Colin Blake

PhD student, Inria MOCQUA

Research ideas

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

Some entries sit near my papers. Others begin with one calculation.

Right Now

Near-term work: talks, qudit fragments, and a few explicit decompositions.

Browse

Each topic links to cards below. Click one for the calculation, test case, or references attached.

Qudit Clifford structure
Current focus

Clifford circuits in prime, even, and composite dimension. Main targets: conventions, normal forms, rewrite rules.

2 ideas Fits algebra plus tooling
Linear optics
Open / exploratory

LOv/LOfi around the Fourier gate; matrices decide the answer.

1 idea Fits explicit matrices and benchmarks
Outreach
Actively looking for help

Playable or printable material for one circuit idea at a time.

1 idea Fits pedagogy, games, or web work
Other algebraic viewpoints
Tomorrow-problem

Exact synthesis, Clifford/geometric algebra, controlled structure, and group presentations. Test each viewpoint on a circuit problem.

5 ideas Fits self-directed algebra work
Equational theories
Current focus

Generators, equations, normal forms, and calculi checked by hand or script.

4 ideas Fits theorems with compiler consequences
ZX/ZH diagrams
Backlog / parked

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

3 ideas Fits implementation-heavy work
Circuit fragments
Current focus

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

4 ideas Fits fragments and measurements

Labels

  • Current focus Close to current papers.
  • Open / exploratory Scope may move.
  • Tomorrow-problem Background work first.
  • Backlog / parked Waiting for the right collaborator.

Qudit Clifford structure

Current focus 2 ideas Fits algebra plus tooling

Clifford circuits in prime, even, and composite dimension. Main targets: conventions, normal forms, rewrite rules.

Linear optics

Open / exploratory 1 idea Fits explicit matrices and benchmarks

LOv/LOfi around the Fourier gate; matrices decide the answer.

Outreach

Actively looking for help 1 idea Fits pedagogy, games, or web work

Playable or printable material for one circuit idea at a time.

Other algebraic viewpoints

Tomorrow-problem 5 ideas Fits self-directed algebra work

Exact synthesis, Clifford/geometric algebra, controlled structure, and group presentations. Test each viewpoint on a circuit problem.

Equational theories

Current focus 4 ideas Fits theorems with compiler consequences

Generators, equations, normal forms, and calculi checked by hand or script.

ZX/ZH diagrams

Backlog / parked 3 ideas Fits implementation-heavy work

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

Circuit fragments

Current focus 4 ideas Fits fragments and measurements

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

Clifford circuit equations beyond prime dimension

Open / exploratory Advanced M2 or PhD Scope: deep

For

Algebraic patience; comfort with convention-hunting.

Background

Stabilizers, Clifford gates, finite rings or modules, normal forms.

Output

Gate conventions, a benchmark list, and one normalization result.

Start

Write the generators and test ten basic identities.

Clifford equations in arbitrary dimension, including composite .

Stabilizer descriptions cover arbitrary dimension. A circuit presentation has to choose phases, generators, normal forms, and tests a program can run.

  • Write the gate conventions on one page.
  • Check them against the standard stabilizer description.
  • Build a script for equality tests on two wires.

Reading: arbitrary-d stabilizers, Clifford ideals, Clifford review

Even-dimensional Clifford circuits

Tomorrow-problem Strong M2 or PhD Scope: deep

For

Careful examples; counterexamples welcome.

Background

Odd-prime Clifford theory, even-dimensional stabilizers, phase conventions.

Output

A list of failed odd-prime identities, plus one fragment that still behaves well.

Start

Run the odd-prime equations in and mark the first failures.

Even dimension: which Clifford equations fail, and why?

Odd-prime proofs often divide by 2. In , that step fails. Separate phase convention from obstruction.

  • Run the odd-prime identities in .
  • Mark failures caused only by phase convention.
  • Keep one surviving fragment and normalize examples inside it.

Reading: Clifford review, Clifford ideals, transmon qudit synthesis

Fourier gates in LOv-calculus

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

For

Matrices, scripts, and convention bookkeeping.

Background

Linear algebra, linear optics conventions, verification by matrices.

Output

A generator, a verifier, and resource counts for a few dimensions.

Start

Generate and check for .

Fourier gates inside LOv, checked by matrices.

fixes the mode conventions. After that, compare resource counts.

  • Reproduce in the chosen LOv/LOfi conventions.
  • Generate candidates for and .
  • Compare matrices and record gate counts.

Reading: LOfi/linear-optical calculus

"Les Chevaliers du Quantique": more levels, better solutions

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

For

Pedagogy, level design, or web/game implementation.

Background

Quantum outreach, educational design, basic web development.

Output

A level pack with short solution notes tied to one circuit rule.

Start

Pick one rule from the printable game and make three levels for it.

New levels for the circuit-simplification game.

"Les Chevaliers du Quantique" is the videogame version of the printable circuit tutorial. Each level pack can teach one rule and show the move used in the solution.

  • 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 new to the paper version.

Reading: playable game, short article

Qutrit exact synthesis with cube-root arithmetic

Open / exploratory Advanced M2 or PhD Scope: deep

For

Exact synthesis driven by computations.

Background

Cyclotomic rings, symbolic algebra, one-qutrit gates.

Output

A qutrit synthesis statement or a documented obstruction.

Start

Test two candidate cube-root rings on sample one-qutrit gates.

A qutrit version of the few-square-roots synthesis result.

The qubit theorem points to a qutrit arithmetic problem: choose rings, sample gates, and look for obstructions.

  • Choose a small candidate ring for one-qutrit unitaries.
  • Search for exact decompositions of a few sample gates.
  • Record the first obstruction that appears.

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

Clifford/geometric algebra as qudit circuit language

Tomorrow-problem Self-directed M2 or PhD Scope: exploratory

For

Survey work with a strict discard test.

Background

Clifford algebras, geometric algebra, qudit gate semantics.

Output

A short survey and one translated gate family.

Start

Translate Fourier, phase, and shift gates in one odd prime dimension.

Geometric algebra as notation for qudit gates.

Translate a gate family. Keep the formalism only when it shortens a calculation.

  • 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.

Reading: qudit Clifford algebras

Promised PROPs for controlled circuits

Tomorrow-problem PhD or very strong M2 Scope: deep

For

Category theory tied to examples.

Background

PROPs, string diagrams, controlled operations, promise problems.

Output

A promise-aware control formalism with one worked circuit example.

Start

Encode one promise fragment and one controlled operation in the same PROP.

Promises and controlled operations in the same PROP.

Controlled structure sits inside the diagram. Promise assumptions usually sit outside. Combining them may change ancilla and control bookkeeping.

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

Reading: polycontrolled PROPs, controlled PROPs

Controlled Lawvere theories

Tomorrow-problem PhD or very strong M2 Scope: deep

For

Categorical definitions checked on finite-field examples.

Background

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

Output

A definition, examples, and one theorem using control structurally.

Start

Define the controlled finite-field function example before generalizing.

Controlled Lawvere theories, starting from finite fields.

Start from examples: finite-field functions and controlled classical gates.

  • Write the definition for the finite-field case first.
  • Test it on -reduced polynomial functions over a finite field.
  • Try one theorem where control appears in the statement.

Reading: controlled PROPs, graphical algebraic geometry

Coxeter shadows of Clifford and qudit fragments

Tomorrow-problem PhD or very strong M2 Scope: exploratory

For

Group presentations checked against circuit examples.

Background

Clifford groups, Coxeter or Artin presentations, exact circuit fragments.

Output

A table of fragments, presentations, normal forms, and failures.

Start

Compare one-qubit Clifford, real Clifford, CNOT-dihedral, and one qutrit fragment.

Coxeter and Artin presentations as a way to sort circuit fragments.

Some qubit fragments have finite-group shadows. Qudit fragments add arithmetic. Put one-qubit Clifford, real Clifford, CNOT-dihedral, and one qutrit fragment in the same table. Record the presentations that explain their normal forms and the ones that fail.

  • 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.

Reading: circuit presentations, CNOT-dihedral presentation, Clifford review

Fewer scaling generators in prime-dimensional calculi

Current focus M1 or M2 Scope: medium

For

Finite rewriting with some algebraic bite.

Background

Finite fields, rewriting, symbolic checks.

Output

A smaller presentation or a countermodel.

Start

Remove most scaling generators from one toy fragment and test the equations.

Scaling generators in prime-dimensional affine/phase calculi.

Remove scaling generators and ask for derivations or countermodels.

  • 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.

Reading: prime affine-diagonal fragments

Affine-plus-diagonal calculi over prime powers or rings

Current focus Ambitious M2 or PhD Scope: deep

For

Algebraic setup before proofs.

Background

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

Output

A ring or prime-power calculus; otherwise a named obstruction.

Start

Choose prime powers or finite rings and write the generator list.

Affine-plus-diagonal calculi over prime powers or rings.

Prime fields hide choices. Prime powers and rings expose them.

  • 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.

Reading: phase polynomials for qudits, prime affine-diagonal fragments

Higher-degree phase fragments for qudits

Current focus Advanced M2 or PhD Scope: deep

For

Phase polynomials, examples, hierarchy questions.

Background

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

Output

Generators and examples for one higher-degree fragment.

Start

Work out degree 4 in one prime dimension and isolate one missing phase.

Higher-degree phase fragments for qudits.

Compute degree 4 by hand in one prime dimension. Missing phase functions become candidate generators.

  • 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.

Reading: qutrit phase gadgets, qudit phase-gadget method, qubit phase polynomials

Two-level swaps, CX, and sparse phase resources

Current focus Advanced M2 or PhD Scope: deep

For

Explicit decompositions in prime-dimensional qudit circuits.

Background

Two-level gates, controlled addition, finite-field phase profiles.

Output

Examples and synthesis statements for the sparse fragment.

Start

For , decompose CZ, CCZ, and one nontrivial controlled phase.

, , and sparse phase resources.

Start from the prime-dimensional phase-affine paper. In , decompose controlled phases using two-level swaps and .

  • Work out and explicitly for .
  • Write one zero-sum diagonal example as a gate decomposition.
  • Compare the decomposition with the phase-affine normal form.

Reading: phase polynomials for qudits, prime affine-diagonal fragments

Circuit extraction from qudit ZX diagrams

Backlog / parked Implementation-heavy M2 or PhD Scope: deep

For

Tool-building with a fixed fragment.

Background

ZX calculus, graph algorithms, benchmarking.

Output

An extractor for one explicit qudit gate set.

Start

Define one extractable diagram class and run five hand-made examples.

Circuit extraction from a chosen qudit ZX fragment.

Choose a fragment, a normal form, and a gate set. Run extraction, then record failures.

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

Reading: complete ZX thesis, finite-dimensional ZX, PyZX

Circuit extraction from qutrit ZX diagrams

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

For

A qutrit extractor before arbitrary dimension.

Background

Qutrit ZX basics, simplification passes, coding.

Output

A qutrit prototype with a failure log.

Start

Extract circuits from a toy qutrit dataset after one simplification pass.

A qutrit-only extractor before arbitrary .

Qutrits have enough structure to expose hard cases before arbitrary dimension.

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

Reading: qutrit ZX stabilizer completeness, ZX graph simplification

A ZX-like calculus for two-level qudit gates

Backlog / parked Strong M2 or PhD Scope: deep

For

Foundational diagrammatics with a hardware-facing gate set.

Background

Diagrammatic reasoning, two-level synthesis, qudit semantics.

Output

Generators, semantics, and the first sound rewrite rules.

Start

Write the semantics for one two-level generator family.

A diagrammatic calculus with two-level gates as primitives.

Two-level operations appear in synthesis and hardware. Make them primitive in the syntax.

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

Reading: qudit ZH, ZH calculus, ZX extraction hardness

Single-qupit Clifford+T and cyclotomic fragments

Current focus M2 Scope: medium

For

One-wire exact synthesis before multi-wire complications.

Background

Exact synthesis, cyclotomic rings, algebraic gate sets.

Output

A normal form and a small exact synthesizer.

Start

Fix one prime dimension and synthesize a short list of target unitaries.

Single-qupit Clifford+T-like exact synthesis.

One wire is enough arithmetic for a first exact-synthesis result. Controls and layout can wait.

  • 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.

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

Qudit controlled-X decompositions in algorithm blocks

Current focus M1/M2 or independent project Scope: medium

For

Benchmarks and arithmetic blocks.

Background

Circuit synthesis, qudit encodings, benchmark scripts.

Output

A comparison of depth, gate count, storage, and ancillas.

Start

Pick one arithmetic block and implement qubit and qudit versions.

Qudit controlled-X decompositions inside algorithm blocks.

Benchmark depth, gate count, storage, and ancillas.

  • 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.

Reading: multi-controlled qudit gates, qudit review

Fault tolerance by construction for qudit circuits

Current focus PhD Scope: deep

For

Rewriting under a fault-tolerance constraint.

Background

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

Output

A qudit-safe rewrite criterion and a tiny pipeline.

Start

Choose one qudit code or noise model and test two rewrite rules.

Fault-tolerance-preserving rewrites for qudit circuits.

A rewrite rule must respect the code or noise model. For qudits, start with one toy model.

  • 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.

Reading: fault tolerance by construction, qudit fault tolerance

Clifford+ancilla minimisation

Current focus Advanced M2 or PhD Scope: deep

For

Optimization, search, Pareto frontiers.

Background

Clifford synthesis, circuit metrics, SAT or exhaustive search.

Output

Lower/upper bounds or a Pareto frontier.

Start

Plot ancillas versus depth for one small Clifford family.

Ancillas versus depth for Clifford circuits.

Set an ancilla budget. Plot depth or gate count against extra workspace.

  • 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.

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