11 Search Results for "Montanaro, Ashley"

Document
DOI: 10.4230/LIPIcs.ITCS.2024.1
A Qubit, a Coin, and an Advice String Walk into a Relational Problem

Authors: Scott Aaronson, Harry Buhrman, and William Kretschmer

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

Abstract
Relational problems (those with many possible valid outputs) are different from decision problems, but it is easy to forget just how different. This paper initiates the study of FBQP/qpoly, the class of relational problems solvable in quantum polynomial-time with the help of polynomial-sized quantum advice, along with its analogues for deterministic and randomized computation (FP, FBPP) and advice (/poly, /rpoly). Our first result is that FBQP/qpoly ≠ FBQP/poly, unconditionally, with no oracle - a striking contrast with what we know about the analogous decision classes. The proof repurposes the separation between quantum and classical one-way communication complexities due to Bar-Yossef, Jayram, and Kerenidis. We discuss how this separation raises the prospect of near-term experiments to demonstrate "quantum information supremacy," a form of quantum supremacy that would not depend on unproved complexity assumptions. Our second result is that FBPP ̸ ⊂ FP/poly - that is, Adleman’s Theorem fails for relational problems - unless PSPACE ⊂ NP/poly. Our proof uses IP = PSPACE and time-bounded Kolmogorov complexity. On the other hand, we show that proving FBPP ̸ ⊂ FP/poly will be hard, as it implies a superpolynomial circuit lower bound for PromiseBPEXP. We prove the following further results: - Unconditionally, FP ≠ FBPP and FP/poly ≠ FBPP/poly (even when these classes are carefully defined). - FBPP/poly = FBPP/rpoly (and likewise for FBQP). For sampling problems, by contrast, SampBPP/poly ≠ SampBPP/rpoly (and likewise for SampBQP).
Cite as

Scott Aaronson, Harry Buhrman, and William Kretschmer. A Qubit, a Coin, and an Advice String Walk into a Relational Problem. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 1:1-1:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{aaronson_et_al:LIPIcs.ITCS.2024.1,
  author =	{Aaronson, Scott and Buhrman, Harry and Kretschmer, William},
  title =	{{A Qubit, a Coin, and an Advice String Walk into a Relational Problem}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{1:1--1:24},
  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.2024.1},
  URN =		{urn:nbn:de:0030-drops-195290},
  doi =		{10.4230/LIPIcs.ITCS.2024.1},
  annote =	{Keywords: Relational problems, quantum advice, randomized advice, FBQP, FBPP}
}
Document
DOI: 10.4230/LIPIcs.CCC.2023.34
The Optimal Depth of Variational Quantum Algorithms Is QCMA-Hard to Approximate

Authors: Lennart Bittel, Sevag Gharibian, and Martin Kliesch

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

Abstract
Variational Quantum Algorithms (VQAs), such as the Quantum Approximate Optimization Algorithm (QAOA) of [Farhi, Goldstone, Gutmann, 2014], have seen intense study towards near-term applications on quantum hardware. A crucial parameter for VQAs is the depth of the variational ansatz used - the smaller the depth, the more amenable the ansatz is to near-term quantum hardware in that it gives the circuit a chance to be fully executed before the system decoheres. In this work, we show that approximating the optimal depth for a given VQA ansatz is intractable. Formally, we show that for any constant ε > 0, it is QCMA-hard to approximate the optimal depth of a VQA ansatz within multiplicative factor N^(1-ε), for N denoting the encoding size of the VQA instance. (Here, Quantum Classical Merlin-Arthur (QCMA) is a quantum generalization of NP.) We then show that this hardness persists in the even "simpler" QAOA-type settings. To our knowledge, this yields the first natural QCMA-hard-to-approximate problems.
Cite as

Lennart Bittel, Sevag Gharibian, and Martin Kliesch. The Optimal Depth of Variational Quantum Algorithms Is QCMA-Hard to Approximate. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 34:1-34:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

Copy BibTex To Clipboard

@InProceedings{bittel_et_al:LIPIcs.CCC.2023.34,
  author =	{Bittel, Lennart and Gharibian, Sevag and Kliesch, Martin},
  title =	{{The Optimal Depth of Variational Quantum Algorithms Is QCMA-Hard to Approximate}},
  booktitle =	{38th Computational Complexity Conference (CCC 2023)},
  pages =	{34:1--34:24},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.34},
  URN =		{urn:nbn:de:0030-drops-183045},
  doi =		{10.4230/LIPIcs.CCC.2023.34},
  annote =	{Keywords: Variational quantum algorithms (VQA), Quantum Approximate Optimization Algorithm (QAOA), circuit depth minimization, Quantum-Classical Merlin-Arthur (QCMA), hardness of approximation, hybrid quantum algorithms}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2023.29
Quantum Majority Vote

Authors: Harry Buhrman, Noah Linden, Laura Mančinska, Ashley Montanaro, and Maris Ozols

Published in: LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

Abstract
Majority vote is a basic method for amplifying correct outcomes that is widely used in computer science and beyond. While it can amplify the correctness of a quantum device with classical output, the analogous procedure for quantum output is not known. We introduce quantum majority vote as the following task: given a product state |ψ_1⟩ ⊗ … ⊗ |ψ_n⟩ where each qubit is in one of two orthogonal states |ψ⟩ or |ψ^⟂⟩, output the majority state. We show that an optimal algorithm for this problem achieves worst-case fidelity of 1/2 + Θ(1/√n). Under the promise that at least 2/3 of the input qubits are in the majority state, the fidelity increases to 1 - Θ(1/n) and approaches 1 as n increases. We also consider the more general problem of computing any symmetric and equivariant Boolean function f: {0,1}ⁿ → {0,1} in an unknown quantum basis, and show that a generalization of our quantum majority vote algorithm is optimal for this task. The optimal parameters for the generalized algorithm and its worst-case fidelity can be determined by a simple linear program of size O(n). The time complexity of the algorithm is O(n⁴ log n) where n is the number of input qubits.
Cite as

Harry Buhrman, Noah Linden, Laura Mančinska, Ashley Montanaro, and Maris Ozols. Quantum Majority Vote. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, p. 29:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

Copy BibTex To Clipboard

@InProceedings{buhrman_et_al:LIPIcs.ITCS.2023.29,
  author =	{Buhrman, Harry and Linden, Noah and Man\v{c}inska, Laura and Montanaro, Ashley and Ozols, Maris},
  title =	{{Quantum Majority Vote}},
  booktitle =	{14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
  pages =	{29:1--29:1},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-263-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{251},
  editor =	{Tauman Kalai, Yael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.29},
  URN =		{urn:nbn:de:0030-drops-175321},
  doi =		{10.4230/LIPIcs.ITCS.2023.29},
  annote =	{Keywords: quantum algorithms, quantum majority vote, Schur-Weyl duality}
}
Document
DOI: 10.4230/LIPIcs.TQC.2022.1
Quantum Algorithms for Learning a Hidden Graph

Authors: Ashley Montanaro and Changpeng Shao

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

Abstract
We study the problem of learning an unknown graph provided via an oracle using a quantum algorithm. We consider three query models. In the first model ("OR queries"), the oracle returns whether a given subset of the vertices contains any edges. In the second ("parity queries"), the oracle returns the parity of the number of edges in a subset. In the third model, we are given copies of the graph state corresponding to the graph. We give quantum algorithms that achieve speedups over the best possible classical algorithms in the OR and parity query models, for some families of graphs, and give quantum algorithms in the graph state model whose complexity is similar to the parity query model. For some parameter regimes, the speedups can be exponential in the parity query model. On the other hand, without any promise on the graph, no speedup is possible in the OR query model. A main technique we use is the quantum algorithm for solving the combinatorial group testing problem, for which a query-efficient quantum algorithm was given by Belovs. Here we additionally give a time-efficient quantum algorithm for this problem, based on the algorithm of Ambainis et al. for a "gapped" version of the group testing problem.
Cite as

Ashley Montanaro and Changpeng Shao. Quantum Algorithms for Learning a Hidden Graph. In 17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 232, pp. 1:1-1:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

Copy BibTex To Clipboard

@InProceedings{montanaro_et_al:LIPIcs.TQC.2022.1,
  author =	{Montanaro, Ashley and Shao, Changpeng},
  title =	{{Quantum Algorithms for Learning a Hidden Graph}},
  booktitle =	{17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022)},
  pages =	{1:1--1:22},
  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.2022.1},
  URN =		{urn:nbn:de:0030-drops-165081},
  doi =		{10.4230/LIPIcs.TQC.2022.1},
  annote =	{Keywords: Quantum algorithms, query complexity, graphs, combinatorial group testing}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2021.55
Quantum Query Complexity with Matrix-Vector Products

Authors: Andrew M. Childs, Shih-Han Hung, and Tongyang Li

Published in: LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

Abstract
We study quantum algorithms that learn properties of a matrix using queries that return its action on an input vector. We show that for various problems, including computing the trace, determinant, or rank of a matrix or solving a linear system that it specifies, quantum computers do not provide an asymptotic speedup over classical computation. On the other hand, we show that for some problems, such as computing the parities of rows or columns or deciding if there are two identical rows or columns, quantum computers provide exponential speedup. We demonstrate this by showing equivalence between models that provide matrix-vector products, vector-matrix products, and vector-matrix-vector products, whereas the power of these models can vary significantly for classical computation.
Cite as

Andrew M. Childs, Shih-Han Hung, and Tongyang Li. Quantum Query Complexity with Matrix-Vector Products. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 55:1-55:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

Copy BibTex To Clipboard

@InProceedings{childs_et_al:LIPIcs.ICALP.2021.55,
  author =	{Childs, Andrew M. and Hung, Shih-Han and Li, Tongyang},
  title =	{{Quantum Query Complexity with Matrix-Vector Products}},
  booktitle =	{48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
  pages =	{55:1--55:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-195-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{198},
  editor =	{Bansal, Nikhil and Merelli, Emanuela and Worrell, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.55},
  URN =		{urn:nbn:de:0030-drops-141242},
  doi =		{10.4230/LIPIcs.ICALP.2021.55},
  annote =	{Keywords: Quantum algorithms, quantum query complexity, matrix-vector products}
}
Document
DOI: 10.4230/LIPIcs.TQC.2021.9
Quantum Probability Oracles & Multidimensional Amplitude Estimation

Authors: Joran van Apeldoorn

Published in: LIPIcs, Volume 197, 16th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2021)

Abstract
We give a multidimensional version of amplitude estimation. Let p be an n-dimensional probability distribution which can be sampled from using a quantum circuit U_p. We show that all coordinates of p can be estimated up to error ε per coordinate using Õ(1/(ε)) applications of U_p and its inverse. This generalizes the normal amplitude estimation algorithm, which solves the problem for n = 2. Our results also imply a Õ(n/ε) query algorithm for 𝓁₁-norm (the total variation distance) estimation and a Õ(√n/ε) query algorithm for 𝓁₂-norm. We also show that these results are optimal up to logarithmic factors.
Cite as

Joran van Apeldoorn. Quantum Probability Oracles & Multidimensional Amplitude Estimation. In 16th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 197, pp. 9:1-9:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

Copy BibTex To Clipboard

@InProceedings{vanapeldoorn:LIPIcs.TQC.2021.9,
  author =	{van Apeldoorn, Joran},
  title =	{{Quantum Probability Oracles \& Multidimensional Amplitude Estimation}},
  booktitle =	{16th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2021)},
  pages =	{9:1--9:11},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-198-6},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{197},
  editor =	{Hsieh, Min-Hsiu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2021.9},
  URN =		{urn:nbn:de:0030-drops-140046},
  doi =		{10.4230/LIPIcs.TQC.2021.9},
  annote =	{Keywords: quantum algorithms, amplitude estimation, monte carlo}
}
Document
DOI: 10.4230/LIPIcs.TQC.2020.1
Exponential Quantum Communication Reductions from Generalizations of the Boolean Hidden Matching Problem

Authors: João F. Doriguello and Ashley Montanaro

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

Abstract
In this work we revisit the Boolean Hidden Matching communication problem, which was the first communication problem in the one-way model to demonstrate an exponential classical-quantum communication separation. In this problem, Alice’s bits are matched into pairs according to a partition that Bob holds. These pairs are compressed using a Parity function and it is promised that the final bit-string is equal either to another bit-string Bob holds, or its complement. The problem is to decide which case is the correct one. Here we generalize the Boolean Hidden Matching problem by replacing the parity function with an arbitrary function f. Efficient communication protocols are presented depending on the sign-degree of f. If its sign-degree is less than or equal to 1, we show an efficient classical protocol. If its sign-degree is less than or equal to 2, we show an efficient quantum protocol. We then completely characterize the classical hardness of all symmetric functions f of sign-degree greater than or equal to 2, except for one family of specific cases. We also prove, via Fourier analysis, a classical lower bound for any function f whose pure high degree is greater than or equal to 2. Similarly, we prove, also via Fourier analysis, a quantum lower bound for any function f whose pure high degree is greater than or equal to 3. These results give a large family of new exponential classical-quantum communication separations.
Cite as

João F. Doriguello and Ashley Montanaro. Exponential Quantum Communication Reductions from Generalizations of the Boolean Hidden Matching Problem. In 15th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 158, pp. 1:1-1:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

Copy BibTex To Clipboard

@InProceedings{doriguello_et_al:LIPIcs.TQC.2020.1,
  author =	{Doriguello, Jo\~{a}o F. and Montanaro, Ashley},
  title =	{{Exponential Quantum Communication Reductions from Generalizations of the Boolean Hidden Matching Problem}},
  booktitle =	{15th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2020)},
  pages =	{1:1--1:16},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2020.1},
  URN =		{urn:nbn:de:0030-drops-120601},
  doi =		{10.4230/LIPIcs.TQC.2020.1},
  annote =	{Keywords: Communication Complexity, Quantum Communication Complexity, Boolean Hidden Matching Problem}
}
Document
DOI: 10.4230/LIPIcs.STACS.2020.45
Improved Bounds on Fourier Entropy and Min-Entropy

Authors: Srinivasan Arunachalam, Sourav Chakraborty, Michal Koucký, Nitin Saurabh, and Ronald de Wolf

Published in: LIPIcs, Volume 154, 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)

Abstract
Given a Boolean function f:{-1,1}ⁿ→ {-1,1}, define the Fourier distribution to be the distribution on subsets of [n], where each S ⊆ [n] is sampled with probability f̂(S)². The Fourier Entropy-Influence (FEI) conjecture of Friedgut and Kalai [E. Friedgut and G. Kalai, 1996] seeks to relate two fundamental measures associated with the Fourier distribution: does there exist a universal constant C>0 such that ℍ(f̂²)≤ C⋅ Inf(f), where ℍ(f̂²) is the Shannon entropy of the Fourier distribution of f and Inf(f) is the total influence of f? In this paper we present three new contributions towards the FEI conjecture: ii) Our first contribution shows that ℍ(f̂²) ≤ 2⋅ aUC^⊕(f), where aUC^⊕(f) is the average unambiguous parity-certificate complexity of f. This improves upon several bounds shown by Chakraborty et al. [S. Chakraborty et al., 2016]. We further improve this bound for unambiguous DNFs. iii) We next consider the weaker Fourier Min-entropy-Influence (FMEI) conjecture posed by O'Donnell and others [R. O'Donnell et al., 2011; R. O'Donnell, 2014] which asks if ℍ_{∞}(f̂²) ≤ C⋅ Inf(f), where ℍ_{∞}(f̂²) is the min-entropy of the Fourier distribution. We show ℍ_{∞}(f̂²) ≤ 2⋅?_{min}^⊕(f), where ?_{min}^⊕(f) is the minimum parity certificate complexity of f. We also show that for all ε ≥ 0, we have ℍ_{∞}(f̂²) ≤ 2log (‖f̂‖_{1,ε}/(1-ε)), where ‖f̂‖_{1,ε} is the approximate spectral norm of f. As a corollary, we verify the FMEI conjecture for the class of read-k DNFs (for constant k). iv) Our third contribution is to better understand implications of the FEI conjecture for the structure of polynomials that 1/3-approximate a Boolean function on the Boolean cube. We pose a conjecture: no flat polynomial (whose non-zero Fourier coefficients have the same magnitude) of degree d and sparsity 2^ω(d) can 1/3-approximate a Boolean function. This conjecture is known to be true assuming FEI and we prove the conjecture unconditionally (i.e., without assuming the FEI conjecture) for a class of polynomials. We discuss an intriguing connection between our conjecture and the constant for the Bohnenblust-Hille inequality, which has been extensively studied in functional analysis.
Cite as

Srinivasan Arunachalam, Sourav Chakraborty, Michal Koucký, Nitin Saurabh, and Ronald de Wolf. Improved Bounds on Fourier Entropy and Min-Entropy. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 45:1-45:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

Copy BibTex To Clipboard

@InProceedings{arunachalam_et_al:LIPIcs.STACS.2020.45,
  author =	{Arunachalam, Srinivasan and Chakraborty, Sourav and Kouck\'{y}, Michal and Saurabh, Nitin and de Wolf, Ronald},
  title =	{{Improved Bounds on Fourier Entropy and Min-Entropy}},
  booktitle =	{37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
  pages =	{45:1--45:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-140-5},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{154},
  editor =	{Paul, Christophe and Bl\"{a}ser, Markus},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.45},
  URN =		{urn:nbn:de:0030-drops-119062},
  doi =		{10.4230/LIPIcs.STACS.2020.45},
  annote =	{Keywords: Fourier analysis of Boolean functions, FEI conjecture, query complexity, polynomial approximation, approximate degree, certificate complexity}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2019.6
Complexity-Theoretic Limitations on Blind Delegated Quantum Computation

Authors: Scott Aaronson, Alexandru Cojocaru, Alexandru Gheorghiu, and Elham Kashefi

Published in: LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

Abstract
Blind delegation protocols allow a client to delegate a computation to a server so that the server learns nothing about the input to the computation apart from its size. For the specific case of quantum computation we know, from work over the past decade, that blind delegation protocols can achieve information-theoretic security (provided the client and the server exchange some amount of quantum information). In this paper we prove, provided certain complexity-theoretic conjectures are true, that the power of information-theoretically secure blind delegation protocols for quantum computation (ITS-BQC protocols) is in a number of ways constrained. In the first part of our paper we provide some indication that ITS-BQC protocols for delegating polynomial-time quantum computations in which the client and the server interact only classically are unlikely to exist. We first show that having such a protocol in which the client and the server exchange O(n^d) bits of communication, implies that BQP subset MA/O(n^d). We conjecture that this containment is unlikely by proving that there exists an oracle relative to which BQP not subset MA/O(n^d). We then show that if an ITS-BQC protocol exists in which the client and the server interact only classically and which allows the client to delegate quantum sampling problems to the server (such as BosonSampling) then there exist non-uniform circuits of size 2^{n - Omega(n/log(n))}, making polynomially-sized queries to an NP^{NP} oracle, for computing the permanent of an n x n matrix. The second part of our paper concerns ITS-BQC protocols in which the client and the server engage in one round of quantum communication and then exchange polynomially many classical messages. First, we provide a complexity-theoretic upper bound on the types of functions that could be delegated in such a protocol by showing that they must be contained in QCMA/qpoly cap coQCMA/qpoly. Then, we show that having such a protocol for delegating NP-hard functions implies coNP^{NP^{NP}} subseteq NP^{NP^{PromiseQMA}}.
Cite as

Scott Aaronson, Alexandru Cojocaru, Alexandru Gheorghiu, and Elham Kashefi. Complexity-Theoretic Limitations on Blind Delegated Quantum Computation. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 6:1-6:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

Copy BibTex To Clipboard

@InProceedings{aaronson_et_al:LIPIcs.ICALP.2019.6,
  author =	{Aaronson, Scott and Cojocaru, Alexandru and Gheorghiu, Alexandru and Kashefi, Elham},
  title =	{{Complexity-Theoretic Limitations on Blind Delegated Quantum Computation}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{6:1--6:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-109-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{132},
  editor =	{Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.6},
  URN =		{urn:nbn:de:0030-drops-105826},
  doi =		{10.4230/LIPIcs.ICALP.2019.6},
  annote =	{Keywords: Quantum cryptography, Complexity theory, Delegated quantum computation, Computing on encrypted data}
}
Document
DOI: 10.4230/LIPIcs.TQC.2019.8
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)

Copy BibTex To Clipboard

@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-dev.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
DOI: 10.4230/LIPIcs.TQC.2018.4
The Quantum Complexity of Computing Schatten p-norms

Authors: Chris Cade and Ashley Montanaro

Published in: LIPIcs, Volume 111, 13th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2018)

Abstract
We consider the quantum complexity of computing Schatten p-norms and related quantities, and find that the problem of estimating these quantities is closely related to the one clean qubit model of computation. We show that the problem of approximating Tr(|A|^p) for a log-local n-qubit Hamiltonian A and p=poly(n), up to a suitable level of accuracy, is contained in DQC1; and that approximating this quantity up to a somewhat higher level of accuracy is DQC1-hard. In some cases the level of accuracy achieved by the quantum algorithm is substantially better than a natural classical algorithm for the problem. The same problem can be solved for arbitrary sparse matrices in BQP. One application of the algorithm is the approximate computation of the energy of a graph.
Cite as

Chris Cade and Ashley Montanaro. The Quantum Complexity of Computing Schatten p-norms. In 13th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 111, pp. 4:1-4:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

Copy BibTex To Clipboard

@InProceedings{cade_et_al:LIPIcs.TQC.2018.4,
  author =	{Cade, Chris and Montanaro, Ashley},
  title =	{{The Quantum Complexity of Computing Schatten p-norms}},
  booktitle =	{13th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2018)},
  pages =	{4:1--4:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-080-4},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{111},
  editor =	{Jeffery, Stacey},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2018.4},
  URN =		{urn:nbn:de:0030-drops-92513},
  doi =		{10.4230/LIPIcs.TQC.2018.4},
  annote =	{Keywords: Schatten p-norm, quantum complexity theory, complexity theory, one clean qubit model}
}
  • Refine by Author
  • 4 Montanaro, Ashley
  • 2 Aaronson, Scott
  • 2 Buhrman, Harry
  • 1 Arunachalam, Srinivasan
  • 1 Bittel, Lennart
  • Show More...

  • Refine by Classification
  • 6 Theory of computation → Quantum complexity theory
  • 3 Theory of computation → Quantum computation theory
  • 2 Theory of computation → Complexity classes
  • 2 Theory of computation → Design and analysis of algorithms
  • 2 Theory of computation → Quantum query complexity
  • Show More...

  • Refine by Keyword
  • 2 Quantum algorithms
  • 2 quantum algorithms
  • 2 query complexity
  • 1 Boolean Hidden Matching Problem
  • 1 Classical simulation
  • Show More...

  • Refine by Type
  • 11 document

  • Refine by Publication Year
  • 2 2019
  • 2 2020
  • 2 2021
  • 2 2023
  • 1 2018
  • Show More...

Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact