A.k.a. VQE. Klassisk-kvante hybridalgoritmer hvis mål er å finne grunntilstanden til et system, og som lar seg kjøre på [[NISQ]]-maskiner, med tvilsom suksess. > While variational quantum algorithms may have some role in the future, it is hard to justify their popularity in terms of concrete performance. In chemical matter simulation, we need empirical evidence that they can outperform classical heuristics. – [Garnet Chan](https://youtu.be/BaC1iMV4h70?si=pGjC3l-RjzBhNlOH&t=5784) Har blitt stilt spørsmål ved om ikke variational algorithms kan overkomme en av de fundamentale barrierene, nemlig [[Barren Plateus]], hvis og bare hvis problemet er klassisk simulerbart, se [X-tråd](https://x.com/mvscerezo/status/1735476510595731755?s=46) og [paper](https://arxiv.org/abs/2312.09121). Så vidt jeg forstår *beviste* de at dette er tilfellet hvis man begynner fra en tilfeldig starttilstand—men som [[Gemini Center on Quantum Technology|Franz Fuchs]] påpekte så gjør man jo sjeldent det. #### Algoritme Så vidt jeg forstår fra Garnet Chans [[Kvantekjemi]]-tutorial ved [[QIP'24]] ([del 2](https://youtu.be/BaC1iMV4h70)): I prinsippet veldig enkelt: Start med energilikningen, $E = \langle 0| U^\dagger(\theta) H U(\theta)|0\rangle$, hvor $\theta$ er en eller annen parameter, og tweak den manuelt til du oppnår minimal energi. Dermed intuitivt hvorfor det er en hybridalgoritme: Du velger en $\theta$, klargjør tilstanden på en [[NISQ]]-maskin, måler energien, og gjør en eller annen klassisk beregning (gradient descent e.l.?) for å velge deg en ny $\theta$, prøver igjen, og sammenligner resultatet. Repeat. For å måle energien, så måler du egenverdiene til Pauli-operatorene til $H$. Dette krever $O(N^4)$ operatorer "in standard basis representation of chemical Hamiltonian." Relatert (men teknisk sett ikke VQE) er Quantum imaginary time evolution, som også er en hybridalgoritme og derfor lar seg kjøre på [[NISQ]]-maskiner. #### Popularitet Fra tutorialen igjen: > VQE has become extremely popular among quantum chemists. Why? I think it is because: > 1. It is easy to understand! You don't need to know anything about quantum computing to use it. > 2. Infinite number of circuit architectures you can put into this, and therefore an infinite number of things to do. You can keep yourself occupied for a very long time. > 3. In principle, it is possible that there are unitaries here that provide compact descriptions of ground states for which there aren't classically compact descriptions of the ground state. Angående det siste poenget, se [[Compact-state conjecture]]. #### Problemer Opinions due to Garnet Chan (igjen fra kvantekjemi-tutorialen, mot slutten av del 2): > For chemists who have tried to participate in the quantum *gold rush*, so to speak, they have just all turned to writing endless VQE papers because, as I say, it is very easy to do, but I have this sense that in the more mathematical quantum information community, this is not viewed particularly favourably, and I share that sort of opinion, because there's so many problems. > > One problem is that you have to measure these things, and theres this enormous amount of measurements ($O(N^4)$), and all these Pauli operators have to be measured to high precision. And you can write down these tiny molecules, like lithium hydrite (LiH) in some small basis, and just count the number of measurements, and it's just some number that seems absolutely enormous (e.g. $10^9$). You can reduce this with heuristics, but there is this very daunting prefactor just to get you started with the method, and that's for a diatonic molecule (?). > > And then in addition you have to optimize the circuit, and if you just choose general unitary circuits, they're very hard to optimize (because they suffer from *barren plateus*, i.e. the gradient vanishes with increasing circuit width/depth). And there is even the idea that if you try to choose a circuit which *doesn't* have barren plateus, [perhaps that means that it is classically tractable](https://arxiv.org/pdf/2312.09121.pdf). > > So, from my perspective, I think VQE combines the worst of quantum algorithms, which is reading out, with the worst of stochastic classical algorithms, which is doing non-linear optimization with noise. It's a little bit mysterious why it's so popular already for these two reasons. > > But I actually think there is a third problem with it, which is that, perhaps all this would be OK if it was (sic) really true that unitary circuits somehow offered this really concise description of problems that we wanted. We can *get over* the measurement problem, we can *find ways* to optimize, *but it's not actually clear that unitary circuits are more expressive than other chemical parameterizations for the states we want to study*. There is surprisingly little evidence in physical problems for this! Igjen, se [[Compact-state conjecture]]. Han forteller deretter at det er "one piece of evidence" i dens favør, nemlig når man studerer 1D lattice problems, hvor alle er enige om hva den mest kompakte klassiske beskrivelsen er (nemlig Matrix Product State (DMRG (?))). I et tidligere arbeid sammenlignet de denne representasjonen med hva man kan oppnå med VQE (hvor de sammenlignet flere forskjellige kretsarkitekturer). > ... And the questions that we wanted to then understand is, are these circuits *more expressive* than the classical description? So, that would mean that you would need *fewer parameters* to represent states. And the second thing we wanted to understand is, are these circuits more *computationally efficient*? In other words, can you achieve the same accuracy with lower cost? (Because it could be that the circuit uses very few parameters, but that it is very expensive to work with.) Resultatet de fant var at kvantebeskrivelsene *var* faktisk i dette tilfellet hakket mer konsise (for de beste arkitekturene de fant), i den forstand at man trengte færre parametre for å oppnå samme nøyaktighet: > So, if you choose the best quantum circuits out of the ones that we studied, they are *a bit* more expressive than classical MPS—not by a huge amount! But they are a little bit more expressive. > > But then, when you look at the cost of working with them... Han gikk tom for tid på dette tidspunktet:( Men konklusjon: > You *can* see very, very weak evidence for *small* polynomial advantage for circuits of physical ground state representation. It's very, very small, but there are at least some examples where it is not *zero*. And that's at least a start. Ikke akkurat earth shattering. Her er sliden: ![[IMG_5431.jpeg]]