Matrix Concentration Inequalities And Efficiency Of Random Universal Sets Of Quantum Gates

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Piotr Dulian^1,2 and Adam Sawicki¹

¹Center for Theoretical Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warsaw, Poland
²Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland

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Abstract

For a random set $mathcal{S} subset U(d)$ of quantum gates we provide bounds on the probability that $mathcal{S}$ forms a $delta$-approximate $t$-design. In particular we have found that for $mathcal{S}$ drawn from an exact $t$-design the probability that it forms a $delta$-approximate $t$-design satisfies the inequality $mathbb{P}left(delta geq x right)leq 2D_t , frac{e^{-|mathcal{S}| x , mathrm{arctanh}(x)}}{(1-x^2)^{|mathcal{S}|/2}} = Oleft( 2D_t left( frac{e^{-x^2}}{sqrt{1-x^2}} right)^{|mathcal{S}|} right)$, where $D_t$ is a sum over dimensions of unique irreducible representations appearing in the decomposition of $U mapsto U^{otimes t}otimes bar{U}^{otimes t}$. We use our results to show that to obtain a $delta$-approximate $t$-design with probability $P$ one needs $O( delta^{-2}(tlog(d)-log(1-P)))$ many random gates. We also analyze how $delta$ concentrates around its expected value $mathbb{E}delta$ for random $mathcal{S}$. Our results are valid for both symmetric and non-symmetric sets of gates.

► BibTeX data

@article{Dulian2023matrixconcentration, doi = {10.22331/q-2023-04-20-983}, url = {https://doi.org/10.22331/q-2023-04-20-983}, title = {Matrix concentration inequalities and efficiency of random universal sets of quantum gates}, author = {Dulian, Piotr and Sawicki, Adam}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{“{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {7}, pages = {983}, month = apr, year = {2023}<br /> }

► References

[1] A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, “Surface codes: Towards practical large-scale quantum computation” Phys. Rev. A 86, 032324 (2012).
https://doi.org/10.1103/PhysRevA.86.032324

[2] J. Preskill “Quantum Computing in the NISQ era and beyond” Quantum 2, 79 (2018).
https://doi.org/10.22331/q-2018-08-06-79

[3] S. Boixo, S. V. Isakov, V. N. Smelyanskiy, R. Babbush, N. Ding, Z. Jiang, M. J. Bremner, J. M. Martinis, and H. Neven, “Characterizing Quantum Supremacy in Near-Term Devices” Nature Physics 14, 595–600 (2018).
https://doi.org/10.1038/s41567-018-0124-x

[4] A. W. Harrowand A. Montanaro “Quantum Computational Supremacy” Nature 549, 203–209 (2017).
https://doi.org/10.1038/nature23458

[5] C. J. Ballance, T. P. Harty, N. M. Linke, M. A. Sepiol, and D. M. Lucas, “High-fidelity quantum logic gates using trapped-ion hyperfine qubits” Phys. Rev. Lett. 117, 060504 (2016).
https://doi.org/10.1103/PhysRevLett.117.060504

[6] R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, C. Neill, P. O`Malley, P. Roushan, A. Vainsencher, J. Wenner, A. N. Korotkov, A. N. Cleland, and John M. Martinis, “Logic gates at the surface code threshold: Superconducting qubits poised for fault-tolerant quantum computing” Nature 508, 500–503 (2014).
https://doi.org/10.1038/nature13171

[7] L. Susskind “Three Lectures on Complexity and Black Holes” Springer Cham (2020).
https://doi.org/10.1007/978-3-030-45109-7
https://arxiv.org/abs/1810.11563

[8] A. Sawickiand K. Karnas “Criteria for universality of quantum gates” Physical Review A 95, 062303 (2017).
https://doi.org/10.1103/physreva.95.062303

[9] A. Sawickiand K. Karnas “Universality of Single-Qudit Gates” Annales Henri PoincarÃ© 18, 3515â3552 (2017).
https://doi.org/10.1007/s00023-017-0604-z

[10] A. Sawicki, L. Mattioli, and Z. Zimborás, “Universality verification for a set of quantum gates” Phys. Rev. A 105, 052602 (2022).
https://doi.org/10.1103/PhysRevA.105.052602
arXiv:2111.03862

[11] M. A. Nielsenand I. L. Chuang “Quantum Computation and Quantum Information: 10th Anniversary Edition” Cambridge University Press (2010).
https://doi.org/10.1017/CBO9780511976667

[12] P. P. Varjú “Random walks in compact groups” Doc. Math. 18, 1137–1175 (2013).
https://doi.org/10.4171/DM/423

[13] M. Oszmaniec, A. Sawicki, and M. Horodecki, “Epsilon-Nets, Unitary Designs, and Random Quantum Circuits” IEEE Transactions on Information Theory 68, 989–1015 (2022).
https://doi.org/10.1109/TIT.2021.3128110

[14] A. Boulandand T. Giurgica-Tiron “Efficient Universal Quantum Compilation: An Inverse-free Solovay-Kitaev Algorithm” arXiv e-prints (2021).
https://doi.org/10.48550/ARXIV.2112.02040
arXiv:2112.02040

[15] A. W. Harrow, B. Recht, and Isaac L. Chuang, “Efficient Discrete Approximations of Quantum Gates” J. Math. Phys. 43, 4445 (2002).
https://doi.org/10.1063/1.1495899

[16] J. M. Epstein, A. W. Cross, E. Magesan, and J. M. Gambetta, “Investigating the limits of randomized benchmarking protocols” Physical Review A 89, 062321 (2014).
https://doi.org/10.1103/physreva.89.062321
arXiv:1308.2928

[17] A. M. Dalzell, N. Hunter-Jones, and F. G. S. L. BrandÃ£o, “Random quantum circuits transform local noise into global white noise” arXiv e-prints (2021).
https://doi.org/10.48550/ARXIV.2111.14907
arXiv:2111.14907

[18] A. Abeyesinghe, I. Devetak, P. Hayden, and A. Winter, “The mother of all protocols: restructuring quantum information’s family tree” Proceedings of the Royal Society of London Series A 465, 2537–2563 (2009).
https://doi.org/10.1098/rspa.2009.0202

[19] J. Radhakrishnan, M. RÃ¶tteler, and P. Sen, “Random measurement bases, quantum state distinction and applications to the hidden subgroup problem” Algorithmica 55, 490â516 (2009).
https://doi.org/10.1007/s00453-008-9231-x

[20] D. A. Robertsand B. Yoshida “Chaos and complexity by design” Journal of High Energy Physics 2017, 121 (2017).
https://doi.org/10.1007/JHEP04(2017)121
arXiv:1610.04903

[21] M. Oszmaniec, M. Horodecki, and N. Hunter-Jones, “Saturation and recurrence of quantum complexity in random quantum circuits” arXiv e-prints (2022).
https://doi.org/10.48550/ARXIV.2205.09734
arXiv:2205.09734

[22] J. Haferkamp, P. Faist, N. B. T. Kothakonda, J. Eisert, and N. Yunger Halpern, “Linear growth of quantum circuit complexity” Nature Physics 18, 528–532 (2022).
https://doi.org/10.1038/s41567-022-01539-6
arXiv:2106.05305

[23] J. Bourgainand A. Gamburd “A spectral gap theorem in SU(d)” J. Eur. Math. Soc. 14, 1455â1511 (2012).
https://doi.org/10.4171/JEMS/337

[24] J Bourgainand Alex Gamburd “On the spectral gap for finitely-generated subgroups of SU(2).” Invent. math. 171, 83â121 (2008).
https://doi.org/10.1007/s00222-007-0072-z

[25] A. Bocharov, Y. Gurevich, and K. M. Svore, “Efficient decomposition of single-qubit gates into V basis circuits” Phys. Rev. A 88, 012313 (2013).
https://doi.org/10.1103/physreva.88.012313

[26] V. Kliuchnikov, A. Bocharov, M. Roetteler, and J. Yard, “A Framework for Approximating Qubit Unitaries” arXiv e-prints (2015).
https://doi.org/10.48550/arXiv.1510.03888
arXiv:1510.03888

[27] V. Kliuchnikov, D. Maslov, and M. Mosca, “Fast and efficient exact synthesis of single qubit unitaries generated by Clifford and T gates” Quantum Information and Computation 13, 607–630 (2013).
https://doi.org/10.26421/QIC13.7-8-4

[28] P. Selinger “Efficient Clifford+T approximation of single-qubit operators” Quantum Information and Computation 15, 159â180 (2015).
https://doi.org/10.26421/QIC15.1-2-10

[29] P. Sarnak “Letter to Scott Aaronson and Andy Pollington on the Solovay-Kitaev theorem” (2015).

[30] A. Lubotzky, R. Phillips, and P. Sarnak, “Hecke operators and distributing points on S2. II” Communications on Pure and Applied Mathematics 40, 401–420 (1987).
https://doi.org/10.1002/cpa.3160400402

[31] J. A. Tropp “An Introduction to Matrix Concentration Inequalities” Now Publishers Inc (2015).
https://doi.org/10.1561/2200000048
https://arxiv.org/abs/1501.01571

[32] M. Abu-Hamedand S. Gelaki “Frobenius-Schur indicators for semisimple Lie algebras” Journal of Algebra 315, 178–191 (2007).
https://doi.org/10.1016/j.jalgebra.2007.06.003

[33] J. Emerson, R. Alicki, and K. Å»yczkowski, “Scalable noise estimation with random unitary operators” Journal of Optics B: Quantum and Semiclassical Optics 7, S347 (2005).
https://doi.org/10.1088/1464-4266/7/10/021

[34] C. Dankert, R. Cleve, J. Emerson, and E. Livine, “Exact and approximate unitary 2-designs and their application to fidelity estimation” Phys. Rev. A 80, 012304 (2009).
https://doi.org/10.1103/PhysRevA.80.012304

[35] Y. Nakata, D. Zhao, T. Okuda, E. Bannai, Y. Suzuki, S. Tamiya, K. Heya, Z. Yan, K. Zuo, S. Tamate, Y. Tabuchi, and Y. Nakamura, “Quantum Circuits for Exact Unitary $t$-Designs and Applications to Higher-Order Randomized Benchmarking” PRX Quantum 2, 030339 (2021).
https://doi.org/10.1103/PRXQuantum.2.030339
arXiv:2102.12617

[36] E. S. Meckes “The Random Matrix Theory of the Classical Compact Groups” Cambridge University Press (2019).
https://doi.org/10.1017/9781108303453

[37] M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator” Phys. Rev. Lett. 73, 58–61 (1994).
https://doi.org/10.1103/PhysRevLett.73.58

[38] A. Sawicki “Universality of beamsplitters” Quantum Information and Computation 16, 291–312 (2016).
https://doi.org/10.26421/QIC16.3-4-6

[39] E. H. Lieb “Convex trace functions and the Wigner-Yanase-Dyson conjecture” Advances in Mathematics 11, 267–288 (1973).
https://doi.org/10.1016/0001-8708(73)90011-X

[40] S. Golden “Lower Bounds for the Helmholtz Function” Phys. Rev. 137, B1127âB1128 (1965).
https://doi.org/10.1103/PhysRev.137.B1127

[41] C. J. Thompson “Inequality with Applications in Statistical Mechanics” J. Math. Phys. 6, 1812â1813 (1965).
https://doi.org/10.1063/1.1704727

[42] B. C. Hall “Lie Groups Lie Algebras and Representations An Elementary Introduction” Springer-Verlag New York (2004).
https://doi.org/10.1007/978-3-319-13467-3

[43] G. Benkart, M. Chakrabarti, T. Halverson, R. Leduc, C. Lee, and J. Stroomer, “Tensor product representations of general linear groups and their connections with Brauer algebras” J. Algebra 166, 529â567 (1994).
https://doi.org/10.1006/jabr.1994.1166

[44] T. Bröckerand T. Dieck “Representations of Compact Lie Groups” Springer Berlin Heidelberg (2003).
https://doi.org/10.1007/978-3-662-12918-0
https://books.google.pl/books?id=AfBzWL5bIIQC

[45] D. Ruiz-Antolinand J. Segura “A new type of sharp bounds for ratios of modified Bessel functions” J. Math. Anal. Appl. 443, 1232–1246 (2016).
https://doi.org/10.1016/j.jmaa.2016.06.011

Cited by

[1] Dmitry Grinko and Maris Ozols, “Linear programming with unitary-equivariant constraints”, arXiv:2207.05713, (2022).

[2] Piotr Dulian and Adam Sawicki, “A random matrix model for random approximate $t$-designs”, arXiv:2210.07872, (2022).

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