + | 1\rangle)

|+\rangle = \frac 1 {\sqrt 2} (| 0 \rangle + | 1\rangle)

How to quantum?

What are quantum gates?

|\psi\rangle = | 0 \rangle

|\psi\rangle = | 0 \rangle

Z|0\rangle = | 0\rangle

Z|0\rangle = | 0\rangle

How does it work again?

What are quantum gates?

|\psi\rangle = | 0 \rangle

|\psi\rangle = | 0 \rangle

|+\rangle = \frac 1 {\sqrt 2} (| 0 \rangle + | 1\rangle)

|+\rangle = \frac 1 {\sqrt 2} (| 0 \rangle + | 1\rangle)

Use the quantum computer to transform

into

Trivial quantum algorithm solves it:

Apply the Haddamard gate

H = \frac 1 {\sqrt 2} \begin{pmatrix} 1 & 1 \\ 1 &- 1\end{pmatrix}

H = \frac 1 {\sqrt 2} \begin{pmatrix} 1 & 1 \\ 1 &- 1\end{pmatrix}

0

H

H

1 qubit is boring, 10 qubits are challenging, 100 qubits are worth

the buzz

What are quantum gates?

|\psi\rangle = | 0 \rangle \otimes |0\rangle

|\psi\rangle = | 0 \rangle \otimes |0\rangle

|+\rangle \otimes |+\rangle = \frac 1 { 2} (| 0 0\rangle + | 01\rangle+|10\rangle + | 11\rangle)

|+\rangle \otimes |+\rangle = \frac 1 { 2} (| 0 0\rangle + | 01\rangle+|10\rangle + | 11\rangle)

Use the quantum computer to transform

into

Apply the Haddamard gate on each qubit

H\otimes H

H\otimes H

0

H

H

0

H

H

What are quantum gates?

|\psi\rangle = | 0 \rangle \otimes |0\rangle

|\psi\rangle = | 0 \rangle \otimes |0\rangle

|\Phi\rangle = \frac 1 { \sqrt 2} (| 0 0\rangle + | 11\rangle)

|\Phi\rangle = \frac 1 { \sqrt 2} (| 0 0\rangle + | 11\rangle)

Use the quantum computer to transform

into

Apply the Haddamard gate on each qubit

H\otimes 1

H\otimes 1

and then apply the controlled-not gate

\text{CNOT} = |0\rangle\langle 0|\otimes 1 + |1\rangle\langle1 |\otimes X

\text{CNOT} = |0\rangle\langle 0|\otimes 1 + |1\rangle\langle1 |\otimes X

H

H

0

+

+

What else is there?

What are quantum algorithms?

|\psi\rangle

|\psi\rangle

|\langle \phi |\psi\rangle|^2

|\langle \phi |\psi\rangle|^2

Use the quantum computer to transform

into their fidelity

|\phi\rangle

|\phi\rangle

and

as an encoded in a measured expectation value

How well does it work?

What is quantum compiling?

Controlled-SWAP

Controlled-SWAP

Use the quantum computer to transform

into the

and

CNOT

CNOT

Controlled-CNOT

Controlled-CNOT

https://quantumcomputing.stackexchange.com/questions/9342/how-to-implement-a-fredkin-gate-using-toffoli-and-cnots

New quantum algorithm for diagonalization

\hat H \mapsto \hat \mathcal H_\ell = \hat \mathcal U_\ell^\dagger \hat H \hat\mathcal U_\ell

\partial_\ell \hat \mathcal H_\ell = [[A(\hat \mathcal H_\ell),B(\hat \mathcal H_\ell)], \hat \mathcal H_\ell]

building useful quantum algorithms

new approach to preparing useful states

building useful variational circuits

tons of fun maths in the appendix

no qubit overheads

no controlled-unitaries

0

arXiv:2206.11772

Quantum algorithms tutorial using QIBO

slides.com/marekgluza

Quantum algorithms tutorial using QIBO

slides.com/marekgluza

What is a quantum algorithm?

What is a quantum computer?

What is a quantum algorithm?

What is a universal quantum computer?

What is our quantum computer?

What is a universal quantum computer?

What are quantum gates?

How to quantum?

What are quantum gates?

How does it work again?

What are quantum gates?

1 qubit is boring, 10 qubits are challenging, 100 qubits are worth

the buzz

What are quantum gates?

What are quantum gates?

What else is there?

What are quantum algorithms?

How well does it work?

What is quantum compiling?

New quantum algorithm for diagonalization

arXiv:2206.11772

Bridging the gap between theoretical quantum algorithms and experimental actualities

Material science?

arXiv:2206.11772