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Documents authored by Bravyi, Sergey

Document
DOI: 10.4230/LIPIcs.TQC.2023.3
Optimal Algorithms for Learning Quantum Phase States

Authors: Srinivasan Arunachalam, Sergey Bravyi, Arkopal Dutt, and Theodore J. Yoder

Published in: LIPIcs, Volume 266, 18th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2023)

Abstract
We analyze the complexity of learning n-qubit quantum phase states. A degree-d phase state is defined as a superposition of all 2ⁿ basis vectors x with amplitudes proportional to (-1)^{f(x)}, where f is a degree-d Boolean polynomial over n variables. We show that the sample complexity of learning an unknown degree-d phase state is Θ(n^d) if we allow separable measurements and Θ(n^{d-1}) if we allow entangled measurements. Our learning algorithm based on separable measurements has runtime poly(n) (for constant d) and is well-suited for near-term demonstrations as it requires only single-qubit measurements in the Pauli X and Z bases. We show similar bounds on the sample complexity for learning generalized phase states with complex-valued amplitudes. We further consider learning phase states when f has sparsity-s, degree-d in its 𝔽₂ representation (with sample complexity O(2^d sn)), f has Fourier-degree-t (with sample complexity O(2^{2t})), and learning quadratic phase states with ε-global depolarizing noise (with sample complexity O(n^{1+ε})). These learning algorithms give us a procedure to learn the diagonal unitaries of the Clifford hierarchy and IQP circuits.
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Srinivasan Arunachalam, Sergey Bravyi, Arkopal Dutt, and Theodore J. Yoder. Optimal Algorithms for Learning Quantum Phase States. In 18th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 266, pp. 3:1-3:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{arunachalam_et_al:LIPIcs.TQC.2023.3,
  author =	{Arunachalam, Srinivasan and Bravyi, Sergey and Dutt, Arkopal and Yoder, Theodore J.},
  title =	{{Optimal Algorithms for Learning Quantum Phase States}},
  booktitle =	{18th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2023)},
  pages =	{3:1--3:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-283-9},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{266},
  editor =	{Fawzi, Omar and Walter, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2023.3},
  URN =		{urn:nbn:de:0030-drops-183139},
  doi =		{10.4230/LIPIcs.TQC.2023.3},
  annote =	{Keywords: Tomography, binary phase states, generalized phase states, IQP circuits}
}
Document
DOI: 10.4230/LIPIcs.TQC.2022.3
The Parametrized Complexity of Quantum Verification

Authors: Srinivasan Arunachalam, Sergey Bravyi, Chinmay Nirkhe, and Bryan O'Gorman

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

Abstract
We initiate the study of parameterized complexity of QMA problems in terms of the number of non-Clifford gates in the problem description. We show that for the problem of parameterized quantum circuit satisfiability, there exists a classical algorithm solving the problem with a runtime scaling exponentially in the number of non-Clifford gates but only polynomially with the system size. This result follows from our main result, that for any Clifford + t T-gate quantum circuit satisfiability problem, the search space of optimal witnesses can be reduced to a stabilizer subspace isomorphic to at most t qubits (independent of the system size). Furthermore, we derive new lower bounds on the T-count of circuit satisfiability instances and the T-count of the W-state assuming the classical exponential time hypothesis (ETH). Lastly, we explore the parameterized complexity of the quantum non-identity check problem.
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Srinivasan Arunachalam, Sergey Bravyi, Chinmay Nirkhe, and Bryan O'Gorman. The Parametrized Complexity of Quantum Verification. In 17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 232, pp. 3:1-3:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{arunachalam_et_al:LIPIcs.TQC.2022.3,
  author =	{Arunachalam, Srinivasan and Bravyi, Sergey and Nirkhe, Chinmay and O'Gorman, Bryan},
  title =	{{The Parametrized Complexity of Quantum Verification}},
  booktitle =	{17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022)},
  pages =	{3:1--3:18},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2022.3},
  URN =		{urn:nbn:de:0030-drops-165104},
  doi =		{10.4230/LIPIcs.TQC.2022.3},
  annote =	{Keywords: parametrized complexity, quantum verification, QMA}
}
Document
DOI: 10.4230/LIPIcs.STACS.2010.2450
Quantum Algorithms for Testing Properties of Distributions

Authors: Sergey Bravyi, Aram W. Harrow, and Avinatan Hassidim

Published in: LIPIcs, Volume 5, 27th International Symposium on Theoretical Aspects of Computer Science (2010)

Abstract
Suppose one has access to oracles generating samples from two unknown probability distributions $p$ and $q$ on some $N$-element set. How many samples does one need to test whether the two distributions are close or far from each other in the $L_1$-norm? This and related questions have been extensively studied during the last years in the field of property testing. In the present paper we study quantum algorithms for testing properties of distributions. It is shown that the $L_1$-distance $\|p-q\|_1$ can be estimated with a constant precision using only $O(N^{1/2})$ queries in the quantum settings, whereas classical computers need $\Omega(N^{1-o(1)})$ queries. We also describe quantum algorithms for testing Uniformity and Orthogonality with query complexity $O(N^{1/3})$. The classical query complexity of these problems is known to be $\Omega(N^{1/2})$. A quantum algorithm for testing Uniformity has been recently independently discovered by Chakraborty et al. \cite{CFMW09}.
Cite as

Sergey Bravyi, Aram W. Harrow, and Avinatan Hassidim. Quantum Algorithms for Testing Properties of Distributions. In 27th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 5, pp. 131-142, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{bravyi_et_al:LIPIcs.STACS.2010.2450,
  author =	{Bravyi, Sergey and Harrow, Aram W. and Hassidim, Avinatan},
  title =	{{Quantum Algorithms for Testing Properties of Distributions}},
  booktitle =	{27th International Symposium on Theoretical Aspects of Computer Science},
  pages =	{131--142},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-16-3},
  ISSN =	{1868-8969},
  year =	{2010},
  volume =	{5},
  editor =	{Marion, Jean-Yves and Schwentick, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2010.2450},
  URN =		{urn:nbn:de:0030-drops-24502},
  doi =		{10.4230/LIPIcs.STACS.2010.2450},
  annote =	{Keywords: Quantum computing, property testing, sampling}
}
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