Research ideas and collaboration directions

Open / exploratory Fit: advanced M2 or PhD Scope: deep

Goal: Extend Clifford-circuit axiomatizations beyond prime p to arbitrary dimension d (including composite d), and produce:

• a clear choice of generators for "arbitrary-d Clifford circuits" (including phase conventions),
• a practical normal form (what is "canonical" in composite d?),
• an implementation strategy (e.g., rewriting + decision procedure) and a benchmark suite.

This matters because many realistic platforms and encodings naturally produce composite dimensions (e.g. d=4), and dimension-generic compilation/verification is a key missing piece for qudit tooling.

State of the art: Already done: constructive stabilizer/Clifford descriptions exist for arbitrary dimensions, and there are "dimension-uniform" viewpoints. Still open: a small, usable circuit rewrite theory that stays manageable in composite dimensions, with a clean normal form and an implementation that works in practice. (quant-ph/0506065, arXiv:1307.5087, arXiv:1810.10259)

Tomorrow-problem Fit: strong M2 or PhD Scope: deep and open-ended

Goal: Pin down what "breaks" in even dimensions, then build an axiomatization (rules + normal form) tailored to even-d Clifford circuits (e.g. ququarts d=4, or more generally d=2^k).

This matters because many experimental qudit platforms naturally provide access to higher transmon levels (often d=4 is a very concrete target), and even-d is where many odd-prime tricks stop being plug-and-play.

State of the art: Already done: general Clifford reviews exist; hardware-motivated compilation work exists for even-d systems. Still open: a rewriting-oriented axiomatization that cleanly handles even-d subtleties and supports automated normalization. (arXiv:1810.10259, arXiv:1307.5087, arXiv:2212.04496)

Open / exploratory Fit: M2/PhD or independent project Scope: medium to deep

Goal: Develop a dimension-generic and rewrite-friendly construction of the generalized Hadamard gate H_d (i.e. the discrete Fourier transform / "mode-mixing" unitary) as an LOv/linear-optical circuit (beam splitters + phase shifters), with a clear story for prime d and composite d.

• Specify the exact convention: which matrix you call H_d (phases / global phase), and which LOv generators you allow.
• Produce explicit circuits for small primes (start with d=2,3,5 and push toward d=7), and identify reusable substructures.
• Implement a generator: input d → output a circuit (plus a verifier that checks the produced matrix matches H_d).
• Extend to composite dimensions by a recursive scheme based on prime-factor "oracles" (e.g. build H_d^k from H_d, and H_dd' from H_d and H_d').
• Benchmark resource counts (beam splitters, phase shifters, depth) vs baseline decompositions.

Why it matters: Fourier/Hadamard mixing is a central primitive behind QFT-like transforms, state preparation, and interferometer constructions. Having a structured, dimension-parametric synthesis (especially for composite d) can become a reusable compiler pass and a good testbed for LOv rewriting.

State of the art: LOv/LOfi-style calculi provide principled rewriting for linear optics, and have also been used as a bridge to complete equational theories for standard quantum circuits. The missing piece here is a specialised, scalable, implementation-ready synthesis story for H_d that exploits Fourier structure rather than treating it as an arbitrary unitary. (arXiv:2402.17693, arXiv:2206.10577, arXiv:2311.07476)

Starting point: PDF notes (draft).

Actively looking for help Fit: side project or collaboration Scope: small to medium

Goal: Build on the existing playable web version with clearly scoped extensions:

• design new levels or challenge packs with explicit learning objectives, including extra rules,
• write/record short solution videos (explain the reasoning, not just the answer),
• improve progression, onboarding, and the guided pedagogical layer around the web version,
• upgrade the quality of the physical game as well potentially,
• optionally run a lightweight evaluation (pre/post quiz or small user study).

I'm not a game developer, so I'll happily take a "make it work, keep it simple" approach but I would really love to have someone co-driving this.

State of the art: the game exists publicly as an educational resource, and the web version is now playable. The open part is extending it with more content, better guided solution material, and (if desired) evidence about learning impact. (Game page)

Open / exploratory Fit: advanced M2 or PhD Scope: deep

Goal: Investigate whether the "few square roots" representability phenomenon has a qutrit analogue (think "few cube roots of unity"). A precise end goal is to produce either:

• a theorem + constructive algorithm for which qutrit unitaries are representable in a small algebraic extension, or
• a clear obstruction (why the analogue fails / needs a different ring).

This matters because these algebraic characterisations often become exact synthesis algorithms (a direct bridge from math to compilers).

State of the art: qutrit exact synthesis work exists for Clifford-cyclotomic gate sets (including normal forms); the open part is a crisp "cube-root" analogue with a clean ring + constructive consequences. (arXiv:2310.14056, arXiv:2303.13644, arXiv:2011.07970, arXiv:2405.08136)

Tomorrow-problem Fit: self-directed M2 or PhD Scope: exploratory

Goal: Survey and extend algebraic frameworks that represent qudit states and gates via generalised Clifford algebras (or geometric-algebra-like formalisms), and connect them to modern tooling: stabilizers, circuit identities, ZX/ZH-style reasoning.

This one is genuinely a "tomorrow problem" for me: I'm interested, but I'm still actively building the background to do it properly. I feel like maybe Spinors might be involved as well, I don't really have the knowledge yet.

State of the art: there is work developing algebraic calculi for multi-qudit computation grounded in generalised Clifford algebras, and broad references on Clifford groups across dimensions. The open part is translating these viewpoints into practical compiler representations or rewrite systems that complement ZX/ZH approaches. (arXiv:1809.07809, arXiv:1810.10259)

Tomorrow-problem Fit: PhD or very strong M2 Scope: deep and foundational

Goal: Compare the promise-style control patterns that show up in Vivien Vandaele's recent comparator / incrementer work with the control-functor viewpoint of controlled PROPs, and see whether there is a clean common framework: call it a promised PROP, a promise-sensitive controlled PROP, or something along those lines.

• identify which promise/control equations are already instances of controlled-PROP structure,
• decide where the extra "promise" data should live: points, annotations on wires, a modality, or something else,
• explain the ancilla-for-control trade as a structural theorem rather than a one-off circuit trick,
• test the framework on a small family such as comparators, incrementers, or multi-controlled constructions.

This matters because several papers now seem to be saying the same conceptual thing in different dialects: control is structured, there are multiple control conventions floating around, and resource tradeoffs such as ancillas versus controls may want a compositional explanation rather than a circuit-by-circuit proof.

State of the art: Delorme and Perdrix give a compact controlled-PROP axiomatization, Vivien's recent work proves a general ancilla/control trade for useful arithmetic blocks, and my own polycontrolled-PROP work already treats whole families of controls uniformly for qudit circuits. The open step is to isolate the right common abstraction and check that it actually leads to usable equations rather than just nicer terminology. (arXiv:2603.12917, arXiv:2508.21756, arXiv:2602.09873)

Current focus Fit: M1 or M2 Scope: medium

Goal: Replace the current family of one-wire scaling generators M_x (one for each nonzero scalar) by a smaller generating set - ideally a single scaling for a primitive element, i.e. one multiplier whose powers generate all nonzero scalings - without losing a clean rewrite theory.

• choose a smaller signature for multiplicative scalings,
• rework the interaction rules with translations, controlled additions, and diagonal phase gates,
• maybe even aim for a minimal presentation.

This matters because current prime-dimensional calculi are already compact, but they still treat every nonzero scalar as a primitive. Compressing that part of the syntax could make rewriting engines and proof-assistant formalizations much lighter.

State of the art: in my current affine-plus-diagonal work, the affine core already has a sound and complete presentation, but it uses the full family of scaling generators. The open step is to shrink that family without losing short local rewrites or constructive normalization. (arXiv:2603.06466, arXiv:1307.5087)

Current focus Fit: ambitious M2 or PhD Scope: deep

Goal: Extend the prime-dimensional affine-plus-diagonal / phase-polynomial story to prime powers and, if possible, to more general finite rings or composite dimensions.

• decide the right mathematical setting (prime-power fields, ring-based models, or something else),
• choose generators for affine updates and diagonal phases that still support local rewrites,
• find the right polynomial / binomial bases and finite-difference identities,
• prove a completeness theorem, or clearly isolate the obstruction if completeness fails.

This matters because a lot of qudit work ultimately wants dimension-generic tooling, and composite or prime-power dimensions show up naturally in encodings and hardware. It is also the clearest route from a prime-only theory to something genuinely "all qudits".

State of the art: my current paper relies on arithmetic over finite fields and on the invertibility of 2 and 6 for the quadratic and cubic layers, but I think that it might easily be generalized (maybe it is direct?). Broader Clifford/stabilizer descriptions exist beyond the prime case, but a compact affine-plus-diagonal rewrite theory with normal forms is still open there. (arXiv:2603.06466, arXiv:1307.5087, arXiv:1810.10259)

Current focus Fit: advanced M2 or PhD Scope: deep

Goal: Push the affine-plus-diagonal fragments beyond the current linear / quadratic / cubic layers, and understand which new generators are forced at higher degree.

• introduce degree-k phase generators (for example based on binomial polynomials C(x,k)) when the dimension allows it,
• determine which extra controlled-diagonal gates are required for closure under affine substitution,
• keep the axiom sets short, local, and usable for rewriting,
• connect the resulting theory to higher-level synthesis / optimization questions such as phase count and hierarchy level.

A particularly interesting subproblem is the qudit analogue of refined-angle constructions: qubits have well-studied "square-root phase" extensions, and it would be great to understand the right higher-dimensional version.

State of the art: my current completeness results stop at degree at most 3 in prime dimension. Above that, we do not yet have a clean account of the necessary generators, normal forms, or controlled-diagonal closure conditions. Existing phase-polynomial and phase-gadget work strongly suggests this direction should matter in practice. (arXiv:2603.06466, arXiv:1701.00140, arXiv:1902.05634, arXiv:2504.12710)

Open / exploratory Fit: advanced M2 or PhD Scope: deep

Goal: Start from the prime-dimensional phase-affine fragment in my recent paper and adjoin the one-wire X^{a,b} gate, i.e. a two-level X gate acting non-trivially only on a chosen pair of levels. Then ask what survives:

• does the resulting fragment still sit inside a natural class of monomial unitaries,
• can the current affine-plus-diagonal normal forms be repaired to include these new generators,
• which interaction rules with translations, scalings, and diagonal phases are forced,
• and is there still any hope for a short local rewrite system?

This matters because two-level gates are very natural from the hardware / synthesis side, so a positive answer would connect the current polynomial-style completeness story to a more realistic gate library. A negative answer would also be useful: it would tell us exactly where monomiality or rewrite-friendliness breaks.

State of the art: the current phase-affine result gives a clean prime-dimensional completeness theorem for affine updates plus diagonal phases. Two-level qudit gates are already a standard synthesis primitive, but it is not clear whether adjoining X^{a,b} keeps the fragment tractable from an equational point of view or forces genuinely new invariants. (arXiv:2603.06466, quant-ph/0509161, arXiv:2212.04496)

Current focus Fit: M2 Scope: medium

Goal: Study a single-qudit analogue of "Clifford+T" (more generally: Clifford-cyclotomic gate sets) on one wire:

• choose a concrete gate set (qutrit or general odd prime dimension),
• define a normal form and a synthesis algorithm,
• turn it into code (small exact synthesizer / normalizer),
• measure "non-Clifford cost" (analogue of T-count).

(arXiv:1803.05047, arXiv:2011.07970, arXiv:2303.13644, arXiv:2405.08136)

Current focus Fit: M1/M2 or independent project Scope: medium

Goal: Use known qudit decompositions (controlled-addition / "CINC"-style gates) to get measurable improvements in concrete algorithms:

• a benchmark suite (arithmetic blocks, oracles, diagonal gadgets),
• alternative encodings (qubits vs qutrits/qudits),
• quantitative tradeoffs (two-qudit gate count, depth, ancillas, runtime/memory).

(arXiv:2008.00959, quant-ph/0509161, arXiv:2303.12979, arXiv:2212.04496)

Current focus Fit: PhD Scope: deep

Goal: Extend "fault-equivalence-preserving rewrite" ideas from qubits to qudits. Deliverables could include: a qudit noise model + fault equivalence notion, qudit-safe rewrites, and a small prototype pipeline.

(arXiv:2506.17181, quant-ph/9802007, arXiv:2307.10095, arXiv:2405.10896)

Current focus Fit: advanced M2 or PhD Scope: deep

Goal: Build optimisation methods for Clifford(-dominant) circuits that treat ancillas as an explicit resource, producing a Pareto tradeoff (CNOT count/depth vs number of ancillas).

(arXiv:2504.00634, Q-Synth (code), arXiv:2305.01674, arXiv:1907.05087, arXiv:1405.6073)

Backlog / parked Fit: implementation-heavy M2 or PhD Scope: deep

Goal: Build an extraction pipeline for qudit ZX diagrams: take a simplified diagram and output an equivalent circuit in a chosen qudit gate set.

• a well-defined "extractable" normal form for qudit diagrams,
• an extraction algorithm with clear complexity/limits,
• a prototype implementation + benchmarks.

State of the art: for qubits, extraction exists in mature tooling (notably PyZX) and is backed by strong simplification. Qudit extraction that is dimension-generic is much less standard, and extraction can be hard in the worst case - so the key is practical fragments. (arXiv:1904.04735, arXiv:1902.03178, arXiv:2202.09194, arXiv:2405.10896)

Backlog / parked Fit: M2+ or independent implementation project Scope: medium to deep

Goal: Build an end-to-end qutrit pipeline: (1) simplify a qutrit ZX diagram, then (2) extract a reasonably efficient qutrit circuit, and (3) benchmark.

State of the art: qutrit ZX is complete for stabilizer quantum mechanics; qutrit diagonal constructions via phase gadgets exist. The open part is robust extraction conventions and tool integration comparable to PyZX. (arXiv:1803.00696, arXiv:2204.13681, arXiv:1904.04735)

Backlog / parked Fit: strong M2 or PhD Scope: deep and foundational

Goal: Develop a diagrammatic rewrite language whose primitive generators match two-level qudit operations (gates acting non-trivially only on a pair of levels).

• define generators + semantics,
• propose core rewrite rules (soundness first; completeness for a useful fragment if feasible),
• show how it plugs into compilation (rewriting → synthesis/extraction in the same gate set).

State of the art: synthesis via two-level/QR decompositions is well studied; hardware-motivated compiler work exists for qudits. The open part is a "ZX-like" calculus that uses these primitives directly and supports simplification/verification workflows. (quant-ph/0509161, arXiv:2212.04496)