12 Search Results for "Christandl, Matthias"

Document
DOI: 10.4230/LIPIcs.TQC.2025.6
Quantum SAT Problems with Finite Sets of Projectors Are Complete for a Plethora of Classes

Authors: Ricardo Rivera Cardoso, Alex Meiburg, and Daniel Nagaj

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

Abstract
Previously, all known variants of the Quantum Satisfiability (QSAT) problem - consisting of determining whether a k-local (k-body) Hamiltonian is frustration-free - could be classified as being either in 𝖯; or complete for NP, MA, or QMA₁. Here, we present new qubit variants of this problem that are complete for BQP₁, coRP, QCMA, PI(coRP,NP), PI(BQP₁,NP), PI(BQP₁,MA), SoPU(coRP,NP), SoPU(BQP₁,NP), and SoPU(BQP₁,MA). Our result implies that a complete classification of quantum constraint satisfaction problems (QCSPs), analogous to Schaefer’s dichotomy theorem for classical CSPs, must either include these 13 classes, or otherwise show that some are equal. Additionally, our result showcases two new types of QSAT problems that can be decided efficiently, as well as the first nontrivial BQP₁-complete problem. We first construct QSAT problems on qudits that are complete for BQP₁, coRP, and QCMA. These are made by restricting the finite set of Hamiltonians to consist of elements similar to H_{init}, H_{prop}, and H_{out}, seen in the circuit-to-Hamiltonian transformation. Usually, these are used to demonstrate hardness of QSAT and Local Hamiltonian problems, and so our proofs of hardness are simple. The difficulty lies in ensuring that all Hamiltonians generated with these three elements can be decided in their respective classes. For this, we build our Hamiltonian terms with high-dimensional data and clock qudits, ternary logic, and either monogamy of entanglement or specific clock encodings. We then show how to express these problems in terms of qubits, by proving that any QCSP can be reduced to a qubit problem while maintaining the same complexity - something not believed possible classically. The remaining six problems are obtained by considering "sums" and "products" of some of the QSAT problems mentioned here. Before this work, the QSAT problems generated in this way resulted in complete problems for PI and SoPU classes that were trivially equal to NP, MA, or QMA₁. We thus commence the study of these new and seemingly nontrivial classes. While [Meiburg, 2021] first sought to prove completeness for coRP, BQP₁, and QCMA, we note that those constructions are flawed. Here, we rework them, provide correct proofs, and obtain improvements on the required qudit dimensionality.
Cite as

Ricardo Rivera Cardoso, Alex Meiburg, and Daniel Nagaj. Quantum SAT Problems with Finite Sets of Projectors Are Complete for a Plethora of Classes. In 20th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 350, pp. 6:1-6:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{riveracardoso_et_al:LIPIcs.TQC.2025.6,
  author =	{Rivera Cardoso, Ricardo and Meiburg, Alex and Nagaj, Daniel},
  title =	{{Quantum SAT Problems with Finite Sets of Projectors Are Complete for a Plethora of Classes}},
  booktitle =	{20th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2025)},
  pages =	{6:1--6:24},
  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.2025.6},
  URN =		{urn:nbn:de:0030-drops-240557},
  doi =		{10.4230/LIPIcs.TQC.2025.6},
  annote =	{Keywords: Quantum complexity theory, quantum satisfiability, circuit-to-Hamiltonian, pairwise union of classes, pairwise intersection of classes}
}
Document
DOI: 10.4230/LIPIcs.TQC.2025.2
A Quantum Cloning Game with Applications to Quantum Position Verification

Authors: Léo Colisson Palais, Llorenç Escolà-Farràs, and Florian Speelman

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

Abstract
We introduce a quantum cloning game in which k separate collaborative parties receive a classical input, determining which of them has to share a maximally entangled state with an additional party (referee). We provide the optimal winning probability of such a game for every number of parties k, and show that it decays exponentially when the game is played n times in parallel. These results have applications to quantum cryptography, in particular in the topic of quantum position verification, where we show security of the routing protocol (played in parallel), and a variant of it, in the random oracle model.
Cite as

Léo Colisson Palais, Llorenç Escolà-Farràs, and Florian Speelman. A Quantum Cloning Game with Applications to Quantum Position Verification. In 20th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 350, pp. 2:1-2:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{colissonpalais_et_al:LIPIcs.TQC.2025.2,
  author =	{Colisson Palais, L\'{e}o and Escol\`{a}-Farr\`{a}s, Lloren\c{c} and Speelman, Florian},
  title =	{{A Quantum Cloning Game with Applications to Quantum Position Verification}},
  booktitle =	{20th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2025)},
  pages =	{2:1--2:17},
  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.2025.2},
  URN =		{urn:nbn:de:0030-drops-240513},
  doi =		{10.4230/LIPIcs.TQC.2025.2},
  annote =	{Keywords: Quantum position verification, cloning game, random oracle, parallel repetition}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.113
Nearly Optimal Circuit Size for Sparse Quantum State Preparation

Authors: Lvzhou Li and Jingquan Luo

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

Abstract
Quantum state preparation is a fundamental and significant subroutine in quantum computing. In this paper, we conduct a systematic investigation of the circuit size (the total count of elementary gates in the circuit) for sparse quantum state preparation. A quantum state is said to be d-sparse if it has only d non-zero amplitudes. For the task of preparing an n-qubit d-sparse quantum state, we obtain the following results: - Without ancillary qubits: Any n-qubit d-sparse quantum state can be prepared by a quantum circuit of size O(nd/(log n) + n) without using ancillary qubits, which improves the previous best results. It is asymptotically optimal when d = poly(n), and this optimality holds for a broader scope under some reasonable assumptions. - With limited ancillary qubits: (i) Based on the first result, we prove for the first time a trade-off between the number of ancillary qubits and the circuit size: any n-qubit d-sparse quantum state can be prepared by a quantum circuit of size O((nd)/(log(n + m)) + n) using m ancillary qubits for any m ∈ O((nd)/(log nd) + n). (ii) We establish a matching lower bound Ω((nd)/(log(n+m))+n) under some reasonable assumptions, and obtain a slightly weaker lower bound Ω((nd)/(log(n+m)+log d) + n) without any assumptions. - With unlimited ancillary qubits: Given an arbitrary amount of ancillary qubits available, the circuit size for preparing n-qubit d-sparse quantum states is Θ((nd)/(log nd) + n).
Cite as

Lvzhou Li and Jingquan Luo. Nearly Optimal Circuit Size for Sparse Quantum State Preparation. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 113:1-113:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{li_et_al:LIPIcs.ICALP.2025.113,
  author =	{Li, Lvzhou and Luo, Jingquan},
  title =	{{Nearly Optimal Circuit Size for Sparse Quantum State Preparation}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{113:1--113:19},
  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.2025.113},
  URN =		{urn:nbn:de:0030-drops-234900},
  doi =		{10.4230/LIPIcs.ICALP.2025.113},
  annote =	{Keywords: Quantum computing, quantum state preparation, circuit complexity}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2025.64
A Universal Sequence of Tensors for the Asymptotic Rank Conjecture

Authors: Petteri Kaski and Mateusz Michałek

Published in: LIPIcs, Volume 325, 16th Innovations in Theoretical Computer Science Conference (ITCS 2025)

Abstract
The exponent σ(T) of a tensor T ∈ 𝔽^d⊗𝔽^d⊗𝔽^d over a field 𝔽 captures the base of the exponential growth rate of the tensor rank of T under Kronecker powers. Tensor exponents are fundamental from the standpoint of algorithms and computational complexity theory; for example, the exponent ω of square matrix multiplication can be characterized as ω = 2σ(MM₂), where MM₂ ∈ 𝔽⁴⊗𝔽⁴⊗𝔽⁴ is the tensor that represents 2×2 matrix multiplication. Strassen [FOCS 1986] initiated a duality theory for spaces of tensors that enables one to characterize the exponent of a tensor via objects in a dual space, called the asymptotic spectrum of the primal (tensor) space. While Strassen’s theory has considerable generality beyond the setting of tensors - Wigderson and Zuiddam [Asymptotic Spectra: Theory, Applications, and Extensions, preprint, 2023] give a recent exposition - progress in characterizing the dual space in the tensor setting has been slow, with the first universal points in the dual identified by Christandl, Vrana, and Zuiddam [J. Amer. Math. Soc. 36 (2023)]. In parallel to Strassen’s theory, the algebraic geometry community has developed a geometric theory of tensors aimed at characterizing the structure of the primal space and tensor exponents therein; the latter study was motivated in particular by an observation of Strassen (implicit in [J. Reine Angew. Math. 384 (1988)]) that matrix-multiplication tensors have limited universality in the sense that σ(𝔽^d⊗𝔽^d⊗𝔽^d) ≤ 2ω/3 = 4/3σ(MM₂) holds for all d ≥ 1. In particular, this limited universality of the tensor MM₂ puts forth the question whether one could construct explicit universal tensors that exactly characterize the worst-case tensor exponent in the primal space. Such explicit universal objects would, among others, give means towards a proof or a disproof of Strassen’s asymptotic rank conjecture [Progr. Math. 120 (1994)]; the former would immediately imply ω = 2 and, among others, refute the Set Cover Conjecture (cf. Björklund and Kaski [STOC 2024] and Pratt [STOC 2024]). Our main result is an explicit construction of a sequence 𝒰_d of zero-one-valued tensors that is universal for the worst-case tensor exponent; more precisely, we show that σ(𝒰_d) = σ(d) where σ(d) = sup_{T ∈ 𝔽^d⊗𝔽^d⊗𝔽^d}σ(T). We also supply an explicit universal sequence 𝒰_Δ localised to capture the worst-case exponent σ(Δ) of tensors with support contained in Δ ⊆ [d]×[d]×[d]; by combining such sequences, we obtain a universal sequence 𝒯_d such that σ(𝒯_d) = 1 holds if and only if Strassen’s asymptotic rank conjecture holds for d. Finally, we show that the limit lim_{d → ∞}σ(d) exists and can be captured as lim_{d → ∞} σ(D_d) for an explicit sequence (D_d)_{d = 1}^∞ of tensors obtained by diagonalisation of the sequences 𝒰_d. As our second result we relate the absence of polynomials of fixed degree vanishing on tensors of low rank, or more generally asymptotic rank, with upper bounds on the exponent σ(d). Using this technique, one may bound asymptotic rank for all tensors of a given format, knowing enough specific tensors of low asymptotic rank.
Cite as

Petteri Kaski and Mateusz Michałek. A Universal Sequence of Tensors for the Asymptotic Rank Conjecture. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 64:1-64:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{kaski_et_al:LIPIcs.ITCS.2025.64,
  author =	{Kaski, Petteri and Micha{\l}ek, Mateusz},
  title =	{{A Universal Sequence of Tensors for the Asymptotic Rank Conjecture}},
  booktitle =	{16th Innovations in Theoretical Computer Science Conference (ITCS 2025)},
  pages =	{64:1--64:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-361-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{325},
  editor =	{Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2025.64},
  URN =		{urn:nbn:de:0030-drops-226925},
  doi =		{10.4230/LIPIcs.ITCS.2025.64},
  annote =	{Keywords: asymptotic rank conjecture, secant variety, Specht module, tensor rank, tensor exponent}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2025.18
Finite Matrix Multiplication Algorithms from Infinite Groups

Authors: Jonah Blasiak, Henry Cohn, Joshua A. Grochow, Kevin Pratt, and Chris Umans

Published in: LIPIcs, Volume 325, 16th Innovations in Theoretical Computer Science Conference (ITCS 2025)

Abstract
The Cohn-Umans (FOCS '03) group-theoretic framework for matrix multiplication produces fast matrix multiplication algorithms from three subsets of a finite group G satisfying a simple combinatorial condition (the Triple Product Property). The complexity of such an algorithm then depends on the representation theory of G. In this paper we extend the group-theoretic framework to the setting of infinite groups. In particular, this allows us to obtain constructions in Lie groups, with favorable parameters, that are provably impossible in finite groups of Lie type (Blasiak, Cohn, Grochow, Pratt, and Umans, ITCS '23). Previously the Lie group setting was investigated purely as an analogue of the finite group case; a key contribution in this paper is a fully developed framework for obtaining bona fide matrix multiplication algorithms directly from Lie group constructions. As part of this framework, we introduce "separating functions" as a necessary new design component, and show that when the underlying group is G = GL_n, these functions are polynomials with their degree being the key parameter. In particular, we show that a construction with "half-dimensional" subgroups and optimal degree would imply ω = 2. We then build up machinery that reduces the problem of constructing optimal-degree separating polynomials to the problem of constructing a single polynomial (and a corresponding set of group elements) in a ring of invariant polynomials determined by two out of the three subgroups that satisfy the Triple Product Property. This machinery combines border rank with the Lie algebras associated with the Lie subgroups in a critical way. We give several constructions illustrating the main components of the new framework, culminating in a construction in a special unitary group that achieves separating polynomials of optimal degree, meeting one of the key challenges. The subgroups in this construction have dimension approaching half the ambient dimension, but (just barely) too slowly. We argue that features of the classical Lie groups make it unlikely that constructions in these particular groups could produce nontrivial bounds on ω unless they prove ω = 2. One way to get ω = 2 via our new framework would be to lift our existing construction from the special unitary group to GL_n, and improve the dimension of the subgroups from (dim G)/2 - Θ(n) to (dim G)/2 - o(n).
Cite as

Jonah Blasiak, Henry Cohn, Joshua A. Grochow, Kevin Pratt, and Chris Umans. Finite Matrix Multiplication Algorithms from Infinite Groups. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 18:1-18:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{blasiak_et_al:LIPIcs.ITCS.2025.18,
  author =	{Blasiak, Jonah and Cohn, Henry and Grochow, Joshua A. and Pratt, Kevin and Umans, Chris},
  title =	{{Finite Matrix Multiplication Algorithms from Infinite Groups}},
  booktitle =	{16th Innovations in Theoretical Computer Science Conference (ITCS 2025)},
  pages =	{18:1--18:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-361-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{325},
  editor =	{Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2025.18},
  URN =		{urn:nbn:de:0030-drops-226460},
  doi =		{10.4230/LIPIcs.ITCS.2025.18},
  annote =	{Keywords: Fast matrix multiplication, representation theory, infinite groups}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2025.11
Rank Lower Bounds on Non-Local Quantum Computation

Authors: Vahid R. Asadi, Eric Culf, and Alex May

Published in: LIPIcs, Volume 325, 16th Innovations in Theoretical Computer Science Conference (ITCS 2025)

Abstract
A non-local quantum computation (NLQC) replaces an interaction between two quantum systems with a single simultaneous round of communication and shared entanglement. We study two classes of NLQC, f-routing and f-BB84, which are of relevance to classical information theoretic cryptography and quantum position-verification. We give the first non-trivial lower bounds on entanglement in both settings, but are restricted to lower bounding protocols with perfect correctness. Within this setting, we give a lower bound on the Schmidt rank of any entangled state that completes these tasks for a given function f(x,y) in terms of the rank of a matrix g(x,y) whose entries are zero when f(x,y) = 0, and strictly positive otherwise. This also leads to a lower bound on the Schmidt rank in terms of the non-deterministic quantum communication complexity of f(x,y). Because of a relationship between f-routing and the conditional disclosure of secrets (CDS) primitive studied in information theoretic cryptography, we obtain a new technique for lower bounding the randomness complexity of CDS.
Cite as

Vahid R. Asadi, Eric Culf, and Alex May. Rank Lower Bounds on Non-Local Quantum Computation. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 11:1-11:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{asadi_et_al:LIPIcs.ITCS.2025.11,
  author =	{Asadi, Vahid R. and Culf, Eric and May, Alex},
  title =	{{Rank Lower Bounds on Non-Local Quantum Computation}},
  booktitle =	{16th Innovations in Theoretical Computer Science Conference (ITCS 2025)},
  pages =	{11:1--11:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-361-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{325},
  editor =	{Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2025.11},
  URN =		{urn:nbn:de:0030-drops-226399},
  doi =		{10.4230/LIPIcs.ITCS.2025.11},
  annote =	{Keywords: Non-local quantum computation, quantum position-verification, conditional disclosure of secrets}
}
Document
Extended Abstract
DOI: 10.4230/LIPIcs.ITCS.2024.20
Discreteness of Asymptotic Tensor Ranks (Extended Abstract)

Authors: Jop Briët, Matthias Christandl, Itai Leigh, Amir Shpilka, and Jeroen Zuiddam

Published in: LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

Abstract
Tensor parameters that are amortized or regularized over large tensor powers, often called "asymptotic" tensor parameters, play a central role in several areas including algebraic complexity theory (constructing fast matrix multiplication algorithms), quantum information (entanglement cost and distillable entanglement), and additive combinatorics (bounds on cap sets, sunflower-free sets, etc.). Examples are the asymptotic tensor rank, asymptotic slice rank and asymptotic subrank. Recent works (Costa-Dalai, Blatter-Draisma-Rupniewski, Christandl-Gesmundo-Zuiddam) have investigated notions of discreteness (no accumulation points) or "gaps" in the values of such tensor parameters. We prove a general discreteness theorem for asymptotic tensor parameters of order-three tensors and use this to prove that (1) over any finite field (and in fact any finite set of coefficients in any field), the asymptotic subrank and the asymptotic slice rank have no accumulation points, and (2) over the complex numbers, the asymptotic slice rank has no accumulation points. Central to our approach are two new general lower bounds on the asymptotic subrank of tensors, which measures how much a tensor can be diagonalized. The first lower bound says that the asymptotic subrank of any concise three-tensor is at least the cube-root of the smallest dimension. The second lower bound says that any concise three-tensor that is "narrow enough" (has one dimension much smaller than the other two) has maximal asymptotic subrank. Our proofs rely on new lower bounds on the maximum rank in matrix subspaces that are obtained by slicing a three-tensor in the three different directions. We prove that for any concise tensor, the product of any two such maximum ranks must be large, and as a consequence there are always two distinct directions with large max-rank.
Cite as

Jop Briët, Matthias Christandl, Itai Leigh, Amir Shpilka, and Jeroen Zuiddam. Discreteness of Asymptotic Tensor Ranks (Extended Abstract). In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 20:1-20:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{briet_et_al:LIPIcs.ITCS.2024.20,
  author =	{Bri\"{e}t, Jop and Christandl, Matthias and Leigh, Itai and Shpilka, Amir and Zuiddam, Jeroen},
  title =	{{Discreteness of Asymptotic Tensor Ranks}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{20:1--20:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-309-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{287},
  editor =	{Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.20},
  URN =		{urn:nbn:de:0030-drops-195483},
  doi =		{10.4230/LIPIcs.ITCS.2024.20},
  annote =	{Keywords: Tensors, Asymptotic rank, Subrank, Slice rank, Restriction, Degeneration, Diagonalization, SLOCC}
}
Document
DOI: 10.4230/LIPIcs.STACS.2023.12
On the Multilinear Complexity of Associative Algebras

Authors: Markus Bläser, Hendrik Mayer, and Devansh Shringi

Published in: LIPIcs, Volume 254, 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)

Abstract
Christandl and Zuiddam [Matthias Christandl and Jeroen Zuiddam, 2019] study the multilinear complexity of d-fold matrix multiplication in the context of quantum communication complexity. Bshouty [Nader H. Bshouty, 2013] investigates the multilinear complexity of d-fold multiplication in commutative algebras to understand the size of so-called testers. The study of bilinear complexity is a classical topic in algebraic complexity theory, starting with the work by Strassen. However, there has been no systematic study of the multilinear complexity of multilinear maps. In the present work, we systematically investigate the multilinear complexity of d-fold multiplication in arbitrary associative algebras. We prove a multilinear generalization of the famous Alder-Strassen theorem, which is a lower bound for the bilinear complexity of the (2-fold) multiplication in an associative algebra. We show that the multilinear complexity of the d-fold multiplication has a lower bound of d ⋅ dim A - (d-1)t, where t is the number of maximal twosided ideals in A. This is optimal in the sense that there are algebras for which this lower bound is tight. Furthermore, we prove the following dichotomy that the quotient algebra A/rad A determines the complexity of the d-fold multiplication in A: When the semisimple algebra A/rad A is commutative, then the multilinear complexity of the d-fold multiplication in A is polynomial in d. On the other hand, when A/rad A is noncommutative, then the multilinear complexity of the d-fold multiplication in A is exponential in d.
Cite as

Markus Bläser, Hendrik Mayer, and Devansh Shringi. On the Multilinear Complexity of Associative Algebras. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 12:1-12:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{blaser_et_al:LIPIcs.STACS.2023.12,
  author =	{Bl\"{a}ser, Markus and Mayer, Hendrik and Shringi, Devansh},
  title =	{{On the Multilinear Complexity of Associative Algebras}},
  booktitle =	{40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)},
  pages =	{12:1--12:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-266-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{254},
  editor =	{Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2023.12},
  URN =		{urn:nbn:de:0030-drops-176645},
  doi =		{10.4230/LIPIcs.STACS.2023.12},
  annote =	{Keywords: Multilinear computations, associative algebras, matrix multiplication, Alder-Strassen theorem}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2022.48
Larger Corner-Free Sets from Combinatorial Degenerations

Authors: Matthias Christandl, Omar Fawzi, Hoang Ta, and Jeroen Zuiddam

Published in: LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

Abstract
There is a large and important collection of Ramsey-type combinatorial problems, closely related to central problems in complexity theory, that can be formulated in terms of the asymptotic growth of the size of the maximum independent sets in powers of a fixed small hypergraph, also called the Shannon capacity. An important instance of this is the corner problem studied in the context of multiparty communication complexity in the Number On the Forehead (NOF) model. Versions of this problem and the NOF connection have seen much interest (and progress) in recent works of Linial, Pitassi and Shraibman (ITCS 2019) and Linial and Shraibman (CCC 2021). We introduce and study a general algebraic method for lower bounding the Shannon capacity of directed hypergraphs via combinatorial degenerations, a combinatorial kind of "approximation" of subgraphs that originates from the study of matrix multiplication in algebraic complexity theory (and which play an important role there) but which we use in a novel way. Using the combinatorial degeneration method, we make progress on the corner problem by explicitly constructing a corner-free subset in F₂ⁿ × F₂ⁿ of size Ω(3.39ⁿ/poly(n)), which improves the previous lower bound Ω(2.82ⁿ) of Linial, Pitassi and Shraibman (ITCS 2019) and which gets us closer to the best upper bound 4^{n - o(n)}. Our new construction of corner-free sets implies an improved NOF protocol for the Eval problem. In the Eval problem over a group G, three players need to determine whether their inputs x₁, x₂, x₃ ∈ G sum to zero. We find that the NOF communication complexity of the Eval problem over F₂ⁿ is at most 0.24n + 𝒪(log n), which improves the previous upper bound 0.5n + 𝒪(log n).
Cite as

Matthias Christandl, Omar Fawzi, Hoang Ta, and Jeroen Zuiddam. Larger Corner-Free Sets from Combinatorial Degenerations. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 48:1-48:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{christandl_et_al:LIPIcs.ITCS.2022.48,
  author =	{Christandl, Matthias and Fawzi, Omar and Ta, Hoang and Zuiddam, Jeroen},
  title =	{{Larger Corner-Free Sets from Combinatorial Degenerations}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{48:1--48:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.48},
  URN =		{urn:nbn:de:0030-drops-156441},
  doi =		{10.4230/LIPIcs.ITCS.2022.48},
  annote =	{Keywords: Corner-free sets, communication complexity, number on the forehead, combinatorial degeneration, hypergraphs, Shannon capacity, eval problem}
}
Document
DOI: 10.4230/LIPIcs.CCC.2019.12
Limits on the Universal Method for Matrix Multiplication

Authors: Josh Alman

Published in: LIPIcs, Volume 137, 34th Computational Complexity Conference (CCC 2019)

Abstract
In this work, we prove limitations on the known methods for designing matrix multiplication algorithms. Alman and Vassilevska Williams [Alman and Williams, 2018] recently defined the Universal Method, which substantially generalizes all the known approaches including Strassen’s Laser Method [V. Strassen, 1987] and Cohn and Umans' Group Theoretic Method [Cohn and Umans, 2003]. We prove concrete lower bounds on the algorithms one can design by applying the Universal Method to many different tensors. Our proofs use new tools for upper bounding the asymptotic slice rank of a wide range of tensors. Our main result is that the Universal method applied to any Coppersmith-Winograd tensor CW_q cannot yield a bound on omega, the exponent of matrix multiplication, better than 2.16805. By comparison, it was previously only known that the weaker "Galactic Method" applied to CW_q could not achieve an exponent of 2. We also study the Laser Method (which is, in principle, a highly special case of the Universal Method) and prove that it is "complete" for matrix multiplication algorithms: when it applies to a tensor T, it achieves omega = 2 if and only if it is possible for the Universal method applied to T to achieve omega = 2. Hence, the Laser Method, which was originally used as an algorithmic tool, can also be seen as a lower bounding tool. For example, in their landmark paper, Coppersmith and Winograd [Coppersmith and Winograd, 1990] achieved a bound of omega <= 2.376, by applying the Laser Method to CW_q. By our result, the fact that they did not achieve omega=2 implies a lower bound on the Universal Method applied to CW_q. Indeed, if it were possible for the Universal Method applied to CW_q to achieve omega=2, then Coppersmith and Winograd’s application of the Laser Method would have achieved omega=2.
Cite as

Josh Alman. Limits on the Universal Method for Matrix Multiplication. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 12:1-12:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{alman:LIPIcs.CCC.2019.12,
  author =	{Alman, Josh},
  title =	{{Limits on the Universal Method for Matrix Multiplication}},
  booktitle =	{34th Computational Complexity Conference (CCC 2019)},
  pages =	{12:1--12:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-116-0},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{137},
  editor =	{Shpilka, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.12},
  URN =		{urn:nbn:de:0030-drops-108347},
  doi =		{10.4230/LIPIcs.CCC.2019.12},
  annote =	{Keywords: Matrix Multiplication, Laser Method, Slice Rank}
}
Document
DOI: 10.4230/LIPIcs.CCC.2019.26
Barriers for Fast Matrix Multiplication from Irreversibility

Authors: Matthias Christandl, Péter Vrana, and Jeroen Zuiddam

Published in: LIPIcs, Volume 137, 34th Computational Complexity Conference (CCC 2019)

Abstract
Determining the asymptotic algebraic complexity of matrix multiplication, succinctly represented by the matrix multiplication exponent omega, is a central problem in algebraic complexity theory. The best upper bounds on omega, leading to the state-of-the-art omega <= 2.37.., have been obtained via the laser method of Strassen and its generalization by Coppersmith and Winograd. Recent barrier results show limitations for these and related approaches to improve the upper bound on omega. We introduce a new and more general barrier, providing stronger limitations than in previous work. Concretely, we introduce the notion of "irreversibility" of a tensor and we prove (in some precise sense) that any approach that uses an irreversible tensor in an intermediate step (e.g., as a starting tensor in the laser method) cannot give omega = 2. In quantitative terms, we prove that the best upper bound achievable is lower bounded by two times the irreversibility of the intermediate tensor. The quantum functionals and Strassen support functionals give (so far, the best) lower bounds on irreversibility. We provide lower bounds on the irreversibility of key intermediate tensors, including the small and big Coppersmith - Winograd tensors, that improve limitations shown in previous work. Finally, we discuss barriers on the group-theoretic approach in terms of "monomial" irreversibility.
Cite as

Matthias Christandl, Péter Vrana, and Jeroen Zuiddam. Barriers for Fast Matrix Multiplication from Irreversibility. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 26:1-26:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{christandl_et_al:LIPIcs.CCC.2019.26,
  author =	{Christandl, Matthias and Vrana, P\'{e}ter and Zuiddam, Jeroen},
  title =	{{Barriers for Fast Matrix Multiplication from Irreversibility}},
  booktitle =	{34th Computational Complexity Conference (CCC 2019)},
  pages =	{26:1--26:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-116-0},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{137},
  editor =	{Shpilka, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.26},
  URN =		{urn:nbn:de:0030-drops-108487},
  doi =		{10.4230/LIPIcs.CCC.2019.26},
  annote =	{Keywords: Matrix multiplication exponent, barriers, laser method}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2017.24
Nondeterministic Quantum Communication Complexity: the Cyclic Equality Game and Iterated Matrix Multiplication

Authors: Harry Buhrman, Matthias Christandl, and Jeroen Zuiddam

Published in: LIPIcs, Volume 67, 8th Innovations in Theoretical Computer Science Conference (ITCS 2017)

Abstract
We study nondeterministic multiparty quantum communication with a quantum generalization of broadcasts. We show that, with number-in-hand classical inputs, the communication complexity of a Boolean function in this communication model equals the logarithm of the support rank of the corresponding tensor, whereas the approximation complexity in this model equals the logarithm of the border support rank. This characterisation allows us to prove a log-rank conjecture posed by Villagra et al. for nondeterministic multiparty quantum communication with message passing. The support rank characterization of the communication model connects quantum communication complexity intimately to the theory of asymptotic entanglement transformation and algebraic complexity theory. In this context, we introduce the graphwise equality problem. For a cycle graph, the complexity of this communication problem is closely related to the complexity of the computational problem of multiplying matrices, or more precisely, it equals the logarithm of the support rank of the iterated matrix multiplication tensor. We employ Strassen’s laser method to show that asymptotically there exist nontrivial protocols for every odd-player cyclic equality problem. We exhibit an efficient protocol for the 5-player problem for small inputs, and we show how Young flattenings yield nontrivial complexity lower bounds.
Cite as

Harry Buhrman, Matthias Christandl, and Jeroen Zuiddam. Nondeterministic Quantum Communication Complexity: the Cyclic Equality Game and Iterated Matrix Multiplication. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 24:1-24:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{buhrman_et_al:LIPIcs.ITCS.2017.24,
  author =	{Buhrman, Harry and Christandl, Matthias and Zuiddam, Jeroen},
  title =	{{Nondeterministic Quantum Communication Complexity: the Cyclic Equality Game and Iterated Matrix Multiplication}},
  booktitle =	{8th Innovations in Theoretical Computer Science Conference (ITCS 2017)},
  pages =	{24:1--24:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-029-3},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{67},
  editor =	{Papadimitriou, Christos H.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2017.24},
  URN =		{urn:nbn:de:0030-drops-81812},
  doi =		{10.4230/LIPIcs.ITCS.2017.24},
  annote =	{Keywords: quantum communication complexity, broadcast channel, number-in-hand, matrix multiplication, support rank}
}
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