Estimating expectation values using approximate quantum states

Marco Paini1, Amir Kalev2, Dan Padilha1, and Brendan Ruck3

1Rigetti Computing, 138 Holborn, London, EC1N 2SW, UK.
2Information Sciences Institute, University of Southern California, Arlington, VA 22203, USA.
3Rigetti Computing, 2919 Seventh St, Berkeley, CA 94710, USA.

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

We introduce an approximate description of an $N$-qubit state, which contains sufficient information to estimate the expectation value of any observable to a precision that is upper bounded by the ratio of a suitably-defined seminorm of the observable to the square root of the number of the system's identical preparations $M$, with no explicit dependence on $N$. We describe an operational procedure for constructing the approximate description of the state that requires, besides the quantum state preparation, only single-qubit rotations followed by single-qubit measurements. We show that following this procedure, the cardinality of the resulting description of the state grows as $3MN$. We test the proposed method on Rigetti's quantum processor unit with 12, 16 and 25 qubits for random states and random observables, and find an excellent agreement with the theory, despite experimental errors.

Density matrices represent our knowledge about state of quantum systems and gives us a way to calculate expectation values and predict experimental results. Estimating the density matrix of an N-body system requires the knowledge of exponentially many numbers in terms of N. The exponentially-expensive representation of quantum states turns into practical difficulties in estimating and storing density matrices, and, moreover, raises theoretical questions about the physical meaning of quantum states. This leads to the important and natural question of whether a protocol that significantly reduces or eliminates the exponential dependence is admissible. In this work, we give an affirmative answer to the question.

Specializing to N-qubit systems as an important case study, we propose an observation-based approximate description of quantum states. The approximate description grows linearly with N (rather than exponentially in N, as for density matrices) and allow us to estimate expectation values of any observable quantity, without the need to go through the density matrix formulation, with an estimation error only dependent on the number of observations and on the observable itself, independently of N. Building the approximate description of the N-qubit state only requires repetitions of single-qubit rotations followed by single-qubit measurements. The N-independence property and the simplicity of building the approximate description has the potential for improving a wide range of quantum protocols and can be considered for implementation on today's noisy quantum computers.

► BibTeX data

► References

[1] G. Vidal. Efficient classical simulation of slightly entangled quantum computations. Phys. Rev. Lett., 91: 147902, Oct 2003. URL https:/​/​doi.org/​10.1103/​PhysRevLett.91.147902.
https:/​/​doi.org/​10.1103/​PhysRevLett.91.147902

[2] S. Aaronson. The learnability of quantum states. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 463 (2088): 3089–3114, 2007. URL https:/​/​doi.org/​10.1098/​rspa.2007.0113.
https:/​/​doi.org/​10.1098/​rspa.2007.0113

[3] M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, and Y.-K. Liu. Efficient quantum state tomography. Nat. Comm., 1: 149, 2010. URL https:/​/​doi.org/​10.1038/​ncomms1147.
https:/​/​doi.org/​10.1038/​ncomms1147

[4] D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, and J. Eisert. Quantum state tomography via compressed sensing. Phys. Rev. Lett., 105 (15): 150401, 2010. URL https:/​/​doi.org/​10.1103/​PhysRevLett.105.150401.
https:/​/​doi.org/​10.1103/​PhysRevLett.105.150401

[5] A. Kalev, R. Kosut, and I. Deutsch. Quantum tomography protocols with positivity are compressed sensing protocols. NPJ Quantum Information, 1: 15018, 2015. URL https:/​/​doi.org/​10.1038/​npjqi.2015.18.
https:/​/​doi.org/​10.1038/​npjqi.2015.18

[6] X. Gao and L.-M. Duan. Efficient representation of quantum many-body states with deep neural networks. Nat. Comm., 8: 662, 2017. URL https:/​/​doi.org/​10.1038/​s41467-017-00705-2.
https:/​/​doi.org/​10.1038/​s41467-017-00705-2

[7] G. Torlai, G. Mazzola, J. Carrasquilla, M. Troyer, R. Melko, and G. Carleo. Neural-network quantum state tomography. Nat. Phys., 18: 447, 2017. URL https:/​/​doi.org/​10.1038/​s41567-018-0048-5.
https:/​/​doi.org/​10.1038/​s41567-018-0048-5

[8] S. Aaronson. Shadow tomography of quantum states. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, pages 325–338, New York, NY, USA, 2018. ACM. ISBN 978-1-4503-5559-9. URL https:/​/​doi.org/​10.1145/​3188745.3188802.
https:/​/​doi.org/​10.1145/​3188745.3188802

[9] D. Gosset and J. Smolin. A compressed classical description of quantum states. In Wim van Dam and Laura Mancinska, editors, 14th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2019), volume 135 of Leibniz International Proceedings in Informatics (LIPIcs), pages 8:1–8:9, Dagstuhl, Germany, 2019. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik. ISBN 978-3-95977-112-2. URL https:/​/​doi.org/​10.4230/​LIPIcs.TQC.2019.8.
https:/​/​doi.org/​10.4230/​LIPIcs.TQC.2019.8

[10] M. Paini. Quantum tomography via group theory. arXiv:quant-ph/​0002078, 2000. URL https:/​/​arxiv.org/​pdf/​quant-ph/​0002078.pdf.
arXiv:quant-ph/0002078

[11] G. M. D'Ariano, L. Maccone, and M. Paini. Spin tomography. J. Opt. B: Quantum and Semiclassical Optics, 5 (1): 77–84, jan 2003. URL https:/​/​doi.org/​10.1088/​1464-4266/​5/​1/​311.
https:/​/​doi.org/​10.1088/​1464-4266/​5/​1/​311

[12] T. Brydges, A. Elben, P. Jurcevic, C. Vermersch, B.and Maier, B. P. Lanyon, P. Zoller, R. Blatt, and C. F. Roos. Probing Rényi entanglement entropy via randomized measurements. Science, 364 (6437): 260–263, 2019. URL https:/​/​doi.org/​10.1126/​science.aau4963.
https:/​/​doi.org/​10.1126/​science.aau4963

[13] A. Elben, J. Yu, G. Zhu, M. Hafezi, F. Pollmann, P. Zoller, and B. Vermersch. Many-body topological invariants from randomized measurements in synthetic quantum matter. Science Advances, 6 (15), 2020a. URL https:/​/​doi.org/​10.1126/​sciadv.aaz3666.
https:/​/​doi.org/​10.1126/​sciadv.aaz3666

[14] A. Elben, B. Vermersch, R. van Bijnen, C. Kokail, T. Brydges, C. Maier, M. K. Joshi, R. Blatt, C. F. Roos, and P. Zoller. Cross-platform verification of intermediate scale quantum devices. Phys. Rev. Lett., 124: 010504, Jan 2020b. URL https:/​/​doi.org/​10.1103/​PhysRevLett.124.010504.
https:/​/​doi.org/​10.1103/​PhysRevLett.124.010504

[15] B. Vermersch, A. Elben, L. M. Sieberer, N. Y. Yao, and P. Zoller. Probing scrambling using statistical correlations between randomized measurements. Phys. Rev. X, 9: 021061, Jun 2019. URL https:/​/​doi.org/​10.1103/​PhysRevX.9.021061.
https:/​/​doi.org/​10.1103/​PhysRevX.9.021061

[16] R. O'Donnell and J. Wright. Efficient quantum tomography. In Proceedings of the Forty-Eighth Annual ACM Symposium on Theory of Computing, STOC '16, page 899–912, New York, NY, USA, 2016. Association for Computing Machinery. ISBN 9781450341325. URL https:/​/​doi.org/​10.1145/​2897518.2897544.
https:/​/​doi.org/​10.1145/​2897518.2897544

[17] J. Haah, A. W. Harrow, Z. Ji, X. Wu, and N. Yu. Sample-optimal tomography of quantum states. In Proceedings of the Forty-Eighth Annual ACM Symposium on Theory of Computing, STOC '16, page 913–925, New York, NY, USA, 2016. Association for Computing Machinery. ISBN 9781450341325. URL https:/​/​doi.org/​10.1145/​2897518.2897585.
https:/​/​doi.org/​10.1145/​2897518.2897585

[18] S. Aaronson and G. N. Rothblum. Gentle measurement of quantum states and differential privacy. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019, page 322–333, New York, NY, USA, 2019. Association for Computing Machinery. ISBN 9781450367059. URL https:/​/​doi.org/​10.1145/​3313276.3316378.
https:/​/​doi.org/​10.1145/​3313276.3316378

[19] H.-Y. Huang, R. Kueng, and J. Preskill. Predicting many properties of a quantum system from very few measurements. Nature Physics, Jun 2020. ISSN 1745-2481. URL https:/​/​doi.org/​10.1038/​s41567-020-0932-7.
https:/​/​doi.org/​10.1038/​s41567-020-0932-7

[20] S. Cheng, J. Chen, and L. Wang. Information perspective to probabilistic modeling: Boltzmann machines versus born machines. Entropy, 20 (8), 2018. URL https:/​/​doi.org/​10.3390/​e20080583.
https:/​/​doi.org/​10.3390/​e20080583

[21] M. Benedetti, E. Lloyd, S. Sack, and M. Fiorentini. Parameterized quantum circuits as machine learning models. Quantum Science and Technology, 4 (4): 043001, nov 2019. URL https:/​/​doi.org/​10.1088/​2058-9565/​ab4eb5.
https:/​/​doi.org/​10.1088/​2058-9565/​ab4eb5

[22] K. E. Cahill and R. J. Glauber. Density operators for fermions. Phys. Rev. A, 59: 1538–1555, Feb 1999. URL https:/​/​doi.org/​10.1103/​PhysRevA.59.1538.
https:/​/​doi.org/​10.1103/​PhysRevA.59.1538

[23] Y. Li. Talk 5 : Averages over the unitary group. In Selected topics in Mathematical Physics : Quantum Information Theory, 2013.

[24] A. Ambainis and J. Emerson. Quantum t-designs: t-wise independence in the quantum world. In Proceedings of the Twenty-Second Annual IEEE Conference on Computational Complexity, CCC '07, pages 129–140, Washington, DC, USA, 2007. IEEE Computer Society. ISBN 0-7695-2780-9. URL https:/​/​doi.org/​10.1109/​CCC.2007.26.
https:/​/​doi.org/​10.1109/​CCC.2007.26

Cited by

[1] Andrew Zhao, Nicholas C. Rubin, and Akimasa Miyake, "Fermionic Partial Tomography via Classical Shadows", Physical Review Letters 127 11, 110504 (2021).

[2] Andreas Ketterer, Satoya Imai, Nikolai Wyderka, and Otfried Gühne, "Statistically significant tests of multiparticle quantum correlations based on randomized measurements", arXiv:2012.12176.

[3] Ivan Henao, Raam Uzdin, and Nadav Katz, "Experimental detection of microscopic environments using thermodynamic observables", arXiv:1908.08968.

[4] Tanmoy Pandit, Alaina M. Green, C. Huerta Alderete, Norbert M. Linke, and Raam Uzdin, "Bounds on the survival probability in periodically driven quantum systems", arXiv:2105.11685.

The above citations are from Crossref's cited-by service (last updated successfully 2021-09-23 03:57:22) and SAO/NASA ADS (last updated successfully 2021-09-23 03:57:23). The list may be incomplete as not all publishers provide suitable and complete citation data.