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Documents authored by Kissinger, Aleks

Document
DOI: 10.4230/LIPIcs.MFCS.2022.80
Higher-Order Causal Theories Are Models of BV-Logic

Authors: Will Simmons and Aleks Kissinger

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

Abstract
The Caus[-] construction takes a compact closed category of basic processes and yields a *-autonomous category of higher-order processes obeying certain signalling/causality constraints, as dictated by the type system in the resulting category. This paper looks at instances where the base category C satisfies additional properties yielding an affine-linear structure on Caus[𝒞] and a substantially richer internal logic. While the original construction only gave multiplicative linear logic, here we additionally obtain additives and a non-commutative, self-dual sequential product yielding a model of Guglielmi’s BV logic. Furthermore, we obtain a natural interpretation for the sequential product as "A can signal to B, but not vice-versa", which sits as expected between the non-signalling tensor and the fully-signalling (i.e. unconstrained) par. Fixing matrices of positive numbers for 𝒞 recovers the BV category structure of probabilistic coherence spaces identified by Blute, Panangaden, and Slavnov, restricted to normalised maps. On the other hand, fixing the category of completely positive maps gives an entirely new model of BV consisting of higher order quantum channels, encompassing recent work in the study of quantum and indefinite causal structures.
Cite as

Will Simmons and Aleks Kissinger. Higher-Order Causal Theories Are Models of BV-Logic. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 80:1-80:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{simmons_et_al:LIPIcs.MFCS.2022.80,
  author =	{Simmons, Will and Kissinger, Aleks},
  title =	{{Higher-Order Causal Theories Are Models of BV-Logic}},
  booktitle =	{47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
  pages =	{80:1--80:14},
  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.2022.80},
  URN =		{urn:nbn:de:0030-drops-168789},
  doi =		{10.4230/LIPIcs.MFCS.2022.80},
  annote =	{Keywords: Causality, linear logic, categorical logic, probabilistic coherence spaces, quantum channels}
}
Document
DOI: 10.4230/LIPIcs.TQC.2022.5
Classical Simulation of Quantum Circuits with Partial and Graphical Stabiliser Decompositions

Authors: Aleks Kissinger, John van de Wetering, and Renaud Vilmart

Published in: LIPIcs, Volume 232, 17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022)

Abstract
Recent developments in classical simulation of quantum circuits make use of clever decompositions of chunks of magic states into sums of efficiently simulable stabiliser states. We show here how, by considering certain non-stabiliser entangled states which have more favourable decompositions, we can speed up these simulations. This is made possible by using the ZX-calculus, which allows us to easily find instances of these entangled states in the simplified diagram representing the quantum circuit to be simulated. We additionally find a new technique of partial stabiliser decompositions that allow us to trade magic states for stabiliser terms. With this technique we require only 2^{α t} stabiliser terms, where α≈ 0.396, to simulate a circuit with T-count t. This matches the α found by Qassim et al. [Qassim et al., 2021], but whereas they only get this scaling in the asymptotic limit, ours applies for a circuit of any size. Our method builds upon a recently proposed scheme for simulation combining stabiliser decompositions and optimisation strategies implemented in the software QuiZX [Kissinger and van de Wetering, 2022]. With our techniques we manage to reliably simulate 50-qubit 1400 T-count hidden shift circuits in a couple of minutes on a consumer laptop.
Cite as

Aleks Kissinger, John van de Wetering, and Renaud Vilmart. Classical Simulation of Quantum Circuits with Partial and Graphical Stabiliser Decompositions. In 17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 232, pp. 5:1-5:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{kissinger_et_al:LIPIcs.TQC.2022.5,
  author =	{Kissinger, Aleks and van de Wetering, John and Vilmart, Renaud},
  title =	{{Classical Simulation of Quantum Circuits with Partial and Graphical Stabiliser Decompositions}},
  booktitle =	{17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022)},
  pages =	{5:1--5:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-237-2},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{232},
  editor =	{Le Gall, Fran\c{c}ois and Morimae, Tomoyuki},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2022.5},
  URN =		{urn:nbn:de:0030-drops-165128},
  doi =		{10.4230/LIPIcs.TQC.2022.5},
  annote =	{Keywords: ZX-calculus, Stabiliser Rank, Quantum Simulation}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2022.119
Circuit Extraction for ZX-Diagrams Can Be #P-Hard

Authors: Niel de Beaudrap, Aleks Kissinger, and John van de Wetering

Published in: LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

Abstract
The ZX-calculus is a graphical language for reasoning about quantum computation using ZX-diagrams, a certain flexible generalisation of quantum circuits that can be used to represent linear maps from m to n qubits for any m,n ≥ 0. Some applications for the ZX-calculus, such as quantum circuit optimisation and synthesis, rely on being able to efficiently translate a ZX-diagram back into a quantum circuit of comparable size. While several sufficient conditions are known for describing families of ZX-diagrams that can be efficiently transformed back into circuits, it has previously been conjectured that the general problem of circuit extraction is hard. That is, that it should not be possible to efficiently convert an arbitrary ZX-diagram describing a unitary linear map into an equivalent quantum circuit. In this paper we prove this conjecture by showing that the circuit extraction problem is #P-hard, and so is itself at least as hard as strong simulation of quantum circuits. In addition to our main hardness result, which relies specifically on the circuit representation, we give a representation-agnostic hardness result. Namely, we show that any oracle that takes as input a ZX-diagram description of a unitary and produces samples of the output of the associated quantum computation enables efficient probabilistic solutions to NP-complete problems.
Cite as

Niel de Beaudrap, Aleks Kissinger, and John van de Wetering. Circuit Extraction for ZX-Diagrams Can Be #P-Hard. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 119:1-119:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{debeaudrap_et_al:LIPIcs.ICALP.2022.119,
  author =	{de Beaudrap, Niel and Kissinger, Aleks and van de Wetering, John},
  title =	{{Circuit Extraction for ZX-Diagrams Can Be #P-Hard}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{119:1--119:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-235-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{229},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.119},
  URN =		{urn:nbn:de:0030-drops-164601},
  doi =		{10.4230/LIPIcs.ICALP.2022.119},
  annote =	{Keywords: ZX-calculus, circuit extraction, quantum circuits, #P}
}
Document
DOI: 10.4230/LIPIcs.FSCD.2016.34
Globular: An Online Proof Assistant for Higher-Dimensional Rewriting

Authors: Krzysztof Bar, Aleks Kissinger, and Jamie Vicary

Published in: LIPIcs, Volume 52, 1st International Conference on Formal Structures for Computation and Deduction (FSCD 2016)

Abstract
This article introduces Globular, an online proof assistant for the formalization and verification of proofs in higher-dimensional category theory. The tool produces graphical visualizations of higher-dimensional proofs, assists in their construction with a point-and-click interface, and performs type checking to prevent incorrect rewrites. Hosted on the web, it has a low barrier to use, and allows hyperlinking of formalized proofs directly from research papers. It allows the formalization of proofs from logic, topology and algebra which are not formalizable by other methods, and we give several examples.
Cite as

Krzysztof Bar, Aleks Kissinger, and Jamie Vicary. Globular: An Online Proof Assistant for Higher-Dimensional Rewriting. In 1st International Conference on Formal Structures for Computation and Deduction (FSCD 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 52, pp. 34:1-34:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{bar_et_al:LIPIcs.FSCD.2016.34,
  author =	{Bar, Krzysztof and Kissinger, Aleks and Vicary, Jamie},
  title =	{{Globular: An Online Proof Assistant for Higher-Dimensional Rewriting}},
  booktitle =	{1st International Conference on Formal Structures for Computation and Deduction (FSCD 2016)},
  pages =	{34:1--34:11},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-010-1},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{52},
  editor =	{Kesner, Delia and Pientka, Brigitte},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2016.34},
  URN =		{urn:nbn:de:0030-drops-59880},
  doi =		{10.4230/LIPIcs.FSCD.2016.34},
  annote =	{Keywords: higher category theory, higher-dimensional rewriting, interactive theorem proving}
}
Document
DOI: 10.4230/LIPIcs.CALCO.2015.171
A First-order Logic for String Diagrams

Authors: Aleks Kissinger and David Quick

Published in: LIPIcs, Volume 35, 6th Conference on Algebra and Coalgebra in Computer Science (CALCO 2015)

Abstract
Equational reasoning with string diagrams provides an intuitive means of proving equations between morphisms in a symmetric monoidal category. This can be extended to proofs of infinite families of equations using a simple graphical syntax called !-box notation. While this does greatly increase the proving power of string diagrams, previous attempts to go beyond equational reasoning have been largely ad hoc, owing to the lack of a suitable logical framework for diagrammatic proofs involving !-boxes. In this paper, we extend equational reasoning with !-boxes to a fully-fledged first order logic with conjunction, implication, and universal quantification over !-boxes. This logic, called !L, is then rich enough to properly formalise an induction principle for !-boxes. We then build a standard model for !L and give an example proof of a theorem for non-commutative bialgebras using !L, which is unobtainable by equational reasoning alone.
Cite as

Aleks Kissinger and David Quick. A First-order Logic for String Diagrams. In 6th Conference on Algebra and Coalgebra in Computer Science (CALCO 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 35, pp. 171-189, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{kissinger_et_al:LIPIcs.CALCO.2015.171,
  author =	{Kissinger, Aleks and Quick, David},
  title =	{{A First-order Logic for String Diagrams}},
  booktitle =	{6th Conference on Algebra and Coalgebra in Computer Science (CALCO 2015)},
  pages =	{171--189},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-84-2},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{35},
  editor =	{Moss, Lawrence S. and Sobocinski, Pawel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CALCO.2015.171},
  URN =		{urn:nbn:de:0030-drops-55335},
  doi =		{10.4230/LIPIcs.CALCO.2015.171},
  annote =	{Keywords: string diagrams, compact closed monoidal categories, abstract tensor systems, first-order logic}
}
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