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arxiv: 2311.14347 · v4 · submitted 2023-11-24 · 💻 cs.PL · cs.LO

Typed compositional quantum computation with lenses

Jacques Garrigue , Takafumi Saikawa This is my paper

Pith reviewed 2026-05-24 06:34 UTC · model grok-4.3

classification 💻 cs.PL cs.LO
keywords quantum circuitslensesCoqtype theorycompositionalitycurryingquantum computationprogram verification
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The pith

Discrete lenses in Coq control currying on quantum states to separate circuit structure from gate content.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a type-theoretic framework in Coq for quantum circuits by exploiting currying on quantum states inside the polymorphic type system. This currying lets gates apply directly inside larger circuits. A discrete lens construct is introduced to manage the currying, which separates the combinatorics of wiring from the actual gate operations. The approach supports defining circuits recursively from smaller parts and establishing their correctness through compositional proofs.

Core claim

We propose a type-theoretic framework for describing and proving properties of quantum computations, in particular those presented as quantum circuits. Our proposal is based on an observation that, in the polymorphic type system of Coq, currying on quantum states allows us to apply quantum gates directly inside a complex circuit. By introducing a discrete notion of lens to control this currying, we are further able to separate the combinatorics of the circuit structure from the computational content of gates. We apply our development to define quantum circuits recursively from the bottom up, and prove their correctness compositionally.

What carries the argument

Discrete lens controlling currying on quantum states in Coq's polymorphic type system.

If this is right

  • Quantum circuits can be defined recursively from the bottom up using the lens construct.
  • Correctness of circuits can be proved compositionally without mixing structure and computation.
  • Gates can be applied directly inside complex circuits through controlled currying.
  • The combinatorics of circuit wiring remain independent of the computational content of individual gates.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The separation of structure and content may reduce the complexity of verifying larger quantum algorithms built from repeated subcircuits.
  • Similar lens-based control could be explored in other typed functional languages that support polymorphic state manipulation for quantum or classical reversible computation.
  • The recursive bottom-up style might integrate with existing categorical models of quantum computation to yield hybrid verification tools.

Load-bearing premise

Currying on quantum states inside Coq's polymorphic type system can be controlled by a discrete lens construct without breaking the underlying quantum semantics or requiring additional axioms.

What would settle it

A concrete quantum circuit whose structure cannot be separated from its gate content using the discrete lens, or whose correctness proof fails to hold compositionally under the lens-controlled currying in Coq.

read the original abstract

We propose a type-theoretic framework for describing and proving properties of quantum computations, in particular those presented as quantum circuits. Our proposal is based on an observation that, in the polymorphic type system of Coq, currying on quantum states allows us to apply quantum gates directly inside a complex circuit. By introducing a discrete notion of lens to control this currying, we are further able to separate the combinatorics of the circuit structure from the computational content of gates. We apply our development to define quantum circuits recursively from the bottom up, and prove their correctness compositionally.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper presents a Coq development for typed quantum computation that introduces a discrete lens construct to control currying of quantum states within Coq's polymorphic type system. This separation of circuit combinatorics from gate semantics enables bottom-up recursive definitions of quantum circuits together with compositional correctness proofs that preserve unitarity by induction on the lens structure.

Significance. If the result holds, the work supplies machine-checked proofs of compositional correctness for recursively defined quantum circuits inside an existing dependently-typed framework (using only standard Coq libraries for vectors and complex numbers). The lens-based separation of structure from semantics is a concrete strength for modularity and reproducibility in formal quantum programming.

minor comments (2)
  1. [Abstract] The abstract states that the lens 'commutes with the standard tensor-product representation' but does not name the precise Coq definition or lemma that establishes this commutation.
  2. [Section 3] Section 3 (or equivalent) would benefit from an explicit statement of the induction hypothesis used for the unitarity preservation proof.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive review of the manuscript and their recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper is a self-contained Coq formalization that defines a discrete lens construct to manage currying of quantum states inside the polymorphic type system, then uses it to separate circuit structure from gate semantics and prove recursive circuit correctness by induction. All steps rely on standard Coq libraries for vectors and complex numbers plus explicit inductive proofs that preserve unitarity; no parameter fitting, self-referential definitions, or load-bearing self-citations appear. The development is machine-checked and externally falsifiable against the tensor-product semantics of quantum states, satisfying the criteria for independent content.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no free parameters, axioms, or invented entities are identifiable.

pith-pipeline@v0.9.0 · 5607 in / 975 out tokens · 22645 ms · 2026-05-24T06:34:27.834861+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Hybrid Path-Sums for Hybrid Quantum Programs

    cs.PL 2026-04 unverdicted novelty 7.0

    Hybrid Path-Sums offer a new symbolic framework with rewriting rules and assertions to represent, simplify, and verify properties of hybrid quantum-classical programs.

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