adding link to overleaf

thesis/link_to_overleaf.txt

+https://fr.overleaf.com/read/tptxydgtnshx

thesis/notes_QAOA.md

+QAOA
+
+# Résumé:
+
+QAOA: combinatorial optim
+
+$p$ higher better approximation
+
+Tests on max-cut
+
+## Intro
+
+$n$ bits, $m$ clauses
+$C$:Objective function is satisfying as many clauses
+
+QC $2^{n}d$ Hilbert space (space defined by unit vector)
+computational basis (?): $|z>$
+Objective function == operator diagonal in the basis.
+
+operator $U(C, \gamma) = \exp{-i\gamma C} = \prod_{\alpha = 1}^{m}{\exp{-i\gamma C_{\alpha}}}$
+
+Commutativity because diagonal
+Term locality == $\alpha$
+
+Integral Eigen-values (?)-> $0 < \gamma < 2\pi$
+
+$B$ sum of $\sigma^{x}$ -> single bit operators (spins of qbits)
+
+$U(B, \beta) = \exp{-i\beta B} = \prod_{\j = 1}^{m}{\exp{-i\beta \sigma^{x}_{\alpha}}}$ with $0 <= \beta <= \pi$
+
+Initial state $|s> = \frac{1}{\sqrt{2^{n}}}\sum_{z}|z>$
+
+Quantum state defines as
+$|\gamma, \beta> = U(B, \beta_{p})U(C, \gamma_{c}) ... U(B,\beta_{1}) U(C, \gamma_{1}) |s>$
+
+This state can be obtained with depth $mp + p$
+
+$F_{p}$ expectation of $C$
+
+$M_{p} = \max_{\gamma, \beta} F_{p}(\gamma, \beta)$
+
+Algo idea :
+(1) fix a $p$
+(2) select $(\gamma , \beta)$ making $F_{p} as large as possible (?).
+(3) run the circuit to get state $|\gamma, \beta>$.
+
+(4) measure in basis for $z$ (?)
+(5) evaluate $C(z)$
+(6) repeat so that $C(z) ~ or > F_{p}(\gamma, \beta)$
+
+With $p !!= n$, grid search for best $(\gamma, \beta)$ on grid over $[0,2\pi]^{p} \times [0, \pi]^{p}$
+
+-> Bounded by $O(m^{2} + mn)$
+
+Run the state for optimized $(\gamma, \beta)$ -> $z$
+Mean of $C(z) == M_{p}$
+
+## Fixed p algo:
+
+For a fixed p, classical preprocessing to find best $\gamma , \beta$.
+
+Example with Maxcut:
+
+qbits only interact in pairs for p = 1.
+
+p fixes the maximum distance at which an edge is taken into account.
+
+Estimation on classical computers indepedent of $n$
+
+difficult stats.
+
+## Concentration
+
+For $v$ and $p$ fixed, distribution of $C(z)$ contracted near its mean.
+
+-> probability and approx ratios
+
+
+## Ring of disagrees
+
+Performances on MaxCut on 2-regular graphs
+-> Graph == ring
+
+for $p <= n/2$, for each edge, subgraph == segment of size 2p+2
+
+For each p, we only have subgraph line of 2p+2
+qbits
+
+We can increase the circuit depth by splitting the edge sum in C into 2 sums over $<j, j+1>$ (?)
+
+### 3-regular graph
+
+Shapes : isolated triangles or squares
+
+better ratio if graph of size 4
+
+
+
+## Relation to Quantum Adiabatic Algorithm
+
+QAA exact solver, can be exponentially long.
+
+## Variant