2 Search Results for "Cleve, Richard"

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-dev.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.ICALP.2017.17
Efficient Quantum Algorithms for Simulating Lindblad Evolution

Authors: Richard Cleve and Chunhao Wang

Published in: LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

Abstract
We consider the natural generalization of the Schrodinger equation to Markovian open system dynamics: the so-called the Lindblad equation. We give a quantum algorithm for simulating the evolution of an n-qubit system for time t within precision epsilon. If the Lindbladian consists of poly(n) operators that can each be expressed as a linear combination of poly(n) tensor products of Pauli operators then the gate cost of our algorithm is O(t polylog(t/epsilon) poly(n)). We also obtain similar bounds for the cases where the Lindbladian consists of local operators, and where the Lindbladian consists of sparse operators. This is remarkable in light of evidence that we provide indicating that the above efficiency is impossible to attain by first expressing Lindblad evolution as Schrodinger evolution on a larger system and tracing out the ancillary system: the cost of such a reduction incurs an efficiency overhead of O(t^2/epsilon) even before the Hamiltonian evolution simulation begins. Instead, the approach of our algorithm is to use a novel variation of the "linear combinations of unitaries" construction that pertains to channels.
Cite as

Richard Cleve and Chunhao Wang. Efficient Quantum Algorithms for Simulating Lindblad Evolution. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 17:1-17:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{cleve_et_al:LIPIcs.ICALP.2017.17,
  author =	{Cleve, Richard and Wang, Chunhao},
  title =	{{Efficient Quantum Algorithms for Simulating Lindblad Evolution}},
  booktitle =	{44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)},
  pages =	{17:1--17:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-041-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{80},
  editor =	{Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.17},
  URN =		{urn:nbn:de:0030-drops-74776},
  doi =		{10.4230/LIPIcs.ICALP.2017.17},
  annote =	{Keywords: quantum algorithms, open quantum systems, Lindblad simulation}
}
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