Ideas

Current focus My interest 95/100 Difficulty 75/100 Importance 85/100

Goal: Design a compact, sound and complete set of circuit identities (generators + relations / rewrite rules) for multi-qupit Clifford circuits (prime dimension p), in a form suitable for automated rewriting.

“Sound and complete” here means: (1) every rule is correct as a matrix identity, and (2) every true equality of qupit Clifford circuits can be derived using the rules.

State of the art: Already done: there is now a complete rewrite rule set for multi-qutrit Clifford circuits (a landmark result for odd prime dimension), and the general Clifford structure for arbitrary dimensions is documented in several foundational works. Still open: a dimension-parametric presentation that works for general prime p (not only p=3), with a normal form and an implementation that scales. (arXiv:2508.14670, quant-ph/0506065, arXiv:1307.5087)

Current focus My interest 90/100 Difficulty 55/100 Importance 70/100

Goal: Build a rewrite-friendly “dictionary” of identities for the 1-qupit Clifford group (prime dimension p). Concretely, the project should deliver:

• a small set of generators (e.g., Fourier/phase-like gates) and a finite set of relations (rewrite rules);
• a canonical normal form (every 1-qupit Clifford reduces to a unique, standard circuit);
• an implementation that takes a 1-wire Clifford circuit and normalizes it automatically.

This matters because 1-wire is the cleanest “sandbox” where you can make proofs rigorous and build tooling that later scales to multi-qudit compilation and verification.

State of the art: Already done: the structure of qudit Clifford operations is well understood at the group-theory level, with constructive characterizations and generating sets. Still open (tooling-wise): turning that understanding into a small, implementation-ready rewrite system + normal form, with automated tests and (ideally) machine-checked proofs. (quant-ph/0506065, arXiv:1307.5087, arXiv:1310.6813)

Open / exploratory My interest 80/100 Difficulty 90/100 Importance 90/100

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)

Open / exploratory My interest 55/100 Difficulty 80/100 Importance 75/100

Goal: Define a qudit analogue of the CNOT-dihedral fragment (roughly: “linear reversible structure + structured phases”), and then:

• either give a finite presentation (generators + relations) with a usable normal form,
• or identify precisely where/why finiteness fails in higher dimension.

This matters because CNOT-dihedral-style fragments capture “phase logic” and directly feed into qutrit/qupit optimizers (especially for diagonal unitaries).

State of the art: for qubits, finite presentations and synthesis methods exist (phase polynomials). For qutrits/qudits, phase-gadget and phase-polynomial optimizers exist, but a clean “qudit CNOT-dihedral” rewrite theory is not yet standard. (arXiv:1701.00140, arXiv:2006.12042, arXiv:1303.2042, arXiv:1902.05634, arXiv:2204.13681)

Tomorrow-problem My interest 65/100 Difficulty 95/100 Importance 80/100

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)

Current focus My interest 95/100 Difficulty 75/100 Importance 75/100

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 My interest 85/100 Difficulty 30/100 Importance 70/100

Goal: Get a minimal but solid web version out (the “bare minimum” I plan to do), and if you're up for it, push further with clearly scoped, student-friendly deliverables:

• design levels or challenge packs with explicit learning objectives, including extra rules,
• write/record short solution videos (explain the reasoning, not just the answer),
• 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 and has been shared as an educational resource. The open part is more content + guided solution material, and (if desired) evidence about learning impact. (Game page)

Open / exploratory My interest 65/100 Difficulty 85/100 Importance 70/100

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 My interest 70/100 Difficulty 80/100 Importance 65/100

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)

Backlog / parked My interest 40/100 Difficulty 85/100 Importance 90/100

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 My interest 45/100 Difficulty 75/100 Importance 85/100

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 My interest 45/100 Difficulty 85/100 Importance 75/100

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)