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Documents authored by Briët, Jop

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Briet, Jop

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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
Extended Abstract
DOI: 10.4230/LIPIcs.ITCS.2024.21
Noisy Decoding by Shallow Circuits with Parities: Classical and Quantum (Extended Abstract)

Authors: Jop Briët, Harry Buhrman, Davi Castro-Silva, and Niels M. P. Neumann

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

Abstract
We consider the problem of decoding corrupted error correcting codes with NC⁰[⊕] circuits in the classical and quantum settings. We show that any such classical circuit can correctly recover only a vanishingly small fraction of messages, if the codewords are sent over a noisy channel with positive error rate. Previously this was known only for linear codes with large dual distance, whereas our result applies to any code. By contrast, we give a simple quantum circuit that correctly decodes the Hadamard code with probability Ω(ε²) even if a (1/2 - ε)-fraction of a codeword is adversarially corrupted. Our classical hardness result is based on an equidistribution phenomenon for multivariate polynomials over a finite field under biased input-distributions. This is proved using a structure-versus-randomness strategy based on a new notion of rank for high-dimensional polynomial maps that may be of independent interest. Our quantum circuit is inspired by a non-local version of the Bernstein-Vazirani problem, a technique to generate "poor man’s cat states" by Watts et al., and a constant-depth quantum circuit for the OR function by Takahashi and Tani.
Cite as

Jop Briët, Harry Buhrman, Davi Castro-Silva, and Niels M. P. Neumann. Noisy Decoding by Shallow Circuits with Parities: Classical and Quantum (Extended Abstract). In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 21:1-21:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{briet_et_al:LIPIcs.ITCS.2024.21,
  author =	{Bri\"{e}t, Jop and Buhrman, Harry and Castro-Silva, Davi and Neumann, Niels M. P.},
  title =	{{Noisy Decoding by Shallow Circuits with Parities: Classical and Quantum (Extended Abstract)}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{21:1--21:11},
  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.2024.21},
  URN =		{urn:nbn:de:0030-drops-195490},
  doi =		{10.4230/LIPIcs.ITCS.2024.21},
  annote =	{Keywords: Coding theory, circuit complexity, quantum complexity theory, higher-order Fourier analysis, non-local games}
}
Document
DOI: 10.4230/LIPIcs.TQC.2022.6
On Converses to the Polynomial Method

Authors: Jop Briët and Francisco Escudero Gutiérrez

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

Abstract
A surprising "converse to the polynomial method" of Aaronson et al. (CCC'16) shows that any bounded quadratic polynomial can be computed exactly in expectation by a 1-query algorithm up to a universal multiplicative factor related to the famous Grothendieck constant. A natural question posed there asks if bounded quartic polynomials can be approximated by 2-query quantum algorithms. Arunachalam, Palazuelos and the first author showed that there is no direct analogue of the result of Aaronson et al. in this case. We improve on this result in the following ways: First, we point out and fix a small error in the construction that has to do with a translation from cubic to quartic polynomials. Second, we give a completely explicit example based on techniques from additive combinatorics. Third, we show that the result still holds when we allow for a small additive error. For this, we apply an SDP characterization of Gribling and Laurent (QIP'19) for the completely-bounded approximate degree.
Cite as

Jop Briët and Francisco Escudero Gutiérrez. On Converses to the Polynomial Method. In 17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 232, pp. 6:1-6:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{briet_et_al:LIPIcs.TQC.2022.6,
  author =	{Bri\"{e}t, Jop and Escudero Guti\'{e}rrez, Francisco},
  title =	{{On Converses to the Polynomial Method}},
  booktitle =	{17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022)},
  pages =	{6:1--6:10},
  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.2022.6},
  URN =		{urn:nbn:de:0030-drops-165139},
  doi =		{10.4230/LIPIcs.TQC.2022.6},
  annote =	{Keywords: Quantum query complexity, polynomial method, completely bounded polynomials}
}
Document
Extended Abstract
DOI: 10.4230/LIPIcs.ITCS.2021.76
High-Entropy Dual Functions and Locally Decodable Codes (Extended Abstract)

Authors: Jop Briët and Farrokh Labib

Published in: LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

Abstract
Locally decodable codes (LDCs) allow any single encoded message symbol to be retrieved from a codeword with good probability by reading only a tiny number of codeword symbols, even if the codeword is partially corrupted. LDCs have surprisingly many applications in computer science and mathematics (we refer to [Yekhanin, 2012; Lovett, 2007] for extensive surveys). But despite their ubiquity, they are poorly understood. Of particular interest is the tradeoff between the codeword length N as a function of message length k when the query complexity - the number of probed codeword symbols - and alphabet size are constant. The Hadamard code is a 2-query LDC of length N = 2^O(k) and this length is optimal in the 2-query regime [Lovett, 2007]. For q ≥ 3, near-exponential gaps persist between the best-known upper and lower bounds. The family of Reed-Muller codes, which generalize the Hadamard code, were for a long time the best-known examples, giving q-query LDCs of length exp(O(k^{1/(q-1)})), until breakthrough constructions of matching vector LDCs of Yekhanin and Efremenko [Yekhanin, 2008; Efremenko, 2012]. In contrast with other combinatorial objects such as expander graphs, the probabilistic method has so far not been successfully used to beat the best explicit LDC constructions. In [Lovett, 2007], a probabilistic framework was given that could in principle yield best-possible LDCs, albeit non-constructively. A special instance of this framework connects LDCs with a probabilistic version of Szemerédi’s theorem. The setup for this is as follows: For a finite abelian group G of size N = |G|, let D ⊆ G be a random subset where each element is present with probability ρ independently of all others. For k ≥ 3 and ε ∈ (0,1), let E be the event that every subset A ⊆ G of size |A| ≥ ε |G| contains a proper k-term arithmetic progression with common difference in D. For fixed ε > 0 and sufficiently large N, it is an open problem to determine the smallest value of ρ - denoted ρ_k - such that Pr[E] ≥ 1/2. In [Lovett, 2007] it is shown that there exist k-query LDCs of message length Ω(ρ_k N) and codeword length O(N). As such, Szemerédi’s theorem with random differences, in particular lower bounds on ρ_k, can be used to show the existence of LDCs. Conversely, this connection indirectly implies the best-known upper bounds on ρ_k for all k ≥ 3 [Lovett, 2007; Lovett, 2007]. However, a conjecture from [Lovett, 2007] states that over ℤ_N we have ρ_k ≤ O_k(N^{-1}log N) for all k, which would be best-possible. Truth of this conjecture would imply that over this group, Szemerédi’s theorem with random differences cannot give LDCs better than the Hadamard code. For finite fields, Altman [Lovett, 2007] showed that this is false. In particular, over 𝔽_pⁿ for p odd, he proved that ρ₃ ≥ Ω(p^{-n} n²); generally, ρ_k ≥ Ω(p^{-n} n^{k-1}) holds when p ≥ k+1 [Lovett, 2007]. In turn, these bounds are conjectured to be optimal for the finite-field setting, which would imply that over finite fields, Szemerédi’s theorem with random differences cannot give LDCs better than Reed-Muller codes. The finite-field conjecture is motivated mainly by the possibility that so-called dual functions can be approximated well by polynomial phases, functions of the form e^{2π i P(x)/p} where P is a multivariate polynomial over 𝔽_p. We show that this is false. Using Yekhanin’s matching-vector-code construction, we give dual functions of order k over 𝔽_pⁿ that cannot be approximated in L_∞-distance by polynomial phases of degree k-1. This answers in the negative a natural finite-field analog of a problem of Frantzikinakis over ℕ [Lovett, 2007].
Cite as

Jop Briët and Farrokh Labib. High-Entropy Dual Functions and Locally Decodable Codes (Extended Abstract). In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 76:1-76:2, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{briet_et_al:LIPIcs.ITCS.2021.76,
  author =	{Bri\"{e}t, Jop and Labib, Farrokh},
  title =	{{High-Entropy Dual Functions and Locally Decodable Codes}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{76:1--76:2},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-177-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{185},
  editor =	{Lee, James R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.76},
  URN =		{urn:nbn:de:0030-drops-136157},
  doi =		{10.4230/LIPIcs.ITCS.2021.76},
  annote =	{Keywords: Higher-order Fourier analysis, dual functions, finite fields, additive combinatorics, coding theory}
}
Document
DOI: 10.4230/LIPIcs.TQC.2020.5
Quasirandom Quantum Channels

Authors: Tom Bannink, Jop Briët, Farrokh Labib, and Hans Maassen

Published in: LIPIcs, Volume 158, 15th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2020)

Abstract
Mixing (or quasirandom) properties of the natural transition matrix associated to a graph can be quantified by its distance to the complete graph. Different mixing properties correspond to different norms to measure this distance. For dense graphs, two such properties known as spectral expansion and uniformity were shown to be equivalent in seminal 1989 work of Chung, Graham and Wilson. Recently, Conlon and Zhao extended this equivalence to the case of sparse vertex transitive graphs using the famous Grothendieck inequality. Here we generalize these results to the non-commutative, or "quantum", case, where a transition matrix becomes a quantum channel. In particular, we show that for irreducibly covariant quantum channels, expansion is equivalent to a natural analog of uniformity for graphs, generalizing the result of Conlon and Zhao. Moreover, we show that in these results, the non-commutative and commutative (resp.) Grothendieck inequalities yield the best-possible constants.
Cite as

Tom Bannink, Jop Briët, Farrokh Labib, and Hans Maassen. Quasirandom Quantum Channels. In 15th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 158, pp. 5:1-5:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bannink_et_al:LIPIcs.TQC.2020.5,
  author =	{Bannink, Tom and Bri\"{e}t, Jop and Labib, Farrokh and Maassen, Hans},
  title =	{{Quasirandom Quantum Channels}},
  booktitle =	{15th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2020)},
  pages =	{5:1--5:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-146-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{158},
  editor =	{Flammia, Steven T.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2020.5},
  URN =		{urn:nbn:de:0030-drops-120642},
  doi =		{10.4230/LIPIcs.TQC.2020.5},
  annote =	{Keywords: Quantum channels, quantum expanders, quasirandomness}
}
Document
DOI: 10.4230/LIPIcs.STACS.2019.12
Bounding Quantum-Classical Separations for Classes of Nonlocal Games

Authors: Tom Bannink, Jop Briët, Harry Buhrman, Farrokh Labib, and Troy Lee

Published in: LIPIcs, Volume 126, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)

Abstract
We bound separations between the entangled and classical values for several classes of nonlocal t-player games. Our motivating question is whether there is a family of t-player XOR games for which the entangled bias is 1 but for which the classical bias goes down to 0, for fixed t. Answering this question would have important consequences in the study of multi-party communication complexity, as a positive answer would imply an unbounded separation between randomized communication complexity with and without entanglement. Our contribution to answering the question is identifying several general classes of games for which the classical bias can not go to zero when the entangled bias stays above a constant threshold. This rules out the possibility of using these games to answer our motivating question. A previously studied set of XOR games, known not to give a positive answer to the question, are those for which there is a quantum strategy that attains value 1 using a so-called Schmidt state. We generalize this class to mod-m games and show that their classical value is always at least 1/m + (m-1)/m t^{1-t}. Secondly, for free XOR games, in which the input distribution is of product form, we show beta(G) >= beta^*(G)^{2^t} where beta(G) and beta^*(G) are the classical and entangled biases of the game respectively. We also introduce so-called line games, an example of which is a slight modification of the Magic Square game, and show that they can not give a positive answer to the question either. Finally we look at two-player unique games and show that if the entangled value is 1-epsilon then the classical value is at least 1-O(sqrt{epsilon log k}) where k is the number of outputs in the game. Our proofs use semidefinite-programming techniques, the Gowers inverse theorem and hypergraph norms.
Cite as

Tom Bannink, Jop Briët, Harry Buhrman, Farrokh Labib, and Troy Lee. Bounding Quantum-Classical Separations for Classes of Nonlocal Games. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 12:1-12:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bannink_et_al:LIPIcs.STACS.2019.12,
  author =	{Bannink, Tom and Bri\"{e}t, Jop and Buhrman, Harry and Labib, Farrokh and Lee, Troy},
  title =	{{Bounding Quantum-Classical Separations for Classes of Nonlocal Games}},
  booktitle =	{36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)},
  pages =	{12:1--12:11},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-100-9},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{126},
  editor =	{Niedermeier, Rolf and Paul, Christophe},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.12},
  URN =		{urn:nbn:de:0030-drops-102512},
  doi =		{10.4230/LIPIcs.STACS.2019.12},
  annote =	{Keywords: Nonlocal games, communication complexity, bounded separations, semidefinite programming, pseudorandomness, Gowers norms}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2018.3
Quantum Query Algorithms are Completely Bounded Forms

Authors: Srinivasan Arunachalam, Jop Briët, and Carlos Palazuelos

Published in: LIPIcs, Volume 94, 9th Innovations in Theoretical Computer Science Conference (ITCS 2018)

Abstract
We prove a characterization of quantum query algorithms in terms of polynomials satisfying a certain (completely bounded) norm constraint. Based on this, we obtain a refined notion of approximate polynomial degree that equals the quantum query complexity, answering a question of Aaronson et al. (CCC'16). Using this characterization, we show that many polynomials of degree at least 4 are far from those coming from quantum query algorithms. Our proof is based on a fundamental result of Christensen and Sinclair (J. Funct. Anal., 1987) that generalizes the well-known Stinespring representation for quantum channels to multilinear forms. We also give a simple and short proof of one of the results of Aaronson et al. showing an equivalence between one-query quantum algorithms and bounded quadratic polynomials.
Cite as

Srinivasan Arunachalam, Jop Briët, and Carlos Palazuelos. Quantum Query Algorithms are Completely Bounded Forms. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 94, pp. 3:1-3:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{arunachalam_et_al:LIPIcs.ITCS.2018.3,
  author =	{Arunachalam, Srinivasan and Bri\"{e}t, Jop and Palazuelos, Carlos},
  title =	{{Quantum Query Algorithms are Completely Bounded Forms}},
  booktitle =	{9th Innovations in Theoretical Computer Science Conference (ITCS 2018)},
  pages =	{3:1--3:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-060-6},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{94},
  editor =	{Karlin, Anna R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2018.3},
  URN =		{urn:nbn:de:0030-drops-83383},
  doi =		{10.4230/LIPIcs.ITCS.2018.3},
  annote =	{Keywords: Quantum query algorithms, operator space theory, polynomial method, approximate degree.}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2017.20
Outlaw Distributions and Locally Decodable Codes

Authors: Jop Briët, Zeev Dvir, and Sivakanth Gopi

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

Abstract
Locally decodable codes (LDCs) are error correcting codes that allow for decoding of a single message bit using a small number of queries to a corrupted encoding. Despite decades of study, the optimal trade-off between query complexity and codeword length is far from understood. In this work, we give a new characterization of LDCs using distributions over Boolean functions whose expectation is hard to approximate (in L_\infty norm) with a small number of samples. We coin the term 'outlaw distributions' for such distributions since they 'defy' the Law of Large Numbers. We show that the existence of outlaw distributions over sufficiently 'smooth' functions implies the existence of constant query LDCs and vice versa. We give several candidates for outlaw distributions over smooth functions coming from finite field incidence geometry and from hypergraph (non)expanders. We also prove a useful lemma showing that (smooth) LDCs which are only required to work on average over a random message and a random message index can be turned into true LDCs at the cost of only constant factors in the parameters.
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Jop Briët, Zeev Dvir, and Sivakanth Gopi. Outlaw Distributions and Locally Decodable Codes. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 20:1-20:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{briet_et_al:LIPIcs.ITCS.2017.20,
  author =	{Bri\"{e}t, Jop and Dvir, Zeev and Gopi, Sivakanth},
  title =	{{Outlaw Distributions and Locally Decodable Codes}},
  booktitle =	{8th Innovations in Theoretical Computer Science Conference (ITCS 2017)},
  pages =	{20:1--20:19},
  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.2017.20},
  URN =		{urn:nbn:de:0030-drops-81888},
  doi =		{10.4230/LIPIcs.ITCS.2017.20},
  annote =	{Keywords: Locally Decodable Code, VC-dimension, Incidence Geometry, Cayley Hypergraphs}
}
Document
DOI: 10.4230/LIPIcs.TQC.2015.206
Round Elimination in Exact Communication Complexity

Authors: Jop Briët, Harry Buhrman, Debbie Leung, Teresa Piovesan, and Florian Speelman

Published in: LIPIcs, Volume 44, 10th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2015)

Abstract
We study two basic graph parameters, the chromatic number and the orthogonal rank, in the context of classical and quantum exact communication complexity. In particular, we consider two types of communication problems that we call promise equality and list problems. For both of these, it was already known that the one-round classical and one-round quantum complexities are characterized by the chromatic number and orthogonal rank of a certain graph, respectively. In a promise equality problem, Alice and Bob must decide if their inputs are equal or not. We prove that classical protocols for such problems can always be reduced to one-round protocols with no extra communication. In contrast, we give an explicit instance of a promise problem that exhibits an exponential gap between the one- and two-round exact quantum communication complexities. Whereas the chromatic number thus captures the complete complexity of promise equality problems, the hierarchy of "quantum chromatic numbers" (starting with the orthogonal rank) giving the quantum communication complexity for every fixed number of communication rounds thus turns out to enjoy a much richer structure. In a list problem, Bob gets a subset of some finite universe, Alice gets an element from Bob's subset, and their goal is for Bob to learn which element Alice was given. The best general lower bound (due to Orlitsky) and upper bound (due to Naor, Orlitsky, and Shor) on the classical communication complexity of such problems differ only by a constant factor. We exhibit an example showing that, somewhat surprisingly, the four-round protocol used in the bound of Naor et al. can in fact be optimal. Finally, we pose a conjecture on the orthogonality rank of a certain graph whose truth would imply an intriguing impossibility of round elimination in quantum protocols for list problems, something that works trivially in the classical case.
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Jop Briët, Harry Buhrman, Debbie Leung, Teresa Piovesan, and Florian Speelman. Round Elimination in Exact Communication Complexity. In 10th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 44, pp. 206-225, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{briet_et_al:LIPIcs.TQC.2015.206,
  author =	{Bri\"{e}t, Jop and Buhrman, Harry and Leung, Debbie and Piovesan, Teresa and Speelman, Florian},
  title =	{{Round Elimination in Exact Communication Complexity}},
  booktitle =	{10th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2015)},
  pages =	{206--225},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-96-5},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{44},
  editor =	{Beigi, Salman and K\"{o}nig, Robert},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2015.206},
  URN =		{urn:nbn:de:0030-drops-55588},
  doi =		{10.4230/LIPIcs.TQC.2015.206},
  annote =	{Keywords: communication complexity, round elimination, quantum communication, protocols, chromatic numbers}
}
Document
DOI: 10.4230/LIPIcs.STACS.2009.1813
Locally Decodable Quantum Codes

Authors: Jop Briet and Ronald de Wolf

Published in: LIPIcs, Volume 3, 26th International Symposium on Theoretical Aspects of Computer Science (2009)

Abstract
We study a quantum analogue of locally decodable error-correcting codes. A $q$-query \emph{locally decodable quantum code} encodes $n$ classical bits in an $m$-qubit state, in such a way that each of the encoded bits can be recovered with high probability by a measurement on at most $q$ qubits of the quantum code, even if a constant fraction of its qubits have been corrupted adversarially. We show that such a quantum code can be transformed into a \emph{classical} $q$-query locally decodable code of the same length that can be decoded well on average (albeit with smaller success probability and noise-tolerance). This shows, roughly speaking, that $q$-query quantum codes are not significantly better than $q$-query classical codes, at least for constant or small $q$.
Cite as

Jop Briet and Ronald de Wolf. Locally Decodable Quantum Codes. In 26th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 3, pp. 219-230, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{briet_et_al:LIPIcs.STACS.2009.1813,
  author =	{Briet, Jop and de Wolf, Ronald},
  title =	{{Locally Decodable Quantum Codes}},
  booktitle =	{26th International Symposium on Theoretical Aspects of Computer Science},
  pages =	{219--230},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-09-5},
  ISSN =	{1868-8969},
  year =	{2009},
  volume =	{3},
  editor =	{Albers, Susanne and Marion, Jean-Yves},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2009.1813},
  URN =		{urn:nbn:de:0030-drops-18134},
  doi =		{10.4230/LIPIcs.STACS.2009.1813},
  annote =	{Keywords: Data structures, Locally decodable codes, Quantum computing}
}

Briët, Jop

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
Extended Abstract
DOI: 10.4230/LIPIcs.ITCS.2024.21
Noisy Decoding by Shallow Circuits with Parities: Classical and Quantum (Extended Abstract)

Authors: Jop Briët, Harry Buhrman, Davi Castro-Silva, and Niels M. P. Neumann

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

Abstract
We consider the problem of decoding corrupted error correcting codes with NC⁰[⊕] circuits in the classical and quantum settings. We show that any such classical circuit can correctly recover only a vanishingly small fraction of messages, if the codewords are sent over a noisy channel with positive error rate. Previously this was known only for linear codes with large dual distance, whereas our result applies to any code. By contrast, we give a simple quantum circuit that correctly decodes the Hadamard code with probability Ω(ε²) even if a (1/2 - ε)-fraction of a codeword is adversarially corrupted. Our classical hardness result is based on an equidistribution phenomenon for multivariate polynomials over a finite field under biased input-distributions. This is proved using a structure-versus-randomness strategy based on a new notion of rank for high-dimensional polynomial maps that may be of independent interest. Our quantum circuit is inspired by a non-local version of the Bernstein-Vazirani problem, a technique to generate "poor man’s cat states" by Watts et al., and a constant-depth quantum circuit for the OR function by Takahashi and Tani.
Cite as

Jop Briët, Harry Buhrman, Davi Castro-Silva, and Niels M. P. Neumann. Noisy Decoding by Shallow Circuits with Parities: Classical and Quantum (Extended Abstract). In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 21:1-21:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{briet_et_al:LIPIcs.ITCS.2024.21,
  author =	{Bri\"{e}t, Jop and Buhrman, Harry and Castro-Silva, Davi and Neumann, Niels M. P.},
  title =	{{Noisy Decoding by Shallow Circuits with Parities: Classical and Quantum (Extended Abstract)}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{21:1--21:11},
  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.2024.21},
  URN =		{urn:nbn:de:0030-drops-195490},
  doi =		{10.4230/LIPIcs.ITCS.2024.21},
  annote =	{Keywords: Coding theory, circuit complexity, quantum complexity theory, higher-order Fourier analysis, non-local games}
}
Document
DOI: 10.4230/LIPIcs.TQC.2022.6
On Converses to the Polynomial Method

Authors: Jop Briët and Francisco Escudero Gutiérrez

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

Abstract
A surprising "converse to the polynomial method" of Aaronson et al. (CCC'16) shows that any bounded quadratic polynomial can be computed exactly in expectation by a 1-query algorithm up to a universal multiplicative factor related to the famous Grothendieck constant. A natural question posed there asks if bounded quartic polynomials can be approximated by 2-query quantum algorithms. Arunachalam, Palazuelos and the first author showed that there is no direct analogue of the result of Aaronson et al. in this case. We improve on this result in the following ways: First, we point out and fix a small error in the construction that has to do with a translation from cubic to quartic polynomials. Second, we give a completely explicit example based on techniques from additive combinatorics. Third, we show that the result still holds when we allow for a small additive error. For this, we apply an SDP characterization of Gribling and Laurent (QIP'19) for the completely-bounded approximate degree.
Cite as

Jop Briët and Francisco Escudero Gutiérrez. On Converses to the Polynomial Method. In 17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 232, pp. 6:1-6:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{briet_et_al:LIPIcs.TQC.2022.6,
  author =	{Bri\"{e}t, Jop and Escudero Guti\'{e}rrez, Francisco},
  title =	{{On Converses to the Polynomial Method}},
  booktitle =	{17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022)},
  pages =	{6:1--6:10},
  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.2022.6},
  URN =		{urn:nbn:de:0030-drops-165139},
  doi =		{10.4230/LIPIcs.TQC.2022.6},
  annote =	{Keywords: Quantum query complexity, polynomial method, completely bounded polynomials}
}
Document
Extended Abstract
DOI: 10.4230/LIPIcs.ITCS.2021.76
High-Entropy Dual Functions and Locally Decodable Codes (Extended Abstract)

Authors: Jop Briët and Farrokh Labib

Published in: LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

Abstract
Locally decodable codes (LDCs) allow any single encoded message symbol to be retrieved from a codeword with good probability by reading only a tiny number of codeword symbols, even if the codeword is partially corrupted. LDCs have surprisingly many applications in computer science and mathematics (we refer to [Yekhanin, 2012; Lovett, 2007] for extensive surveys). But despite their ubiquity, they are poorly understood. Of particular interest is the tradeoff between the codeword length N as a function of message length k when the query complexity - the number of probed codeword symbols - and alphabet size are constant. The Hadamard code is a 2-query LDC of length N = 2^O(k) and this length is optimal in the 2-query regime [Lovett, 2007]. For q ≥ 3, near-exponential gaps persist between the best-known upper and lower bounds. The family of Reed-Muller codes, which generalize the Hadamard code, were for a long time the best-known examples, giving q-query LDCs of length exp(O(k^{1/(q-1)})), until breakthrough constructions of matching vector LDCs of Yekhanin and Efremenko [Yekhanin, 2008; Efremenko, 2012]. In contrast with other combinatorial objects such as expander graphs, the probabilistic method has so far not been successfully used to beat the best explicit LDC constructions. In [Lovett, 2007], a probabilistic framework was given that could in principle yield best-possible LDCs, albeit non-constructively. A special instance of this framework connects LDCs with a probabilistic version of Szemerédi’s theorem. The setup for this is as follows: For a finite abelian group G of size N = |G|, let D ⊆ G be a random subset where each element is present with probability ρ independently of all others. For k ≥ 3 and ε ∈ (0,1), let E be the event that every subset A ⊆ G of size |A| ≥ ε |G| contains a proper k-term arithmetic progression with common difference in D. For fixed ε > 0 and sufficiently large N, it is an open problem to determine the smallest value of ρ - denoted ρ_k - such that Pr[E] ≥ 1/2. In [Lovett, 2007] it is shown that there exist k-query LDCs of message length Ω(ρ_k N) and codeword length O(N). As such, Szemerédi’s theorem with random differences, in particular lower bounds on ρ_k, can be used to show the existence of LDCs. Conversely, this connection indirectly implies the best-known upper bounds on ρ_k for all k ≥ 3 [Lovett, 2007; Lovett, 2007]. However, a conjecture from [Lovett, 2007] states that over ℤ_N we have ρ_k ≤ O_k(N^{-1}log N) for all k, which would be best-possible. Truth of this conjecture would imply that over this group, Szemerédi’s theorem with random differences cannot give LDCs better than the Hadamard code. For finite fields, Altman [Lovett, 2007] showed that this is false. In particular, over 𝔽_pⁿ for p odd, he proved that ρ₃ ≥ Ω(p^{-n} n²); generally, ρ_k ≥ Ω(p^{-n} n^{k-1}) holds when p ≥ k+1 [Lovett, 2007]. In turn, these bounds are conjectured to be optimal for the finite-field setting, which would imply that over finite fields, Szemerédi’s theorem with random differences cannot give LDCs better than Reed-Muller codes. The finite-field conjecture is motivated mainly by the possibility that so-called dual functions can be approximated well by polynomial phases, functions of the form e^{2π i P(x)/p} where P is a multivariate polynomial over 𝔽_p. We show that this is false. Using Yekhanin’s matching-vector-code construction, we give dual functions of order k over 𝔽_pⁿ that cannot be approximated in L_∞-distance by polynomial phases of degree k-1. This answers in the negative a natural finite-field analog of a problem of Frantzikinakis over ℕ [Lovett, 2007].
Cite as

Jop Briët and Farrokh Labib. High-Entropy Dual Functions and Locally Decodable Codes (Extended Abstract). In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 76:1-76:2, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{briet_et_al:LIPIcs.ITCS.2021.76,
  author =	{Bri\"{e}t, Jop and Labib, Farrokh},
  title =	{{High-Entropy Dual Functions and Locally Decodable Codes}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{76:1--76:2},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-177-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{185},
  editor =	{Lee, James R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.76},
  URN =		{urn:nbn:de:0030-drops-136157},
  doi =		{10.4230/LIPIcs.ITCS.2021.76},
  annote =	{Keywords: Higher-order Fourier analysis, dual functions, finite fields, additive combinatorics, coding theory}
}
Document
DOI: 10.4230/LIPIcs.TQC.2020.5
Quasirandom Quantum Channels

Authors: Tom Bannink, Jop Briët, Farrokh Labib, and Hans Maassen

Published in: LIPIcs, Volume 158, 15th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2020)

Abstract
Mixing (or quasirandom) properties of the natural transition matrix associated to a graph can be quantified by its distance to the complete graph. Different mixing properties correspond to different norms to measure this distance. For dense graphs, two such properties known as spectral expansion and uniformity were shown to be equivalent in seminal 1989 work of Chung, Graham and Wilson. Recently, Conlon and Zhao extended this equivalence to the case of sparse vertex transitive graphs using the famous Grothendieck inequality. Here we generalize these results to the non-commutative, or "quantum", case, where a transition matrix becomes a quantum channel. In particular, we show that for irreducibly covariant quantum channels, expansion is equivalent to a natural analog of uniformity for graphs, generalizing the result of Conlon and Zhao. Moreover, we show that in these results, the non-commutative and commutative (resp.) Grothendieck inequalities yield the best-possible constants.
Cite as

Tom Bannink, Jop Briët, Farrokh Labib, and Hans Maassen. Quasirandom Quantum Channels. In 15th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 158, pp. 5:1-5:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bannink_et_al:LIPIcs.TQC.2020.5,
  author =	{Bannink, Tom and Bri\"{e}t, Jop and Labib, Farrokh and Maassen, Hans},
  title =	{{Quasirandom Quantum Channels}},
  booktitle =	{15th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2020)},
  pages =	{5:1--5:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-146-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{158},
  editor =	{Flammia, Steven T.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2020.5},
  URN =		{urn:nbn:de:0030-drops-120642},
  doi =		{10.4230/LIPIcs.TQC.2020.5},
  annote =	{Keywords: Quantum channels, quantum expanders, quasirandomness}
}
Document
DOI: 10.4230/LIPIcs.STACS.2019.12
Bounding Quantum-Classical Separations for Classes of Nonlocal Games

Authors: Tom Bannink, Jop Briët, Harry Buhrman, Farrokh Labib, and Troy Lee

Published in: LIPIcs, Volume 126, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)

Abstract
We bound separations between the entangled and classical values for several classes of nonlocal t-player games. Our motivating question is whether there is a family of t-player XOR games for which the entangled bias is 1 but for which the classical bias goes down to 0, for fixed t. Answering this question would have important consequences in the study of multi-party communication complexity, as a positive answer would imply an unbounded separation between randomized communication complexity with and without entanglement. Our contribution to answering the question is identifying several general classes of games for which the classical bias can not go to zero when the entangled bias stays above a constant threshold. This rules out the possibility of using these games to answer our motivating question. A previously studied set of XOR games, known not to give a positive answer to the question, are those for which there is a quantum strategy that attains value 1 using a so-called Schmidt state. We generalize this class to mod-m games and show that their classical value is always at least 1/m + (m-1)/m t^{1-t}. Secondly, for free XOR games, in which the input distribution is of product form, we show beta(G) >= beta^*(G)^{2^t} where beta(G) and beta^*(G) are the classical and entangled biases of the game respectively. We also introduce so-called line games, an example of which is a slight modification of the Magic Square game, and show that they can not give a positive answer to the question either. Finally we look at two-player unique games and show that if the entangled value is 1-epsilon then the classical value is at least 1-O(sqrt{epsilon log k}) where k is the number of outputs in the game. Our proofs use semidefinite-programming techniques, the Gowers inverse theorem and hypergraph norms.
Cite as

Tom Bannink, Jop Briët, Harry Buhrman, Farrokh Labib, and Troy Lee. Bounding Quantum-Classical Separations for Classes of Nonlocal Games. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 12:1-12:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bannink_et_al:LIPIcs.STACS.2019.12,
  author =	{Bannink, Tom and Bri\"{e}t, Jop and Buhrman, Harry and Labib, Farrokh and Lee, Troy},
  title =	{{Bounding Quantum-Classical Separations for Classes of Nonlocal Games}},
  booktitle =	{36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)},
  pages =	{12:1--12:11},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-100-9},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{126},
  editor =	{Niedermeier, Rolf and Paul, Christophe},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.12},
  URN =		{urn:nbn:de:0030-drops-102512},
  doi =		{10.4230/LIPIcs.STACS.2019.12},
  annote =	{Keywords: Nonlocal games, communication complexity, bounded separations, semidefinite programming, pseudorandomness, Gowers norms}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2018.3
Quantum Query Algorithms are Completely Bounded Forms

Authors: Srinivasan Arunachalam, Jop Briët, and Carlos Palazuelos

Published in: LIPIcs, Volume 94, 9th Innovations in Theoretical Computer Science Conference (ITCS 2018)

Abstract
We prove a characterization of quantum query algorithms in terms of polynomials satisfying a certain (completely bounded) norm constraint. Based on this, we obtain a refined notion of approximate polynomial degree that equals the quantum query complexity, answering a question of Aaronson et al. (CCC'16). Using this characterization, we show that many polynomials of degree at least 4 are far from those coming from quantum query algorithms. Our proof is based on a fundamental result of Christensen and Sinclair (J. Funct. Anal., 1987) that generalizes the well-known Stinespring representation for quantum channels to multilinear forms. We also give a simple and short proof of one of the results of Aaronson et al. showing an equivalence between one-query quantum algorithms and bounded quadratic polynomials.
Cite as

Srinivasan Arunachalam, Jop Briët, and Carlos Palazuelos. Quantum Query Algorithms are Completely Bounded Forms. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 94, pp. 3:1-3:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{arunachalam_et_al:LIPIcs.ITCS.2018.3,
  author =	{Arunachalam, Srinivasan and Bri\"{e}t, Jop and Palazuelos, Carlos},
  title =	{{Quantum Query Algorithms are Completely Bounded Forms}},
  booktitle =	{9th Innovations in Theoretical Computer Science Conference (ITCS 2018)},
  pages =	{3:1--3:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-060-6},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{94},
  editor =	{Karlin, Anna R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2018.3},
  URN =		{urn:nbn:de:0030-drops-83383},
  doi =		{10.4230/LIPIcs.ITCS.2018.3},
  annote =	{Keywords: Quantum query algorithms, operator space theory, polynomial method, approximate degree.}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2017.20
Outlaw Distributions and Locally Decodable Codes

Authors: Jop Briët, Zeev Dvir, and Sivakanth Gopi

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

Abstract
Locally decodable codes (LDCs) are error correcting codes that allow for decoding of a single message bit using a small number of queries to a corrupted encoding. Despite decades of study, the optimal trade-off between query complexity and codeword length is far from understood. In this work, we give a new characterization of LDCs using distributions over Boolean functions whose expectation is hard to approximate (in L_\infty norm) with a small number of samples. We coin the term 'outlaw distributions' for such distributions since they 'defy' the Law of Large Numbers. We show that the existence of outlaw distributions over sufficiently 'smooth' functions implies the existence of constant query LDCs and vice versa. We give several candidates for outlaw distributions over smooth functions coming from finite field incidence geometry and from hypergraph (non)expanders. We also prove a useful lemma showing that (smooth) LDCs which are only required to work on average over a random message and a random message index can be turned into true LDCs at the cost of only constant factors in the parameters.
Cite as

Jop Briët, Zeev Dvir, and Sivakanth Gopi. Outlaw Distributions and Locally Decodable Codes. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 20:1-20:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{briet_et_al:LIPIcs.ITCS.2017.20,
  author =	{Bri\"{e}t, Jop and Dvir, Zeev and Gopi, Sivakanth},
  title =	{{Outlaw Distributions and Locally Decodable Codes}},
  booktitle =	{8th Innovations in Theoretical Computer Science Conference (ITCS 2017)},
  pages =	{20:1--20:19},
  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.2017.20},
  URN =		{urn:nbn:de:0030-drops-81888},
  doi =		{10.4230/LIPIcs.ITCS.2017.20},
  annote =	{Keywords: Locally Decodable Code, VC-dimension, Incidence Geometry, Cayley Hypergraphs}
}
Document
DOI: 10.4230/LIPIcs.TQC.2015.206
Round Elimination in Exact Communication Complexity

Authors: Jop Briët, Harry Buhrman, Debbie Leung, Teresa Piovesan, and Florian Speelman

Published in: LIPIcs, Volume 44, 10th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2015)

Abstract
We study two basic graph parameters, the chromatic number and the orthogonal rank, in the context of classical and quantum exact communication complexity. In particular, we consider two types of communication problems that we call promise equality and list problems. For both of these, it was already known that the one-round classical and one-round quantum complexities are characterized by the chromatic number and orthogonal rank of a certain graph, respectively. In a promise equality problem, Alice and Bob must decide if their inputs are equal or not. We prove that classical protocols for such problems can always be reduced to one-round protocols with no extra communication. In contrast, we give an explicit instance of a promise problem that exhibits an exponential gap between the one- and two-round exact quantum communication complexities. Whereas the chromatic number thus captures the complete complexity of promise equality problems, the hierarchy of "quantum chromatic numbers" (starting with the orthogonal rank) giving the quantum communication complexity for every fixed number of communication rounds thus turns out to enjoy a much richer structure. In a list problem, Bob gets a subset of some finite universe, Alice gets an element from Bob's subset, and their goal is for Bob to learn which element Alice was given. The best general lower bound (due to Orlitsky) and upper bound (due to Naor, Orlitsky, and Shor) on the classical communication complexity of such problems differ only by a constant factor. We exhibit an example showing that, somewhat surprisingly, the four-round protocol used in the bound of Naor et al. can in fact be optimal. Finally, we pose a conjecture on the orthogonality rank of a certain graph whose truth would imply an intriguing impossibility of round elimination in quantum protocols for list problems, something that works trivially in the classical case.
Cite as

Jop Briët, Harry Buhrman, Debbie Leung, Teresa Piovesan, and Florian Speelman. Round Elimination in Exact Communication Complexity. In 10th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 44, pp. 206-225, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{briet_et_al:LIPIcs.TQC.2015.206,
  author =	{Bri\"{e}t, Jop and Buhrman, Harry and Leung, Debbie and Piovesan, Teresa and Speelman, Florian},
  title =	{{Round Elimination in Exact Communication Complexity}},
  booktitle =	{10th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2015)},
  pages =	{206--225},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-96-5},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{44},
  editor =	{Beigi, Salman and K\"{o}nig, Robert},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2015.206},
  URN =		{urn:nbn:de:0030-drops-55588},
  doi =		{10.4230/LIPIcs.TQC.2015.206},
  annote =	{Keywords: communication complexity, round elimination, quantum communication, protocols, chromatic numbers}
}
Document
DOI: 10.4230/LIPIcs.STACS.2009.1813
Locally Decodable Quantum Codes

Authors: Jop Briet and Ronald de Wolf

Published in: LIPIcs, Volume 3, 26th International Symposium on Theoretical Aspects of Computer Science (2009)

Abstract
We study a quantum analogue of locally decodable error-correcting codes. A $q$-query \emph{locally decodable quantum code} encodes $n$ classical bits in an $m$-qubit state, in such a way that each of the encoded bits can be recovered with high probability by a measurement on at most $q$ qubits of the quantum code, even if a constant fraction of its qubits have been corrupted adversarially. We show that such a quantum code can be transformed into a \emph{classical} $q$-query locally decodable code of the same length that can be decoded well on average (albeit with smaller success probability and noise-tolerance). This shows, roughly speaking, that $q$-query quantum codes are not significantly better than $q$-query classical codes, at least for constant or small $q$.
Cite as

Jop Briet and Ronald de Wolf. Locally Decodable Quantum Codes. In 26th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 3, pp. 219-230, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{briet_et_al:LIPIcs.STACS.2009.1813,
  author =	{Briet, Jop and de Wolf, Ronald},
  title =	{{Locally Decodable Quantum Codes}},
  booktitle =	{26th International Symposium on Theoretical Aspects of Computer Science},
  pages =	{219--230},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-09-5},
  ISSN =	{1868-8969},
  year =	{2009},
  volume =	{3},
  editor =	{Albers, Susanne and Marion, Jean-Yves},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2009.1813},
  URN =		{urn:nbn:de:0030-drops-18134},
  doi =		{10.4230/LIPIcs.STACS.2009.1813},
  annote =	{Keywords: Data structures, Locally decodable codes, Quantum computing}
}
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