5 Search Results for "Nayak, Ashwin"

Document
DOI: 10.4230/LIPIcs.ITCS.2024.97
Quantum Event Learning and Gentle Random Measurements

Authors: Adam Bene Watts and John Bostanci

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

Abstract
We prove the expected disturbance caused to a quantum system by a sequence of randomly ordered two-outcome projective measurements is upper bounded by the square root of the probability that at least one measurement in the sequence accepts. We call this bound the Gentle Random Measurement Lemma. We then extend the techniques used to prove this lemma to develop protocols for problems in which we are given sample access to an unknown state ρ and asked to estimate properties of the accepting probabilities Tr[M_i ρ] of a set of measurements {M₁, M₂, … , M_m}. We call these types of problems Quantum Event Learning Problems. In particular, we show randomly ordering projective measurements solves the Quantum OR problem, answering an open question of Aaronson. We also give a Quantum OR protocol which works on non-projective measurements and which outperforms both the random measurement protocol analyzed in this paper and the protocol of Harrow, Lin, and Montanaro. However, this protocol requires a more complicated type of measurement, which we call a Blended Measurement. Given additional guarantees on the set of measurements {M₁, …, M_m}, we show the random and blended measurement Quantum OR protocols developed in this paper can also be used to find a measurement M_i such that Tr[M_i ρ] is large. We call the problem of finding such a measurement Quantum Event Finding. We also show Blended Measurements give a sample-efficient protocol for Quantum Mean Estimation: a problem in which the goal is to estimate the average accepting probability of a set of measurements on an unknown state. Finally we consider the Threshold Search Problem described by O'Donnell and Bădescu where, given given a set of measurements {M₁, …, M_m} along with sample access to an unknown state ρ satisfying Tr[M_i ρ] ≥ 1/2 for some M_i, the goal is to find a measurement M_j such that Tr[M_j ρ] ≥ 1/2 - ε. By building on our Quantum Event Finding result we show that randomly ordered (or blended) measurements can be used to solve this problem using O(log²(m) / ε²) copies of ρ. This matches the performance of the algorithm given by O'Donnell and Bădescu, but does not require injected noise in the measurements. Consequently, we obtain an algorithm for Shadow Tomography which matches the current best known sample complexity (i.e. requires Õ(log²(m)log(d)/ε⁴) samples). This algorithm does not require injected noise in the quantum measurements, but does require measurements to be made in a random order, and so is no longer online.
Cite as

Adam Bene Watts and John Bostanci. Quantum Event Learning and Gentle Random Measurements. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 97:1-97:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{watts_et_al:LIPIcs.ITCS.2024.97,
  author =	{Watts, Adam Bene and Bostanci, John},
  title =	{{Quantum Event Learning and Gentle Random Measurements}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{97:1--97:22},
  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.2024.97},
  URN =		{urn:nbn:de:0030-drops-196254},
  doi =		{10.4230/LIPIcs.ITCS.2024.97},
  annote =	{Keywords: Event learning, gentle measurments, random measurements, quantum or, threshold search, shadow tomography}
}
Document
DOI: 10.4230/LIPIcs.CCC.2020.7
Quantum Lower Bounds for Approximate Counting via Laurent Polynomials

Authors: Scott Aaronson, Robin Kothari, William Kretschmer, and Justin Thaler

Published in: LIPIcs, Volume 169, 35th Computational Complexity Conference (CCC 2020)

Abstract
We study quantum algorithms that are given access to trusted and untrusted quantum witnesses. We establish strong limitations of such algorithms, via new techniques based on Laurent polynomials (i.e., polynomials with positive and negative integer exponents). Specifically, we resolve the complexity of approximate counting, the problem of multiplicatively estimating the size of a nonempty set S ⊆ [N], in two natural generalizations of quantum query complexity. Our first result holds in the standard Quantum Merlin - Arthur (QMA) setting, in which a quantum algorithm receives an untrusted quantum witness. We show that, if the algorithm makes T quantum queries to S, and also receives an (untrusted) m-qubit quantum witness, then either m = Ω(|S|) or T = Ω(√{N/|S|}). This is optimal, matching the straightforward protocols where the witness is either empty, or specifies all the elements of S. As a corollary, this resolves the open problem of giving an oracle separation between SBP, the complexity class that captures approximate counting, and QMA. In our second result, we ask what if, in addition to a membership oracle for S, a quantum algorithm is also given "QSamples" - i.e., copies of the state |S⟩ = 1/√|S| ∑_{i ∈ S} |i⟩ - or even access to a unitary transformation that enables QSampling? We show that, even then, the algorithm needs either Θ(√{N/|S|}) queries or else Θ(min{|S|^{1/3},√{N/|S|}}) QSamples or accesses to the unitary. Our lower bounds in both settings make essential use of Laurent polynomials, but in different ways.
Cite as

Scott Aaronson, Robin Kothari, William Kretschmer, and Justin Thaler. Quantum Lower Bounds for Approximate Counting via Laurent Polynomials. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 7:1-7:47, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{aaronson_et_al:LIPIcs.CCC.2020.7,
  author =	{Aaronson, Scott and Kothari, Robin and Kretschmer, William and Thaler, Justin},
  title =	{{Quantum Lower Bounds for Approximate Counting via Laurent Polynomials}},
  booktitle =	{35th Computational Complexity Conference (CCC 2020)},
  pages =	{7:1--7:47},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-156-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{169},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.7},
  URN =		{urn:nbn:de:0030-drops-125593},
  doi =		{10.4230/LIPIcs.CCC.2020.7},
  annote =	{Keywords: Approximate counting, Laurent polynomials, QSampling, query complexity}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2020.82
Quantum Distributed Complexity of Set Disjointness on a Line

Authors: Frédéric Magniez and Ashwin Nayak

Published in: LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

Abstract
Given x,y ∈ {0,1}ⁿ, Set Disjointness consists in deciding whether x_i = y_i = 1 for some index i ∈ [n]. We study the problem of computing this function in a distributed computing scenario in which the inputs x and y are given to the processors at the two extremities of a path of length d. Each vertex of the path has a quantum processor that can communicate with each of its neighbours by exchanging O(log n) qubits per round. We are interested in the number of rounds required for computing Set Disjointness with constant probability bounded away from 1/2. We call this problem "Set Disjointness on a Line". Set Disjointness on a Line was introduced by Le Gall and Magniez [Le Gall and Magniez, 2018] for proving lower bounds on the quantum distributed complexity of computing the diameter of an arbitrary network in the CONGEST model. However, they were only able to provide a lower bound when the local memory used by the processors on the intermediate vertices of the path is severely limited. More precisely, their bound applies only when the local memory of each intermediate processor consists of O(log n) qubits. In this work, we prove an unconditional lower bound of Ω̃(∛{n d²} + √n) rounds for Set Disjointness on a Line with d + 1 processors. This is the first non-trivial lower bound when there is no restriction on the memory used by the processors. The result gives us a new lower bound of Ω̃ (∛{nδ²} + √n) on the number of rounds required for computing the diameter δ of any n-node network with quantum messages of size O(log n) in the CONGEST model. We draw a connection between the distributed computing scenario above and a new model of query complexity. In this model, an algorithm computing a bi-variate function f (such as Set Disjointness) has access to the inputs x and y through two separate oracles 𝒪_x and 𝒪_y, respectively. The restriction is that the algorithm is required to alternately make d queries to 𝒪_x and d queries to 𝒪_y, with input-independent computation in between queries. The model reflects a "switching delay" of d queries between a "round" of queries to x and the following "round" of queries to y. The technique we use for deriving the round lower bound for Set Disjointness on a Line also applies to this query model. We provide an algorithm for Set Disjointness in this query model with query complexity that matches the round lower bound stated above, up to a polylogarithmic factor. In this sense, the round lower bound we show for Set Disjointness on a Line is optimal.
Cite as

Frédéric Magniez and Ashwin Nayak. Quantum Distributed Complexity of Set Disjointness on a Line. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 82:1-82:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{magniez_et_al:LIPIcs.ICALP.2020.82,
  author =	{Magniez, Fr\'{e}d\'{e}ric and Nayak, Ashwin},
  title =	{{Quantum Distributed Complexity of Set Disjointness on a Line}},
  booktitle =	{47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)},
  pages =	{82:1--82:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-138-2},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{168},
  editor =	{Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.82},
  URN =		{urn:nbn:de:0030-drops-124892},
  doi =		{10.4230/LIPIcs.ICALP.2020.82},
  annote =	{Keywords: Quantum distributed computing, Set Disjointness, communication complexity, query complexity}
}
Document
DOI: 10.4230/LIPIcs.ESA.2019.16
Quantum Algorithms for Classical Probability Distributions

Authors: Aleksandrs Belovs

Published in: LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)

Abstract
We study quantum algorithms working on classical probability distributions. We formulate four different models for accessing a classical probability distribution on a quantum computer, which are derived from previous work on the topic, and study their mutual relationships. Additionally, we prove that quantum query complexity of distinguishing two probability distributions is given by their inverse Hellinger distance, which gives a quadratic improvement over classical query complexity for any pair of distributions. The results are obtained by using the adversary method for state-generating input oracles and for distinguishing probability distributions on input strings.
Cite as

Aleksandrs Belovs. Quantum Algorithms for Classical Probability Distributions. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 16:1-16:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{belovs:LIPIcs.ESA.2019.16,
  author =	{Belovs, Aleksandrs},
  title =	{{Quantum Algorithms for Classical Probability Distributions}},
  booktitle =	{27th Annual European Symposium on Algorithms (ESA 2019)},
  pages =	{16:1--16:11},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-124-5},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{144},
  editor =	{Bender, Michael A. and Svensson, Ola and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2019.16},
  URN =		{urn:nbn:de:0030-drops-111370},
  doi =		{10.4230/LIPIcs.ESA.2019.16},
  annote =	{Keywords: quantum query complexity, quantum adversary method, distinguishing probability distributions, Hellinger distance}
}
Document
DOI: 10.4230/LIPIcs.CCC.2017.23
Augmented Index and Quantum Streaming Algorithms for DYCK(2)

Authors: Ashwin Nayak and Dave Touchette

Published in: LIPIcs, Volume 79, 32nd Computational Complexity Conference (CCC 2017)

Abstract
We show how two recently developed quantum information theoretic tools can be applied to obtain lower bounds on quantum information complexity. We also develop new tools with potential for broader applicability, and use them to establish a lower bound on the quantum information complexity for the Augmented Index function on an easy distribution. This approach allows us to handle superpositions rather than distributions over inputs, the main technical challenge faced previously. By providing a quantum generalization of the argument of Jain and Nayak [IEEE TIT'14], we leverage this to obtain a lower bound on the space complexity of multi-pass, unidirectional quantum streaming algorithms for the DYCK(2) language.
Cite as

Ashwin Nayak and Dave Touchette. Augmented Index and Quantum Streaming Algorithms for DYCK(2). In 32nd Computational Complexity Conference (CCC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 79, pp. 23:1-23:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{nayak_et_al:LIPIcs.CCC.2017.23,
  author =	{Nayak, Ashwin and Touchette, Dave},
  title =	{{Augmented Index and  Quantum Streaming Algorithms for DYCK(2)}},
  booktitle =	{32nd Computational Complexity Conference (CCC 2017)},
  pages =	{23:1--23:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-040-8},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{79},
  editor =	{O'Donnell, Ryan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2017.23},
  URN =		{urn:nbn:de:0030-drops-75437},
  doi =		{10.4230/LIPIcs.CCC.2017.23},
  annote =	{Keywords: Quantum streaming algorithms, Space complexity, Quantum communication complexity, Quantum information cost, DYCK(2), Augmented Index}
}
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