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On the Entanglement Cost of One-Shot Compression




Shima Bab Hadiashar and Ashwin Nayak

Department of Combinatorics and Optimization, and Institute for Quantum Computing, University of Waterloo, 200 University Ave. W., Waterloo, ON, N2L 3G1, Canada.

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We revisit the task of visible compression of an ensemble of quantum states with entanglement assistance in the one-shot setting. The protocols achieving the best compression use many more qubits of shared entanglement than the number of qubits in the states in the ensemble. Other compression protocols, with potentially larger communication cost, have entanglement cost bounded by the number of qubits in the given states. This motivates the question as to whether entanglement is truly necessary for compression, and if so, how much of it is needed.
Motivated by questions in communication complexity, we lift certain restrictions that are placed on compression protocols in tasks such as state-splitting and channel simulation. We show that an ensemble of the form designed by Jain, Radhakrishnan, and Sen (ICALP’03) saturates the known bounds on the sum of communication and entanglement costs, even with the relaxed compression protocols we study.
The ensemble and the associated one-way communication protocol have several remarkable properties. The ensemble is incompressible by more than a constant number of qubits without shared entanglement, even when constant error is allowed. Moreover, in the presence of shared entanglement, the communication cost of compression can be arbitrarily smaller than the entanglement cost. The quantum information cost of the protocol can thus be arbitrarily smaller than the cost of compression without shared entanglement. The ensemble can also be used to show the impossibility of reducing, via compression, the shared entanglement used in two-party protocols for computing Boolean functions.

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► References

[1] Anurag Anshu and Rahul Jain. Efficient methods for one-shot quantum communication. Technical Report arXiv:1809.07056 [quant-ph],, https:/​/​​abs/​1809.07056, September 2018.

[2] Anurag Anshu, Rahul Jain, Priyanka Mukhopadhyay, Ala Shayeghi, and Penghui Yao. New one shot quantum protocols with application to communication complexity. IEEE Transactions on Information Theory, 62 (12): 7566–7577, 2016. 10.1109/​TIT.2016.2616125.

[3] Anurag Anshu, Dave Touchette, Penghui Yao, and Nengkun Yu. Exponential separation of quantum communication and classical information. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, pages 277–288, New York, NY, USA, 2017. ACM. ISBN 978-1-4503-4528-6. 10.1145/​3055399.3055401.

[4] Shima Bab Hadiashar, Ashwin Nayak, and Renato Renner. Communication complexity of one-shot remote state preparation. IEEE Transactions on Information Theory, 64 (7): 4709–4728, July 2018. 10.1109/​TIT.2018.2811509.

[5] Howard Barnum, Carlton M. Caves, Christopher A. Fuchs, Richard Jozsa, and Benjamin Schumacher. On quantum coding for ensembles of mixed states. Journal of Physics A: Mathematical and General, 34 (35): 6767–6785, August 2001. 10.1088/​0305-4470/​34/​35/​304.

[6] Charles H. Bennett, Igor Devetak, Aram W. Harrow, Peter W. Shor, and Andreas Winter. The quantum reverse Shannon theorem and resource tradeoffs for simulating quantum channels. IEEE Transactions on Information Theory, 60 (5): 2926–2959, May 2014. ISSN 0018-9448. 10.1109/​TIT.2014.2309968.

[7] Mario Berta, Matthias Christandl, and Renato Renner. The quantum reverse Shannon theorem based on one-shot information theory. Communications in Mathematical Physics, 306 (3): 579–615, August 2011. ISSN 1432-0916. 10.1007/​s00220-011-1309-7.

[8] Mario Berta, Matthias Christandl, and Dave Touchette. Smooth entropy bounds on one-shot quantum state redistribution. IEEE Transactions on Information Theory, 62 (3): 1425–1439, March 2016. 10.1109/​TIT.2016.2516006.

[9] Igor Devetak. Triangle of dualities between quantum communication protocols. Physical Review Letters, 97: 140503, Oct 2006. 10.1103/​PhysRevLett.97.140503.

[10] Igor Devetak and Jon Yard. Exact cost of redistributing multipartite quantum states. Physical Review Letters, 100 (230501), June 2008. 10.1103/​PhysRevLett.100.230501.

[11] Christopher A. Fuchs and Jeroen van de Graaf. Cryptographic distinguishability measures for quantum-mechanical states. IEEE Transactions on Information Theory, 45 (4): 1216–1227, May 1999. ISSN 1557-9654. 10.1109/​18.761271.

[12] Alexei Gilchrist, Nathan K. Langford, and Michael A. Nielsen. Distance measures to compare real and ideal quantum processes. Physical Review A, 71: 062310, Jun 2005. 10.1103/​PhysRevA.71.062310.

[13] Carl W. Helstrom. Detection theory and quantum mechanics. Information and Control, 10 (3): 254–291, 1967. 10.1016/​S0019-9958(67)90302-6.

[14] Alexander S. Holevo. An analogue of statistical decision theory and noncommutative probability theory. Trudy Moskovskogo Matematicheskogo Obshchestva, 26: 133–149, 1972.

[15] Michał Horodecki, Jonathan Oppenheim, and Andreas Winter. Partial quantum information. Nature, 436 (7051): 673–676, August 2005. 10.1038/​nature03909.

[16] Michał Horodecki, Jonathan Oppenheim, and Andreas Winter. Quantum state merging and negative information. Communications in Mathematical Physics, 269 (1): 107–136, January 2007. 10.1007/​s00220-006-0118-x.

[17] Rahul Jain, Jaikumar Radhakrishnan, and Pranab Sen. A direct sum theorem in communication complexity via message compression. In Jos C. M. Baeten, Jan Karel Lenstra, Joachim Parrow, and Gerhard J. Woeginger, editors, Automata, Languages and Programming, volume 2719 of Lecture Notes in Computer Science, pages 300–315, Berlin, Heidelberg, 2003. Springer Berlin Heidelberg. ISBN 978-3-540-45061-0. 10.1007/​3-540-45061-0_26.

[18] Rahul Jain, Jaikumar Radhakrishnan, and Pranab Sen. Prior entanglement, message compression and privacy in quantum communication. In Proceedings of the 20th Annual IEEE Conference on Computational Complexity, pages 285–296. IEEE Computer Society, 2005. 10.1109/​CCC.2005.24.

[19] Rahul Jain, Pranab Sen, and Jaikumar Radhakrishnan. Optimal direct sum and privacy trade-off results for quantum and classical communication complexity. Technical Report arXiv:0807.1267v1 [cs.DC],, https:/​/​​abs/​0807.1267, July 2008.

[20] Zahra B. Khanian and Andreas Winter. Entanglement-assisted quantum data compression. In 2019 IEEE International Symposium on Information Theory (ISIT), pages 1147–1151, 2019. 10.1109/​ISIT.2019.8849352.

[21] Troy Lee and Adi Shraibman. Lower bounds in communication complexity. Foundations and Trends in Theoretical Computer Science, 3 (4): 263–399, 2009. ISSN 1551-305X. 10.1561/​0400000040.

[22] Zi-Wen Liu, Christopher Perry, Yechao Zhu, Dax Enshan Koh, and Scott Aaronson. Doubly infinite separation of quantum information and communication. Phys. Rev. A, 93: 012347, Jan 2016. 10.1103/​PhysRevA.93.012347.

[23] Zhicheng Luo and Igor Devetak. Channel simulation with quantum side information. IEEE Transactions on Information Theory, 55 (3): 1331–1342, March 2009. ISSN 0018-9448. 10.1109/​TIT.2008.2011424.

[24] Jiří Matoušek. Lectures on Discrete Geometry, volume 212 of Graduate Texts in Mathematics. Springer-Verlag New York, 1st edition, 2002. ISBN 978-0-387-95373-1. 10.1007/​978-1-4613-0039-7.

[25] Elizabeth S. Meckes. The Random Matrix Theory of the Classical Compact Groups, volume 218 of Cambridge Tracts in Mathematics. Cambridge University Press, July 2019. 10.1017/​9781108303453.

[26] Vitali D. Milman and Gideon Schechtman. Asymptotic Theory of Finite Dimensional Normed Spaces, volume 1200 of Lecture notes in mathematics. Springer-Verlag Berlin Heidelberg, 1986. 10.1007/​978-3-540-38822-7.

[27] Marco Tomamichel, Roger Colbeck, and Renato Renner. Duality between smooth min- and max-entropies. IEEE Transactions on Information Theory, 56 (9): 4674–4681, September 2010. ISSN 0018-9448, 1557-9654. 10.1109/​TIT.2010.2054130.

[28] Dave Touchette. Quantum information complexity. In Proceedings of the Forty-seventh Annual ACM Symposium on Theory of Computing, STOC ’15, pages 317–326, New York, NY, USA, 2015. ACM. ISBN 978-1-4503-3536-2. 10.1145/​2746539.2746613.

[29] John Watrous. The Theory of Quantum Information. Cambridge University Press, May 2018. 10.1017/​9781316848142.

[30] Jon T. Yard and Igor Devetak. Optimal quantum source coding with quantum side information at the encoder and decoder. IEEE Transactions on Information Theory, 55 (11): 5339–5351, November 2009. ISSN 0018-9448. 10.1109/​TIT.2009.2030494.

Cited by

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What can you do in 48 hours?




Have you ever wondered what can be done in 48 hours? For instance, our heart beats around 200 000 times. One of the biggest supercomputers crunches petabytes (peta = 1015) of numbers to simulate an experiment that took Google’s quantum processor only 300 seconds to run. In 48 hours, one can also participate in the Sciathon with almost 500 young researchers from more than 80 countries! 

Two weeks ago I participated in a scientific marathon, the Sciathon. The structure of this event roughly resembled a hackathon. I am sure many readers are familiar with the idea of a hackathon from personal experience. For those unfamiliar — a hackathon is an intense collaborative event, usually organized over the weekend, during which people with different backgrounds work in groups to create prototypes of functioning software or hardware. For me, it was the very first time to have firsthand experience with a hackathon-like event!

The Sciathon was organized by the Lindau Nobel Laureate Meetings (more about the meetings with Nobel laureates, which happen annually in the lovely German town of Lindau, in another blogpost, I promise!) This year, unfortunately, the face-to-face meeting in Lindau was postponed until the summer of 2021. Instead, the Lindau Nobel Laureate Meetings alumni and this year’s would-be attendees had an opportunity to gather for the Sciathon, as well as the Online Science Days earlier this week, during which the best Sciathon projects were presented.

The participants of the Sciathon could choose to contribute new views, perspectives and solutions to three main topics: Lindau Guidelines, Communicating Climate Change and Capitalism After Corona. The first topic concerned an open, cooperative science community where data and knowledge are freely shared, the second — how scientists could show that the climate crisis is just as big a threat as the SARS-CoV-19 virus, and the last — how to remodel our current economic systems so that they are more robust to unexpected sudden crises. More detailed descriptions of each topic can be found on the official Sciathon webpage.

My group of ten eager scientists, mostly physicists, from master students to postdoctoral researchers, focused on the first topic. In particular, our goal was to develop a method of familiarizing high school students with the basics of quantum information and computation. We envisioned creating an online notebook, where an engaging story would be intertwined with interactive blocks of Python code utilizing the open-source quantum computing toolkit Qiskit. This hands-on approach would enable students to play with quantum systems described in the story-line by simply running the pre-programmed commands with a click of the mouse and then observe how “experiment” matches “the theory”. We decided to work with a system comprising one or two qubits and explain such fundamental concepts in quantum physics as superposition, entanglement and measurement. The last missing part was a captivating story.

The story we came up with involved two good friends from the lab, Miss Schrödinger and Miss Pauli, as well as their kittens, Alice and Bob. At first, Alice and Bob seemed to be ordinary cats, however whenever they sipped quantum milk, they would turn into quantum cats, or as quantum physicists would say — kets. Do I have to remind the reader that a quantum cat, unlike an ordinary one, could be both awake and asleep at the same time?

Miss Schrödinger was a proud cat owner who not only loved her cat, but also would take hundreds of pictures of Alice and eagerly upload them on social media. Much to Miss Schrödinger’s surprise, none of the pictures showed Alice partly awake and partly asleep — the ket would always collapse to the cat awake or the cat asleep! Every now and then, Miss Pauli would come to visit Miss Schrödinger and bring her own cat Bob. While the good friends were chit-chatting over a cup of afternoon tea, the cats sipped a bit of quantum milk and started to play with a ball of wool, resulting in a cute mess of two kittens tangled up in wool. Every time after coming back home, Miss Pauli would take a picture of Bob and share it with Miss Schrödinger, who would obviously also take a picture of Alice. After a while, the young scientists started to notice some strange correlations between the states of their cats… 

The adventures of Miss Schrödinger and her cat continue! For those interested, you can watch a short video about our project! 

Overall, I can say that I had a lot of fun participating in the Sciathon. It was an intense yet extremely gratifying event. In addition to the obvious difficulty of racing against the clock, our group also had to struggle with coordinating video calls between group members scattered across three almost equidistant time zones — Eastern Australian, Central European and Central US! During the Sciathon I had a chance to interact with other science enthusiasts from different backgrounds and work on something from outside my area of expertise. I would strongly encourage anyone to participate in hackathon-like events to break the daily routine, particularly monotonous during the lockdown, and unleash one’s creative spirit. Such events can also be viewed as an opportunity to communicate science and scientific progress to the public. Lastly, I would like to thank other members of my team — collaborating with you during the Sciathon was a blast!

During the Sciathon, we had many brainstorming sessions. You can see most of the members of my group in this video call (from left to right, top to bottom): Shuang, myself, Martin, Kyle, Hadewijch, Saskia, Michael and Bartłomiej. The team also included Ahmed and Watcharaphol.


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Optimal probes and error-correction schemes in multi-parameter quantum metrology




Wojciech Górecki1, Sisi Zhou2,3,4, Liang Jiang2,3,4, and Rafał Demkowicz-Dobrzański1

1Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland
2Departments of Applied Physics and Physics, Yale University, New Haven, Connecticut 06511, USA
3Yale Quantum Institute, Yale University, New Haven, Connecticut 06511, USA
4Pritzker School of Molecular Engineering, University of Chicago, Chicago, IL 60637, USA

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We derive a necessary and sufficient condition for the possibility of achieving the Heisenberg scaling in general adaptive multi-parameter estimation schemes in presence of Markovian noise. In situations where the Heisenberg scaling is achievable, we provide a semidefinite program to identify the optimal quantum error correcting (QEC) protocol that yields the best estimation precision. We overcome the technical challenges associated with potential incompatibility of the measurement optimally extracting information on different parameters by utilizing the Holevo Cramér-Rao (HCR) bound for pure states. We provide examples of significant advantages offered by our joint-QEC protocols, that sense all the parameters utilizing a single error-corrected subspace, over separate-QEC protocols where each parameter is effectively sensed in a separate subspace.

► BibTeX data

► References

[1] V. Giovannetti, S. Lloyd, and L. Maccone, Quantum metrology, Phys. Rev. Lett. 96, 010401 (2006).

[2] M. G. A. Paris, Quantum estimation for quantum technologies, Int. J. Quantum Inf. 07, 125 (2009).

[3] V. Giovannetti, S. Lloyd, and L. Maccone, Advances in quantum metrology, Nat. Photonics 5, 222 (2011).

[4] G. Toth and I. Apellaniz, Quantum metrology from a quantum information science perspective, J. Phys. A: Math. Theor. 47, 424006 (2014).

[5] R. Demkowicz-Dobrzański, M. Jarzyna, and J. Kołodyński, in Prog. Optics, Vol. 60, edited by E. Wolf (Elsevier, 2015) pp. 345–435.

[6] R. Schnabel, Squeezed states of light and their applications in laser interferometers, Phys. Rep. 684, 1 (2017).

[7] C. L. Degen, F. Reinhard, and P. Cappellaro, Quantum sensing, Rev. Mod. Phys. 89, 035002 (2017).

[8] L. Pezzè, A. Smerzi, M. K. Oberthaler, R. Schmied, and P. Treutlein, Quantum metrology with nonclassical states of atomic ensembles, Rev. Mod. Phys. 90, 035005 (2018).

[9] S. Pirandola, B. R. Bardhan, T. Gehring, C. Weedbrook, and S. Lloyd, Advances in photonic quantum sensing, Nat. Photonics 12, 724 (2018).

[10] C. M. Caves, Quantum-mechanical noise in an interferometer, Phys. Rev. D 23, 1693 (1981).

[11] M. Holland and K. Burnett, Interferometric detection of optical phase shifts at the heisenberg limit, Phys. Rev. Lett. 71, 1355 (1993).

[12] H. Lee, P. Kok, and J. P. Dowling, A quantum rosetta stone for interferometry, J. Mod. Optic. 49, 2325 (2002).

[13] D. Wineland, J. Bollinger, W. Itano, F. Moore, and D. Heinzen, Spin squeezing and reduced quantum noise in spectroscopy, Phys. Rev. A 46, R6797 (1992).

[14] K. McKenzie, D. A. Shaddock, D. E. McClelland, B. C. Buchler, and P. K. Lam, Experimental demonstration of a squeezing-enhanced power-recycled michelson interferometer for gravitational wave detection, Phys. Rev. Lett. 88, 231102 (2002).

[15] J. Bollinger, W. M. Itano, D. Wineland, and D. Heinzen, Optimal frequency measurements with maximally correlated states, Phys. Rev. A 54, R4649 (1996).

[16] D. Leibfried, M. Barrett, T. Schaetz, J. Britton, J. Chiaverini, W. Itano, J. Jost, C. Langer, and D. Wineland, Toward heisenberg-limited spectroscopy with multiparticle entangled states, Science 304, 1476 (2004).

[17] V. Giovannetti, S. Lloyd, and L. Maccone, Quantum-enhanced measurements: beating the standard quantum limit, Science 306, 1330 (2004).

[18] S. F. Huelga, C. Macchiavello, T. Pellizzari, A. K. Ekert, M. B. Plenio, and J. I. Cirac, Improvement of frequency standards with quantum entanglement, Phys. Rev. Lett. 79, 3865 (1997).

[19] D. W. Berry and H. M. Wiseman, Optimal states and almost optimal adaptive measurements for quantum interferometry, Phys. Rev. Lett. 85, 5098 (2000).

[20] M. de Burgh and S. D. Bartlett, Quantum methods for clock synchronization: Beating the standard quantum limit without entanglement, Phys. Rev. A 72, 042301 (2005).

[21] A. Fujiwara and H. Imai, A fibre bundle over manifolds of quantum channels and its application to quantum statistics, J. Phys. A: Math. Theor. 41, 255304 (2008).

[22] R. Demkowicz-Dobrzański, U. Dorner, B. Smith, J. Lundeen, W. Wasilewski, K. Banaszek, and I. Walmsley, Quantum phase estimation with lossy interferometers, Phys. Rev. A 80, 013825 (2009).

[23] B. Escher, R. de Matos Filho, and L. Davidovich, General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology, Nat. Phys. 7, 406 (2011).

[24] R. Demkowicz-Dobrzański, J. Kołodyński, and M. Guţă, The elusive heisenberg limit in quantum-enhanced metrology, Nat. Commun. 3, 1063 (2012).

[25] J. Kołodyński and R. Demkowicz-Dobrzański, Efficient tools for quantum metrology with uncorrelated noise, New J. Phys. 15, 073043 (2013).

[26] S. I. Knysh, E. H. Chen, and G. A. Durkin, True limits to precision via unique quantum probe, arXiv:1402.0495 (2014).

[27] R. Demkowicz-Dobrzański and L. Maccone, Using entanglement against noise in quantum metrology, Phys. Rev. Lett. 113, 250801 (2014).

[28] E. M. Kessler, I. Lovchinsky, A. O. Sushkov, and M. D. Lukin, Quantum error correction for metrology, Phys. Rev. Lett. 112, 150802 (2014).

[29] W. Dür, M. Skotiniotis, F. Froewis, and B. Kraus, Improved quantum metrology using quantum error correction, Phys. Rev. Lett. 112, 080801 (2014).

[30] R. Ozeri, Heisenberg limited metrology using quantum error-correction codes. arXiv:1310.3432 (2013).

[31] G. Arrad, Y. Vinkler, D. Aharonov, and A. Retzker, Increasing sensing resolution with error correction, Phys. Rev. Lett. 112, 150801 (2014).

[32] T. Unden, P. Balasubramanian, D. Louzon, Y. Vinkler, M. B. Plenio, M. Markham, D. Twitchen, A. Stacey, I. Lovchinsky, A. O. Sushkov, et al., Quantum metrology enhanced by repetitive quantum error correction, Phys. Rev. Lett. 116, 230502 (2016).

[33] F. Reiter, A. S. Sørensen, P. Zoller, and C. A. Muschik, Dissipative quantum error correction and application to quantum sensing with trapped ions, Nat. Commun. 8, 1822 (2017).

[34] P. Sekatski, M. Skotiniotis, J. Kołodyński, and W. Dür, Quantum metrology with full and fast quantum control, Quantum 1, 27 (2017).

[35] R. Demkowicz-Dobrzański, J. Czajkowski, and P. Sekatski, Adaptive quantum metrology under general markovian noise, Phys. Rev. X 7, 041009 (2017).

[36] S. Zhou, M. Zhang, J. Preskill, and L. Jiang, Achieving the heisenberg limit in quantum metrology using quantum error correction, Nat. Commun. 9, 78 (2018).

[37] D. Layden and P. Cappellaro, Spatial noise filtering through error correction for quantum sensing, npj Quantum Inf. 4, 30 (2018).

[38] D. Layden, S. Zhou, P. Cappellaro, and L. Jiang, Ancilla-free quantum error correction codes for quantum metrology, Phys. Rev. Lett. 122, 040502 (2019).

[39] T. Kapourniotis and A. Datta, Fault-tolerant quantum metrology, Phys. Rev. A 100, 022335 (2019).

[40] K. C. Tan, S. Omkar, and H. Jeong, Quantum-error-correction-assisted quantum metrology without entanglement, Phys. Rev. A 100, 022312 (2019).

[41] S. Zhou and L. Jiang, The theory of entanglement-assisted metrology for quantum channels, arXiv:2003.10559 (2020a).

[42] Y. Chen, H. Chen, J. Liu, Z. Miao, and H. Yuan, Fluctuation-enhanced quantum metrology, arXiv:2003.13010 (2020).

[43] T. Baumgratz and A. Datta, Quantum enhanced estimation of a multidimensional field, Phys. Rev. Lett. 116, 030801 (2016).

[44] M. Tsang, R. Nair, and X.-M. Lu, Quantum theory of superresolution for two incoherent optical point sources, Phys. Rev. X 6, 031033 (2016).

[45] P. C. Humphreys, M. Barbieri, A. Datta, and I. A. Walmsley, Quantum enhanced multiple phase estimation, Phys. Rev. Lett. 111, 070403 (2013).

[46] M. Gessner, L. Pezzè, and A. Smerzi, Sensitivity bounds for multiparameter quantum metrology, Phys. Rev. Lett. 121, 130503 (2018).

[47] M. Tsang, H. M. Wiseman, and C. M. Caves, Fundamental quantum limit to waveform estimation, Phys. Rev. Lett. 106, 090401 (2011).

[48] D. W. Berry, M. J. W. Hall, and H. M. Wiseman, Stochastic heisenberg limit: Optimal estimation of a fluctuating phase, Phys. Rev. Lett. 111, 113601 (2013).

[49] K. Matsumoto, A new approach to the cramér-rao-type bound of the pure-state model, J. Phys. A.: Math. Theor. 35, 3111 (2002).

[50] M. G. Genoni, M. G. A. Paris, G. Adesso, H. Nha, P. L. Knight, and M. S. Kim, Optimal estimation of joint parameters in phase space, Phys. Rev. A 87, 012107 (2013).

[51] S. Ragy, M. Jarzyna, and R. Demkowicz-Dobrzański, Compatibility in multiparameter quantum metrology, Phys. Rev. A 94, 052108 (2016).

[52] H. Yuan, Sequential feedback scheme outperforms the parallel scheme for hamiltonian parameter estimation, Phys. Rev. Lett. 117, 160801 (2016).

[53] N. Kura and M. Ueda, Finite-error metrological bounds on multiparameter hamiltonian estimation, Phys. Rev. A 97, 012101 (2018).

[54] J. Liu and H. Yuan, Control-enhanced multiparameter quantum estimation, Phys. Rev. A 96, 042114 (2017).

[55] R. Nichols, P. Liuzzo-Scorpo, P. A. Knott, and G. Adesso, Multiparameter gaussian quantum metrology, Phys. Rev. A 98, 012114 (2018).

[56] W. Ge, K. Jacobs, Z. Eldredge, A. V. Gorshkov, and M. Foss-Feig, Distributed quantum metrology with linear networks and separable inputs, Phys. Rev. Lett. 121, 043604 (2018).

[57] S. L. Braunstein and C. M. Caves, Statistical distance and the geometry of quantum states, Phys. Rev. Lett. 72, 3439 (1994).

[58] C. W. Helstrom, Quantum detection and estimation theory (Academic press, 1976).

[59] A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory (North Holland, Amsterdam, 1982).

[60] R. Demkowicz-Dobrzanski, W. Gorecki, and M. Guta, Multi-parameter estimation beyond quantum fisher information, Journal of Physics A: Mathematical and Theoretical (2020).

[61] H. Nagaoka and M. Hayashi, Asymptotic Theory of Quantum Statistical Inference (World Scientific Singapore, 2005) Chap. 8.

[62] J. Suzuki, Explicit formula for the holevo bound for two-parameter qubit-state estimation problem, J. Math. Phys. 57, 042201 (2016).

[63] M. Guţă and A. Jenčová, Local asymptotic normality in quantum statistics, Comm. Math. Phys. 276, 341 (2007).

[64] K. Yamagata, A. Fujiwara, R. D. Gill, et al., Quantum local asymptotic normality based on a new quantum likelihood ratio, Ann. Statist. 41, 2197 (2013).

[65] A. Fujiwara, Multi-parameter pure state estimation based on the right logarithmic derivative, Math. Eng. Tech. Rep 94, 94 (1994).

[66] F. Albarelli, J. F. Friel, and A. Datta, Evaluating the holevo cramér-rao bound for multiparameter quantum metrology, Phys. Rev. Lett. 123, 200503 (2019).

[67] G. Lindblad, On the generators of quantum dynamical semigroups, Comm. Math. Phys. 48, 119 (1976).

[68] V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, Completely positive dynamical semigroups of n-level systems, J. Math. Phys. 17, 821 (1976).

[69] H.-P. Breuer, F. Petruccione, et al., The theory of open quantum systems (Oxford University Press on Demand, 2002).

[70] S. M. Kay, Fundamentals of statistical signal processing: estimation theory (Prentice Hall, 1993).

[71] R. Gill and S. Massar, State estimation for large ensembles, Phys. Rev. A 61, 042312 (2000).

[72] E. Knill and R. Laflamme, Theory of quantum error-correcting codes, Phys. Rev. A 55, 900 (1997).

[73] M. Grant and S. Boyd, Cvx: Matlab software for disciplined convex programming,.

[74] S. Zhou and L. Jiang, Optimal approximate quantum error correction for quantum metrology, Phys. Rev. Research 2, 013235 (2020b).

[75] W. Górecki, R. Demkowicz-Dobrzański, H. M. Wiseman, and D. W. Berry, ${pi}$-corrected heisenberg limit, Phys. Rev. Lett. 124, 030501 (2020).

[76] D. A. Lidar, I. L. Chuang, and K. B. Whaley, Decoherence-free subspaces for quantum computation, Phys. Rev. Lett. 81, 2594 (1998).

Cited by

[1] Philippe Faist, Sepehr Nezami, Victor V. Albert, Grant Salton, Fernando Pastawski, Patrick Hayden, and John Preskill, “Continuous symmetries and approximate quantum error correction”, arXiv:1902.07714.

[2] Francesco Albarelli, Jamie F. Friel, and Animesh Datta, “Evaluating the Holevo Cramér-Rao Bound for Multiparameter Quantum Metrology”, Physical Review Letters 123 20, 200503 (2019).

[3] Francesco Albarelli, Mankei Tsang, and Animesh Datta, “Upper bounds on the Holevo Cramér-Rao bound for multiparameter quantum parametric and semiparametric estimation”, arXiv:1911.11036.

[4] F. Albarelli, M. Barbieri, M. G. Genoni, and I. Gianani, “A perspective on multiparameter quantum metrology: From theoretical tools to applications in quantum imaging”, Physics Letters A 384, 126311 (2020).

[5] Yingkai Ouyang, Nathan Shettell, and Damian Markham, “Robust quantum metrology with explicit symmetric states”, arXiv:1908.02378.

[6] Emanuele Polino, Mauro Valeri, Nicolò Spagnolo, and Fabio Sciarrino, “Photonic Quantum Metrology”, arXiv:2003.05821.

[7] Sisi Zhou and Liang Jiang, “Optimal approximate quantum error correction for quantum metrology”, Physical Review Research 2 1, 013235 (2020).

[8] Rafal Demkowicz-Dobrzanski, Wojciech Gorecki, and Madalin Guta, “Multi-parameter estimation beyond Quantum Fisher Information”, arXiv:2001.11742.

[9] Sisi Zhou and Liang Jiang, “The theory of entanglement-assisted metrology for quantum channels”, arXiv:2003.10559.

[10] Aleksander Kubica and Rafal Demkowicz-Dobrzanski, “Using Quantum Metrological Bounds in Quantum Error Correction: A Simple Proof of the Approximate Eastin-Knill Theorem”, arXiv:2004.11893.

[11] Alexander Predko, Francesco Albarelli, and Alessio Serafini, “Time-local optimal control for parameter estimation in the Gaussian regime”, Physics Letters A 384, 126268 (2020).

[12] Le Bin Ho, Hideaki Hakoshima, Yuichiro Matsuzaki, Masayuki Matsuzaki, and Yasushi Kondo, “Multiparameter quantum estimation under dephasing noise”, arXiv:2004.00720.

The above citations are from SAO/NASA ADS (last updated successfully 2020-07-02 13:02:52). The list may be incomplete as not all publishers provide suitable and complete citation data.

Could not fetch Crossref cited-by data during last attempt 2020-07-02 13:02:50: Could not fetch cited-by data for 10.22331/q-2020-07-02-288 from Crossref. This is normal if the DOI was registered recently.


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Efficient Quantum Walk Circuits for Metropolis-Hastings Algorithm




Jessica Lemieux1, Bettina Heim2, David Poulin1,3, Krysta Svore2, and Matthias Troyer2

1Département de Physique & Institut Quantique, Université de Sherbrooke, Québec, Canada
2Quantum Architecture and Computation Group, Microsoft Research, Redmond, WA 98052, USA
3Canadian Institute for Advanced Research, Toronto, Ontario, Canada M5G 1Z8

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We present a detailed circuit implementation of Szegedy’s quantization of the Metropolis-Hastings walk. This quantum walk is usually defined with respect to an oracle. We find that a direct implementation of this oracle requires costly arithmetic operations. We thus reformulate the quantum walk, circumventing its implementation altogether by closely following the classical Metropolis-Hastings walk. We also present heuristic quantum algorithms that use the quantum walk in the context of discrete optimization problems and numerically study their performances. Our numerical results indicate polynomial quantum speedups in heuristic settings.

► BibTeX data

► References

[1] Dorit Aharonov and Amnon Ta-Shma. Adiabatic quantum state generation and statistical zero knowledge. In Proceedings of the thirty-fifth ACM symposium on Theory of computing – STOC ’03, page 20, New York, New York, USA, 2003. ACM Press. ISBN 1581136749. 10.1145/​780542.780546.

[2] Andris Ambainis. Quantum walk algorithm for element distinctness. In Proceedings – Annual IEEE Symposium on Foundations of Computer Science, FOCS, pages 22–31, 2004. 10.1109/​focs.2004.54.

[3] Francisco Barahona. On the computational complexity of Ising spin glass models. Journal of Physics A: Mathematical and General, 15: 3241–3253, 1982. 10.1088/​0305-4470/​15/​10/​028.

[4] Adriano Barenco, Charles H. Bennett, Richard Cleve, David P. Divincenzo, Norman Margolus, Peter Shor, Tycho Sleator, John A. Smolin, and Harald Weinfurter. Elementary gates for quantum computation. Physical Review A, 52 (5): 3457–3467, nov 1995. ISSN 10502947. 10.1103/​PhysRevA.52.3457.

[5] Alex Bocharov, Martin Roetteler, and Krysta M. Svore. Efficient synthesis of probabilistic quantum circuits with fallback. Physical Review A – Atomic, Molecular, and Optical Physics, 91 (5): 052317, may 2015. ISSN 10941622. 10.1103/​PhysRevA.91.052317.

[6] S. Boixo, E. Knill, and R. D. Somma. Fast quantum algorithms for traversing paths of eigenstates. may 2010. URL https:/​/​​abs/​1005.3034.

[7] Sergio Boixo, Emanuel Knill, and Rolando Somma. Eigenpath traversal by phase randomization. Quantum Information and Computation, 9 (9&10): 0833, 2009. URL http:/​/​​abs/​0903.1652.

[8] 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, nov 2012. ISSN 1367-2630. 10.1088/​1367-2630/​14/​11/​115023.

[9] Edward Farhi, Jeffrey Goldstone, Sam Gutmann, and Michael Sipser. Quantum Computation by Adiabatic Evolution. jan 2000. URL http:/​/​​abs/​quant-ph/​0001106.

[10] Roy J. Glauber. Time-dependent statistics of the Ising model. Journal of Mathematical Physics, 4 (2): 294–307, feb 1963. ISSN 00222488. 10.1063/​1.1703954.

[11] Jeongwan Haah. Product Decomposition of Periodic Functions in Quantum Signal Processing. jun 2018. 10.22331/​q-2019-10-07-190.

[12] W K Hastings. Monte Carlo sampling methods using Markov chains and their applications. Biometrika, 57 (1): 97–109, apr 1970. ISSN 0006-3444. 10.1093/​biomet/​57.1.97.

[13] Janus Collaboration, F. Belletti, M. Cotallo, A. Cruz, L. A. Fernández, A. Gordillo, M. Guidetti, A. Maiorano, F. Mantovani, E. Marinari, V. Martín-Mayor, A. Muñoz-Sudupe, D. Navarro, G. Parisi, S. Pérez-Gaviro, M. Rossi, J. J. Ruiz-Lorenzo, S. F. Schifano, D. Sciretti, A. Tarancón, R. Tripiccione, and J. L. Velasco. JANUS: an FPGA-based System for High Performance Scientific Computing. Computing in Science & Engineering, 11 (1): 48–58, 2009. 10.1109/​MCSE.2009.11.

[14] Janus Collaboration, M. Baity-Jesi, R. A. Banos, A. Cruz, L. A. Fernandez, J. M. Gil-Narvion, A. Gordillo-Guerrero, M. Guidetti, D. Iniguez, A. Maiorano, F. Mantovani, E. Marinari, V. Martin-Mayor, J. Monforte-Garcia, A. Munoz Sudupe, D. Navarro, G. Parisi, M. Pivanti, S. Perez-Gaviro, F. Ricci-Tersenghi, J. J. Ruiz-Lorenzo, S. F. Schifano, B. Seoane, A. Tarancon, P. Tellez, R. Tripiccione, and D. Yllanes. Reconfigurable computing for Monte Carlo simulations: results and prospects of the Janus project. The European Physical Journal Special Topics, 210 (33), 2012. 10.1140/​epjst/​e2012-01636-9.

[15] S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi. Optimization by simulated annealing. Science, 220 (4598): 671–680, 1983. ISSN 00368075. 10.1126/​science.220.4598.671.

[16] A. Yu. Kitaev. Quantum measurements and the Abelian Stabilizer Problem. nov 1995. URL http:/​/​​abs/​quant-ph/​9511026.

[17] Jessica Lemieux, Guillaume Duclos-Cianci, David Sénéchal, and David Poulin. Resource estimate for quantum many-body ground state preparation on a quantum computer. 2020. URL https:/​/​​abs/​2006.04650.

[18] Guang Hao Low and Isaac L. Chuang. Hamiltonian Simulation by Qubitization. oct 2016. 10.22331/​q-2019-07-12-163.

[19] Guang Hao Low and Isaac L. Chuang. Optimal Hamiltonian Simulation by Quantum Signal Processing. Physical Review Letters, 118 (1): 010501, jan 2017. ISSN 10797114. 10.1103/​PhysRevLett.118.010501.

[20] Guang Hao Low, Theodore J. Yoder, and Isaac L. Chuang. Methodology of resonant equiangular composite quantum gates. Physical Review X, 6 (4): 041067, dec 2016. ISSN 21603308. 10.1103/​PhysRevX.6.041067.

[21] F. Magniez, A. Nayak, J. Roland, and M. Santha. Search via quantum walk. SIAM Journal on Computing, 40: 142–164. 10.1137/​090745854.

[22] Chris Marriott and John Watrous. Quantum Arthur-Merlin games. In Computational Complexity, volume 14, pages 122–152. Springer, jun 2005. 10.1007/​s00037-005-0194-x.

[23] Nicholas Metropolis, Arianna W. Rosenbluth, Marshall N. Rosenbluth, Augusta H. Teller, and Edward Teller. Equation of state calculations by fast computing machines. The Journal of Chemical Physics, 21 (6): 1087–1092, jun 1953. ISSN 00219606. 10.1063/​1.1699114.

[24] Troels F. Rønnow, Zhihui Wang, Joshua Job, Sergio Boixo, Sergei V. Isakov, David Wecker, John M. Martinis, Daniel A. Lidar, and Matthias Troyer. Defining and detecting quantum speedup. Science, 345 (6195): 420–424, jul 2014. ISSN 10959203. 10.1126/​science.1252319.

[25] Neil J Ross and Peter Selinger. Optimal ancilla-free Clifford+T approximation of Z-rotations. Quantum Information and Computation, 16 (11&12): 0901, 2016. URL http:/​/​​abs/​1403.2975.

[26] Terry Rudolph and Lov Grover. A 2 rebit gate universal for quantum computing. oct 2002. URL https:/​/​​abs/​quant-ph/​0210187.

[27] R. D. Somma, S. Boixo, H. Barnum, and E. Knill. Quantum simulations of classical annealing processes. Physical Review Letters, 101 (13): 130504, sep 2008. ISSN 00319007. 10.1103/​PhysRevLett.101.130504.

[28] Mario Szegedy. Quantum speed-up of Markov Chain based algorithms. In Proceedings – Annual IEEE Symposium on Foundations of Computer Science, FOCS, pages 32–41, 2004. 10.1109/​focs.2004.53.

[29] K. Temme, T. J. Osborne, K. G. Vollbrecht, D. Poulin, and F. Verstraete. Quantum Metropolis sampling. Nature, 471 (7336): 87–90, mar 2011. ISSN 00280836. 10.1038/​nature09770.

[30] Marija Vucelja. Lifting—A nonreversible Markov chain Monte Carlo algorithm. American Journal of Physics, 84 (12): 958–968, dec 2016. ISSN 0002-9505. 10.1119/​1.4961596.

[31] Man-Hong Yung and Alán Aspuru-Guzik. A quantum-quantum Metropolis algorithm. Proceedings of the National Academy of Sciences of the United States of America, 109 (3): 754–9, jan 2012. ISSN 1091-6490. 10.1073/​pnas.1111758109.

Cited by

[1] Jessica Lemieux, Guillaume Duclos-Cianci, David Sénéchal, and David Poulin, “Resource estimate for quantum many-body ground state preparation on a quantum computer”, arXiv:2006.04650.

The above citations are from SAO/NASA ADS (last updated successfully 2020-06-29 12:29:48). The list may be incomplete as not all publishers provide suitable and complete citation data.

Could not fetch Crossref cited-by data during last attempt 2020-06-29 12:29:47: Could not fetch cited-by data for 10.22331/q-2020-06-29-287 from Crossref. This is normal if the DOI was registered recently.


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