## A QIG seminar on “the Polynomial Hierarchy” by Friederike Dziemba

June 6, 2016

In this post I’d like to share a video we’ve uploaded of a recent talk in the quantum information group seminar series by Friederike Dziemba where she presents a physicist-friendly overview of the polynomial hierarchy, a central idea in computational complexity theory:

If you’d like to see more content like this then please do hit the like button. And if you’d like to keep up to date with content from the quantum information group, Hannover, then please don’t hesitate to subscribe to our channel and my own channel.

## The theory of quantum noise and decoherence, lecture 10

January 27, 2016

The video of my 10th lecture on the theory of quantum noise and decoherence is now available.
Here in lecture 10 I continue the discussion of master equations for quantum dots and introduce a model for the continuous measurement of position:

## Presentations are more important than papers: my first youtube video

July 27, 2015

During the past 5 years or so I have come to believe that presentations are actually more important than scientific papers. As a consequence, I have recently spent quite a lot of energy learning how to give better presentations. This is a truly fascinating and rewarding topic. While I find it is difficult, if not downright impossible, to master good public speaking, I’ve very much enjoyed trying to improve how I give my presentations.

Today I’d like to annouce the appearance of my first youtube video:

This is a recording of a talk I recently gave to graduate students here in Hannover. The objective of the talk was to share and channel advice I’ve received in the past years on how to give a good presentation. While I don’t claim to be especially good at giving good talks myself (the excruciating experience of watching myself on video for essentially the first time only serves to underline this!) I have learnt a great deal from other excellent speakers, and I hope that I can at least share a couple of the tips and tricks I’ve learnt.

Depending on the reception to this video it might signal a change in the way I will go about communicating our research. I’ve recently noticed that I am spending an increasing amount of time watching videos of talks at conferences, video tutorials, and miscellaneous other videos (cat videos, unboxings, etc. 🙂 ). It truly is a supremely powerful medium of communication, combining both visual and auditory modes of delivery, and, given that you can pause and skip, I have found it to often be superior to attending talks.

I am still passionate about open science, and open notebook science, and I am always contemplating better and more efficient ways to implement at least some core principles of openness. This is why this blog and my twitter account have become so neglected as of late: I’ve just found that github provides an amazingly useful, superior, and simple tool to achieve this. If you want to know what I’m doing on any given day then you can check out my activity there. (Basically all of my notes are now stored openly there.)

However, github is not the right tool for communicating and sharing ideas. Here I think video is superior, and youtube a natural platform. We’ll see.

I do hope you enjoy watching my video; any comments, suggestions, and criticisms are (actually) welcome!

## Antrittsvorlesung

June 9, 2011

On Tuesday June 7th I gave my Antrittsvorlesung (= inaugural lecture) entitled “Größenverhältnisse und Information” (= “Scale and information”). You can find the slides here. The german text of the lecture can be found here and the english text here.

## QUEST special lecture 3 and the COQUIT conference

November 23, 2010

I’ve been a little distracted with a conference we hosted last week (see here for the program), so I didn’t get around to loading the slides for the last of the QUEST special lectures. My dilly-dallying has meant that I can kill two birds with one stone: the talk I gave at the COQUIT conference was essentially identical to the third special lecture. I attach the slides here, if you’re interested.

## QUEST special lecture 2

November 4, 2010

Yesterday I gave the second of three QUEST special lectures. I discussed the variational principle in the context of quantum spin systems and spoke about the problems and some solutions associated with its application. You can find the slides here.

## QUEST special lecture 1

October 28, 2010

Yesterday I gave the first of three QUEST special lectures. You can find the slides here. Readers of this blog will be familiar with the content: I talked about the simulation problem and hamiltonian complexity and ended with the result that the dynamics of a 1D quantum spin system can be efficiently approximated (by a quantum cellular automaton) for $|t| \sim \log(n)$.

In the next lecture I’ll show how to turn this result around and use quantum circuits to simulate the statics and dynamics of strongly interacting quantum systems via the variational principle.

## What I’ve been doing

April 26, 2010

After a hiatus of 6 months or so, I’m hoping to get back to posting again.

During the past half year I have been living in Berlin as a fellow of the Wissenschaftskolleg zu Berlin. During this time I’ve been involved in plenty of activities both research and non-research. Here is a brief description.

At the WiKo I’ve had the sincere pleasure to meet a great number of extremely talented people from all walks of academic life, from musicians and humanists to social scientists and natural scientists. The most striking thing in all these interactions has been the communication problem that we face in talking to our fellow academics. In listening to the colloquia of the fellows it is remarkable how broadly even basic methodologies differ from discipline to discipline. For example, in the humanist tradition, in contrast to the natural sciences, it is standard practice to sit and read the entire presentation with no visual cues. Also, it is often that case that such things as stating the hypothesis of an investigation is not done in certain disciplines (which causes no small amount of frustration for the natural scientists!). Conversely, many of the humanist and social scientists find the natural scientists “practical approach” somewhat off-putting and over-simplifying 🙂

Things I’ve learnt: (i) the value of a narrative in a presentation; (ii) everything traces back to Aristotle and Plato, and what doesn’t most certainly can be found in the works of Darwin; (iii) a sense of humility.

I also gave a colloquium on my on work, which can be found here: Dienstagskolloquium 2010. In designing the presentation I hoped to made it possible for all of the fellows to understand. I’m not sure if I completely achieved that goal; please judge for yourself.

On the research front it was my objective to immerse myself in a problem I’ve been idly contemplating for many years; my thinking being that this year will be the last chance I’ll get for a while to work on highly speculative research. I’m not sure that it has been entirely successful, but I certainly did learn a lot of interesting stuff.

I’ll briefly sketch the idea here. Basically I was trying to develop a “canonical” procedure to quantise algebraic geometric spaces in positive characteristic. What does this mean? To explain this it is helpful to recall what geometric quantisation means. Here the input is a symplectic manifold (or a Poisson manifold), with a closed two-form which can be thought of as generating hamiltonian flow. Roughly speaking: I really just mean some classical dynamical system. The output of this procedure is a hilbert space ${\mathcal{H}}$, and a hamiltonian operator ${H}$ on that space which is meant to be associated in some canonical way with the input classical system. The best and most well-known example of this procedure in action is known as Schrödinger quantisation which takes the symplectic manifold ${\mathbb{R}^n\times \mathbb{R}^n}$ with linear dynamics as input and outputs the quantisation ${\mathcal{H} \equiv L^2(\mathbb{R}^n)}$.

What I initially wanted to do was to replace ${X}$ by an algebraic variety over a field of positive characteristic (actually, a general symplectic scheme) and find out what ${\mathcal{H}}$ ought to be. I believe I pretty much succeeded in this initial goal: the result is what it ought to be in the case of the symplectic scheme ${X = k^n\times k^n}$, namely, ${\mathcal{H} = L^2(k^n)}$. Linear dynamics quantise to something familiar in the case of ${k = \mathbb{Z}/2\mathbb{Z}}$, namely (tensor products of) the pauli matrices. Rather interesting stuff happens for even the simplest nontrivial varieties, but I won’t elaborate here as it is lengthy and assumes rather a lot of background.

Why would anyone want attempt such a baroque thing? Well I’ve had hopes for many years that a natural quantisation program would be a first step in a program to provide another proof of the Weil conjectures in algebraic geometry from the perspective Polya-Hilbert approach. Needless to say, I have not been successful in this objective.

Lately I’ve been thinking about much more physical problems. After a break of several years I’ve been working again on matrix product states and all that stuff. And also thinking a lot about disorder and topologically ordered systems. I hope to report more on all of this in future posts.

## Hamiltonian complexity talk

April 1, 2009

On Tuesday I gave a talk at the IMA conference at the IMS. I’ve put the slides here (it is a huge file, alas, as I included lots of pictures…)

In this talk I took the opportunity to popularise a problem that I hope will provide a fruitful avenue for future research in the emergent field of hamiltonian complexity.

Before I describe this problem, I’d like to say a couple of words about hamiltonian complexity itself: this field, which gained considerable momentum last year thanks to a couple of tightly focussed workshops at Leiden and Santa Fe, is (roughly speaking) aimed at exploring the interplay between theoretical physics and computational complexity theory. The central question of hamiltonian complexity, I would argue, is: “how hard is it to simulate a physical system?” To actually answer this question quantitatively requires that we be rather careful about what we mean by “physical system”, “simulation”, and “hardness”.

For “physical system” it has been convenient for quantum information theorists to focus on quantum spin systems (i.e., a set of $n$ quantum spins arrayed in a low-dimensional lattice). Thus we say the size of such a system is $n$. (Note, however, that the dimension of the hilbert space of such a system scales at least as fast as $2^n$.)

For “simulation” we typically mean a classical computer carrying out some algorithm, although one could widen this definition. To count as a simulation the algorithm in question needs to be able to calculate a prediction for some physical observable.

For “hardness”, the field of computational complexity provides a now-standard framework to make this precise: one quantifies the temporal and spatial resources (measured in arithmetic operations and bits, respectively) required by a computation to carry out the desired task.

With these ingredients in hand we can now state the main problem of hamiltonian complexity (well, of theoretical physics, really):

The simulation problem (equilibrium).

Input: A hamiltonian $H$ for a system of size $n$; an observable $A$; a tolerance $\epsilon$; a temperature $T$.

Output: an approximation $\alpha$ to the expectation value $\langle A\rangle=\mbox{tr}(Ae^{-H/kT}/\mathcal{Z})$ which satisfies $|\alpha- \langle A \rangle| \le \epsilon$.

The simulation problem, as stated, is simply the task of making a prediction for some observable. Hamiltonian complexity theory adds another dimension to the simulation problem by asking how hard it is to actually calculate such a prediction. This is clearly an important question: after all, suppose it took $2^n$ fundamental arithmetic operations to make a prediction for some experimentally accessible observable $A$. This would be a disaster; even for 50 spins, one would need to perform $\approx2^{50}$ operations, which is clearly unfeasible. So we’d have to give up on making any predictions whatsoever for such systems!

Obviously physicists are very good at making predictions, so this nightmare scenario never really occurs in practice. Hamiltonian complexity is thus aimed at explaining why this is the case. This is far from trivial because it is now relatively straightforward to write down 1D quantum spin systems whose equilibrium properties really do require something like $2^n$ arithmetic operations to approximate to within fairly reasonable tolerances! (We heard a lot about the state of the art of these results on Tuesday.)

Hamiltonian complexity is broken up into many sub-fields aimed at understanding variations on the theme of simulation and complexity. There are many subtle and profound techniques that have been developed to construct semi-realistic systems whose physical properties are hard to approximate. And there is yet another set of approaches aimed at actually simulating real (and imaginary) physical systems.

My talk was aimed at describing some situations where we actually can make predictions about quantum spin systems. I described two different situations where physical properties can be efficiently (on a classical computer) predicted. Typically — and the results I described are no exception — a proof that a physical property can be efficiently simulated actually provides a provably efficient algorithm to calculate the prediction.

What I feel is the most interesting open question now is to understand better the interplay between the observables $A$ one wishes to simulate and the hamiltonian $H$ to be simulated. I think this adds a new layer of richness to the complexity-theoretic landscape for quantum spin systems: most work to date has focussed on the LOCAL HAMILTONIAN PROBLEM, which is a special case of the SIMULATION PROBLEM, above (by only demanding predictions for the observable $H$ itself). However, it is completely plausible that if you aren’t interested in the energy, but some other observable $A$, the LOCAL HAMILTONIAN PROBLEM may become easier or harder!

## Visit to to the AMOPP group at UCL

February 11, 2009

Today I was invited to give a talk to the AMOPP group at UCL. While I was visiting I was given a lab tour of the LCN, and I got to see all their fancy stainless steel lab equipment including big magnets and other stuff to move and measure small cold things.  During the visit I heard about the latest stuff they are doing there, including some very exciting implementations of electrical spin-trap readout of coherence in silicon.

I heard from the group of Sougato Bose, and I learnt about some counterintuitive results showing that after a quench of interactions in the XXZ chain there can be a substantial generation of entanglement from an initial thermal state. I also heard about the some other results about entanglement generation after a quench, but this time where only one or two interactions are quenched. Both of these papers are intimately linked with how quasiparticles propagate through interacting quantum spin systems and, by focussing on entanglement generation, expose subtle transport physics in these models. Eg. check out figure 4 in this paper where some kind of interesting transition (probably not a quantum phase transition) is occurring at $\Delta = -0.5$.

The slides for the talk I gave are here. I spoke about the some of the known results concerning the computational complexity of simulating interacting quantum systems, in particular, quantum spin systems.