Colin Blake

PhD student, Inria MOCQUA

Research ideas

A PhD-student page: problems for discussion, collaboration, or a concrete side project.

Some ideas are close to my current work. Others are rougher and need a first pass before they deserve a bigger claim.

Right Now

My near-term energy is mostly going into talks and current qudit-fragment work. If I had to pick one research thread to sharpen next, it would be this one.

Browse

Each topic links to cards below. The cards are intentionally short; click one for the actual question and first checks.

Qudit Clifford structure
Current focus

Clifford circuits in prime, even, and composite dimension. The practical targets are conventions, normal forms, and rewrite rules that can actually be used.

2 ideas Best for advanced students or researchers who like both algebra and tooling
Linear optics
Open / exploratory

Small constructive questions in LOv/LOfi, with Fourier gates as the main test case.

1 idea Best for people who like explicit matrices and benchmarks
Outreach
Actively looking for help

Playable or printable material that teaches a real circuit idea without pretending to be a full course.

1 idea Best for collaborators who enjoy pedagogy, games, or web work
Other algebraic viewpoints
Tomorrow-problem

Exact synthesis, Clifford/geometric algebra, controlled structure, and group presentations. A viewpoint stays on the list only if it makes a circuit problem clearer.

5 ideas Best for self-directed people with strong algebra taste
Equational theories
Current focus

Generators, equations, normal forms, and the places where a calculus becomes small enough to reason with.

4 ideas Best for people who want theorem work with compiler consequences
ZX/ZH diagrams
Backlog / parked

Extraction and rewrite tooling for qudit diagrammatic calculi. Lower priority for now.

3 ideas Best for implementation-heavy collaborators
Circuit fragments
Current focus

Small gate sets, expressivity, fault-tolerance constraints, and quantitative tradeoffs.

4 ideas Best for people who like concrete fragments and measurements

Labels

  • Current focus I am already close to this.
  • Open / exploratory Good question; scope may move.
  • Tomorrow-problem Worth keeping warm; not my main work this week.
  • Backlog / parked Needs a driven collaborator.

Qudit Clifford structure

Current focus 2 ideas Best for advanced students or researchers who like both algebra and tooling

Clifford circuits in prime, even, and composite dimension. The practical targets are conventions, normal forms, and rewrite rules that can actually be used.

Linear optics

Open / exploratory 1 idea Best for people who like explicit matrices and benchmarks

Small constructive questions in LOv/LOfi, with Fourier gates as the main test case.

Outreach

Actively looking for help 1 idea Best for collaborators who enjoy pedagogy, games, or web work

Playable or printable material that teaches a real circuit idea without pretending to be a full course.

Other algebraic viewpoints

Tomorrow-problem 5 ideas Best for self-directed people with strong algebra taste

Exact synthesis, Clifford/geometric algebra, controlled structure, and group presentations. A viewpoint stays on the list only if it makes a circuit problem clearer.

Equational theories

Current focus 4 ideas Best for people who want theorem work with compiler consequences

Generators, equations, normal forms, and the places where a calculus becomes small enough to reason with.

ZX/ZH diagrams

Backlog / parked 3 ideas Best for implementation-heavy collaborators

Extraction and rewrite tooling for qudit diagrammatic calculi. Lower priority for now.

Circuit fragments

Current focus 4 ideas Best for people who like concrete fragments and measurements

Small gate sets, expressivity, fault-tolerance constraints, and quantitative tradeoffs.

Axiomatizing Clifford circuits for qudits in arbitrary dimension

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

Who this fits

For someone who can live with conventions for a while before the theorem statement becomes clean.

Background

Stabilizers, Clifford gates, finite rings or modules, and a taste for normal forms.

Likely output

A usable convention, a small benchmark set, and a first normal-form or rewriting result.

First milestone

Fix the gate conventions for one composite dimension, then test the first identities by hand or script.

Question: Can we write Clifford circuit equations for all without hiding composite dimensions under casework?

Why this is here: Abstract descriptions of stabilizers and Clifford groups already cover arbitrary dimension. The missing piece is a circuit-level account: conventions, generators, normal forms, and small tests that a compiler could use.

First checks:

  • Choose one composite test case, probably , and fix the phase and gate conventions.
  • Compare the resulting gates with the arbitrary-dimensional stabilizer descriptions.
  • Build a tiny verifier for equalities before trying to prove a general theorem.

Starting points: arbitrary-d stabilizers , Clifford ideals , Clifford review

What exactly goes wrong for Clifford circuits in even dimension?

Tomorrow-problem Fit: strong M2 or PhD Scope: deep

Who this fits

Good for someone who enjoys finding the small counterexample that explains the larger obstruction.

Background

Odd-prime Clifford theory, even-dimensional stabilizers, and careful examples.

Likely output

A short map of the obstructions, or one even-dimensional fragment that rewrites cleanly.

First milestone

Work in d = 4 and list the identities from odd prime dimension that stop being true.

Question: Which odd-prime Clifford arguments fail in even dimension, and which failures are only convention problems?

Why this is here: Even-dimensional qudits show up naturally in hardware and encodings. Many odd-prime tricks stop being automatic there. A useful output would name the obstruction precisely.

First checks:

  • Work in and list the first identities that fail.
  • Separate phase-convention issues from genuine algebraic obstructions.
  • Look for one even-dimensional subfragment that still has a clean rewrite story.

Starting points: Clifford review , Clifford ideals , transmon qudit synthesis

Fourier gates in LOv-calculus, in prime and composite dimension

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

Who this fits

A good match for someone who likes explicit matrices and small verification scripts.

Background

Linear algebra, linear optics conventions, and some coding.

Likely output

A generator, a verifier, and a small table of resource counts.

First milestone

Generate and verify H_d circuits for d = 2, 3, and 5.

Question: Can the Fourier gate be generated and certified inside LOv for both prime and composite ?

Why this is here: Fourier gates are a good stress test. They force the conventions to be explicit, and they give quick resource counts once the construction works.

First checks:

  • Reproduce the case inside the chosen LOv/LOfi conventions.
  • Generate candidate circuits for and .
  • Check each circuit by matrix comparison and record the resource count.

Starting points: LOfi/linear-optical calculus

Extend "Les Chevaliers du Quantique" with new levels and guided solutions

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

Who this fits

Best for someone who likes making a small learning tool feel clear and playable.

Background

Web development, educational design, or quantum outreach.

Likely output

A polished level pack, with solutions that explain the circuit idea being practiced.

First milestone

Ship one short sequence of levels around a single circuit rule.

Question: Which new levels would teach the printable circuit-game rules without turning the videogame into a textbook?

Why this is here: "Les Chevaliers du Quantique" is the videogame version of a simplified circuit tutorial game. New levels should carry one visible learning goal and a guided solution.

First checks:

  • Pick one rule from the printable circuit game and make a three-to-five-level sequence around it.
  • Write short guided solutions that explain the circuit move behind each answer.
  • Playtest with someone who has not seen the paper version.

Starting points: playable game , short article

Can the few-square-roots exact synthesis story work for qutrits?

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

Who this fits

A theorem-first project that starts with computations.

Background

Exact synthesis, cyclotomic rings, and symbolic experiments.

Likely output

Either a constructive qutrit statement or a clear reason this analogy breaks.

First milestone

Test a few cube-root rings on small one-qutrit unitaries.

Question: Is there a qutrit analogue of the few-square-roots exact synthesis story, using cube roots instead?

Why this is here: The qubit result suggests a qutrit test. Start with the arithmetic: small rings, small one-qutrit unitaries, and the first obstruction that shows up.

First checks:

  • Choose a small candidate ring for one-qutrit unitaries.
  • Search for exact decompositions of a few sample gates.
  • Record the first obstruction instead of forcing the qubit proof to fit.

Starting points: few square roots , qutrit cyclotomic ideas , single-qutrit Clifford+T

Do Clifford or geometric algebras help with qudit circuits?

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

Who this fits

Best for someone who is happy to survey, translate, and then discard what is not useful.

Background

Algebra, Clifford algebras, and qudit gate semantics.

Likely output

A short survey plus one translation that either simplifies something or proves it does not.

First milestone

Translate one familiar qudit gate family and see what becomes shorter.

Question: Does geometric algebra make any qudit circuit calculation shorter, or is it mostly a change of language?

Why this is here: A new formalism has to shorten something: a normal form, a synthesis step, or a proof. A translation exercise would make that visible quickly.

First checks:

  • Translate one familiar qudit gate family into the formalism.
  • Compare the length of one standard calculation before and after translation.
  • Write down what the formalism cannot see easily.

Starting points: qudit Clifford algebras

Promised PROPs for controlled and promise-style circuit structure

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

Who this fits

For category-minded work grounded in actual circuits.

Background

PROPs, string diagrams, controlled operations, and promise problems.

Likely output

A small formalism with equations and one worked circuit example.

First milestone

Translate one promise-style fragment into controlled-PROP language.

Question: Can promise-style assumptions and controlled operations be handled in one PROP-level language?

Why this is here: Controlled structure appears everywhere in circuits. Promise assumptions usually sit outside the diagram. Bringing them into the same language may simplify some transformations.

First checks:

  • Pick one small promise fragment and write its controlled operations diagrammatically.
  • Check whether ancilla-for-control conversions become structural.
  • Keep only the equations that survive a concrete circuit example.

Starting points: polycontrolled PROPs , controlled PROPs

Controlled Lawvere theories

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

Who this fits

For categorical definitions with a finite-field test case nearby.

Background

Lawvere theories, cartesian PROPs, finite-field functions, and controlled gates.

Likely output

A definition, a few examples, and one place where the controlled structure does real work.

First milestone

Write the minimal definition and check it on small finite-field functions.

Question: What is the smallest useful definition of a controlled Lawvere theory?

Why this is here: A definition here needs a small example quickly. Finite-field functions and controlled classical gates are the first test case.

First checks:

  • Write the minimal definition, without trying to solve every variant at once.
  • Test it on -reduced polynomial functions over a finite field.
  • Look for one theorem that is cleaner because control is part of the structure.

Starting points: controlled PROPs , graphical algebraic geometry

Coxeter shadows of Clifford and qudit circuit fragments

Tomorrow-problem Fit: PhD or very strong M2 Scope: exploratory

Who this fits

For group-theory taste, with the pattern earned from small circuit examples.

Background

Clifford groups, Coxeter or Artin presentations, finite reflection groups, and exact circuit fragments.

Likely output

A careful table of circuit fragments and the group-theoretic structures that really explain their normal forms.

First milestone

Compare one-qubit Clifford, real Clifford, CNOT-dihedral, and one qutrit or qupit analogue.

Question: When a circuit fragment has a good normal form, is there a Coxeter, Artin, triangle, or reflection-group presentation hiding behind it?

Why this is here: Small qubit fragments sometimes have finite-group shadows. Qudit fragments add arithmetic that may break the Coxeter picture. A useful project would compare a few fragments side by side and ask which presentations explain their normal forms.

First checks:

  • Start with one-qubit Clifford and real subfragments, recording which generators are involutions and which relations are braid-like.
  • Add CNOT-dihedral or phase-polynomial fragments and separate true group relations from rewrite conveniences.
  • Repeat the exercise for a qutrit or qupit fragment and write down exactly where the Coxeter picture breaks.

Starting points: circuit presentations , CNOT-dihedral presentation , Clifford review

Can prime-dimensional affine/phase calculi use fewer scaling generators?

Current focus Fit: M1 or M2 Scope: medium

Who this fits

A concrete entry project if you want a completeness question with a finite amount of algebra.

Background

Finite fields, rewriting, and small symbolic calculations.

Likely output

A smaller presentation, or a proof that the extra generators are doing real work.

First milestone

Replace the scaling family in one toy fragment and check which rewrites survive.

Question: Do prime-dimensional affine/phase calculi really need a full family of scaling generators?

Why this is here: The scaling family is a compact place to test redundancy. Derivable scalings would simplify the presentation; a counterexample would say what the fragment is remembering.

First checks:

  • Pick one toy fragment and remove most of the scaling family.
  • Check which local rewrites still work.
  • Try to derive the missing scalings or produce a small countermodel.

Starting points: prime affine-diagonal fragments

Affine-plus-diagonal calculi beyond prime dimension

Current focus Fit: ambitious M2 or PhD Scope: deep

Who this fits

For someone who is comfortable choosing the right algebraic setting before proving anything.

Background

Finite fields, finite rings, phase functions, and completeness proofs.

Likely output

A calculus over prime powers or rings, or a precise obstruction.

First milestone

Choose prime powers or rings as the first target and write the smallest generator list.

Question: What should affine-plus-diagonal calculi look like over prime powers or finite rings?

Why this is here: Prime fields make many proofs clean. Composite settings are messier and cannot stay as an afterthought.

First checks:

  • Choose prime powers or rings as the first target.
  • Write the smallest possible generator list.
  • Test whether the prime-dimensional normal form has a recognizable replacement.

Starting points: phase polynomials for qudits , prime affine-diagonal fragments

Higher-degree phase fragments for qudits

Current focus Fit: advanced M2 or PhD Scope: deep

Who this fits

A good fit if you like phase polynomials and examples that slowly turn into structure.

Background

Phase gadgets, finite-field polynomials, and qudit circuit fragments.

Likely output

A new fragment with generators, examples, and the first hierarchy result.

First milestone

Work out degree 4 in one prime dimension and identify the first forced generator.

Question: Which new generators appear when qudit phase functions move beyond low degree?

Why this is here: Low-degree phase fragments are already useful. Higher-degree behavior should say more about the hierarchy of qudit resources.

First checks:

  • Work out degree 4 in one prime dimension.
  • Find the first phase function that cannot be built from the smaller fragment.
  • Turn that example into a proposed generator or obstruction.

Starting points: qutrit phase gadgets , qudit phase-gadget method , qubit phase polynomials

How much do two-level swaps, CX, and a few phases express?

Current focus Fit: advanced M2 or PhD Scope: deep

Who this fits

For someone who likes explicit decompositions and turning notes into a theorem-level story.

Background

Prime-dimensional qudits, two-level gates, controlled addition, and phase profiles.

Likely output

A polished account of the fragment, with examples and synthesis statements.

First milestone

Turn the current notes into a clean story for p >= 5.

Question: How much of the prime-dimensional phase-affine world is already generated by , , and a few phases?

Why this is here: These are my current notes: a sparse fragment with explicit decompositions, zero-sum diagonal examples, and controlled phases that should become synthesis statements.

First checks:

  • Clean the current decomposition notes for .
  • Make the zero-sum diagonal examples readable.
  • Explain where , , and controlled phases enter.

Starting points: phase polynomials for qudits , prime affine-diagonal fragments

Extract circuits from qudit ZX diagrams

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

Who this fits

Best for someone who wants to build tooling and can make pragmatic fragment choices.

Background

ZX calculus, graph algorithms, and benchmarking.

Likely output

An extraction pipeline for one explicit qudit gate set.

First milestone

Define one extractable normal form and run it on small diagrams.

Question: Can we extract circuits from qudit ZX diagrams without first solving every possible diagrammatic fragment?

Why this is here: A full arbitrary-dimensional extraction theory is a lot. A useful prototype can be narrower: pick one normal form, one target gate set, and one benchmark family.

First checks:

  • Choose a small extractable class of diagrams.
  • Define the target qudit gate set.
  • Run extraction on hand-made examples before touching random benchmarks.

Starting points: complete ZX thesis , finite-dimensional ZX , PyZX

Extract circuits from qutrit ZX diagrams

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

Who this fits

A smaller implementation project than arbitrary-d extraction.

Background

Qutrit ZX basics and practical coding.

Likely output

A qutrit-only prototype with a few benchmarks.

First milestone

Adapt one qutrit simplification pass and extract circuits on a toy dataset.

Question: Can a qutrit-only extractor be made simple enough to be a real tool?

Why this is here: Qutrits are a good middle ground: richer than qubits, much less sprawling than arbitrary .

First checks:

  • Pick one qutrit simplification pass.
  • Define a tiny extracted gate set.
  • Build a toy dataset and measure where extraction fails.

Starting points: qutrit ZX stabilizer completeness , ZX graph simplification

A ZX-like calculus for two-level qudit gates

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

Who this fits

For someone who wants a foundational calculus project with a hardware-facing gate set.

Background

Diagrammatic reasoning and hardware-aware qudit synthesis.

Likely output

A candidate calculus with semantics and first rewrite rules.

First milestone

Choose generators and semantics for a strict two-level fragment.

Question: What would a diagrammatic calculus look like if two-level qudit gates were primitive?

Why this is here: Two-level operations are natural in synthesis and hardware. A ZX-like syntax has to use them without collapsing into ordinary circuit notation.

First checks:

  • Choose the two-level generators and their semantics.
  • Write the first local rewrite rules.
  • Compare against a small hardware-native synthesis example.

Starting points: qudit ZH , ZH calculus , ZX extraction hardness

Single-qupit Clifford+T and cyclotomic fragments

Current focus Fit: M2 Scope: medium

Who this fits

A contained exact-synthesis project if you want one wire before many wires.

Background

Exact synthesis, cyclotomic rings, and algebraic gate sets.

Likely output

A normal form and a small exact synthesizer.

First milestone

Fix one gate set and implement exact synthesis on one wire.

Question: What is the clean one-wire exact synthesis story for single-qupit Clifford+T-like fragments?

Why this is here: One wire is the right place to be strict about arithmetic before adding controls, entanglement, and layout.

First checks:

  • Fix one prime dimension and one gate set.
  • Implement exact synthesis for a small sample of unitaries.
  • Compare the normal form with known qutrit and qudit exact-synthesis results.

Starting points: single-qudit Clifford+T , single-qutrit Clifford+T , multi-qutrit exact synthesis

Use qudit controlled-X decompositions in actual algorithms

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

Who this fits

Best for someone who likes benchmarks and concrete tradeoffs.

Background

Circuit synthesis, small algorithms, and enough coding to compare encodings.

Likely output

A benchmark study of compression, depth, and storage tradeoffs.

First milestone

Pick one arithmetic or oracle block and compare qubit and qudit encodings.

Question: Do generalized qudit controlled-X decompositions improve any real algorithm block?

Why this is here: Use benchmarks here. The point is to find where qudit encodings save depth, gates, or storage.

First checks:

  • Choose one arithmetic, oracle, or lookup block.
  • Implement the qubit and qudit versions with the same metric conventions.
  • Plot the first compression/depth/storage tradeoff.

Starting points: multi-controlled qudit gates , qudit review

Fault tolerance by construction for qudit circuits

Current focus Fit: PhD Scope: deep

Who this fits

A high-risk project for someone interested in both rewriting and fault tolerance.

Background

Noise models, rewrite systems, and fault-tolerant circuit theory.

Likely output

A qudit-safe rewrite criterion and a first prototype pipeline.

First milestone

Define a tiny fault-equivalence notion and test it on a small rewrite set.

Question: Can fault-tolerance-by-construction ideas be stated for qudit rewrite systems?

Why this is here: Rewrite rules are only useful in a fault-tolerant setting if they respect the code or noise model being used. For qudits, those choices are less settled.

First checks:

  • Choose one tiny qudit noise or code model.
  • Define what it means for one rewrite to be safe.
  • Test the definition on a small set of circuit equations.

Starting points: fault tolerance by construction , qudit fault tolerance

Clifford+ancilla minimisation

Current focus Fit: advanced M2 or PhD Scope: deep

Who this fits

For someone who likes optimization, solver thinking, and quantitative tradeoffs.

Background

Circuit metrics, Clifford synthesis, and preferably SAT or search.

Likely output

A Pareto-style optimizer, or strong lower and upper bounds.

First milestone

Model one small Clifford family and plot the first ancilla-versus-depth frontier.

Question: How much depth or gate count can Clifford circuits save if we allow a fixed number of ancillas?

Why this is here: The output should be a frontier: plots or bounds for ancillas versus depth or gate count.

First checks:

  • Pick one small Clifford family and one metric pair, such as ancillas versus depth.
  • Model the search with SAT, dynamic programming, or brute force for very small sizes.
  • Compare the first frontier with known CNOT and Clifford optimization results.

Starting points: Q-Synth , depth-optimal Clifford SAT , CNOT space-depth tradeoff , ancilla cost tradeoff