13th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2018), TQC 2018, July 16, 2018, Sydney, Australia
TQC 2018
July 16, 2018
Sydney, Australia
Conference on the Theory of Quantum Computation, Communication and Cryptography
TQC
https://www.tqcconference.org/
https://dblp.org/db/conf/tqc
Leibniz International Proceedings in Informatics
LIPIcs
https://www.dagstuhl.de/dagpub/1868-8969
https://dblp.org/db/series/lipics
1868-8969
Stacey
Jeffery
Stacey Jeffery
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
111
2018
978-3-95977-080-4
https://www.dagstuhl.de/dagpub/978-3-95977-080-4
Front Matter, Table of Contents
Front Matter, Table of Contents
Front Matter
Table of Contents
0:i-0:vi
Front Matter
Stacey
Jeffery
Stacey Jeffery
10.4230/LIPIcs.TQC.2018.0
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Quantum Ciphertext Authentication and Key Recycling with the Trap Code
We investigate quantum authentication schemes constructed from quantum error-correcting codes. We show that if the code has a property called purity testing, then the resulting authentication scheme guarantees the integrity of ciphertexts, not just plaintexts. On top of that, if the code is strong purity testing, the authentication scheme also allows the encryption key to be recycled, partially even if the authentication rejects. Such a strong notion of authentication is useful in a setting where multiple ciphertexts can be present simultaneously, such as in interactive or delegated quantum computation. With these settings in mind, we give an explicit code (based on the trap code) that is strong purity testing but, contrary to other known strong-purity-testing codes, allows for natural computation on ciphertexts.
quantum authentication
ciphertext authentication
trap code
purity-testing codes
quantum computing on encrypted data
Theory of computation~Cryptographic protocols
Theory of computation~Error-correcting codes
Security and privacy~Information-theoretic techniques
Security and privacy~Symmetric cryptography and hash functions
Theory of computation~Quantum information theory
1:1-1:17
Regular Paper
https://arxiv.org/abs/1804.02237
Yfke
Dulek
Yfke Dulek
Qusoft, Centrum voor Wiskunde en Informatica, Amsterdam, the Netherlands
Florian
Speelman
Florian Speelman
QMATH, Department of Mathematical Sciences, University of Copenhagen, Denmark
European Research Council (ERC Grant Agreement no 337603), the Danish Council for Independent Research (Sapare Aude), Qubiz Quantum Innovation Center, and VILLUM FONDEN via the QMATH Centre of Excellence (Grant No. 10059)
10.4230/LIPIcs.TQC.2018.1
Dorit Aharonov, Michael Ben-Or, and Elad Eban. Interactive proofs for quantum computations. arXiv preprint arXiv:0810.5375, 2008.
Gorjan Alagic, Yfke Dulek, Christian Schaffner, and Florian Speelman. Quantum fully homomorphic encryption with verification. In Advances in Cryptology - ASIACRYPT 2017, pages 438-467, Cham, 2017. Springer International Publishing. URL: http://dx.doi.org/10.1007/978-3-319-70694-8_16.
http://dx.doi.org/10.1007/978-3-319-70694-8_16
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Gorjan Alagic and Christian Majenz. Quantum non-malleability and authentication. In Advances in Cryptology - CRYPTO 2017, pages 310-341, Cham, 2017. Springer International Publishing. URL: http://dx.doi.org/10.1007/978-3-319-63715-0_11.
http://dx.doi.org/10.1007/978-3-319-63715-0_11
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http://dx.doi.org/10.1109/FOCS.2016.13
Anne Broadbent and Evelyn Wainewright. Efficient simulation for quantum message authentication. In Information Theoretic Security, pages 72-91, Cham, 2016. Springer International Publishing. URL: http://dx.doi.org/10.1007/978-3-319-49175-2_4.
http://dx.doi.org/10.1007/978-3-319-49175-2_4
Christoph Dankert, Richard Cleve, Joseph Emerson, and Etera Livine. Exact and approximate unitary 2-designs and their application to fidelity estimation. Physical Review A, 80(1):012304, 2009.
Frédéric Dupuis, Jesper Buus Nielsen, and Louis Salvail. Actively secure two-party evaluation of any quantum operation. In Advances in Cryptology - CRYPTO 2012, volume 7417, pages 794-811. Springer International Publishing, 2012. Full version on IACR eprint archive: eprint.iacr.org/2012/304. URL: http://dx.doi.org/10.1007/978-3-642-32009-5_46.
http://dx.doi.org/10.1007/978-3-642-32009-5_46
Serge Fehr and Louis Salvail. Quantum authentication and encryption with key recycling. In Annual International Conference on the Theory and Applications of Cryptographic Techniques, pages 311-338. Springer, 2017.
Sumegha Garg, Henry Yuen, and Mark Zhandry. New security notions and feasibility results for authentication of quantum data. In Jonathan Katz and Hovav Shacham, editors, Advances in Cryptology - CRYPTO 2017, pages 342-371, Cham, 2017. Springer International Publishing. URL: http://dx.doi.org/10.1007/978-3-319-63715-0_12.
http://dx.doi.org/10.1007/978-3-319-63715-0_12
Patrick Hayden, Debbie W Leung, and Dominic Mayers. The universal composable security of quantum message authentication with key recyling. arXiv preprint arXiv:1610.09434, 2016.
Jonathan Oppenheim and Michał Horodecki. How to reuse a one-time pad and other notes on authentication, encryption, and protection of quantum information. Phys. Rev. A, 72:042309, Oct 2005. URL: http://dx.doi.org/10.1103/PhysRevA.72.042309.
http://dx.doi.org/10.1103/PhysRevA.72.042309
Christopher Portmann. Quantum authentication with key recycling. In Annual International Conference on the Theory and Applications of Cryptographic Techniques, pages 339-368. Springer, 2017.
John Preskill. Quantum computation, 1997. URL: http://www.theory.caltech.edu/people/preskill/ph229/index.html.
http://www.theory.caltech.edu/people/preskill/ph229/index.html
Mark N. Wegman and J. Lawrence Carter. New hash functions and their use in authentication and set equality. Journal of Computer and System Sciences, 22(3):265-279, 1981. URL: http://dx.doi.org/10.1016/0022-0000(81)90033-7.
http://dx.doi.org/10.1016/0022-0000(81)90033-7
Yfke Dulek and Florian Speelman
Creative Commons Attribution 3.0 Unported license
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On the Complexity of Two Dimensional Commuting Local Hamiltonians
The complexity of the commuting local Hamiltonians (CLH) problem still remains a mystery after two decades of research of quantum Hamiltonian complexity; it is only known to be contained in NP for few low parameters. Of particular interest is the tightly related question of understanding whether groundstates of CLHs can be generated by efficient quantum circuits. The two problems touch upon conceptual, physical and computational questions, including the centrality of non-commutation in quantum mechanics, quantum PCP and the area law. It is natural to try to address first the more physical case of CLHs embedded on a 2D lattice, but this problem too remained open apart from some very specific cases [Aharonov and Eldar, 2011; Hastings, 2012; Schuch, 2011]. Here we consider a wide class of two dimensional CLH instances; these are k-local CLHs, for any constant k; they are defined on qubits set on the edges of any surface complex, where we require that this surface complex is not too far from being "Euclidean". Each vertex and each face can be associated with an arbitrary term (as long as the terms commute). We show that this class is in NP, and moreover that the groundstates have an efficient quantum circuit that prepares them. This result subsumes that of Schuch [Schuch, 2011] which regarded the special case of 4-local Hamiltonians on a grid with qubits, and by that it removes the mysterious feature of Schuch's proof which showed containment in NP without providing a quantum circuit for the groundstate and considerably generalizes it. We believe this work and the tools we develop make a significant step towards showing that 2D CLHs are in NP.
local Hamiltonian complexity
commuting Hamiltonians
local Hamiltonian problem
trivial states
toric code
ground states
quantum NP
QMA
topological order
multiparticle entanglement
logical operators
ribbon
Theory of computation~Quantum complexity theory
2:1-2:21
Regular Paper
The authors are grateful for the generous funding of the ERC grant number 280157, and of the Simon grant number 385590
https://arxiv.org/abs/1803.02213
Dorit
Aharonov
Dorit Aharonov
School of Computer Science and Engineering, The Hebrew University, Jerusalem, Israel
Oded
Kenneth
Oded Kenneth
School of Computer Science and Engineering, The Hebrew University, Jerusalem, Israel
Itamar
Vigdorovich
Itamar Vigdorovich
Faculty of Mathematics and Computer Science, Weizmann Institute of Science, Rehovot, Israel
10.4230/LIPIcs.TQC.2018.2
Miguel Aguado and Guifre Vidal. Entanglement renormalization and topological order. Physical review letters, 100(7):070404, 2008.
Dorit Aharonov, Itai Arad, Zeph Landau, and Umesh Vazirani. The detectability lemma and quantum gap amplification. In Proceedings of the forty-first annual ACM symposium on Theory of computing, pages 417-426. ACM, 2009.
Dorit Aharonov, Itai Arad, and Thomas Vidick. Guest column: the quantum pcp conjecture. Acm sigact news, 44(2):47-79, 2013.
Dorit Aharonov and Lior Eldar. On the complexity of commuting local hamiltonians, and tight conditions for topological order in such systems. In Foundations of Computer Science (FOCS), 2011 IEEE 52nd Annual Symposium on, pages 334-343. IEEE, 2011.
Dorit Aharonov and Lior Eldar. The commuting local hamiltonian problem on locally expanding graphs is approximable in $$1mathsf NP np. Quantum Information Processing, 14(1):83-101, 2015.
Dorit Aharonov, Oded Kenneth, and Itamar Vigdorovich. On the complexity of two dimensional commuting local hamiltonians. arXiv preprint arXiv:1803.02213, 2018.
Dorit Aharonov and Tomer Naveh. Quantum np-a survey. arXiv preprint quant-ph/0210077, 2002.
S Bravyi, MB Hastings, and F Verstraete. Lieb-robinson bounds and the generation of correlations and topological quantum order. Physical review letters, 97(5):050401, 2006.
Sergey Bravyi and Mikhail Vyalyi. Commutative version of the k-local hamiltonian problem and common eigenspace problem. arXiv preprint quant-ph/0308021, 2003.
Sergey B Bravyi and A Yu Kitaev. Quantum codes on a lattice with boundary. arXiv preprint quant-ph/9811052, 1998.
Dmitri Burago, Yurĭ Dmitrievich Burago, and Sergeĭ Ivanov. A course in metric geometry. American Mathematical Society, 2001. URL: http://dx.doi.org/10.1090/gsm/033.
http://dx.doi.org/10.1090/gsm/033
Reinhard Diestel. Graph theory. Springer Publishing Company, Incorporated, 2017.
Michael H Freedman and Matthew B Hastings. Quantum systems on non-k-hyperfinite complexes: A generalization of classical statistical mechanics on expander graphs. arXiv preprint arXiv:1301.1363, 2013.
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Matthew B Hastings. Matrix product operators and central elements: Classical description of a quantum state. Geometry &Topology Monographs, 18(115-160):276, 2012.
Matthew B Hastings. Trivial low energy states for commuting hamiltonians, and the quantum pcp conjecture. arXiv preprint arXiv:1201.3387, 2012.
Allen Hatcher. Algebraic topology. Cambridge University Press, 2002.
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Dorit Aharonov, Oded Kenneth, and Itamar Vigdorovich
Creative Commons Attribution 3.0 Unported license
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Quantum Lower Bounds for Tripartite Versions of the Hidden Shift and the Set Equality Problems
In this paper, we study quantum query complexity of the following rather natural tripartite generalisations (in the spirit of the 3-sum problem) of the hidden shift and the set equality problems, which we call the 3-shift-sum and the 3-matching-sum problems.
The 3-shift-sum problem is as follows: given a table of 3 x n elements, is it possible to circularly shift its rows so that the sum of the elements in each column becomes zero? It is promised that, if this is not the case, then no 3 elements in the table sum up to zero. The 3-matching-sum problem is defined similarly, but it is allowed to arbitrarily permute elements within each row. For these problems, we prove lower bounds of Omega(n^{1/3}) and Omega(sqrt n), respectively. The second lower bound is tight.
The lower bounds are proven by a novel application of the dual learning graph framework and by using representation-theoretic tools from [Belovs, 2018].
Adversary Bound
Dual Learning Graphs
Quantum Query Complexity
Representation Theory
Theory of computation~Quantum query complexity
3:1-3:15
Regular Paper
https://arxiv.org/abs/1712.10194
Aleksandrs
Belovs
Aleksandrs Belovs
Faculty of Computing, University of Latvia, Raina 19, Riga, Latvia
This research is partially supported by the ERDF project number 1.1.1.2/I/16/113. Part of this work was done while supported by the ERC Advanced Grant MQC.
Ansis
Rosmanis
Ansis Rosmanis
Centre for Quantum Technologies, National University of Singapore, Block S15, 3 Science Drive 2, Singapore
This research is partially funded by the Singapore Ministry of Education and the National Research Foundation under grant R-710-000-012-135.
10.4230/LIPIcs.TQC.2018.3
Scott Aaronson and Andris Ambainis. Forrelation: A problem that optimally separates quantum from classical computing. In Proc. of 47th ACM STOC, pages 307-316, 2015.
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Aleksandrs Belovs. Applications of the Adversary Method in Quantum Query Algorithms. PhD thesis, University of Latvia, 2014.
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Aleksandrs Belovs and Ansis Rosmanis. Quantum lower bounds for tripartite versions of the hidden shift and the set equality problems. Full version. Avaiable at arXiv:1712.10194, 2018.
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Aleksandrs Belovs and Ansis Rosmanis
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
The Quantum Complexity of Computing Schatten p-norms
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.
Schatten p-norm
quantum complexity theory
complexity theory
one clean qubit model
Theory of computation~Quantum complexity theory
Theory of computation~Complexity classes
4:1-4:20
Regular Paper
CC was supported by the EPSRC. AM was supported by an EPSRC Early Career Fellowship (EP/L021005/1). No new data were created during this study.
https://arxiv.org/abs/1706.09279
Chris
Cade
Chris Cade
School of Mathematics, University of Bristol, UK
Ashley
Montanaro
Ashley Montanaro
School of Mathematics, University of Bristol, UK
10.4230/LIPIcs.TQC.2018.4
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Chris Cade and Ashley Montanaro
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Subset Sum Quantumly in 1.17^n
We study the quantum complexity of solving the subset sum problem, where the elements a_1, ..., a_n are randomly chosen from Z_{2^{l(n)}} and t = sum_i a_i in Z_{2^{l(n)}} is a sum of n/2 elements. In 2013, Bernstein, Jeffery, Lange and Meurer constructed a quantum subset sum algorithm with heuristic time complexity 2^{0.241n}, by enhancing the classical subset sum algorithm of Howgrave-Graham and Joux with a quantum random walk technique. We improve on this by defining a quantum random walk for the classical subset sum algorithm of Becker, Coron and Joux. The new algorithm only needs heuristic running time and memory 2^{0.226n}, for almost all random subset sum instances.
Subset sum
Quantum walk
Representation technique
Theory of computation~Quantum complexity theory
5:1-5:15
Regular Paper
Alexander
Helm
Alexander Helm
Horst Görtz Institute for IT-Security, Ruhr University Bochum, Germany
Founded by NRW Research Training Group SecHuman.
Alexander
May
Alexander May
Horst Görtz Institute for IT-Security, Ruhr University Bochum, Germany
10.4230/LIPIcs.TQC.2018.5
Dorit Aharonov, Andris Ambainis, Julia Kempe, and Umesh Vazirani. Quantum walks on graphs. In Proceedings of the thirty-third annual ACM symposium on Theory of computing, pages 50-59. ACM, 2001.
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http://dx.doi.org/10.1007/978-3-642-38616-9_2
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http://arxiv.org/abs/1703.00263
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Alexander Helm and Alexander May
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Trading Inverses for an Irrep in the Solovay-Kitaev Theorem
The Solovay-Kitaev theorem states that universal quantum gate sets can be exchanged with low overhead. More specifically, any gate on a fixed number of qudits can be simulated with error epsilon using merely polylog(1/epsilon) gates from any finite universal quantum gate set G. One drawback to the theorem is that it requires the gate set G to be closed under inversion. Here we show that this restriction can be traded for the assumption that G contains an irreducible representation of any finite group G. This extends recent work of Sardharwalla et al. [Sardharwalla et al., 2016], and applies also to gates from the special linear group. Our work can be seen as partial progress towards the long-standing open problem of proving an inverse-free Solovay-Kitaev theorem [Dawson and Nielsen, 2006; Kuperberg, 2015].
Solovay-Kitaev theorem
quantum gate sets
gate set compilation
Theory of computation~Quantum computation theory
6:1-6:15
Regular Paper
Adam
Bouland
Adam Bouland
Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, Berkeley, CA, USA
https://orcid.org/0000-0002-8556-8337
AB was partially supported by the NSF GRFP under Grant No. 1122374, by the Vannevar Bush Fellowship from the US Department of Defense, by ARO Grant W911NF-12-1-0541, by NSF Grant CCF-1410022, and by the NSF Waterman award under grant number 1249349.
Maris
Ozols
Maris Ozols
QuSoft and University of Amsterdam, Amsterdam, Netherlands
https://orcid.org/0000-0002-3238-8594
Part of this work was done while MO was at the University of Cambridge where he was supported by a Leverhulme Trust Early Career Fellowship (ECF-2015-256). MO also acknowledges hospitality of MIT where this project was initiated.
10.4230/LIPIcs.TQC.2018.6
Scott Aaronson and Alex Arkhipov. The computational complexity of linear optics. In Proceedings of the forty-third annual ACM symposium on Theory of computing, pages 333-342. ACM, 2011. URL: http://dx.doi.org/10.1145/1993636.1993682.
http://dx.doi.org/10.1145/1993636.1993682
Dorit Aharonov, Itai Arad, Elad Eban, and Zeph Landau. Polynomial quantum algorithms for additive approximations of the Potts model and other points of the Tutte plane, 2007. URL: http://arxiv.org/abs/quant-ph/0702008.
http://arxiv.org/abs/quant-ph/0702008
Dorit Aharonov and Michael Ben-Or. Fault-tolerant quantum computation with constant error rate. SIAM Journal on Computing, 38(4):1207-1282, 2008. URL: http://dx.doi.org/10.1137/S0097539799359385.
http://dx.doi.org/10.1137/S0097539799359385
Dorit Aharonov, Vaughan Jones, and Zeph Landau. A polynomial quantum algorithm for approximating the Jones polynomial. Algorithmica, 55(3):395-421, Nov 2009. URL: http://dx.doi.org/10.1007/s00453-008-9168-0.
http://dx.doi.org/10.1007/s00453-008-9168-0
Johannes Bausch, Toby Cubitt, and Maris Ozols. The complexity of translationally invariant spin chains with low local dimension. Annales Henri Poincaré, 18(11):3449-3513, Nov 2017. URL: http://dx.doi.org/10.1007/s00023-017-0609-7.
http://dx.doi.org/10.1007/s00023-017-0609-7
Johannes Bausch and Stephen Piddock. The complexity of translationally invariant low-dimensional spin lattices 3D. Journal of Mathematical Physics, 58(11):111901, 2017. URL: http://dx.doi.org/10.1063/1.5011338.
http://dx.doi.org/10.1063/1.5011338
Dominic W. Berry, Andrew M. Childs, and Robin Kothari. Hamiltonian simulation with nearly optimal dependence on all parameters. In Foundations of Computer Science (FOCS), 2015 IEEE 56th Annual Symposium on, pages 792-809. IEEE, Oct 2015. URL: http://dx.doi.org/10.1109/FOCS.2015.54.
http://dx.doi.org/10.1109/FOCS.2015.54
Alex Bocharov, Martin Roetteler, and Krysta M. Svore. Efficient synthesis of probabilistic quantum circuits with fallback. Phys. Rev. A, 91(5):052317, May 2015. URL: http://dx.doi.org/10.1103/PhysRevA.91.052317.
http://dx.doi.org/10.1103/PhysRevA.91.052317
Alex Bocharov, Martin Roetteler, and Krysta M. Svore. Efficient synthesis of universal repeat-until-success quantum circuits. Phys. Rev. Lett., 114(8):080502, Feb 2015. URL: http://dx.doi.org/10.1103/PhysRevLett.114.080502.
http://dx.doi.org/10.1103/PhysRevLett.114.080502
Adam Bouland, Joseph F. Fitzsimons, and Dax E. Koh. Complexity classification of conjugated Clifford circuits. Proc. 33rd Computational Complexity Conference (CCC), 2018. URL: http://arxiv.org/abs/1709.01805.
http://arxiv.org/abs/1709.01805
Adam Bouland, Laura Mančinska, and Xue Zhang. Complexity classification of two-qubit commuting Hamiltonians. In Ran Raz, editor, 31st Conference on Computational Complexity (CCC 2016), volume 50 of Leibniz International Proceedings in Informatics (LIPIcs), pages 28:1-28:33, Dagstuhl, Germany, 2016. Schloss Dagstuhl-Leibniz-Zentrum für Informatik. URL: http://dx.doi.org/10.4230/LIPIcs.CCC.2016.28.
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Michael J. Bremner, Richard Jozsa, and Dan J. Shepherd. Classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 2010. URL: http://dx.doi.org/10.1098/rspa.2010.0301.
http://dx.doi.org/10.1098/rspa.2010.0301
Andrew M. Childs. Fourier analysis in nonabelian groups. Lecture notes at University of Waterloo, 2013. URL: http://www.cs.umd.edu/~amchilds/teaching/w13/l06.pdf.
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Andrew M. Childs. Lecture notes on quantum algorithms. Lecture notes at University of Maryland, 2017. URL: http://www.cs.umd.edu/~amchilds/qa/qa.pdf.
http://www.cs.umd.edu/~amchilds/qa/qa.pdf
Andrew M. Childs, Debbie Leung, Laura Mančinska, and Maris Ozols. Characterization of universal two-qubit Hamiltonians. Quantum Information &Computation, 11(1&2):19-39, Jan 2011. URL: http://arxiv.org/abs/1004.1645.
http://arxiv.org/abs/1004.1645
Christopher M. Dawson and Michael A. Nielsen. The Solovay-Kitaev algorithm. Quantum Information &Computation, 6(1):81-95, Jan 2006. URL: http://arxiv.org/abs/quant-ph/0505030.
http://arxiv.org/abs/quant-ph/0505030
Daniel Gottesman and Sandy Irani. The quantum and classical complexity of translationally invariant tiling and Hamiltonian problems. Theory of Computing, 9(2):31-116, 2013. URL: http://dx.doi.org/10.4086/toc.2013.v009a002.
http://dx.doi.org/10.4086/toc.2013.v009a002
Aram W. Harrow, Benjamin Recht, and Isaac L. Chuang. Efficient discrete approximations of quantum gates. Journal of Mathematical Physics, 43(9):4445-4451, 2002. URL: http://dx.doi.org/10.1063/1.1495899.
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Alexei Yu. Kitaev. Quantum computations: algorithms and error correction. Russian Mathematical Surveys, 52(6):1191-1249, 1997. URL: http://dx.doi.org/10.1070/RM1997v052n06ABEH002155.
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Vadym Kliuchnikov. Synthesis of unitaries with Clifford+T circuits, 2013. URL: http://arxiv.org/abs/1306.3200.
http://arxiv.org/abs/1306.3200
Vadym Kliuchnikov, Dmitri Maslov, and Michele Mosca. Practical approximation of single-qubit unitaries by single-qubit quantum Clifford and T circuits. IEEE Transactions on Computers, 65(1):161-172, Jan 2016. URL: http://dx.doi.org/10.1109/TC.2015.2409842.
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Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information. Cambridge University Press, 2010. URL: https://books.google.com/books?id=-s4DEy7o-a0C.
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Māris Ozols. The Solovay-Kitaev theorem. Essay at University of Waterloo, 2009. URL: http://home.lu.lv/~sd20008/papers/essays/Solovay-Kitaev.pdf.
http://home.lu.lv/~sd20008/papers/essays/Solovay-Kitaev.pdf
Adam Paetznick and Krysta M. Svore. Repeat-until-success: Non-deterministic decomposition of single-qubit unitaries. Quantum Information &Computation, 14(15&16):1277-1301, 2014. URL: http://arxiv.org/abs/1311.1074.
http://arxiv.org/abs/1311.1074
Ori Parzanchevski and Peter Sarnak. Super-Golden-Gates for PU(2). Advances in Mathematics, 327:869-901, 2018. Special volume honoring David Kazhdan. URL: http://dx.doi.org/10.1016/j.aim.2017.06.022.
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Neil J. Ross and Peter Selinger. Optimal ancilla-free Clifford+T approximation of Z-rotations. Quantum Information &Computation, 16(11&12):0901-0953, 2016. URL: http://dx.doi.org/10.26421/QIC16.11-12.
http://dx.doi.org/10.26421/QIC16.11-12
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http://arxiv.org/abs/1602.07963
Peter Sarnak. Letter to Aaronson and Pollington on the Solvay-Kitaev Theorem and Golden Gates, 2015. URL: http://publications.ias.edu/sarnak/paper/2637.
http://publications.ias.edu/sarnak/paper/2637
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http://dx.doi.org/10.1007/s00023-017-0604-z
Peter Selinger. Efficient Clifford+T approximation of single-qubit operators. Quantum Information &Computation, 15(1&2):159-180, 2015. URL: http://arxiv.org/abs/1212.6253.
http://arxiv.org/abs/1212.6253
Jean-Pierre Serre. Linear Representations of Finite Groups. Springer, 2012. URL: https://books.google.com/books?id=9mT1BwAAQBAJ.
https://books.google.com/books?id=9mT1BwAAQBAJ
Peter W. Shor. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Journal on Computing, 26(5):1484-1509, 1997. URL: http://dx.doi.org/10.1137/S0097539795293172.
http://dx.doi.org/10.1137/S0097539795293172
Péter Pál Varjú. Random walks in compact groups. Documenta Mathematica, 18:1137-1175, 2013. URL: http://arxiv.org/abs/1209.1745.
http://arxiv.org/abs/1209.1745
Y. Zhiyenbayev, V. M. Akulin, and A. Mandilara. Quantum compiling with diffusive sets of gates, 2017. URL: http://arxiv.org/abs/1708.08909.
http://arxiv.org/abs/1708.08909
Adam Bouland and Māris Ozols
Creative Commons Attribution 3.0 Unported license
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Two-qubit Stabilizer Circuits with Recovery I: Existence
In this paper, we further investigate the many ways of using stabilizer operations to generate a single qubit output from a two-qubit state. In particular, by restricting the input to certain product states, we discover probabilistic operations capable of transforming stabilizer circuit outputs back into stabilizer circuit inputs. These secondary operations are ideally suited for recovery purposes and require only one extra resource input to proceed. As a result of reusing qubits in this manner, we present an alternative to the original state preparation process that can lower the overall costs of executing a two-qubit stabilizer procedure involving non-stabilizer resources.
stabilizer circuit
recovery circuit
magic state
Theory of computation~Quantum computation theory
7:1-7:15
Regular Paper
Wim
van Dam
Wim van Dam
Department of Computer Science, Department of Physics, University of California, Santa Barbara, CA, USA
https://orcid.org/0000-0001-7852-6158
Raymond
Wong
Raymond Wong
Department of Computer Science, University of California, Santa Barbara, CA, USA
10.4230/LIPIcs.TQC.2018.7
Alex Bocharov, Martin Roetteler, and Krysta M. Svore. Efficient synthesis of probabilistic quantum circuits with fallback. Phys. Rev. A, 91:052317, May 2015. URL: http://dx.doi.org/10.1103/PhysRevA.91.052317.
http://dx.doi.org/10.1103/PhysRevA.91.052317
Alex Bocharov, Martin Roetteler, and Krysta M. Svore. Efficient Synthesis of Universal Repeat-Until-Success Quantum Circuits. Phys. Rev. Lett., 114:080502, Feb 2015. URL: http://dx.doi.org/10.1103/PhysRevLett.114.080502.
http://dx.doi.org/10.1103/PhysRevLett.114.080502
Sergey Bravyi and Jeongwan Haah. Magic-state distillation with low overhead. Phys. Rev. A, 86:052329, Nov 2012. URL: http://dx.doi.org/10.1103/PhysRevA.86.052329.
http://dx.doi.org/10.1103/PhysRevA.86.052329
Sergey Bravyi and Alexei Kitaev. Universal quantum computation with ideal clifford gates and noisy ancillas. Phys. Rev. A, 71:022316, Feb 2005. URL: http://dx.doi.org/10.1103/PhysRevA.71.022316.
http://dx.doi.org/10.1103/PhysRevA.71.022316
Earl T. Campbell and Mark Howard. Unified framework for magic state distillation and multiqubit gate synthesis with reduced resource cost. Phys. Rev. A, 95:022316, Feb 2017. URL: http://dx.doi.org/10.1103/PhysRevA.95.022316.
http://dx.doi.org/10.1103/PhysRevA.95.022316
Earl T. Campbell and Mark Howard. Unifying Gate Synthesis and Magic State Distillation. Phys. Rev. Lett., 118:060501, Feb 2017. URL: http://dx.doi.org/10.1103/PhysRevLett.118.060501.
http://dx.doi.org/10.1103/PhysRevLett.118.060501
Earl T Campbell and Joe O’Gorman. An efficient magic state approach to small angle rotations. Quantum Science and Technology, 1(1):015007, 2016. URL: http://stacks.iop.org/2058-9565/1/i=1/a=015007.
http://stacks.iop.org/2058-9565/1/i=1/a=015007
Guillaume Duclos-Cianci and David Poulin. Reducing the quantum-computing overhead with complex gate distillation. Phys. Rev. A, 91:042315, Apr 2015. URL: http://dx.doi.org/10.1103/PhysRevA.91.042315.
http://dx.doi.org/10.1103/PhysRevA.91.042315
Guillaume Duclos-Cianci and Krysta M. Svore. Distillation of nonstabilizer states for universal quantum computation. Phys. Rev. A, 88:042325, Oct 2013. URL: http://dx.doi.org/10.1103/PhysRevA.88.042325.
http://dx.doi.org/10.1103/PhysRevA.88.042325
Austin G. Fowler, Matteo Mariantoni, John M. Martinis, and Andrew N. Cleland. Surface codes: Towards practical large-scale quantum computation. Phys. Rev. A, 86:032324, Sep 2012. URL: http://dx.doi.org/10.1103/PhysRevA.86.032324.
http://dx.doi.org/10.1103/PhysRevA.86.032324
Brett Giles and Peter Selinger. Exact synthesis of multiqubit Clifford+T circuits. Phys. Rev. A, 87:032332, Mar 2013. URL: http://dx.doi.org/10.1103/PhysRevA.87.032332.
http://dx.doi.org/10.1103/PhysRevA.87.032332
Jeongwan Haah, Matthew B. Hastings, D. Poulin, and D. Wecker. Magic State Distillation with Low Space Overhead and Optimal Asymptotic Input Count. Quantum, 1:31, Oct 2017. URL: http://dx.doi.org/10.22331/q-2017-10-03-31.
http://dx.doi.org/10.22331/q-2017-10-03-31
N Cody Jones, James D Whitfield, Peter L McMahon, Man-Hong Yung, Rodney Van Meter, Alán Aspuru-Guzik, and Yoshihisa Yamamoto. Faster quantum chemistry simulation on fault-tolerant quantum computers. New Journal of Physics, 14(11):115023, 2012. URL: http://stacks.iop.org/1367-2630/14/i=11/a=115023.
http://stacks.iop.org/1367-2630/14/i=11/a=115023
J.G. Kemény and J.L. Snell. Finite markov chains. University series in undergraduate mathematics. Springer-Verlag New York, 1976.
Vadym Kliuchnikov, Dmitri Maslov, and Michele Mosca. Asymptotically Optimal Approximation of Single Qubit Unitaries by Clifford and T Circuits Using a Constant Number of Ancillary Qubits. Phys. Rev. Lett., 110:190502, May 2013. URL: http://dx.doi.org/10.1103/PhysRevLett.110.190502.
http://dx.doi.org/10.1103/PhysRevLett.110.190502
Vadym Kliuchnikov, Dmitri Maslov, and Michele Mosca. Fast and efficient exact synthesis of single qubit unitaries generated by Clifford and T gates. Quantum Information and Computation, 13(7-8):607-630, 2013. URL: http://arxiv.org/abs/1206.5236.
http://arxiv.org/abs/1206.5236
E. Knill. Quantum computing with realistically noisy devices. Nature, 434, Mar 2005. URL: http://dx.doi.org/10.1038/nature03350.
http://dx.doi.org/10.1038/nature03350
Andrew J. Landahl and Chris Cesare. Complex instruction set computing architecture for performing accurate quantum Z rotations with less magic, Feb 2013. URL: http://arxiv.org/abs/1302.3240.
http://arxiv.org/abs/1302.3240
Adam Meier, Bryan Eastin, and Emanuel Knill. Magic-state distillation with the four-qubit code. Quantum Information and Computation, 13:195-209, 2013. URL: http://arxiv.org/abs/1204.4221.
http://arxiv.org/abs/1204.4221
Adam Paetznick and Krysta M. Svore. Repeat-Until-Success: Non-determistic decomposition of single-qubit unitaries. Quantum Info. Comput., 14(15-16):1277-1301, Nov 2014. URL: http://arxiv.org/abs/1311.1074.
http://arxiv.org/abs/1311.1074
Ben Reichardt. Quantum universality by state distillation. Quantum Information and Computation, 9:1030-1052, 2009. URL: http://arxiv.org/abs/quant-ph/0608085v2.
http://arxiv.org/abs/quant-ph/0608085v2
Neil J. Ross. Optimal ancilla-free Clifford+V approximation of z-rotations. Quantum Information and Computation, 15(11-12):932-950, 2015. URL: http://arxiv.org/abs/1409.4355.
http://arxiv.org/abs/1409.4355
Neil J. Ross and Peter Selinger. Optimal ancilla-free Clifford+T approximation of z-rotations. Quantum Information and Computation, 16(11-12):901-953, 2016. URL: http://www.rintonpress.com/xxqic16/qic-16-1112/0901-0953.pdf.
http://www.rintonpress.com/xxqic16/qic-16-1112/0901-0953.pdf
Wim van Dam and Raymond Wong
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Two-qubit Stabilizer Circuits with Recovery II: Analysis
We study stabilizer circuits that use non-stabilizer qubits and Z-measurements to produce other non-stabilizer qubits. These productions are successful when the correct measurement outcome occurs, but when the opposite outcome is observed, the non-stabilizer input qubit is potentially destroyed. In preceding work [arXiv:1803.06081 (2018)] we introduced protocols able to recreate the expensive non-stabilizer input qubit when the two-qubit stabilizer circuit has an unsuccessful measurement outcome. Such protocols potentially allow a deep computation to recover from such failed measurements without the need to repeat the whole prior computation. Possible complications arise when the recovery protocol itself suffers from a failed measurement. To deal with this, we need to use nested recovery protocols. Here we give a precise analysis of the potential advantage of such recovery protocols as we examine its optimal nesting depth. We show that if the expensive input qubit has cost d, then typically a depth O(log d) recovery protocol is optimal, while a certain special case has optimal depth O(sqrt{d}). We also show that the recovery protocol can achieve a cost reduction by a factor of at most two over circuits that do not use recovery.
stabilizer circuit
recovery circuit
magic state
Theory of computation~Quantum computation theory
8:1-8:21
Regular Paper
This material is based upon work supported by the National Science Foundation under Grants No. 0917244 and 1719118.
Wim
van Dam
Wim van Dam
Department of Computer Science, Department of Physics, University of California, Santa Barbara, CA, USA
https://orcid.org/0000-0001-7852-6158
Raymond
Wong
Raymond Wong
Department of Computer Science, University of California, Santa Barbara, CA, USA
10.4230/LIPIcs.TQC.2018.8
Alex Bocharov, Yuri Gurevich, and Krysta M. Svore. Efficient decomposition of single-qubit gates into V basis circuits. Phys. Rev. A, 88:012313, Jul 2013. URL: http://dx.doi.org/10.1103/PhysRevA.88.012313.
http://dx.doi.org/10.1103/PhysRevA.88.012313
Alex Bocharov, Martin Roetteler, and Krysta M. Svore. Efficient synthesis of probabilistic quantum circuits with fallback. Phys. Rev. A, 91:052317, May 2015. URL: http://dx.doi.org/10.1103/PhysRevA.91.052317.
http://dx.doi.org/10.1103/PhysRevA.91.052317
Alex Bocharov, Martin Roetteler, and Krysta M. Svore. Efficient Synthesis of Universal Repeat-Until-Success Quantum Circuits. Phys. Rev. Lett., 114:080502, Feb 2015. URL: http://dx.doi.org/10.1103/PhysRevLett.114.080502.
http://dx.doi.org/10.1103/PhysRevLett.114.080502
Sergey Bravyi and Jeongwan Haah. Magic-state distillation with low overhead. Phys. Rev. A, 86:052329, Nov 2012. URL: http://dx.doi.org/10.1103/PhysRevA.86.052329.
http://dx.doi.org/10.1103/PhysRevA.86.052329
Sergey Bravyi and Alexei Kitaev. Universal quantum computation with ideal clifford gates and noisy ancillas. Phys. Rev. A, 71:022316, Feb 2005. URL: http://dx.doi.org/10.1103/PhysRevA.71.022316.
http://dx.doi.org/10.1103/PhysRevA.71.022316
Earl T Campbell and Joe O’Gorman. An efficient magic state approach to small angle rotations. Quantum Science and Technology, 1(1):015007, 2016. URL: http://stacks.iop.org/2058-9565/1/i=1/a=015007.
http://stacks.iop.org/2058-9565/1/i=1/a=015007
Peter G. Doyle and J. Laurie Snell. Random walks and electric networks, 2006. URL: https://math.dartmouth.edu/~doyle/docs/walks/walks.pdf.
https://math.dartmouth.edu/~doyle/docs/walks/walks.pdf
Guillaume Duclos-Cianci and David Poulin. Reducing the quantum-computing overhead with complex gate distillation. Phys. Rev. A, 91:042315, Apr 2015. URL: http://dx.doi.org/10.1103/PhysRevA.91.042315.
http://dx.doi.org/10.1103/PhysRevA.91.042315
Guillaume Duclos-Cianci and Krysta M. Svore. Distillation of nonstabilizer states for universal quantum computation. Phys. Rev. A, 88:042325, Oct 2013. URL: http://dx.doi.org/10.1103/PhysRevA.88.042325.
http://dx.doi.org/10.1103/PhysRevA.88.042325
Austin G. Fowler, Matteo Mariantoni, John M. Martinis, and Andrew N. Cleland. Surface codes: Towards practical large-scale quantum computation. Phys. Rev. A, 86:032324, Sep 2012. URL: http://dx.doi.org/10.1103/PhysRevA.86.032324.
http://dx.doi.org/10.1103/PhysRevA.86.032324
Jeongwan Haah, Matthew B. Hastings, D. Poulin, and D. Wecker. Magic State Distillation with Low Space Overhead and Optimal Asymptotic Input Count. Quantum, 1:31, Oct 2017. URL: http://dx.doi.org/10.22331/q-2017-10-03-31.
http://dx.doi.org/10.22331/q-2017-10-03-31
N Cody Jones, James D Whitfield, Peter L McMahon, Man-Hong Yung, Rodney Van Meter, Alán Aspuru-Guzik, and Yoshihisa Yamamoto. Faster quantum chemistry simulation on fault-tolerant quantum computers. New Journal of Physics, 14(11):115023, 2012. URL: http://stacks.iop.org/1367-2630/14/i=11/a=115023.
http://stacks.iop.org/1367-2630/14/i=11/a=115023
J.G. Kemény and J.L. Snell. Finite markov chains. University series in undergraduate mathematics. Springer-Verlag New York, 1976.
Vadym Kliuchnikov, Dmitri Maslov, and Michele Mosca. Fast and efficient exact synthesis of single qubit unitaries generated by Clifford and T gates. Quantum Information and Computation, 13(7-8):607-630, 2013. URL: http://arxiv.org/abs/1206.5236.
http://arxiv.org/abs/1206.5236
Andrew J. Landahl and Chris Cesare. Complex instruction set computing architecture for performing accurate quantum Z rotations with less magic, Feb 2013. URL: http://arxiv.org/abs/1302.3240.
http://arxiv.org/abs/1302.3240
Adam Meier, Bryan Eastin, and Emanuel Knill. Magic-state distillation with the four-qubit code. Quantum Information and Computation, 13:195-209, 2013. URL: http://arxiv.org/abs/1204.4221.
http://arxiv.org/abs/1204.4221
Ben Reichardt. Quantum universality by state distillation. Quantum Information and Computation, 9:1030-1052, 2009. URL: http://arxiv.org/abs/quant-ph/0608085v2.
http://arxiv.org/abs/quant-ph/0608085v2
Neil J. Ross. Optimal ancilla-free Clifford+V approximation of z-rotations. Quantum Information and Computation, 15(11-12):932-950, 2015. URL: http://arxiv.org/abs/1409.4355.
http://arxiv.org/abs/1409.4355
Neil J. Ross and Peter Selinger. Optimal ancilla-free Clifford+T approximation of z-rotations. Quantum Information and Computation, 16(11-12):901-953, 2016. URL: http://www.rintonpress.com/xxqic16/qic-16-1112/0901-0953.pdf.
http://www.rintonpress.com/xxqic16/qic-16-1112/0901-0953.pdf
Peter Selinger. Efficient Clifford+T approximation of single-qubit operators. Quantum Information and Computation, 15(1-2):159-180, 2015. URL: http://arxiv.org/abs/1212.6253.
http://arxiv.org/abs/1212.6253
Wim van Dam and Raymond Wong. Two-qubit Stabilizer Circuits with Recovery I: Existence, Mar 2018. URL: http://arxiv.org/abs/1803.06081.
http://arxiv.org/abs/1803.06081
Wim van Dam and Raymond Wong
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Efficient Population Transfer via Non-Ergodic Extended States in Quantum Spin Glass
Quantum tunneling has been proposed as a physical mechanism for solving binary optimization problems on a quantum computer because it provides an alternative to simulated annealing by directly connecting deep local minima of the energy landscape separated by large Hamming distances. However, classical simulations using Quantum Monte Carlo (QMC) were found to efficiently simulate tunneling transitions away from local minima if the tunneling is effectively dominated by a single path. We analyze a new computational role of coherent multi-qubit tunneling that gives rise to bands of non-ergodic extended (NEE) quantum states each formed by a superposition of a large number of deep local minima with similar energies. NEE provide a coherent pathway for population transfer (PT) between computational states with similar energies. In this regime, PT cannot be efficiently simulated by QMC. PT can serve as a new quantum subroutine for quantum search, quantum parallel tempering and reverse annealing optimization algorithms. We study PT resulting from quantum evolution under a transverse field of an n-spin system that encodes the energy function E(z) of an optimization problem over the set of bit configurations z. Transverse field is rapidly switched on in the beginning of algorithm, kept constant for sufficiently long time and switched off at the end. Given an energy function of a binary optimization problem and an initial bit-string with atypically low energy, PT protocol searches for other bitstrings at energies within a narrow window around the initial one. We provide an analytical solution for PT in a simple yet nontrivial model: M randomly chosen marked bit-strings are assigned energies E(z) within a narrow strip [-n -W/2, n + W/2], while the rest of the states are assigned energy 0. The PT starts at a marked state and ends up in a superposition of L marked states inside the narrow energy window whose width is smaller than W. The best known classical algorithm for finding another marked state is the exhaustive search. We find that the scaling of a typical PT runtime with n and L is the same as that in the multi-target Grover's quantum search algorithm, except for a factor that is equal to exp(n /(2B^2)) for finite transverse field B >>1. Unlike the Hamiltonians used in analog quantum unstructured search algorithms known so far, the model we consider is non-integrable and the transverse field delocalizes the marked states. As a result, our PT protocol is not exponentially sensitive in n to the weight of the driver Hamiltonian and may be initialized with a computational basis state. We develop the microscopic theory of PT by constructing a down-folded dense Hamiltonian acting in the space of marked states of dimension M. It belongs to the class of preferred basis Levy matrices (PBLM) with heavy-tailed distribution of the off-diagonal matrix elements. Under certain conditions, the band of the marked states splits into minibands of non-ergodic delocalized states. We obtain an explicit form of the heavy-tailed distribution of PT times by solving cavity equations for the ensemble of down-folded Hamiltonians. We study numerically the PT subroutine as a part of quantum parallel tempering algorithm for a number of examples of binary optimization problems on fully connected graphs.
Quantum algorithms
Discrete optimization
Quantum spin glass
Machine learning
Theory of computation~Quantum computation theory
Theory of computation~Discrete optimization
Theory of computation~Machine learning theory
9:1-9:16
Regular Paper
Kostyantyn
Kechedzhi
Kostyantyn Kechedzhi
QuAIL and USRA, NASA Ames Research Center, Moffett Field, CA 94035, USA , Google Inc., Venice, CA 90291, USA
K.K. acknowledges support by NASA Academic Mission Services, contract number NNA16BD14C. This research is based upon work supported in part by the AFRL Information Directorate under grant F4HBKC4162G001 and the Office of the Director of National Intelligence (ODNI) and the Intelligence Advanced Research Projects Activity (IARPA), via IAA 145483. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of ODNI, IARPA, AFRL, or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purpose notwithstanding any copyright annotation thereon.
Vadim
Smelyanskiy
Vadim Smelyanskiy
Google Inc., Venice, CA 90291, USA
Jarrod R.
McClean
Jarrod R. McClean
Google Inc., Venice, CA 90291, USA
Vasil S.
Denchev
Vasil S. Denchev
Google Inc., Venice, CA 90291, USA
Masoud
Mohseni
Masoud Mohseni
Google Inc., Venice, CA 90291, USA
Sergei
Isakov
Sergei Isakov
Google Inc., Venice, CA 90291, USA
Sergio
Boixo
Sergio Boixo
Google Inc., Venice, CA 90291, USA
Boris
Altshuler
Boris Altshuler
Physics Department, Columbia University, 538 West 120th Street, New York, New York 10027 , USA
Hartmut
Neven
Hartmut Neven
Google Inc., Venice, CA 90291, USA
10.4230/LIPIcs.TQC.2018.9
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Kostyantyn Kechedzhi, Vadim Smelyanskiy, Jarrod R. McClean, Vasil S. Denchev, Masoud Mohseni, Sergei Isakov, Sergio Boixo, and Hartmut Neven
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Quantum Network Code for Multiple-Unicast Network with Quantum Invertible Linear Operations
This paper considers the communication over a quantum multiple-unicast network where r sender-receiver pairs communicate independent quantum states. We concretely construct a quantum network code for the quantum multiple-unicast network as a generalization of the code [Song and Hayashi, arxiv:1801.03306, 2018] for the quantum unicast network. When the given node operations are restricted to invertible linear operations between bit basis states and the rates of transmissions and interferences are restricted, our code certainly transmits a quantum state for each sender-receiver pair by n-use of the network asymptotically, which guarantees no information leakage to the other users. Our code is implemented only by the coding operation in the senders and receivers and employs no classical communication and no manipulation of the node operations. Several networks that our code can be applied are also given.
Quantum network code
Multiple-unicast quantum network
Quantum invertible linear operation
Hardware~Quantum communication and cryptography
10:1-10:20
Regular Paper
Seunghoan
Song
Seunghoan Song
Graduate School of Mathematics, Nagoya University, Nagoya, Japan
Masahito
Hayashi
Masahito Hayashi
Graduate School of Mathematics, Nagoya University, Nagoya, Japan, Centre for Quantum Technologies, National University of Singapore, Singapore, Singapore, Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology, Shenzhen, China
10.4230/LIPIcs.TQC.2018.10
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Masahito Hayashi. Group Representation for Quantum Theory. Springer, 2017.
Masahito Hayashi. Group Theoretic Approach to Quantum Information. Springer, 2017.
Masahito Hayashi, Kazuo Iwama, Harumichi Nishimura, Rudy Raymond, and Shigeru Yamashita. Quantum network coding. In Annual Symposium on Theoretical Aspects of Computer Science, pages 610-621. Springer, 2007.
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Go Kato, Masaki Owari, and Masahito Hayashi. Single-shot secure quantum network coding for general multiple unicast network with free public communication. In International Conference on Information Theoretic Security, pages 166-187. Springer, 2017.
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Seunghoan Song and Masahito Hayashi
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