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Registers

n bits register	n qubits register
$\text{color}red{2}^n\text{ possible states }\mathbf{\text{once at a time}}$	$\text{color}green{2}^n\text{possible states }\mathbf{\text{linearly superposed}}$
evaluable	partially evaluable
independant copies	no copies
individually erasable	non individualy erasable
non destructive readout	value changed after readout
deterministic	probabilistic

Gates

Classical logic gates

Irreversible gates:

• NAND
• NOR
• AND
• OR

Quelle est leur consequence ?

Comme on perd un bit, on a une perte d'energie
Decouverte par Rolf Landauer

Des gens travaillent aujourd'hui pour creer une informatique classique sans perte d'energie

Quantum gates

Matrix based reversible unitary transformations

• NOT: rotation
  $X$
• Rotation
  $Y$

$$\left[\begin{array}{cc}0& -i\\ i& 0\end{array}\right]$$

• Pauli-Z: rotation
  $Z$
• Hadamard: superposition

Porte CNOT

On va changer la valeur d'un qubit en fonction d'un autre

Mathematiquement, a quoi ca ressemble ?

$$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 1& 0\end{array}\right]$$

Si on intrique des qubits a des portes a 2 qubits, est-ce que ca reste ?

Oui

C2NOT

$$\left[\begin{array}{cccccccc}1& 0& 0& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0& 0& 1& 0\end{array}\right]$$

SWAP

$$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 1\end{array}\right]$$

Fredkin

Conditional SWAP

$$\left[\begin{array}{cccccccc}1& 0& 0& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 1\end{array}\right]$$

Single qubit operations visualization

CNOT gate effect

$$\begin{array}{ccc}\text{color}blue\text{control qubit}& & \text{color}blue\text{tensor product of control and target qubits before CNOT}\\ {\alpha }_1|0⟩& & {\alpha }_1{\alpha }_1|00⟩+{\alpha }_1{\beta }_2|01⟩+{\alpha }_2{\beta }_1|10⟩+{\beta }_1{\beta }_2|11⟩\\ ⨂& \Rightarrow & \text{CNOT}\\ {\alpha }_2|0⟩+{\beta }_2|1⟩& & {\alpha }_1{\alpha }_1|00⟩+{\alpha }_1{\beta }_2|01⟩+{\alpha }_2{\beta }_1|11⟩+{\beta }_1{\beta }_2|10⟩\\ \text{color}blue\text{target qubit}& & \text{color}blue\text{control and target qubits state after CNOT}\\ \text{color}blue\text{control qubit is }|0⟩\\ {\alpha }_1=1& & {\alpha }_2|00⟩+{\beta }_2|01⟩\\ & \Rightarrow & \text{CNOT}\\ {\beta }_1=0& & {\alpha }_2|00⟩+{\beta }_2|01⟩\end{array}$$

CNOT is not changing the qubit

The EPR pair entanglemet building block

Put control qubit into superposition state, then future gates act on 2 states simultaneously

$$\frac{|0⟩+|1⟩}{\sqrt{2}}$$

$$\}\frac{|00⟩+|11⟩}{\sqrt{2}}$$

Subsenquently, flipping a qubit in an entangled state modifies all of tis components

Control-U gate

On prend une porte U qui est une porte arbitraire

$$\left[\begin{array}{cccc}1& \dots & \dots & \dots \\ \dots & 1& \dots & \dots \\ \dots & \dots & U_{11}& U_{12}\\ \dots & \dots & U_{21}& U_{22}\end{array}\right]$$

Qubit lifecycle

• Initialization
  • $|0⟩$
• Hadamard gate
  • $\frac{|0⟩+|1⟩}{\sqrt{2}}$
• Other gate
  • aubit vector turning around in Bloch sphere
• Measurement
  • Measurement returns
    $|0⟩$ qith a probability
    ${\alpha }^2$ depending on the qubit state, then qubit state becomes
    $|0⟩$
  • Measurement returns
    $|1⟩$ with a probability
    ${\beta }^2$

Universal gates sets

Jeu de portes universel
Jeu de portes simples qu'on peut combiner pour recreer toutes les transformations unitaires

Ex: CNOT peut etre recree avec HZH
Three CNOT gates: one SWAP gate

Universal quantum computing requires a T gate (

$\frac{\pi }{4}$ rotation)

Getting confused with phase rotations

• One round =
  $2\pi$
• $S=$ one quarter round
  $=\frac{\pi }{2}$
• $T=$ one eight roung

Solovay-Kitaev theorem

Theorem

Any desired gate can be approximated by a sequence of gates from an universal gates set.

A quantum circuit of

$m$ constant-qubit gates can be approximated to

$\epsilon$ error by a quantum circuit of

$O(m{\log }^c(\frac{m}{\epsilon }))$ gates from a desired finite universal gate set with

$c=3,97$

For example, creating a

$R_{15}$ gate requires

$127$ H/Z/T gates

In other words

On veut appliquer a

$n$ qubits n'importe quelle operation generique

$U$, on enchaine une serie de transformations unitaires.

$SU(2^n)$ - Space of unitaries on

$n$ qubits

Espace contenant toutes les transformations

On reversibility

All quantum gates are mathematically reversible, this is a property of the matrix linear transformations

We could theortically run an algorithm and rewinf it entirely to return to the initial state, which could help recover port of the energy spent in the system

On a practical basis:

• The gates are not physically and thermodynamically reversible due to some irreversible processes like micro-wave generations and DACs (digital analog converters)
• The whole digital process taking place before micro-wave generation and after their readout conversion back to digital could be implemented in classical adiabatic\thermodynamically reversible fashion
• Currently being investigated at Sandia Labs, Wisconsin University and with SeeQC

Inputs and outputs

Probabilistic or deterministic readouts ?

A single qubit measurement is probabilistic, ie: a qubit registered after a Hadamard gate applied to all qubits is a simple random numbers generator

On a practical basis:

• the algorithm is executed many times, up to 8000 for IBM Q Experience
• an average of qubits results is computed, producing a real number
• the averahed result is theoratically deterministic
• modulo the error generated by noise and decoherence

Basis, pure and mixed states

Examples

Normalement vous avez rien compris
Olivier EzrattyTue, Oct 5, 2021 3:55 PM

L'origine aleatoire du photon provient de la physique classique et non quantique

Single qubit mixed state

Toying with density matrices

Qubits measurement

Measurement is using a collection

$M_m$ of operators acting on the measured system state space

$|\psi ⟩$, with probability of

$m$ being:

$$p(m)=⟨\psi |M_m^{✝}M+m|\psi ⟩$$

System state after measurement becomes:

$$\frac{M_m|\psi ⟩}{\sqrt{⟨\psi |M_m^{✝}M+m|\psi ⟩}}$$

with:

$$\sum _m{M}_m^{✝}M+m=1$$

Various qubits measurement methods

Computing semantics summary

5 DiVienzo criteria (IBM, 2000)

Main qubit types

From lab to packaged computers

Les ordinateurs quantiques actuels d'IBM:

L'ordinateur version commerciale:

Il y a un cube derriere qui contient l'ordinateur

IBM pense atteindre

$1000$ qubits d'ici 2 ans, mais ca a pas trop l'air possible car au-dessus de

$28$ qubits il y a une enorme perte de qualite.

Inside a typical quantum computer

En resume: 4 composantes

Avec des atomes froids, on n'aurait pas des compresseurs mais des pompes a ultra-vide.

Chez Google

Pourquoi les fils tournent ?

Pour passer plus de temps dans le froid ?

Systeme de dilatation thermique du au changement de temperature hardcore

• Refroidit: contracte
• Rechauffement: dilate

Pourquoi plusieurs etages ?

On est a

$300K$ a l'exterieur, on veut minimiser plusieurs poches
Chaque etage = une temperature
Chaque disque a une taille plus petite en descendant les etages, pour faire passer le moins de chaleur possible
Chaque etage est isole de celui au-dessus
Les fils sont des attenuateurs de puissance mais ils generent de la chaleur

C'est l'isolation thermique

Quantum computer architecture

Physical layout example

Error correction

Each quantum gate and readout generate significant errors

Coming form decoherence generated by:

• flip, phase and leakage error
• calibration errors
• thermal noise
• electric and magnetic noise
• gravity
• radioactivty
• vacuum quantum fluctuations
• cosmical rays

It accumulates with the number of quantum gates and qubits

QEC zoo