19 Search Results for "Perdrix, Simon"

Document
DOI: 10.4230/LIPIcs.MFCS.2025.85
Cutoff Theorems for the Equivalence of Parameterized Quantum Circuits

Authors: Neil J. Ross and Scott Wesley

Published in: LIPIcs, Volume 345, 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)

Abstract
Many promising quantum algorithms in economics, medical science, and material science rely on circuits that are parameterized by a large number of angles. To ensure that these algorithms are efficient, these parameterized circuits must be heavily optimized. However, most quantum circuit optimizers are not verified, so this procedure is known to be error-prone. For this reason, there is growing interest in the design of equivalence checking algorithms for parameterized quantum circuits. In this paper, we define a generalized class of parameterized circuits with arbitrary rotations and show that this problem is decidable for cyclotomic gate sets. We propose a cutoff-based procedure which reduces the problem of verifying the equivalence of parameterized quantum circuits to the problem of verifying the equivalence of finitely many parameter-free quantum circuits. Because the number of parameter-free circuits grows exponentially with the number of parameters, we also propose a probabilistic variant of the algorithm for cases when the number of parameters is intractably large. We show that our techniques extend to equivalence modulo global phase, and describe an efficient angle sampling procedure for cyclotomic gate sets.
Cite as

Neil J. Ross and Scott Wesley. Cutoff Theorems for the Equivalence of Parameterized Quantum Circuits. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 85:1-85:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{ross_et_al:LIPIcs.MFCS.2025.85,
  author =	{Ross, Neil J. and Wesley, Scott},
  title =	{{Cutoff Theorems for the Equivalence of Parameterized Quantum Circuits}},
  booktitle =	{50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
  pages =	{85:1--85:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-388-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{345},
  editor =	{Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.85},
  URN =		{urn:nbn:de:0030-drops-241921},
  doi =		{10.4230/LIPIcs.MFCS.2025.85},
  annote =	{Keywords: Quantum Circuits, Parameterized Equivalence Checking}
}
Artifact
Software
DOI: 10.4230/artifacts.23042
LU-equals-LC-up-to-19qubits

Authors: Nathan Claudet

Abstract
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Nathan Claudet. LU-equals-LC-up-to-19qubits (Software, Script). Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@misc{dagstuhl-artifact-23042,
   title = {{LU-equals-LC-up-to-19qubits}}, 
   author = {Claudet, Nathan},
   note = {Software, swhId: \href{https://archive.softwareheritage.org/swh:1:dir:9a2dae69bfe44d00e0a28d0b426596c4a56dc9c5;origin=https://github.com/nathanclaudet/LU-equals-LC-up-to-19qubits;visit=swh:1:snp:efcecacbc1ef9eeba4aff90dd6a7c0b086d86fe4;anchor=swh:1:rev:6125c5b225af36983a59586045f2150698f21470}{\texttt{swh:1:dir:9a2dae69bfe44d00e0a28d0b426596c4a56dc9c5}} (visited on 2025-06-30)},
   url = {https://github.com/nathanclaudet/LU-equals-LC-up-to-19qubits},
   doi = {10.4230/artifacts.23042},
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.59
Deciding Local Unitary Equivalence of Graph States in Quasi-Polynomial Time

Authors: Nathan Claudet and Simon Perdrix

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract
We describe an algorithm with quasi-polynomial runtime n^{log₂(n)+O(1)} for deciding local unitary (LU) equivalence of graph states. The algorithm builds on a recent graphical characterisation of LU-equivalence via generalised local complementation. By first transforming the corresponding graphs into a standard form using usual local complementations, LU-equivalence reduces to the existence of a single generalised local complementation that maps one graph to the other. We crucially demonstrate that this reduces to solving a system of quasi-polynomially many linear equations, avoiding an exponential blow-up. As a byproduct, we generalise Bouchet’s algorithm for deciding local Clifford (LC) equivalence of graph states by allowing the addition of arbitrary linear constraints. We also improve existing bounds on the size of graph states that are LU- but not LC-equivalent. While the smallest known examples involve 27 qubits, and it is established that no such examples exist for up to 8 qubits, we refine this bound by proving that LU- and LC-equivalence coincide for graph states involving up to 19 qubits.
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Nathan Claudet and Simon Perdrix. Deciding Local Unitary Equivalence of Graph States in Quasi-Polynomial Time. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 59:1-59:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{claudet_et_al:LIPIcs.ICALP.2025.59,
  author =	{Claudet, Nathan and Perdrix, Simon},
  title =	{{Deciding Local Unitary Equivalence of Graph States in Quasi-Polynomial Time}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{59:1--59:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.59},
  URN =		{urn:nbn:de:0030-drops-234367},
  doi =		{10.4230/LIPIcs.ICALP.2025.59},
  annote =	{Keywords: Quantum computing, Graph theory, Entanglement, Local complementation}
}
Document
DOI: 10.4230/LIPIcs.STACS.2025.41
Residue Domination in Bounded-Treewidth Graphs

Authors: Jakob Greilhuber, Philipp Schepper, and Philip Wellnitz

Published in: LIPIcs, Volume 327, 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)

Abstract
For the vertex selection problem (σ,ρ)-DomSet one is given two fixed sets σ and ρ of integers and the task is to decide whether we can select vertices of the input graph such that, for every selected vertex, the number of selected neighbors is in σ and, for every unselected vertex, the number of selected neighbors is in ρ [Telle, Nord. J. Comp. 1994]. This framework covers many fundamental graph problems such as Independent Set and Dominating Set. We significantly extend the recent result by Focke et al. [SODA 2023] to investigate the case when σ and ρ are two (potentially different) residue classes modulo m ≥ 2. We study the problem parameterized by treewidth and present an algorithm that solves in time m^tw ⋅ n^O(1) the decision, minimization and maximization version of the problem. This significantly improves upon the known algorithms where for the case m ≥ 3 not even an explicit running time is known. We complement our algorithm by providing matching lower bounds which state that there is no (m-ε)^pw ⋅ n^O(1)-time algorithm parameterized by pathwidth pw, unless SETH fails. For m = 2, we extend these bounds to the minimization version as the decision version is efficiently solvable.
Cite as

Jakob Greilhuber, Philipp Schepper, and Philip Wellnitz. Residue Domination in Bounded-Treewidth Graphs. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 41:1-41:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{greilhuber_et_al:LIPIcs.STACS.2025.41,
  author =	{Greilhuber, Jakob and Schepper, Philipp and Wellnitz, Philip},
  title =	{{Residue Domination in Bounded-Treewidth Graphs}},
  booktitle =	{42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)},
  pages =	{41:1--41:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-365-2},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{327},
  editor =	{Beyersdorff, Olaf and Pilipczuk, Micha{\l} and Pimentel, Elaine and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2025.41},
  URN =		{urn:nbn:de:0030-drops-228675},
  doi =		{10.4230/LIPIcs.STACS.2025.41},
  annote =	{Keywords: Parameterized Complexity, Treewidth, Generalized Dominating Set, Strong Exponential Time Hypothesis}
}
Document
DOI: 10.4230/LIPIcs.STACS.2025.27
Local Equivalence of Stabilizer States: A Graphical Characterisation

Authors: Nathan Claudet and Simon Perdrix

Published in: LIPIcs, Volume 327, 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)

Abstract
Stabilizer states form a ubiquitous family of quantum states that can be graphically represented through the graph state formalism. A fundamental property of graph states is that applying a local complementation - a well-known and extensively studied graph transformation - results in a graph that represents the same entanglement as the original. In other words, the corresponding graph states are LU-equivalent. This property served as the cornerstone for capturing non-trivial quantum properties in a simple graphical manner, in the study of quantum entanglement but also for developing protocols and models based on graph states and stabilizer states, such as measurement-based quantum computing, secret sharing, error correction, entanglement distribution... However, local complementation fails short to fully characterise entanglement: there exist pairs of graph states that are LU-equivalent but cannot be transformed one into the other using local complementations. Only few is known about the equivalence of graph states beyond local complementation. We introduce a generalisation of local complementation which graphically characterises the LU-equivalence of graph states. We use this characterisation to show the existence of a strict infinite hierarchy of equivalences of graph states. Our approach is based on minimal local sets, which are subsets of vertices that are known to cover any graph, and to be invariant under local complementation and even LU-equivalence. We use these structures to provide a type to each vertex of a graph, leading to a natural standard form in which the LU-equivalence can be exhibited and captured by means of generalised local complementation.
Cite as

Nathan Claudet and Simon Perdrix. Local Equivalence of Stabilizer States: A Graphical Characterisation. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 27:1-27:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{claudet_et_al:LIPIcs.STACS.2025.27,
  author =	{Claudet, Nathan and Perdrix, Simon},
  title =	{{Local Equivalence of Stabilizer States: A Graphical Characterisation}},
  booktitle =	{42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)},
  pages =	{27:1--27:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-365-2},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{327},
  editor =	{Beyersdorff, Olaf and Pilipczuk, Micha{\l} and Pimentel, Elaine and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2025.27},
  URN =		{urn:nbn:de:0030-drops-228527},
  doi =		{10.4230/LIPIcs.STACS.2025.27},
  annote =	{Keywords: Quantum computing, Graph theory, Entanglement, Local complementation}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.36
Vertex-Minor Universal Graphs for Generating Entangled Quantum Subsystems

Authors: Maxime Cautrès, Nathan Claudet, Mehdi Mhalla, Simon Perdrix, Valentin Savin, and Stéphan Thomassé

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
We study the notion of k-stabilizer universal quantum state, that is, an n-qubit quantum state, such that it is possible to induce any stabilizer state on any k qubits, by using only local operations and classical communications. These states generalize the notion of k-pairable states introduced by Bravyi et al., and can be studied from a combinatorial perspective using graph states and k-vertex-minor universal graphs. First, we demonstrate the existence of k-stabilizer universal graph states that are optimal in size with n = Θ(k²) qubits. We also provide parameters for which a random graph state on Θ(k²) qubits is k-stabilizer universal with high probability. Our second contribution consists of two explicit constructions of k-stabilizer universal graph states on n = O(k⁴) qubits. Both rely upon the incidence graph of the projective plane over a finite field 𝔽_q. This provides a major improvement over the previously known explicit construction of k-pairable graph states with n = O(2^{3k}), bringing forth a new and potentially powerful family of multipartite quantum resources.
Cite as

Maxime Cautrès, Nathan Claudet, Mehdi Mhalla, Simon Perdrix, Valentin Savin, and Stéphan Thomassé. Vertex-Minor Universal Graphs for Generating Entangled Quantum Subsystems. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 36:1-36:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{cautres_et_al:LIPIcs.ICALP.2024.36,
  author =	{Cautr\`{e}s, Maxime and Claudet, Nathan and Mhalla, Mehdi and Perdrix, Simon and Savin, Valentin and Thomass\'{e}, St\'{e}phan},
  title =	{{Vertex-Minor Universal Graphs for Generating Entangled Quantum Subsystems}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{36:1--36:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.36},
  URN =		{urn:nbn:de:0030-drops-201796},
  doi =		{10.4230/LIPIcs.ICALP.2024.36},
  annote =	{Keywords: Quantum networks, graph states, vertex-minors, k-pairability}
}
Document
DOI: 10.4230/LIPIcs.CSL.2024.20
Quantum Circuit Completeness: Extensions and Simplifications

Authors: Alexandre Clément, Noé Delorme, Simon Perdrix, and Renaud Vilmart

Published in: LIPIcs, Volume 288, 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024)

Abstract
Although quantum circuits have been ubiquitous for decades in quantum computing, the first complete equational theory for quantum circuits has only recently been introduced. Completeness guarantees that any true equation on quantum circuits can be derived from the equational theory. We improve this completeness result in two ways: (i) We simplify the equational theory by proving that several rules can be derived from the remaining ones. In particular, two out of the three most intricate rules are removed, the third one being slightly simplified. (ii) The complete equational theory can be extended to quantum circuits with ancillae or qubit discarding, to represent respectively quantum computations using an additional workspace, and hybrid quantum computations. We show that the remaining intricate rule can be greatly simplified in these more expressive settings, leading to equational theories where all equations act on a bounded number of qubits. The development of simple and complete equational theories for expressive quantum circuit models opens new avenues for reasoning about quantum circuits. It provides strong formal foundations for various compiling tasks such as circuit optimisation, hardware constraint satisfaction and verification.
Cite as

Alexandre Clément, Noé Delorme, Simon Perdrix, and Renaud Vilmart. Quantum Circuit Completeness: Extensions and Simplifications. In 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 288, pp. 20:1-20:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{clement_et_al:LIPIcs.CSL.2024.20,
  author =	{Cl\'{e}ment, Alexandre and Delorme, No\'{e} and Perdrix, Simon and Vilmart, Renaud},
  title =	{{Quantum Circuit Completeness: Extensions and Simplifications}},
  booktitle =	{32nd EACSL Annual Conference on Computer Science Logic (CSL 2024)},
  pages =	{20:1--20:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-310-2},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{288},
  editor =	{Murano, Aniello and Silva, Alexandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2024.20},
  URN =		{urn:nbn:de:0030-drops-196639},
  doi =		{10.4230/LIPIcs.CSL.2024.20},
  annote =	{Keywords: Quantum Circuits, Completeness, Graphical Language}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2022.24
Complete ZX-Calculi for the Stabiliser Fragment in Odd Prime Dimensions

Authors: Robert I. Booth and Titouan Carette

Published in: LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

Abstract
We introduce a family of ZX-calculi which axiomatise the stabiliser fragment of quantum theory in odd prime dimensions. These calculi recover many of the nice features of the qubit ZX-calculus which were lost in previous proposals for higher-dimensional systems. We then prove that these calculi are complete, i.e. provide a set of rewrite rules which can be used to prove any equality of stabiliser quantum operations. Adding a discard construction, we obtain a calculus complete for mixed state stabiliser quantum mechanics in odd prime dimensions, and this furthermore gives a complete axiomatisation for the related diagrammatic language for affine co-isotropic relations.
Cite as

Robert I. Booth and Titouan Carette. Complete ZX-Calculi for the Stabiliser Fragment in Odd Prime Dimensions. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 24:1-24:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{booth_et_al:LIPIcs.MFCS.2022.24,
  author =	{Booth, Robert I. and Carette, Titouan},
  title =	{{Complete ZX-Calculi for the Stabiliser Fragment in Odd Prime Dimensions}},
  booktitle =	{47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
  pages =	{24:1--24:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-256-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{241},
  editor =	{Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.24},
  URN =		{urn:nbn:de:0030-drops-168225},
  doi =		{10.4230/LIPIcs.MFCS.2022.24},
  annote =	{Keywords: ZX-calculus, completeness, quantum, stabiliser, qudits}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2022.35
LO_v-Calculus: A Graphical Language for Linear Optical Quantum Circuits

Authors: Alexandre Clément, Nicolas Heurtel, Shane Mansfield, Simon Perdrix, and Benoît Valiron

Published in: LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

Abstract
We introduce the LO_v-calculus, a graphical language for reasoning about linear optical quantum circuits with so-called vacuum state auxiliary inputs. We present the axiomatics of the language and prove its soundness and completeness: two LO_v-circuits represent the same quantum process if and only if one can be transformed into the other with the rules of the LO_v-calculus. We give a confluent and terminating rewrite system to rewrite any polarisation-preserving LO_v-circuit into a unique triangular normal form, inspired by the universal decomposition of Reck et al. (1994) for linear optical quantum circuits.
Cite as

Alexandre Clément, Nicolas Heurtel, Shane Mansfield, Simon Perdrix, and Benoît Valiron. LO_v-Calculus: A Graphical Language for Linear Optical Quantum Circuits. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 35:1-35:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{clement_et_al:LIPIcs.MFCS.2022.35,
  author =	{Cl\'{e}ment, Alexandre and Heurtel, Nicolas and Mansfield, Shane and Perdrix, Simon and Valiron, Beno\^{i}t},
  title =	{{LO\underlinev-Calculus: A Graphical Language for Linear Optical Quantum Circuits}},
  booktitle =	{47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
  pages =	{35:1--35:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-256-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{241},
  editor =	{Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.35},
  URN =		{urn:nbn:de:0030-drops-168334},
  doi =		{10.4230/LIPIcs.MFCS.2022.35},
  annote =	{Keywords: Quantum Computing, Graphical Language, Linear Optical Circuits, Linear Optical Quantum Computing, Completeness}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2022.36
Resource Optimisation of Coherently Controlled Quantum Computations with the PBS-Calculus

Authors: Alexandre Clément and Simon Perdrix

Published in: LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

Abstract
Coherent control of quantum computations can be used to improve some quantum protocols and algorithms. For instance, the complexity of implementing the permutation of some given unitary transformations can be strictly decreased by allowing coherent control, rather than using the standard quantum circuit model. In this paper, we address the problem of optimising the resources of coherently controlled quantum computations. We refine the PBS-calculus, a graphical language for coherent control which is inspired by quantum optics. In order to obtain a more resource-sensitive language, it manipulates abstract gates - that can be interpreted as queries to an oracle - and more importantly, it avoids the representation of useless wires by allowing unsaturated polarising beam splitters. Technically the language forms a coloured PROP. The language is equipped with an equational theory that we show to be sound, complete, and minimal. Regarding resource optimisation, we introduce an efficient procedure to minimise the number of oracle queries of a given diagram. We also consider the problem of minimising both the number of oracle queries and the number of polarising beam splitters. We show that this optimisation problem is NP-hard in general, but introduce an efficient heuristic that produces optimal diagrams when at most one query to each oracle is required.
Cite as

Alexandre Clément and Simon Perdrix. Resource Optimisation of Coherently Controlled Quantum Computations with the PBS-Calculus. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 36:1-36:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{clement_et_al:LIPIcs.MFCS.2022.36,
  author =	{Cl\'{e}ment, Alexandre and Perdrix, Simon},
  title =	{{Resource Optimisation of Coherently Controlled Quantum Computations with the PBS-Calculus}},
  booktitle =	{47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
  pages =	{36:1--36:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-256-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{241},
  editor =	{Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.36},
  URN =		{urn:nbn:de:0030-drops-168348},
  doi =		{10.4230/LIPIcs.MFCS.2022.36},
  annote =	{Keywords: Quantum computing, Graphical language, Coherent control, Completeness, Resource optimisation, NP-hardness}
}
Document
DOI: 10.4230/LIPIcs.FSCD.2022.13
Addition and Differentiation of ZX-Diagrams

Authors: Emmanuel Jeandel, Simon Perdrix, and Margarita Veshchezerova

Published in: LIPIcs, Volume 228, 7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022)

Abstract
The ZX-calculus is a powerful framework for reasoning in quantum computing. It provides in particular a compact representation of matrices of interests. A peculiar property of the ZX-calculus is the absence of a formal sum allowing the linear combinations of arbitrary ZX-diagrams. The universality of the formalism guarantees however that for any two ZX-diagrams, the sum of their interpretations can be represented by a ZX-diagram. We introduce a general, inductive definition of the addition of ZX-diagrams, relying on the construction of controlled diagrams. Based on this addition technique, we provide an inductive differentiation of ZX-diagrams. Indeed, given a ZX-diagram with variables in the description of its angles, one can differentiate the diagram according to one of these variables. Differentiation is ubiquitous in quantum mechanics and quantum computing (e.g. for solving optimization problems). Technically, differentiation of ZX-diagrams is strongly related to summation as witnessed by the product rules. We also introduce an alternative, non inductive, differentiation technique rather based on the isolation of the variables. Finally, we apply our results to deduce a diagram for an Ising Hamiltonian.
Cite as

Emmanuel Jeandel, Simon Perdrix, and Margarita Veshchezerova. Addition and Differentiation of ZX-Diagrams. In 7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 228, pp. 13:1-13:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{jeandel_et_al:LIPIcs.FSCD.2022.13,
  author =	{Jeandel, Emmanuel and Perdrix, Simon and Veshchezerova, Margarita},
  title =	{{Addition and Differentiation of ZX-Diagrams}},
  booktitle =	{7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022)},
  pages =	{13:1--13:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-233-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{228},
  editor =	{Felty, Amy P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2022.13},
  URN =		{urn:nbn:de:0030-drops-162946},
  doi =		{10.4230/LIPIcs.FSCD.2022.13},
  annote =	{Keywords: ZX calculus, Addition of ZX diagrams, Diagrammatic differentiation}
}
Document
DOI: 10.4230/LIPIcs.FSCD.2022.21
Linear Lambda-Calculus is Linear

Authors: Alejandro Díaz-Caro and Gilles Dowek

Published in: LIPIcs, Volume 228, 7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022)

Abstract
We prove a linearity theorem for an extension of linear logic with addition and multiplication by a scalar: the proofs of some propositions in this logic are linear in the algebraic sense. This work is part of a wider research program that aims at defining a logic whose proof language is a quantum programming language.
Cite as

Alejandro Díaz-Caro and Gilles Dowek. Linear Lambda-Calculus is Linear. In 7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 228, pp. 21:1-21:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{diazcaro_et_al:LIPIcs.FSCD.2022.21,
  author =	{D{\'\i}az-Caro, Alejandro and Dowek, Gilles},
  title =	{{Linear Lambda-Calculus is Linear}},
  booktitle =	{7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022)},
  pages =	{21:1--21:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-233-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{228},
  editor =	{Felty, Amy P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2022.21},
  URN =		{urn:nbn:de:0030-drops-163024},
  doi =		{10.4230/LIPIcs.FSCD.2022.21},
  annote =	{Keywords: Proof theory, Lambda calculus, Linear logic, Quantum computing}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2021.22
Coherent Control and Distinguishability of Quantum Channels via PBS-Diagrams

Authors: Cyril Branciard, Alexandre Clément, Mehdi Mhalla, and Simon Perdrix

Published in: LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

Abstract
Even though coherent control of quantum operations appears to be achievable in practice, it is still not yet well understood. Among theoretical challenges, standard completely positive trace preserving (CPTP) maps are known not to be appropriate to represent coherently controlled quantum channels. We introduce here a graphical language for coherent control of general quantum channels inspired by practical quantum optical setups involving polarising beam splitters (PBS). We consider different situations of coherent control and disambiguate CPTP maps by considering purified channels, an extension of Stinespring’s dilation. First, we show that in classical control settings, the observational equivalence classes of purified channels correspond to the standard definition of quantum channels (CPTP maps). Then, we propose a refinement of this equivalence class generalising the "half quantum switch" situation, where one is allowed to coherently control which quantum channel is applied; in this case, quantum channel implementations can be distinguished using a so-called transformation matrix. A further refinement characterising observational equivalence with general extended PBS-diagrams as contexts is also obtained. Finally, we propose a refinement that could be used for more general coherent control settings.
Cite as

Cyril Branciard, Alexandre Clément, Mehdi Mhalla, and Simon Perdrix. Coherent Control and Distinguishability of Quantum Channels via PBS-Diagrams. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 22:1-22:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{branciard_et_al:LIPIcs.MFCS.2021.22,
  author =	{Branciard, Cyril and Cl\'{e}ment, Alexandre and Mhalla, Mehdi and Perdrix, Simon},
  title =	{{Coherent Control and Distinguishability of Quantum Channels via PBS-Diagrams}},
  booktitle =	{46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)},
  pages =	{22:1--22:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-201-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{202},
  editor =	{Bonchi, Filippo and Puglisi, Simon J.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.22},
  URN =		{urn:nbn:de:0030-drops-144629},
  doi =		{10.4230/LIPIcs.MFCS.2021.22},
  annote =	{Keywords: Quantum Computing, Diagrammatic Language, Quantum Control, Polarising Beam Splitter, Categorical Quantum Mechanics, Quantum Switch}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2020.24
PBS-Calculus: A Graphical Language for Coherent Control of Quantum Computations

Authors: Alexandre Clément and Simon Perdrix

Published in: LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

Abstract
We introduce the PBS-calculus to represent and reason on quantum computations involving coherent control of quantum operations. Coherent control, and in particular indefinite causal order, is known to enable multiple computational and communication advantages over classically ordered models like quantum circuits. The PBS-calculus is inspired by quantum optics, in particular the polarising beam splitter (PBS for short). We formalise the syntax and the semantics of the PBS-diagrams, and we equip the language with an equational theory, which is proved to be sound and complete: two diagrams are representing the same quantum evolution if and only if one can be transformed into the other using the rules of the PBS-calculus. Moreover, we show that the equational theory is minimal. Finally, we consider applications like the implementation of controlled permutations and the unrolling of loops.
Cite as

Alexandre Clément and Simon Perdrix. PBS-Calculus: A Graphical Language for Coherent Control of Quantum Computations. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 24:1-24:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{clement_et_al:LIPIcs.MFCS.2020.24,
  author =	{Cl\'{e}ment, Alexandre and Perdrix, Simon},
  title =	{{PBS-Calculus: A Graphical Language for Coherent Control of Quantum Computations}},
  booktitle =	{45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
  pages =	{24:1--24:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-159-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{170},
  editor =	{Esparza, Javier and Kr\'{a}l', Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.24},
  URN =		{urn:nbn:de:0030-drops-126921},
  doi =		{10.4230/LIPIcs.MFCS.2020.24},
  annote =	{Keywords: Quantum Computing, Diagrammatic Language, Completeness, Quantum Control, Polarising Beam Splitter, Categorical Quantum Mechanics, Quantum Switch}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2019.55
SZX-Calculus: Scalable Graphical Quantum Reasoning

Authors: Titouan Carette, Dominic Horsman, and Simon Perdrix

Published in: LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

Abstract
We introduce the Scalable ZX-calculus (SZX-calculus for short), a formal and compact graphical language for the design and verification of quantum computations. The SZX-calculus is an extension of the ZX-calculus, a powerful framework that captures graphically the fundamental properties of quantum mechanics through its complete set of rewrite rules. The ZX-calculus is, however, a low level language, with each wire representing a single qubit. This limits its ability to handle large and elaborate quantum evolutions. We extend the ZX-calculus to registers of qubits and allow compact representation of sub-diagrams via binary matrices. We show soundness and completeness of the SZX-calculus and provide two examples of applications, for graph states and error correcting codes.
Cite as

Titouan Carette, Dominic Horsman, and Simon Perdrix. SZX-Calculus: Scalable Graphical Quantum Reasoning. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 55:1-55:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{carette_et_al:LIPIcs.MFCS.2019.55,
  author =	{Carette, Titouan and Horsman, Dominic and Perdrix, Simon},
  title =	{{SZX-Calculus: Scalable Graphical Quantum Reasoning}},
  booktitle =	{44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
  pages =	{55:1--55:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-117-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{138},
  editor =	{Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.55},
  URN =		{urn:nbn:de:0030-drops-109999},
  doi =		{10.4230/LIPIcs.MFCS.2019.55},
  annote =	{Keywords: Quantum computing, categorical quantum mechanics, completeness, scalability}
}
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