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Registers
| n bits register | n qubits register |
|---|---|
| $\color{red}{2^n\text{ possible states } \textbf{once at a time}}$ | $\color{green}{ 2^n \text{possible states }\textbf{linearly superposed}}$ |
| 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 $Y$
- 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 ?
\[\begin{bmatrix} 1&0&0&0\\ 0&1&0&0\\ 0&0&0&1\\ 0&0&1&0 \end{bmatrix}\]
Si on intrique des qubits a des portes a 2 qubits, est-ce que ca reste ?
Oui
C2NOT
SWAP
Fredkin
Conditional SWAP
Single qubit operations visualization
CNOT gate effect
\[\begin{matrix} \color{blue}{\text{control qubit}} &&\color{blue}{\text{tensor product of control and target qubits before CNOT}}\\ \alpha_1\vert0\rangle &&\alpha_1\alpha_1\vert00\rangle+\alpha_1\beta_2\vert01\rangle + \alpha_2\beta_1\vert10\rangle+\beta_1\beta_2\vert11\rangle\\ \bigotimes&\Rightarrow&\text{CNOT}\\ \alpha_2\vert0\rangle+\beta_2\vert1\rangle&&\alpha_1\alpha_1\vert00\rangle+\alpha_1\beta_2\vert01\rangle + \alpha_2\beta_1\vert11\rangle+\beta_1\beta_2\vert10\rangle\\ \color{blue}{\text{target qubit}}&&\color{blue}{\text{control and target qubits state after CNOT}}\\ \color{blue}{\text{control qubit is }\vert0\rangle}\\ \alpha_1=1&&\alpha_2\vert00\rangle+\beta_2\vert01\rangle\\ &\Rightarrow&\text{CNOT}\\ \beta_1=0&&\alpha_2\vert00\rangle+\beta_2\vert01\rangle\\ \end{matrix}\]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
\[\frac{\vert0\rangle+\vert1\rangle}{\sqrt 2}\]
\(\biggr\}\frac{\vert00\rangle+\vert11\rangle}{\sqrt{2}}\)
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
Qubit lifecycle
- Initialization
- $\vert0\rangle$
- Hadamard gate
- $\frac{\vert0\rangle + \vert1\rangle}{\sqrt{2}}$
- Other gate
- aubit vector turning around in Bloch sphere
- Measurement
- Measurement returns $\vert 0\rangle$ qith a probability $\alpha^2$ depending on the qubit state, then qubit state becomes $\vert0\rangle$
- Measurement returns $\vert1\rangle$ 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 $\varepsilon$ error by a quantum circuit of $O(m\log^c(\frac{m}{\varepsilon}))$ 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 [name=Olivier Ezratty] [time=Tue, Oct 5, 2021 3:55 PM] [color=#907bf7]
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 $\vert\psi\rangle$, with probability of $m$ being:
\[p(m)=\langle\psi\vert M_m^✝M+m\vert\psi\rangle\]System state after measurement becomes:
\[\frac{M_m\vert\psi\rangle}{\sqrt{\langle\psi\vert M_m^✝M+m\vert\psi\rangle}}\]with:
\[\sum_mM_m^✝M+m=1\]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
