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Documents authored by Samperton, Eric

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DOI: 10.4230/LIPIcs.FSTTCS.2025.34
On the Hardness of Approximating Distances of Quantum Codes

Authors: Elena Grigorescu, Vatsal Jha, and Eric Samperton

Published in: LIPIcs, Volume 360, 45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025)

Abstract
The problem of computing distances of error-correcting codes is fundamental in both the classical and quantum settings. While hardness for the classical version of these problems has been known for some time (in both the exact and approximate settings), it was only recently that Kapshikar and Kundu showed these problems are also hard in the quantum setting. As our first main result, we reprove this using arguably simpler arguments based on hypergraph product codes. In particular, we get a direct reduction to CSS codes, the most commonly used type of quantum code, from the minimum distance problem for classical linear codes. Our second set of results considers the distance of a graph state, which is a key parameter for quantum codes obtained via the codeword stabilized formalism. We show that it is NP-hard to compute/approximate the distance of a graph state when the adjacency matrix of the graph is the input. In fact, we show this is true even if we only consider X-type errors of a graph state. Our techniques moreover imply an interesting classical consequence: the hardness of computing or approximating the distance of classical codes with rate equal to 1/2. One of the main motivations of the present work is a question raised by Kapshikar and Kundu concerning the NP-hardness of approximation when there is an additive error proportional to a quantum code’s length. We show that no such hardness can hold for hypergraph product codes. These observations suggest the possibility of a new kind of square root barrier.
Cite as

Elena Grigorescu, Vatsal Jha, and Eric Samperton. On the Hardness of Approximating Distances of Quantum Codes. In 45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 360, pp. 34:1-34:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{grigorescu_et_al:LIPIcs.FSTTCS.2025.34,
  author =	{Grigorescu, Elena and Jha, Vatsal and Samperton, Eric},
  title =	{{On the Hardness of Approximating Distances of Quantum Codes}},
  booktitle =	{45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025)},
  pages =	{34:1--34:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-406-2},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{360},
  editor =	{Aiswarya, C. and Mehta, Ruta and Roy, Subhajit},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2025.34},
  URN =		{urn:nbn:de:0030-drops-251152},
  doi =		{10.4230/LIPIcs.FSTTCS.2025.34},
  annote =	{Keywords: quantum codes, minimum distance problem, NP-hardness, graph state distance}
}
Document
DOI: 10.4230/LIPIcs.TQC.2025.5
Towards a Complexity-Theoretic Dichotomy for TQFT Invariants

Authors: Nicolas Bridges and Eric Samperton

Published in: LIPIcs, Volume 350, 20th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2025)

Abstract
We show that for any fixed (2+1)-dimensional TQFT over ℂ of either Turaev-Viro-Barrett-Westbury or Reshetikhin-Turaev type, the problem of (exactly) computing its invariants on closed 3-manifolds is either solvable in polynomial time, or else it is #𝖯-hard to (exactly) contract certain tensors that are built from the TQFT’s fusion category. Our proof is an application of a dichotomy result of Cai and Chen [J. ACM, 2017] concerning weighted constraint satisfaction problems over ℂ. We leave for future work the issue of reinterpreting the conditions of Cai and Chen that distinguish between the two cases (i.e. #𝖯-hard tensor contractions vs. polynomial time invariants) in terms of fusion categories. We expect that with more effort, our reduction can be improved so that one gets a dichotomy directly for TQFTs' invariants of 3-manifolds rather than more general tensors built from TQFTs' fusion categories.
Cite as

Nicolas Bridges and Eric Samperton. Towards a Complexity-Theoretic Dichotomy for TQFT Invariants. In 20th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 350, pp. 5:1-5:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bridges_et_al:LIPIcs.TQC.2025.5,
  author =	{Bridges, Nicolas and Samperton, Eric},
  title =	{{Towards a Complexity-Theoretic Dichotomy for TQFT Invariants}},
  booktitle =	{20th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2025)},
  pages =	{5:1--5:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-392-8},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{350},
  editor =	{Fefferman, Bill},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2025.5},
  URN =		{urn:nbn:de:0030-drops-240548},
  doi =		{10.4230/LIPIcs.TQC.2025.5},
  annote =	{Keywords: Complexity, topological quantum field theory, dichotomy theorems, constraint satisfaction problems, tensor categories}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.38
An Algorithm for Tambara-Yamagami Quantum Invariants of 3-Manifolds, Parameterized by the First Betti Number

Authors: Colleen Delaney, Clément Maria, and Eric Samperton

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
Quantum topology provides various frameworks for defining and computing invariants of manifolds inspired by quantum theory. One such framework of substantial interest in both mathematics and physics is the Turaev-Viro-Barrett-Westbury state sum construction, which uses the data of a spherical fusion category to define topological invariants of triangulated 3-manifolds via tensor network contractions. In this work we analyze the computational complexity of state sum invariants of 3-manifolds derived from Tambara-Yamagami categories. While these categories are the simplest source of state sum invariants beyond finite abelian groups (whose invariants can be computed in polynomial time) their computational complexities are yet to be fully understood. We first establish that the invariants arising from even the smallest Tambara-Yamagami categories are #P-hard to compute, so that one expects the same to be true of the whole family. Our main result is then the existence of a fixed parameter tractable algorithm to compute these 3-manifold invariants, where the parameter is the first Betti number of the 3-manifold with ℤ/2ℤ coefficients. Contrary to other domains of computational topology, such as graphs on surfaces, very few hard problems in 3-manifold topology are known to admit FPT algorithms with a topological parameter. However, such algorithms are of particular interest as their complexity depends only polynomially on the combinatorial representation of the input, regardless of size or combinatorial width. Additionally, in the case of Betti numbers, the parameter itself is computable in polynomial time. Thus while one generally expects quantum invariants to be hard to compute classically, our results suggest that the hardness of computing state sum invariants from Tambara-Yamagami categories arises from classical 3-manifold topology rather than the quantum nature of the algebraic input.
Cite as

Colleen Delaney, Clément Maria, and Eric Samperton. An Algorithm for Tambara-Yamagami Quantum Invariants of 3-Manifolds, Parameterized by the First Betti Number. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 38:1-38:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{delaney_et_al:LIPIcs.SoCG.2025.38,
  author =	{Delaney, Colleen and Maria, Cl\'{e}ment and Samperton, Eric},
  title =	{{An Algorithm for Tambara-Yamagami Quantum Invariants of 3-Manifolds, Parameterized by the First Betti Number}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{38:1--38:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.38},
  URN =		{urn:nbn:de:0030-drops-231901},
  doi =		{10.4230/LIPIcs.SoCG.2025.38},
  annote =	{Keywords: 3-manifold, quantum invariant, fixed parameter tractable algorithm, topological parameter, Gauss sums, topological quantum field theory}
}
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