41 Search Results for "Harrow, Aram"


Volume

LIPIcs, Volume 27

9th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2014)

TQC 2014, May 21-23, 2014, Singapore

Editors: Steven T. Flammia and Aram W. Harrow

Document
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.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
On the Power of Nonstandard Quantum Oracles

Authors: Roozbeh Bassirian, Bill Fefferman, and Kunal Marwaha

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


Abstract
We study how the choices made when designing an oracle affect the complexity of quantum property testing problems defined relative to this oracle. We encode a regular graph of even degree as an invertible function f, and present f in different oracle models. We first give a one-query QMA protocol to test if a graph encoded in f has a small disconnected subset. We then use representation theory to show that no classical witness can help a quantum verifier efficiently decide this problem relative to an in-place oracle. Perhaps surprisingly, a simple modification to the standard oracle prevents a quantum verifier from efficiently deciding this problem, even with access to an unbounded witness.

Cite as

Roozbeh Bassirian, Bill Fefferman, and Kunal Marwaha. On the Power of Nonstandard Quantum Oracles. In 18th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 266, pp. 11:1-11:25, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


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@InProceedings{bassirian_et_al:LIPIcs.TQC.2023.11,
  author =	{Bassirian, Roozbeh and Fefferman, Bill and Marwaha, Kunal},
  title =	{{On the Power of Nonstandard Quantum Oracles}},
  booktitle =	{18th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2023)},
  pages =	{11:1--11:25},
  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.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2023.11},
  URN =		{urn:nbn:de:0030-drops-183215},
  doi =		{10.4230/LIPIcs.TQC.2023.11},
  annote =	{Keywords: quantum complexity, QCMA, expander graphs, representation theory}
}
Document
A Distribution Testing Oracle Separating QMA and QCMA

Authors: Anand Natarajan and Chinmay Nirkhe

Published in: LIPIcs, Volume 264, 38th Computational Complexity Conference (CCC 2023)


Abstract
It is a long-standing open question in quantum complexity theory whether the definition of non-deterministic quantum computation requires quantum witnesses (QMA) or if classical witnesses suffice (QCMA). We make progress on this question by constructing a randomized classical oracle separating the respective computational complexity classes. Previous separations [Aaronson and Kuperberg, 2007; Bill Fefferman and Shelby Kimmel, 2018] required a quantum unitary oracle. The separating problem is deciding whether a distribution supported on regular un-directed graphs either consists of multiple connected components (yes instances) or consists of one expanding connected component (no instances) where the graph is given in an adjacency-list format by the oracle. Therefore, the oracle is a distribution over n-bit boolean functions.

Cite as

Anand Natarajan and Chinmay Nirkhe. A Distribution Testing Oracle Separating QMA and QCMA. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 22:1-22:27, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


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@InProceedings{natarajan_et_al:LIPIcs.CCC.2023.22,
  author =	{Natarajan, Anand and Nirkhe, Chinmay},
  title =	{{A Distribution Testing Oracle Separating QMA and QCMA}},
  booktitle =	{38th Computational Complexity Conference (CCC 2023)},
  pages =	{22:1--22:27},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-282-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{264},
  editor =	{Ta-Shma, Amnon},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.22},
  URN =		{urn:nbn:de:0030-drops-182928},
  doi =		{10.4230/LIPIcs.CCC.2023.22},
  annote =	{Keywords: quantum non-determinism, complexity theory}
}
Document
Quantum Search-To-Decision Reductions and the State Synthesis Problem

Authors: Sandy Irani, Anand Natarajan, Chinmay Nirkhe, Sujit Rao, and Henry Yuen

Published in: LIPIcs, Volume 234, 37th Computational Complexity Conference (CCC 2022)


Abstract
It is a useful fact in classical computer science that many search problems are reducible to decision problems; this has led to decision problems being regarded as the de facto computational task to study in complexity theory. In this work, we explore search-to-decision reductions for quantum search problems, wherein a quantum algorithm makes queries to a classical decision oracle to output a desired quantum state. In particular, we focus on search-to-decision reductions for QMA, and show that there exists a quantum polynomial-time algorithm that can generate a witness for a QMA problem up to inverse polynomial precision by making one query to a PP decision oracle. We complement this result by showing that QMA-search does not reduce to QMA-decision in polynomial-time, relative to a quantum oracle. We also explore the more general state synthesis problem, in which the goal is to efficiently synthesize a target state by making queries to a classical oracle encoding the state. We prove that there exists a classical oracle with which any quantum state can be synthesized to inverse polynomial precision using only one oracle query and to inverse exponential precision using two oracle queries. This answers an open question of Aaronson [Aaronson, 2016], who presented a state synthesis algorithm that makes O(n) queries to a classical oracle to prepare an n-qubit state, and asked if the query complexity could be made sublinear.

Cite as

Sandy Irani, Anand Natarajan, Chinmay Nirkhe, Sujit Rao, and Henry Yuen. Quantum Search-To-Decision Reductions and the State Synthesis Problem. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 5:1-5:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


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@InProceedings{irani_et_al:LIPIcs.CCC.2022.5,
  author =	{Irani, Sandy and Natarajan, Anand and Nirkhe, Chinmay and Rao, Sujit and Yuen, Henry},
  title =	{{Quantum Search-To-Decision Reductions and the State Synthesis Problem}},
  booktitle =	{37th Computational Complexity Conference (CCC 2022)},
  pages =	{5:1--5:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-241-9},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{234},
  editor =	{Lovett, Shachar},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.5},
  URN =		{urn:nbn:de:0030-drops-165674},
  doi =		{10.4230/LIPIcs.CCC.2022.5},
  annote =	{Keywords: Search-to-decision, state synthesis, quantum computing}
}
Document
Robust Quantum Entanglement at (Nearly) Room Temperature

Authors: Lior Eldar

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


Abstract
We formulate an average-case analog of the NLTS conjecture of Freedman and Hastings (QIC 2014) by asking whether there exist topologically ordered systems with corresponding local Hamiltonians for which the thermal Gibbs state for constant temperature cannot even be approximated by shallow quantum circuits. We then prove this conjecture for nearly optimal parameters: we construct a quantum error correcting code whose corresponding (log) local Hamiltonian has the following property: for nearly constant temperature (temperature decays as 1/log²log(n)) the thermal Gibbs state of that Hamiltonian cannot be approximated by any circuit of depth less than log(n), and it is highly entangled in a well-defined way. This implies that appropriately chosen local Hamiltonians can give rise to ground-state long-range entanglement which can survive without active error correction at temperatures which are nearly independent of the system size: thereby improving exponentially upon previously known bounds.

Cite as

Lior Eldar. Robust Quantum Entanglement at (Nearly) Room Temperature. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 49:1-49:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)


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@InProceedings{eldar:LIPIcs.ITCS.2021.49,
  author =	{Eldar, Lior},
  title =	{{Robust Quantum Entanglement at (Nearly) Room Temperature}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{49:1--49:20},
  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.2021.49},
  URN =		{urn:nbn:de:0030-drops-135886},
  doi =		{10.4230/LIPIcs.ITCS.2021.49},
  annote =	{Keywords: Quantum error-correcting codes, Quantum Entanglement, Quantum Locally-Testable Codes, Local Hamiltonians, quantum PCP, NLTS}
}
Document
Time-Space Lower Bounds for Simulating Proof Systems with Quantum and Randomized Verifiers

Authors: Abhijit S. Mudigonda and R. Ryan Williams

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


Abstract
A line of work initiated by Fortnow in 1997 has proven model-independent time-space lower bounds for the SAT problem and related problems within the polynomial-time hierarchy. For example, for the SAT problem, the state-of-the-art is that the problem cannot be solved by random-access machines in n^c time and n^o(1) space simultaneously for c < 2cos(π/7) ≈ 1.801. We extend this lower bound approach to the quantum and randomized domains. Combining Grover’s algorithm with components from SAT time-space lower bounds, we show that there are problems verifiable in O(n) time with quantum Merlin-Arthur protocols that cannot be solved in n^c time and n^o(1) space simultaneously for c < (3+√3)/2 ≈ 2.366, a super-quadratic time lower bound. This result and the prior work on SAT can both be viewed as consequences of a more general formula for time lower bounds against small-space algorithms, whose asymptotics we study in full. We also show lower bounds against randomized algorithms: there are problems verifiable in O(n) time with (classical) Merlin-Arthur protocols that cannot be solved in n^c randomized time and O(log n) space simultaneously for c < 1.465, improving a result of Diehl. For quantum Merlin-Arthur protocols, the lower bound in this setting can be improved to c < 1.5.

Cite as

Abhijit S. Mudigonda and R. Ryan Williams. Time-Space Lower Bounds for Simulating Proof Systems with Quantum and Randomized Verifiers. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 50:1-50:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)


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@InProceedings{mudigonda_et_al:LIPIcs.ITCS.2021.50,
  author =	{Mudigonda, Abhijit S. and Williams, R. Ryan},
  title =	{{Time-Space Lower Bounds for Simulating Proof Systems with Quantum and Randomized Verifiers}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{50:1--50:20},
  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.2021.50},
  URN =		{urn:nbn:de:0030-drops-135897},
  doi =		{10.4230/LIPIcs.ITCS.2021.50},
  annote =	{Keywords: Time-space tradeoffs, lower bounds, QMA}
}
Document
Universality of EPR Pairs in Entanglement-Assisted Communication Complexity, and the Communication Cost of State Conversion

Authors: Matthew Coudron and Aram W. Harrow

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


Abstract
In this work we consider the role of entanglement assistance in quantum communication protocols, focusing, in particular, on whether the type of shared entangled state can affect the quantum communication complexity of a function. This question is interesting because in some other settings in quantum information, such as non-local games, or tasks that involve quantum communication between players and referee, or simulating bipartite unitaries or communication channels, maximally entangled states are known to be less useful as a resource than some partially entangled states. By contrast, we prove that the bounded-error entanglement-assisted quantum communication complexity of a partial or total function cannot be improved by more than a constant factor by replacing maximally entangled states with arbitrary entangled states. In particular, we show that every quantum communication protocol using Q qubits of communication and arbitrary shared entanglement can be epsilon-approximated by a protocol using O(Q/epsilon+log(1/epsilon)/epsilon) qubits of communication and only EPR pairs as shared entanglement. This conclusion is opposite of the common wisdom in the study of non-local games, where it has been shown, for example, that the I3322 inequality has a non-local strategy using a non-maximally entangled state, which surpasses the winning probability achievable by any strategy using a maximally entangled state of any dimension [Vidick and Wehner, 2011]. We leave open the question of how much the use of a shared maximally entangled state can reduce the quantum communication complexity of a function. Our second result concerns an old question in quantum information theory: How much quantum communication is required to approximately convert one pure bipartite entangled state into another? We give simple and efficiently computable upper and lower bounds. Given two bipartite states |chi> and |upsilon>, we define a natural quantity, d_{infty}(|chi>, |upsilon>), which we call the l_{infty} Earth Mover’s distance, and we show that the communication cost of converting between |chi> and |upsilon> is upper bounded by a constant multiple of d_{infty}(|chi>, |upsilon>). Here d_{infty}(|chi>, |upsilon>) may be informally described as the minimum over all transports between the log of the Schmidt coefficients of |chi> and those of |upsilon>, of the maximum distance that any amount of mass must be moved in that transport. A precise definition is given in the introduction. Furthermore, we prove a complementary lower bound on the cost of state conversion by the epsilon-Smoothed l_{infty}-Earth Mover’s Distance, which is a natural smoothing of the l_{infty}-Earth Mover’s Distance that we will define via a connection with optimal transport theory.

Cite as

Matthew Coudron and Aram W. Harrow. Universality of EPR Pairs in Entanglement-Assisted Communication Complexity, and the Communication Cost of State Conversion. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 20:1-20:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{coudron_et_al:LIPIcs.CCC.2019.20,
  author =	{Coudron, Matthew and Harrow, Aram W.},
  title =	{{Universality of EPR Pairs in Entanglement-Assisted Communication Complexity, and the Communication Cost of State Conversion}},
  booktitle =	{34th Computational Complexity Conference (CCC 2019)},
  pages =	{20:1--20:25},
  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.2019.20},
  URN =		{urn:nbn:de:0030-drops-108421},
  doi =		{10.4230/LIPIcs.CCC.2019.20},
  annote =	{Keywords: Entanglement, quantum communication complexity}
}
Document
Size-Degree Trade-Offs for Sums-of-Squares and Positivstellensatz Proofs

Authors: Albert Atserias and Tuomas Hakoniemi

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


Abstract
We show that if a system of degree-k polynomial constraints on n Boolean variables has a Sums-of-Squares (SOS) proof of unsatisfiability with at most s many monomials, then it also has one whose degree is of the order of the square root of n log s plus k. A similar statement holds for the more general Positivstellensatz (PS) proofs. This establishes size-degree trade-offs for SOS and PS that match their analogues for weaker proof systems such as Resolution, Polynomial Calculus, and the proof systems for the LP and SDP hierarchies of Lovász and Schrijver. As a corollary to this, and to the known degree lower bounds, we get optimal integrality gaps for exponential size SOS proofs for sparse random instances of the standard NP-hard constraint optimization problems. We also get exponential size SOS lower bounds for Tseitin and Knapsack formulas. The proof of our main result relies on a zero-gap duality theorem for pre-ordered vector spaces that admit an order unit, whose specialization to PS and SOS may be of independent interest.

Cite as

Albert Atserias and Tuomas Hakoniemi. Size-Degree Trade-Offs for Sums-of-Squares and Positivstellensatz Proofs. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 24:1-24:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{atserias_et_al:LIPIcs.CCC.2019.24,
  author =	{Atserias, Albert and Hakoniemi, Tuomas},
  title =	{{Size-Degree Trade-Offs for Sums-of-Squares and Positivstellensatz Proofs}},
  booktitle =	{34th Computational Complexity Conference (CCC 2019)},
  pages =	{24:1--24:20},
  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.2019.24},
  URN =		{urn:nbn:de:0030-drops-108464},
  doi =		{10.4230/LIPIcs.CCC.2019.24},
  annote =	{Keywords: Proof complexity, semialgebraic proof systems, Sums-of-Squares, Positivstellensatz, trade-offs, lower bounds, monomial size, degree}
}
Document
Average-Case Quantum Advantage with Shallow Circuits

Authors: François Le Gall

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


Abstract
Recently Bravyi, Gosset and König (Science 2018) proved an unconditional separation between the computational powers of small-depth quantum and classical circuits for a relation. In this paper we show a similar separation in the average-case setting that gives stronger evidence of the superiority of small-depth quantum computation: we construct a computational task that can be solved on all inputs by a quantum circuit of constant depth with bounded-fanin gates (a "shallow" quantum circuit) and show that any classical circuit with bounded-fanin gates solving this problem on a non-negligible fraction of the inputs must have logarithmic depth. Our results are obtained by introducing a technique to create quantum states exhibiting global quantum correlations from any graph, via a construction that we call the extended graph. Similar results have been very recently (and independently) obtained by Coudron, Stark and Vidick (arXiv:1810.04233}), and Bene Watts, Kothari, Schaeffer and Tal (STOC 2019).

Cite as

François Le Gall. Average-Case Quantum Advantage with Shallow Circuits. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 21:1-21:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{legall:LIPIcs.CCC.2019.21,
  author =	{Le Gall, Fran\c{c}ois},
  title =	{{Average-Case Quantum Advantage with Shallow Circuits}},
  booktitle =	{34th Computational Complexity Conference (CCC 2019)},
  pages =	{21:1--21:20},
  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.2019.21},
  URN =		{urn:nbn:de:0030-drops-108432},
  doi =		{10.4230/LIPIcs.CCC.2019.21},
  annote =	{Keywords: Quantum computing, circuit complexity, constant-depth circuits}
}
Document
On the Qubit Routing Problem

Authors: Alexander Cowtan, Silas Dilkes, Ross Duncan, Alexandre Krajenbrink, Will Simmons, and Seyon Sivarajah

Published in: LIPIcs, Volume 135, 14th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2019)


Abstract
We introduce a new architecture-agnostic methodology for mapping abstract quantum circuits to realistic quantum computing devices with restricted qubit connectivity, as implemented by Cambridge Quantum Computing’s t|ket> compiler. We present empirical results showing the effectiveness of this method in terms of reducing two-qubit gate depth and two-qubit gate count, compared to other implementations.

Cite as

Alexander Cowtan, Silas Dilkes, Ross Duncan, Alexandre Krajenbrink, Will Simmons, and Seyon Sivarajah. On the Qubit Routing Problem. In 14th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 135, pp. 5:1-5:32, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{cowtan_et_al:LIPIcs.TQC.2019.5,
  author =	{Cowtan, Alexander and Dilkes, Silas and Duncan, Ross and Krajenbrink, Alexandre and Simmons, Will and Sivarajah, Seyon},
  title =	{{On the Qubit Routing Problem}},
  booktitle =	{14th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2019)},
  pages =	{5:1--5:32},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-112-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{135},
  editor =	{van Dam, Wim and Man\v{c}inska, Laura},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2019.5},
  URN =		{urn:nbn:de:0030-drops-103972},
  doi =		{10.4230/LIPIcs.TQC.2019.5},
  annote =	{Keywords: Quantum Computing, Qubit routing, Compilation}
}
Document
A Compressed Classical Description of Quantum States

Authors: David Gosset and John Smolin

Published in: LIPIcs, Volume 135, 14th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2019)


Abstract
We show how to approximately represent a quantum state using the square root of the usual amount of classical memory. The classical representation of an n-qubit state psi consists of its inner products with O(sqrt{2^n}) stabilizer states. A quantum state initially specified by its 2^n entries in the computational basis can be compressed to this form in time O(2^n poly(n)), and, subsequently, the compressed description can be used to additively approximate the expectation value of an arbitrary observable. Our compression scheme directly gives a new protocol for the vector in subspace problem with randomized one-way communication complexity that matches (up to polylogarithmic factors) the optimal upper bound, due to Raz. We obtain an exponential improvement over Raz’s protocol in terms of computational efficiency.

Cite as

David Gosset and John Smolin. A Compressed Classical Description of Quantum States. In 14th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 135, pp. 8:1-8:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{gosset_et_al:LIPIcs.TQC.2019.8,
  author =	{Gosset, David and Smolin, John},
  title =	{{A Compressed Classical Description of Quantum States}},
  booktitle =	{14th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2019)},
  pages =	{8:1--8:9},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-112-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{135},
  editor =	{van Dam, Wim and Man\v{c}inska, Laura},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2019.8},
  URN =		{urn:nbn:de:0030-drops-104005},
  doi =		{10.4230/LIPIcs.TQC.2019.8},
  annote =	{Keywords: Quantum computation, Quantum communication complexity, Classical simulation}
}
Document
Algorithms, Bounds, and Strategies for Entangled XOR Games

Authors: Adam Bene Watts, Aram W. Harrow, Gurtej Kanwar, and Anand Natarajan

Published in: LIPIcs, Volume 124, 10th Innovations in Theoretical Computer Science Conference (ITCS 2019)


Abstract
Entangled games are a quantum analog of constraint satisfaction problems and have had important applications to quantum complexity theory, quantum cryptography, and the foundations of quantum mechanics. Given a game, the basic computational problem is to compute its entangled value: the supremum success probability attainable by a quantum strategy. We study the complexity of computing the (commuting-operator) entangled value omega^* of entangled XOR games with any number of players. Based on a duality theory for systems of operator equations, we introduce necessary and sufficient criteria for an XOR game to have omega^* = 1, and use these criteria to derive the following results: 1) An algorithm for symmetric games that decides in polynomial time whether omega^* = 1 or omega^* < 1, a task that was not previously known to be decidable, together with a simple tensor-product strategy that achieves value 1 in the former case. The only previous candidate algorithm for this problem was the Navascués-Pironio-Acín (also known as noncommutative Sum of Squares or ncSoS) hierarchy, but no convergence bounds were known. 2) A family of games with three players and with omega^* < 1, where it takes doubly exponential time for the ncSoS algorithm to witness this. By contrast, our algorithm runs in polynomial time. 3) Existence of an unsatisfiable phase for random (non-symmetric) XOR games. We show that there exists a constant C_k^{unsat} depending only on the number k of players, such that a random k-XOR game over an alphabet of size n has omega^* < 1 with high probability when the number of clauses is above C_k^{unsat} n. 4) A lower bound of Omega(n log(n)/log log(n)) on the number of levels in the ncSoS hierarchy required to detect unsatisfiability for most random 3-XOR games. This is in contrast with the classical case where the (3n)^{th} level of the sum-of-squares hierarchy is equivalent to brute-force enumeration of all possible solutions.

Cite as

Adam Bene Watts, Aram W. Harrow, Gurtej Kanwar, and Anand Natarajan. Algorithms, Bounds, and Strategies for Entangled XOR Games. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 10:1-10:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)


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@InProceedings{benewatts_et_al:LIPIcs.ITCS.2019.10,
  author =	{Bene Watts, Adam and Harrow, Aram W. and Kanwar, Gurtej and Natarajan, Anand},
  title =	{{Algorithms, Bounds, and Strategies for Entangled XOR Games}},
  booktitle =	{10th Innovations in Theoretical Computer Science Conference (ITCS 2019)},
  pages =	{10:1--10:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-095-8},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{124},
  editor =	{Blum, Avrim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2019.10},
  URN =		{urn:nbn:de:0030-drops-101032},
  doi =		{10.4230/LIPIcs.ITCS.2019.10},
  annote =	{Keywords: Nonlocal games, XOR Games, Pseudotelepathy games, Multipartite entanglement}
}
Document
Lower Bound on Expected Communication Cost of Quantum Huffman Coding

Authors: Anurag Anshu, Ankit Garg, Aram W. Harrow, and Penghui Yao

Published in: LIPIcs, Volume 61, 11th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2016)


Abstract
Data compression is a fundamental problem in quantum and classical information theory. A typical version of the problem is that the sender Alice receives a (classical or quantum) state from some known ensemble and needs to transmit them to the receiver Bob with average error below some specified bound. We consider the case in which the message can have a variable length and the goal is to minimize its expected length. For classical messages this problem has a well-known solution given by Huffman coding. In this scheme, the expected length of the message is equal to the Shannon entropy of the source (with a constant additive factor) and the scheme succeeds with zero error. This is a single-shot result which implies the asymptotic result, viz. Shannon's source coding theorem, by encoding each state sequentially. For the quantum case, the asymptotic compression rate is given by the von-Neumann entropy. However, we show that there is no one-shot scheme which is able to match this rate, even if interactive communication is allowed. This is a relatively rare case in quantum information theory when the cost of a quantum task is significantly different than the classical analogue. Our result has implications for direct sum theorems in quantum communication complexity and one-shot formulations of Quantum Reverse Shannon theorem.

Cite as

Anurag Anshu, Ankit Garg, Aram W. Harrow, and Penghui Yao. Lower Bound on Expected Communication Cost of Quantum Huffman Coding. In 11th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 61, pp. 3:1-3:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)


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@InProceedings{anshu_et_al:LIPIcs.TQC.2016.3,
  author =	{Anshu, Anurag and Garg, Ankit and Harrow, Aram W. and Yao, Penghui},
  title =	{{Lower Bound on Expected Communication Cost of Quantum Huffman Coding}},
  booktitle =	{11th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2016)},
  pages =	{3:1--3:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-019-4},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{61},
  editor =	{Broadbent, Anne},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2016.3},
  URN =		{urn:nbn:de:0030-drops-66843},
  doi =		{10.4230/LIPIcs.TQC.2016.3},
  annote =	{Keywords: Quantum information, quantum communication, expected communica- tion cost, huffman coding}
}
Document
Tight SoS-Degree Bounds for Approximate Nash Equilibria

Authors: Aram Harrow, Anand V. Natarajan, and Xiaodi Wu

Published in: LIPIcs, Volume 50, 31st Conference on Computational Complexity (CCC 2016)


Abstract
Nash equilibria always exist, but are widely conjectured to require time to find that is exponential in the number of strategies, even for two-player games. By contrast, a simple quasi-polynomial time algorithm, due to Lipton, Markakis and Mehta (LMM), can find approximate Nash equilibria, in which no player can improve their utility by more than epsilon by changing their strategy. The LMM algorithm can also be used to find an approximate Nash equilibrium with near-maximal total welfare. Matching hardness results for this optimization problem re found assuming the hardness of the planted-clique problem (by Hazan and Krauthgamer) and assuming the Exponential Time Hypothesis (by Braverman, Ko and Weinstein). In this paper we consider the application of the sum-squares (SoS) algorithm from convex optimization to the problem of optimizing over Nash equilibria. We show the first unconditional lower bounds on the number of levels of SoS needed to achieve a constant factor approximation to this problem. While it may seem that Nash equilibria do not naturally lend themselves to convex optimization, we also describe a simple LP (linear programming) hierarchy that can find an approximate Nash equilibrium in time comparable to that of the LMM algorithm, although neither algorithm is obviously a generalization of the other. This LP can be viewed as arising from the SoS algorithm at log(n) levels - matching our lower bounds. The lower bounds involve a modification of the Braverman-Ko-Weinstein embedding of CSPs into strategic games and techniques from sum-of-squares proof systems. The upper bound (i.e. analysis of the LP) uses information-theory techniques that have been recently applied to other linear- and semidefinite-programming hierarchies.

Cite as

Aram Harrow, Anand V. Natarajan, and Xiaodi Wu. Tight SoS-Degree Bounds for Approximate Nash Equilibria. In 31st Conference on Computational Complexity (CCC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 50, pp. 22:1-22:25, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)


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@InProceedings{harrow_et_al:LIPIcs.CCC.2016.22,
  author =	{Harrow, Aram and Natarajan, Anand V. and Wu, Xiaodi},
  title =	{{Tight SoS-Degree Bounds for Approximate Nash Equilibria}},
  booktitle =	{31st Conference on Computational Complexity (CCC 2016)},
  pages =	{22:1--22:25},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-008-8},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{50},
  editor =	{Raz, Ran},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2016.22},
  URN =		{urn:nbn:de:0030-drops-58565},
  doi =		{10.4230/LIPIcs.CCC.2016.22},
  annote =	{Keywords: Approximate Nash Equilibrium, Sum of Squares, LP, SDP}
}
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