# Smart person needed for a PhD!

I am now looking for a PhD student to work with me, starting next year (December 2022 at the latest). I have the money, I just need a smart person. The ideal topic, described in more detail in the offer below, would be tensor networks and their use in non-perturbative quantum field theory. However, I am open to inquiries on any other of my past subjects, so if there is something you are interested in, feel free to contact me!

The best is to start with an internship, to get a feel for the subject. This is why it is important to contact me asap if you’re interested, even before the offer is formally advertised on the website of the école doctorale.

# Various news

1- Since October 4th, I am back in Paris with a position at Mines Paristech, which is a French “grande école” part of Paris Science et Lettres (PSL). I am also a member of the Quantic group, which is a joint venture with Inria and École Normale Supérieure exploring superconducting circuits for quantum information processing (theoretically and experimentally). I am very happy about this move back to Paris after 5 great years in Munich where I learned immensely and became more of an “adult” theoretical physicist. In Paris I will keep an interest in my old subjects: quantum field theory with variational methods, many body physics, quantum foundations (and others..). But I will try to develop tighter connection between state compression methods (tensor networks) and what is done in the group especially with cat qubits and error correction. For all these subjects, I will probably have some money soon from various sources. Students interested in any of these themes (even the exotic stuff) should feel free to contact me in advance to discuss possible PhDs or internships.

2- I gave a rather conceptual talk on collapse models at the Quantum Boundaries workshop. The talk was recorded:

3- For those who are in Paris, I will speak at Foundation 2020 (which was delayed because of covid) on Thursday October 28th. I will present our latest results with Howard Wiseman on the equivalence between non-Markovian collapse models and Bohmian theories (this preprint).

# Notes on continuous tensor networks

For the European Tensor Network school 2021 at ICCUB in Barcelona, I started to write some lecture notes on continuous tensor network states. So far they cover only continuous matrix product states, which is a pretty stabilized subject. I tried to make them very detailed about the computations which can be scary for beginners. I plan (or rather hope) to extend the notes to include more recent developments like relativistic extensions in 1 space dimension, and non-relativistic extensions in 2 space dimensions and more.

# Une discussion sur la gravité (non)-quantique

J’ai discuté il y a quelques jours avec Vincent Debierre (du podcast Libaca) du problème de l’unification (ou de la jonction) entre modèle standard et gravité. J’ai surtout essayé de clarifier la logique : qu’est-ce qu’on cherche à faire, qu’est-ce qui est a priori possible et qu’est-ce qui ne l’est pas ?

Pour ceux qui peuvent passer outre ma grammaire floue et mon élocution un peu non-linéaire, c’est un moyen d’avoir une idée de ce domaine d’un point de vue peut-être hétérodoxe. Je suis aussi preneur de commentaires éventuellement critiques sur la présentation du problème, car je ne suis évidemment pas sans biais.

À la fin, je mentionne aussi les réseaux de tenseurs. C’est une nouvelle méthode pour résoudre numériquement des modèles existants difficiles (et non un nouveau modèle de la nature) qui est assez à la mode et qui m’occupe aujourd’hui davantage que la gravité. Je suis curieux de savoir si la philosophie en est compréhensible en quelques phrases.

# Progress in the variational method for QFT

Today I put online a major update of my pair of papers on the variational method for quantum field theory (short here, long here). The idea is still to use the same class of variational wave functions (relativistic continuous matrix product states) to find the ground state of (so far bosonic) quantum field theories in 1+1 dimensions. The novelty comes from the algorithm I now use to compute expectation values, that has a cost only proportional to $D^3$ where $D$ is the bond dimension. Using backpropagation techniques, the cost of computing the gradient of observables with respect to the parameters is also only $D^3$. This is basically the same asymptotic scaling as standard continuous matrix product states.

Previously, the method was a nice theoretical advance as it worked without any cutoff, but it was not numerically competitive compared to bruteforce discretization + standard tensor methods (at least not competitive for most observables insensitive to the UV). With the new algorithm with improved scaling, I can go fairly easily from $D=9$ to $D=32$, which gives $\simeq 10^{-5}$ relative error for the renormalized energy density at a coupling of order 1. Crucially, the error really seems to decrease exponentially as a function of the bond dimension, and thus with only a slightly higher numerical effort (say 100 times more) one could probably get close to machine precision. Already at $10^{-5}$ relative error for the renormalized energy density, that is a quantity where the leading lattice contribution has been subtracted, I doubt methods relying on a discretization can compete. This makes me a bit more confident that methods working directly in the continuum are a promising way forward even if only for numerics.

Now, let me go a bit more into the technicalities. Computing expectation values of local functions of the field $\phi$ with such a low cost $D^3$ seems difficult at first because the ansatz is not written in terms of local functions of $\phi$. Naively this should at the very least square the cost to $D^6$. The main idea to obtain the cheap scaling is to realize that the expectation value of vertex operators, i.e. operators of the form $: \exp( \alpha \phi):$, can be computed by solving an ordinary differential equation (ODE) where the generator has a cost $D^3$. Basically, computing vertex operators for relativistic CMPS is as expensive as computing field expectation values for standard non-translation-invariant CMPS. To solve this ODE, one can use powerful method with extremely quickly decaying errors as a function of the discretization step (e.g. very high order Runge-Kutta). So vertex operators are, in fact cheap. But local powers of the field are merely differentials of vertex operators, and thus can be computed as well for the same cost. Finally, to get the gradient, one can differentiate through the ODE with backpropagation, and obtain the result for only twice the cost. This allows the full variational optimization for all well-defined bosonic Hamiltonians with polynomial and exponential potentials.

# Two recent articles on tensor networks: Monte-Carlo chimera and the phi^4 supremacy

There are 2 recent preprints on tensor networks I found interesting.

The first is Collective Monte Carlo updates through tensor network renormalization (arxiv:2104.13264) by Miguel Frias-Perez, Michael Marien, David Perez Garcia, Mari Carmen Banuls and Sofyan Iblisdir. I am certainly a bit biased because two authors come from our group at MPQ, and I heard about the result twice in seminars, but I think it is a genuinely new and interesting use of tensor networks. The main idea is to reduce the rejection rate of Markov Chain Monte Carlo (MCMC) by crudely estimating the probability distribution to be sampled from with tensor network renormalization. This combines the advantage of the Monte Carlo method (it is exact asymptotically, but slow especially for frustrated systems) with the advantage of tensor renormalization (it is very fast for approximations).

A quick word of caution for purists, technically they use the tensor renormalization group (TRG), which is the simplest approximate contraction method for 2d tensor networks. Tensor network renormalization (TNR) can refer to a specific (and more subtle) renormalization method introduced by Evenbly and Vidal (arxiv:1412.0732) that is in fact closer to what the Wilsonian RG does. This is just a matter of terminology.

Terminology aside, it seems the hybrid Monte Carlo method they obtain is dramatically faster than standard algorithms, at least in number of steps to thermalize the Markov chain (which remains of order 1 even for reasonably low bond dimensions). These steps are however necessarily more costly than for typical methods because sampling from the approximate distribution generated with TRG gets increasingly expensive as the bond dimension increases. As a result, for a large frustrated system, it is not clear to me how much faster the method can be in true computing time. I think in the future it would be really interesting to have a benchmark to see if the method can crush the most competitive Markov chain heuristics on genuine hard problems, like spin glasses or hard spheres for example. It is known that TRG struggles to get precise results for such frustrated problems, and so the approximation will surely degrade. But perhaps there is a sweet spot where the crude approximation with TRG (of high but manageable bond dimension) is sufficient to have a rejection rate in the MCMC at 10^(-4) instead of 10^(-15) for other methods in a hard-to-sample phase.

The second one is Entanglement scaling for $\lambda\phi^4_2$ by Bram Vanhecke, Frank Verstraete, and Karel Van Acoleyen. I am of course interested because it is on $\lambda \phi^4$, my favorite toy theory of relativistic quantum field theory. There are many ways to solve this non-trivial theory numerically, and I really find them all interesting. Recently, I explored ways to work straight in the continuum, which is the most robust or rigorous perhaps, but so far not the most efficient for most observables.

So far, the most efficient method is to discretize the model, use the most powerful lattice tensor techniques to solve it, and then extrapolate the results to the continuum limit. The extrapolation step is crucial. Indeed, such relativistic field theories are free conformal field theories at short distance. This implies that for a fixed error, the bond dimension of the tensor representation (and thus the computational cost) explodes as the lattice spacing goes to zero. One can take the lattice spacing small, but not arbitrarily small.

With Clément Delcamp, we explored this strategy in the most naive way. We had used a very powerful and to some extent subtle algorithm (tensor network renormalization with graph independent local truncation, aka GILT-TNR) to solve the lattice theory. We pushed the bond dimension to the max that could run on our cluster, plotted 10 points and extrapolated. At the smallest lattice spacing, we were already very close to the real continuum theory, and a fairly easy 1-dimensional fit with 3 parameters gave us accurate continuum limit results. A crucial parameter for $\lambda \phi^4$ theory is the value of the critical coupling $f_c$, and our estimate was $f_c = 11.0861(90)$. We estimated it naively: by putting our lattice models exactly criticality, and extrapolating the line of results to the continuum limit. This was in 2020, and back then this was the best estimate. It was the best because of the exceptional accuracy of GILT-TNR, which allowed us to simulate well systems even near criticality, and allowed us to go closer to the continuum limit than Monte Carlo people.

Bram Vanhecke and collaborators follow a philosophically different strategy. They use an arguably simpler method to solve the lattice theory, boundary matrix product states. Compared to GILT-TNR, this method is simpler but a priori less efficient to solve an exactly critical problem. But, spoiler, they ultimately get better estimates of the critical coupling than we did. How is that possible? Instead of simulating exactly the critical theory with the highest possible bond dimensions, the authors simulate many points away from criticality and with different bond dimension (see image below). Then they use an hypothesis about how the results should scale as a function of the bond dimension and distance from criticality. This allows them to fit the whole theory manifold around the critical point without ever simulating it. This approach is much more data driven in a way. It swallows the bullet that some form of extrapolation is going to be needed anyway and turns it into an advantage. Since the extrapolation is done with respect to more parameters (not just lattice spacing), the whole manifold of models is fitted instead of simply a critical line. This gives the fit more rigidity. By putting sufficiently many points, which are cheap to get since they are not exactly at criticality or with maximal D, one can get high precision estimates. They find $f_c = 11.09698(31)$, which if their error estimate is correct, is about 1 and a half digits better than our earlier estimate (which is a posteriori confirmed, since the new result is about 1 standard deviation away from the mean we had proposed).

Naturally, an extrapolation is only as good as the hypothesis one has for the functional form of the fit. There, I like to hope that our earlier paper with Clément helped. By looking at the critical scaling as a function of the lattice spacing, and going closer to the continuum limit than before, we could see that the best fit clearly contained logarithms and was not just a very high order polynomial (see above). I think it was the first time one could be 100 % sure that such non-obvious log corrections existed (previous Monte Carlo papers had ruled it out after suspecting it), the $\chi^2$ left no doubt. Interestingly, Bram and his Ghent collaborators had tried their smart extrapolations techniques back in 2019 (arxiv:1907.08603) but their ansatz for the functional form of the critical coupling as a function of the lattice spacing was polynomial. As a result their estimates looked precise, between 11.06 and 11.07, but were in fact underestimating the true value (like almost all papers in the literature did).

Could this scaling have been found theoretically? Perhaps. In this new paper, Vanhecke and collaborators provide a perturbative justification. My understanding (which may be updated) is that such a derivation is only a hint of the functional form because the critical point is deep in the non-perturbative regime and thus all diagrams contribute with a comparable weight. For example, individual log contributions from infinitely many diagrams could get re-summed into polynomial ones, e.g. $\sum_k \log(\lambda)^k/k! = \lambda$ non-perturbatively. In any case, I think (or rather hope…) that precise but “less extrapolated” methods will remain complementary and even guide the hypothesis for more data driven estimates.

In the end, congratulations to Bram, Frank, and Karel for holding the critical $\phi^4_2$ crown now. I will dearly miss it, and the bar is now so high that I am not sure we can contest it…

# Variational method in field theory – videos

A bit more than a month ago, I put on arxiv two preprints (short and long) that summarize my recent work on applying the variational method in relativistic quantum field theory. I am happy that they were (so far) well received, and I got the chance to present the corresponding results in a few seminars already. The last two, at the University of Helsinki and EPFL, were recorded. I had a better microphone at EPFL, and expanded more on tensor networks, so I embed the video below (or direct link here).

I skipped the introduction to the basics of relativistic quantum field theory at EPFL and so you may prefer the Helsinki recording if this is not well known to you (as I was a basically two years ago). As the typical Frenchman, I say “eeeeuuuh” a lot when presenting, which makes the whole experience particularly awful for people listening. Now that I am painfully aware of it, I will try to work on it ifor future presentations…

# Random news

The preprint on subcritical reactors I had written about two months ago has finally appeared on arxiv. Apparently it was “on hold” for that long because the primary category (applied physics) was not the right one, and the moderators thought “instrumentation and detectors” was better. Honestly, I am not sure they are right, given how theoretical the paper is, but I am just glad the paper appeared (I would have just preferred it to be faster).

I also have one article written with Vincent Vennin and one interview by Philippe Pajot in the latest edition of La Recherche. Both are, of course, in French. This edition, with a new format, features many interesting articles, in particular an interview of the president of the Max Planck Society.

# Sustaining the chain reaction without criticality

I have found a neat nuclear physics problem, and have written a draft of article about it. Below I explain how I came to be interested in this problem, give some context most physicists may not be familiar with, and explain the result briefly.

A while back, I took a small part in the organization of the French public debate on radioactive waste management. My work was not very technical, but got me interested in the rich physics involved. The debate itself didn’t allow me to do real research on the subject: it was certainly not what was asked from me, and there was already so much to learn about the practical details.

After the debate ended, I kept on reading about the subject on the side. In particular, I read more on advanced reactor designs, where I found a neat theoretical question which I believe was unanswered.

# The sound of quantum jumps

Before I finally go on holidays, I put on arxiv an essay on quantum jumps, in fact rather on collapse models, that I initially submitted to the FQXI essay contest.

In this essay/paper, I just make a simple point which I have made orally for years at conferences. Every time, people looked quite surprised and so I thought it made sense to write it down.

The argument is simple enough that I can try to reproduce it here. Collapse models are stochastic non-linear modifications of the Schrödinger equation. The modification is meant to solve the measurement problem. The measurement problem is the fact that in ordinary quantum mechanics, what happens in measurement situations is postulated rather than derived from the dynamics. This is a real problem (contrary to what some may say), the dynamics should say what can be measured and how, it makes no sense to have an independent axiom (and it could bring contradictions). Decoherence explains why the measurement problem does not bring contradictions for all practical purposes, but again contrary to what some may say, it certainly does not solve the measurement problem. So the measurement problem is a real problem and collapse models provide a solution that works.

The stochastic non-linearity brought by collapse models creates minor deviations from the standard quantum mechanical predictions (it makes sense, the dynamics has been modified). This is often seen, paradoxically perhaps, as a good thing, because it makes the approach falsifiable. It is true that collapse models are falsifiable. What is not true, is that collapse models modify the predictions of quantum mechanics understand broadly. This is what is more surprising, sometimes seems contradictory with the previous point, and is the subject of my essay.

How is it possible? Collapse models are non-linear and stochastic, surely ordinary quantum mechanics cannot reproduce that? But in fact it can. As was understood when collapse models were constructed in the eighties, the non-linearity of collapse models, which is useful to solve the measurement problem, has to vanish upon averaging the randomness away. Since we have no a priori access to this randomness, all the things we can measure in practice can be deduced from linear equations, even in the context of collapse models. This linear equation is not the Schrödinger equation, but one that does not preserve purity, the Lindblad equation. However, it is also known that by enlarging the Hilbert space (essentially assuming hidden particles), Lindblad dynamics can be reproduced by Schrödinger dynamics. Hence, the predictions of collapse models can always be reproduced exactly by a purely quantum theory (linear and deterministic) at the price enlarging the Hilbert space with extra degrees of freedom. Collapse models do not deviate from quantum theory, they deviate from the Standard Model of particle physics, which is an instantiation of quantum theory. Even if experiments showed precisely the kind of deviations predicted by collapse models, one could still defend orthodox quantum mechanics (not that it would necessarily be advisable to do so).

Collapse models are still useful in that they solve the measurement problem, which is an ontological problem (what the theory says the world is like or what the world is made of). However, the empirical content of collapse models (what the theory predicts) is less singular that one might think. In the essay, I essentially make this point in a more precise way, and illustrate it on what I believe is the most shocking example, the sound of quantum jumps, borrowed from a paper by Feldmann and Tumulka. I doesn’t make sense to write more here since I will end up paraphrasing the essay, but I encourage whoever is interested to read it here.