Niels Bohr Quantum Summer School

The following is a brief report on my experience as a part of the 2026 Niels Bohr Quantum Summer School in Odense, Denmark. I will summarise the various lecture series and what I took away from each topic, in roughly chronological order, mixing in some of the other activities and social events in between. All together it was a wonderful experience. It is a rare thing to get to be surrounded by 70 other PhD students all working on the many sides of quantum computing. I truly appreciated getting to meet wonderful students, and getting to learn from world class lecturers, I cannot recommend the opportunity highly enough. I have done the best I could with my notes and the available lecture slides to recreate what I have learned, but if I have not accurately conveyed parts of what was covered then such feedback is greatly appreciated.

Welcome Address

After arriving in Odense, and making my way to the Danish Institute for Advanced Studies, housed on the campus of Southern Denmark University (SDU), I sat in the lecture theatre amongst my fellow PhD students. We were greeted by the director for the Centre for Quantum Mathematics, Jørgen Andersen as well as the Rector of SDU, Jens Ringsmose. Prof. Anderson discussed both the history of the Centre for Quantum Mathematics as well as some of his own research and notable recent working coming out of the centre. Specifically, he he discussed applications of quantum boson sampling. I was vaguely familiar with the idea that there are some classes of sampling problems with a quantum advantage distinct from Shor or Grover type advantages, but I didn’t know much more than that. I was honestly surprised, not just at the theoretical interest of boson sampling but that it is in fact already being deployed by a financial firm where it is out performing existing classical techniques. Prof. Andersen then went on to introduce the focus for this year’s summers school - Quantum Algorithms and Quantum Software. Last year’s had been focused on hardware, so I felt that I had definitely chosen the right year for me! After that we heard from the Rector. Prof. Ringsmose doesn’t come from the deep STEM world, but his introduction was pleasant and welcoming, it certainly underscored the size of the investment the university is putting into quantum research and the networking of industry and academic partners they are building. All good signs for the many young students!

Quantum Error Correction

Daniel Eric Gottesman, University of Maryland

The first lecture series was a real treat. It is always a privilege to get to learn from a pioneer of any field. While I already had experience using error correction, it was still useful to start from the theoretical basics and build up towards the current QEC frontier of surface codes, concatenation and high-rate LDPC codes. This was also the first time I really understood the diversity of students present at the summer school, as those coming from other disciplines such as chemistry, computer science and mathematics had a different knowledge base and also a different way of engaging with the content. I appreciated the thoroughness of going through the QEC basics with more rigour than I had seen before. I had also never encountered concatenation prior, although it seems such an obvious idea in hindsight. The challenge of designing optimal QECCs for given hardware is an interesting one with lots of fertile research soil. While there is progress to be made on the theory side, QEC strikes me more as a way to magnify hardware improvements rather than bypass hardware issues.

Rethinking Quantum Circuits

Steven Rayan, University of Saskatchewan

Prof. Rayan’s lectures were a useful introduction to first understanding quantum circuits for those who were less familiar, as many the audience came from such diverse backgrounds this was a useful central starting point. I found the circuit design and error correction content less novel for me personally however I had never encountered ZX calculus or ZX diagrams before, so once we started with that I found myself appreciating the introductory pacing. The ZX diagram approach initially seemed further divorced from the ”physical” notion of a qubit and quantum circuit in the sense that many of the gates that a quantum computer would perform are abstracted away and grouped into more elegant diagram rules. The optimisation and simplification that the ZX approach creates is well-ordered, the fact that it is computationally difficult to turn such diagrams back into instruction-set-architecture does pose some challenges for generic hardware interfacing. It is definitely useful as an intermediary abstraction layer that a higher level programming language could interface with instead. Additionally, the tail end of the lecture series focused on taking the ZX approach to surface code error correction, and using this to design novel hyperbolic qubit lattices for superconducting qubits which have even been demonstrated experimentally. The advantage of a hyperbolic QECC family as opposed to Euclidean lattice surface codes is the change to the rate-distance tradeoff. The hyperbolic case allows the encoding rate, the number of logical qubits k per physical qubit n, to become constant at the cost of decreasing code distance from square root in n to logarithmic in n. I recommend readers take a look at the complete lecture notes here. The ZX approach additionally builds some intuition into the nature of quantum state evolution that I appreciated and I would like to practice a few examples of my own to further that along. Like some other circuit abstract that was presented, to my understanding this approach ”tracks all of the branches” so to speak, rather than illuminating the branch selection aspect of measurement, however this could be my misunderstanding or simply something we did not have time to cover in this short lecture series.

Student Challenge

Several days into the first week we were presented with a list of 9 challenge problems to choose from. We selected our favourite 2 and were assigned into small groups accordingly. The problems to choose from are roughly described as follows:

1. Calculate expectation value estimates using Gaussian Boson Sampling
2. Quantum simulations of nonlinear PDEs
3. Apply Quantum machine learning for satellite imagery detection
4. Develop a hardware agnostic quantum state tomography suite
5. Apply quantum computing to filtering problems
6. Use quantum computing for finance portfolio optimisation
7. Use analogue neutral atom quantum computers for multiple hypothesis testing problems
8. Quantum approaches to Coupled-cluster problems
9. Topological quantum computing gate compilation.

I ended up selecting the neutral atom problem as my first choice and the topological QC problem as my second choice. Last year I participated in Quera’s neutral atom QC hackathon, and given recent breakthrough Neutral Atom results coming out of the Lukin group at Harvard and the Endres lab at Caltech, I wanted to learn more about what might be the future dominant hardware paradigm. I was surprised when the choice distribution was shared later that the topological QC problem was the most popular by far, something like 35% of all first choices, perhaps a testament to Prof. Wang’s engaging lectures. I was also surprised that virtually no-one was interested in the quantum computing for finance problem, given that we had seen evidence of real commercial utility.

Our specific challenge was to map the multiple hypothesis testing (MHT) problem, to the maximum weight independent set problem (MWISP), and then to map that to an Ising type evolution on a neutral atom quantum computer. The MHT problem, in a simple case involves tracking the path of multiple unlabelled objects across noisy frames of data. Instead of assigning definite object labels in each frame, all possible labels are tracked as hypotheses, that are then updated with new information, and eventually ”pruned” to a global set of maximum likelihood non-conflicting hypotheses, which is where MWISP comes in. Each hypothesis is assigned a node, with a weight given by its likelihood score, and edges are placed between conflicting hypotheses, e.g. claims by different objects to the same detection event in a given frame. The idea then, instead of the combinatorially difficult task of selecting subsets that are non-conflicting with the highest combine likelihood, we could encode the graph as atoms in a neutral atom QC, and use the Rydberg blockade interaction to represent an edge. The system could then evolve adiabatically to select the strongest global hypothesis set. Our full report is available for those who are interested in learning more about our approach and results. I am very grateful to my group members, who were a joy to work with and from whom I learned a lot! I had the privilege of getting to to present our poster and we were awarded honourable mentions for our efforts.

Topological Fault Tolerance

Zhenghan Wang, University of California Santa Barbara

At the opposite end of the spectrum from the ground-up, rigorous approach taken by some of the other lecture series was Prof. Wang’s topological fault tolerance series. I was vaguely familiar with anyon braiding and some of the ideas behind topological quantum computing, not least because of the contro- versial claims coming out of Microsoft’s hardware team. The lectures themselves were not at all focused on practical implementation or hardware but instead on the deep mathematical ideas behind topologi- cal quantum field theories, error correction and category theory. Category theory has perennially been something I wish to learn and this lecture series only added fuel to that fire. I was excited when Prof. Wang began the first lecture with a brief discussion of the quantum extended Church-Turing thesis and some speculation about complexity classes, before settling into the basics of anyon models. Fusion rules, F matrices, functors and pentagon equations were all new to me and I had a difficult time keeping up however he was an engaging lecturer that did his best to explain things intuitively, including some helpful statements such as ”the ground state of a Topological QFT is always a QECC” and ”CSS codes are gen- eralised Topological QFTs”. His final lecture began with determining which anyon model can be realised by a general local gapped Hamiltonian, which was closer to ”practical” implementation than the rest of his content.

He eventually ended his series with his more speculative claims which were captivating. He referenced some old work on quantum logic by Birkhoff and von Neumann, and speculated that in some sense logic should be empirical, and that we need a logic and a mathematics inspired by quantum physics. Several years ago while working in data science research at Columbia I was interested in the idea of quantum logic. I thought I had better begin with understanding classical logic in the way that a philosopher would approach the subject. I worked my way through the oft-recommended 1978 work Philosophy of Logics by the late Susan Haack. This was equal parts interesting and reassuring that I belong in a physics department rather than a philosophy one. Haack specifically addresses quantum logic under the category of ”deviant logics”, which she motivates with a discussion of Reichenbach’s argument for a 3-valued logic as well as mentioning the work of Birkhoff and von Neumann in a favourable sense. I followed up with her 1996 work, Deviant Logic, in which she write, ”I have heard it argued that the question, whether any of the arguments offered in this context in favour of a non-standard logic, are acceptable is now a dead one”. She takes a somewhat dissenting tone to this, although it is clear that much of the classical logic field are opposed to such ideas. There is even a choice quote by Karl Popper in there that upon re-reading made me laugh, ”If we do not use it (classical 2-valued logic) but retreat into the use of some weaker logic…. then I assert we are not critical enough; it is a sign that something is rotten in the state of Denmark”. Though an honorary New Zealander himself, it is clear that he never developed a taste for the ”Spirit of Copenhagen” that his fellow Kiwi, Ernest Rutherford had. Prof. Wang’s enthusiasm for a quantum logic and the fact that it is seemingly at odds with the beliefs of many in the classical logic world is wondrous puzzle.

Prof. Wang further speculated that Hilbert spaces and tensor products may not be the ”natural” way to build quantum mechanics highlighting two reasons. The first reason being that even in a single qubit system, the state can require incomputable values to be represented, which had me thinking about what would actually be required computationally to simulate the entire universe, and the converse of what prop- erties must the universe have in order to for it to be simulatable under differing resource assumptions. The second reason was that entanglement does not require Hilbert spaces or tensor product structures. This reminded my of the lectures with my adviser Prof. Bergou on the Bell experiment and how it is possible to demonstrate entanglement, or more generally non-classical correlation, by only using expec- tation values and the tools of classical probability theory. This also made me think of Prof. Susskind’s lecture at CUNY that I attended where he showed the nature of vacuum fluctuation entanglement in con- formal QFTs. After this lecture Prof. Wang kindly indulged my questions, as he previously mentioned conformal QFTs briefly alongside topological QFTs, I wanted to know similar mappings between QECCs and conformal QFTs existed, to which enthusiastically answered that he did not know and that it was something he is actively working on. I thoroughly enjoyed this series and I look forward to reading Prof. Wang’s topological quantum computation monograph.

Efficient Quantum Learning

Sitan Chen, Harvard University

While I was aware of quantum state tomography, and in particular the difficulties of realising it, from some quantum cryptographic protocols such as the original entanglement protocol by Reichardt, Unger, and Vazirani, I had never heard of ”shadow tomography”. The idea is appealing in that the problem is easy to understand; given some observables and some copies of a state, can we estimate the the expectation values to some degree of accuracy? The challenge of determining efficiency, optimisation, conditions and bounds is not so straightforward. Prof. Chen’s lectures were clear, methodical and engaging. The approach to the problem was interesting to me as it ties together nicely with rigidity theorems used in entanglement-based cryptographic protocols and I was left wondering how to formulate such situations as essentially special cases of shadow tomography. I am also interested in understanding shadow tomography given noisy or imperfect copies of the initial state, which could be a new research direction to pursue.

Spirit of Copenhagen

As a break from the lecture days we took a trip up to Copenhagen to the famed Niels Bohr Institute. We were graced with an excellent history lesson by the resident science historian Christan Joas, where we got to see pictures of the founders of quantum mechanics in the very room in which we sat. I was interested to hear about Niels Bohr’s first grant proposal, exactly one sentence long, for a trip to England. From what I recall, Bohr, didn’t fit in so well with his first position at the J.J. Thomson lab at Cambridge, in part due to the insular class attitudes of the time but found his place working with the great Kiwi physicist of humble beginnings, Sir Ernest Rutherford in Manchester. Perhaps it was this experience that Niels Bohr took with him to create the non-hierarchical teaching style that came to define the Copenhagen Spirit. Beyond the history of the institute, we got to hear from the director and a current experimental researcher on the new and exciting work still being done in Copenhagen. Later in the day, we visited some of the labs as well as the offices of Atom Computing and QuNorth. Unfortunately, we did not get to see the neutral atom QC that was being setup however we did get to hear an enjoyable lecture on Ytterbium based neutral atom QCs from another New Zealander, Alex Elliott who is an AMO theorist at Atom Computing. The day also including some sightseeing and getting our own taste for the Copenhagen Spirit, which as I understand, is also synonymous with ice cream.

Quantum Programming

Mingsheng Ying, University of Technology, Sydney

Prof. Ying’s lecture series was also in the vein of abstracting away from the circuit model. Specifically, He introduced the theoretical foundations for quantum programming, using Hoare triples, which I had not encountered before. As I understand it, a motivation for moving beyond the circuit model to quantum programming is due to loops and conditionals that depend on measurements. Prof. Ying introduced some formal semantics for quantum programs, as well as some theorems for soundness and completeness. Trying to define correctness of quantum programs was interesting as Ying had to introduce two notions of partial and total correctness to account for quantum programs that may never terminate. There was a fair bit that I didn’t fully grasp and that I would like to learn more about, in particular I don’t think I have a good understanding of the halting problem in either the classical or the quantum case, nor how the quantum case relates to trace non-unitarity. I did enjoy that the formalism Ying developed could address questions of composability in an organised way, as this is often a property of interest in cryptographic protocols, this connection would be interesting to explore.

Quantum Simulations

Patrick Emonts, Ulm University

It was a change of pace to get to learn about classical simulation methods from Prof. Emonts. While I’d used tensor based ML libraries such as PyTorch before, I honestly didn’t really understand what a tensor network was or how it was useful for representing quantum states. This series began with the motivating question of how do we efficiently represent a 2N sized quantum on a classical system? Of course there are useful heuristics that work in certain cases or there are ways we can make some simplifying approximations but the general problem is intractable. Instead, we can use tensor networks to do the best we can for low entanglement states. The approach includes a graphical notation that was fun to learn and use as well. The second part of the series focused on semi-definite programming relaxation hierarchies that can serve to find a lower bound of a ground state energy. While this series was practical and useful, I found it theoretically interesting as it shined some light on the broader question of what is it about a quantum state that makes it difficult to represent classically.

Conditional Quantum Gate Compilation

Jens Palsberg, University of California, Los Angeles

As I alluded to earlier, the question of optimal circuit compilation is both a theoretically interesting one, and a practically useful one, given the state of current hardware. Prof. Palsberg’s lecture series began with a comparative demonstration of his own novel benchmarking tool with a series of competitors. The key result being the scalability of other tools being much worse, which is an issue for low physical error rates, which we ideally want. The second portion of the series focused on optimally compiling the Toffoli gate into two qubit operations. The proof of which was long and technical. Efficiently compiling a circuit into an optimal set of gates is a difficult task. While a linear scaling result was achieved, the coefficient was several hundred times larger than the lower bound. The architect for the proof of the lower bound happened to be sitting in the audience also! This compilation process was done without the use of ”helper qubits”, so I am curious how the results change by allowing such resources. It had me thinking about entanglement catalysts and what role a reusable helper qubit resource could play in optimizing circuit compilation.

The Best of Odense

The town of Odense was lovely, filled with a range of architectural eras and Scandinavian design, excellent public transportation, bike-friendly infrastructure, walkable public spaces and mixed-use buildings. In short, everything that makes a place nice to live in. It also happens to be the birthplace of Hans Christian Andersen, and the town was not eager to let you forget that fact. Many statues, a light show, a museum and his birth house were all dedicated to him and his children’s stories. While in Odense, we got to experience an 83% partial solar eclipse, and the town’s annual flower festival, which were both delightful. We were also treated to an extravagant prix-fixe dinner at Det Glade Vanvid, which was an excellent chance to get to know our fellow students outside of the lecture hall setting.

Circuit Abstraction and Rig Categories

Chris Heunen, University of Edinburgh

My earlier desire to learn category theory was furthered in the lecture series by Prof. Heunen. This series also fell into the ”category” of abstracting away from the circuit model of quantum computation. As with Prof. Ying’s lectures, the struggle for circuit models to represent loops, reuse and more complex data structures was a point of emphasis, as was the parallel with classical computing development, where we certainly don’t think just in terms of logic gates anymore. In addition to the standard ways of understanding composition of quantum gates, namely sequential matrix multiplication or tensor products, an additional composition operation was introduced to represent control. I remember as an undergraduate trying to factor a CNOT gate into the identity and an X gate only to realise that doesn’t work. At the time I did not know it but I was in fact looking for a control composition instead. I enjoyed the idea of viewing quantum computing as a completion of classical computing. The inclusion of measurement into this Rig category framework was interesting. This was done by including the ideas of ”Garbage” and ”Heap”, from which allocation and discarding can be defined. To my understanding, this was also a ”track all branches” type approach, and I would be curious if branch selection could be included naturally into this formulation. The need for building more abstract ways to interact and control quantum devices is clear, but I am unsure of a way to assess different approaches to the task. It would be interesting to learn more though, and it no doubt will be used more in the future.

Trapped Ion Quantum Computing

Dave Hayes, Quantinuum

Given the exciting state of the field, it was refreshing to get to hear from the industry perspective. Admittedly, I knew vary little about the trapped ion paradigm but thankfully Dr. Hayes aimed his lectures broadly. I particularly enjoyed learning about the proposed integrated photonics designs that have tremendous potential to improve scalability. Last year, I remember reading about some similar approaches being trialled for neutral atom contexts too. An illuminating fact about the experimental engineering process was made clear in the fact that Quantinuum systems originally used ytterbium for qubits and barium for sympathetic cooling, however after actually building and trialling the device, they discovered that the barium ions actually work better for qubits. The takeaway for me is that theory is really only going to go so far and eventually you have to just play around with the actual experiment and see what happens. His last lecture, discussed random circuit sampling and gate streaming, which is pretty near an application of my own work on verifiable delegated quantum computing as you could get. Dr. Hayes kindly entertained my questions afterwards regarding circuit delegation and shared entanglement, despite overhearing my earlier less than favourable comparison between neutral atoms and trapped ions on the train.

Student Posters

As a part of the summer school, all the students were required to bring a poster of their own research to present. I brought a poster of my work on verifiable delegated multiparty quantum computing, which was a fun time getting to introduce people to the quantum cryptography world. I was thoroughly impressed with the quality and diversity of the students in attendance. There were too many excellent pieces of research on display ranging from applied quantum chemistry to pure math to have any hope of getting to see them all let alone understand each of them. Some personal highlights included posters on trapped ion modes, resource theory approaches to quantum communication, coherent cat state qubit design, and spin interferometry. I felt lucky to be in such a stimulating environment.

Conclusion

I cannot speak highly enough of my experience as a part of the 2026 Niels Bohr quantum summer school. It was a wonderful opportunity to learn from leaders in the field both in academia and industry, as well as to get to learn from other PhD students from around the world, all deeply engaged in their own fascinating research. I am taking back with me bucketfuls of enthusiasm, many ideas for new research projects and wonderful friendships. I am grateful to the Danish government, the University of Southern Denmark, DeiC, and the many other sponsors and organisers who put the summer school together.