Anticoncentration theorems for schemes showing a quantum speedup

Dominik Hangleiter, Juan Bermejo-Vega, Martin Schwarz, and Jens Eisert

Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany

One of the main milestones in quantum information science is to realise quantum devices that exhibit an exponential computational advantage over classical ones without being universal quantum computers, a state of affairs dubbed quantum speedup, or sometimes "quantum computational supremacy". The known schemes heavily rely on mathematical assumptions that are plausible but unproven, prominently results on anticoncentration of random prescriptions. In this work, we aim at closing the gap by proving two anticoncentration theorems and accompanying hardness results, one for circuit-based schemes, the other for quantum quench-type schemes for quantum simulations. Compared to the few other known such results, these results give rise to a number of comparably simple, physically meaningful and resource-economical schemes showing a quantum speedup in one and two spatial dimensions. At the heart of the analysis are tools of unitary designs and random circuits that allow us to conclude that universal random circuits anticoncentrate as well as an embedding of known circuit-based schemes in a 2D translation-invariant architecture.

Most near-term proposals for demonstrating a quantum speedup are based on sampling from the output probability distribution of certain random circuits. The technique most often used to prove the speedup for these tasks requires complexity-theoretic conjectures about the sampled distribution to be assumed, one of them being that the distribution `anticoncentrates'. Here, we prove this conjecture for two-designs as well as for a scheme based on the constant-time evolution of a translation-invariant Ising model. Our results give rise to a range of schemes exhibiting a quantum speedup that satisfy this condition by construction, covering a number of experimentally relevant settings, including random universal circuits, diagonal unitaries, Clifford+T circuits and instances of quantum simulation schemes.

► BibTeX data

► References

[1] J. Preskill, Bull. Am. Phys. Soc. 58 (2013), arXiv:1203.5813.
arXiv:1203.5813
http:/​/​meetings.aps.org/​link/​BAPS.2013.APR.P1.2

[2] S. Trotzky, Y.-A. Chen, A. Flesch, I. P. McCulloch, U. Schollwöck, J. Eisert, and I. Bloch, Nature Phys. 8, 325 (2012), arXiv:1101.2659.
https://doi.org/doi:10.1038/nphys2232
arXiv:1101.2659

[3] J.-Y. Choi, S. Hild, J. Zeiher, P. Schauß, A. Rubio-Abadal, T. Yefsah, V. Khemani, D. A. Huse, I. Bloch, and C. Gross, Science 352, 1547 (2016), arXiv:1604.04178.
https://doi.org/10.1126/science.aaf8834
arXiv:1604.04178

[4] S. Braun, M. Friesdorf, S. S. Hodgman, M. Schreiber, J. P. Ronzheimer, A. Riera, M. del Rey, I. Bloch, J. Eisert, and U. Schneider, Proc. Natl. Ac. Sc. 112, 3641 (2015), arXiv:1403.7199.
https://doi.org/10.1073/pnas.1408861112
arXiv:1403.7199

[5] S. Aaronson and A. Arkhipov, Th. Comp. 9, 143 (2013), arXiv:1011.3245.
arXiv:1011.3245

[6] M. J. Bremner, A. Montanaro, and D. J. Shepherd, Phys. Rev. Lett. 117, 080501 (2016), arXiv:1504.07999.
https://doi.org/10.1103/PhysRevLett.117.080501
arXiv:1504.07999

[7] M. J. Bremner, A. Montanaro, and D. J. Shepherd, Quantum 1, 8 (2017).
https://doi.org/10.22331/q-2017-04-25-8

[8] S. Boixo, S. V. Isakov, V. N. Smelyanskiy, R. Babbush, N. Ding, Z. Jiang, M. J. Bremner, J. M. Martinis, and H. Neven, Nature Physics , 1 (2018), arXiv:1608.00263.
https://doi.org/10.1038/s41567-018-0124-x
arXiv:1608.00263

[9] X. Gao, S.-T. Wang, and L.-M. Duan, Phys. Rev. Lett. 118, 040502 (2017), arXiv:1607.04947.
https://doi.org/10.1103/PhysRevLett.118.040502
arXiv:1607.04947

[10] J. Bermejo-Vega, D. Hangleiter, M. Schwarz, R. Raussendorf, and J. Eisert, Phys. Rev. X 8, 021010 (2018), arXiv:1703.00466.
https://doi.org/10.1103/PhysRevX.8.021010
arXiv:1703.00466

[11] T. Morimae, Phys. Rev. A 96, 040302 (2017), arXiv:1704.03640.
https://doi.org/10.1103/PhysRevA.96.040302
arXiv:1704.03640

[12] J. Miller, S. Sanders, and A. Miyake, Phys. Rev. A 96, 062320 (2017), arXiv:1703.11002.
https://doi.org/10.1103/PhysRevA.96.062320
arXiv:1703.11002

[13] C. Gogolin, M. Kliesch, L. Aolita, and J. Eisert, ``Boson sampling in the light of sample complexity,'' arXiv:1306.3995.
arXiv:1306.3995

[14] S. Aaronson and A. Arkhipov, ``BosonSampling is far from uniform,'' arXiv:1309.7460.
arXiv:1309.7460

[15] D. Hangleiter, M. Kliesch, M. Schwarz, and J. Eisert, Quantum Sci. Technol. 2, 015004 (2017), arXiv:1602.00703.
https://doi.org/10.1088/2058-9565/2/1/015004
arXiv:1602.00703

[16] T. Kapourniotis and A. Datta, (2017), arXiv:1703.09568.
arXiv:1703.09568

[17] L. Stockmeyer, Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing, STOC '83, 118 (1983).
https://doi.org/10.1145/800061.808740

[18] A. P. Lund, M. J. Bremner, and T. C. Ralph, npj Quant. Inf. 3, 15 (2017), arXiv:1702.03061.
https://doi.org/10.1038/s41534-017-0018-2
arXiv:1702.03061

[19] M. Schwarz and M. V. den Nest, (2013), arXiv:1310.6749.
arXiv:1310.6749

[20] R. Jozsa and M. V. d. Nest, Quant. Inf. Comp 14, 0633–0648 (2014), arXiv:1305.6190.
arXiv:1305.6190

[21] Y. Nakata, M. Koashi, and M. Murao, New J. Phys. 16, 053043 (2014), arXiv:1311.1128.
https://doi.org/10.1088/1367-2630/16/5/053043
arXiv:1311.1128

[22] D. Gross, K. Audenaert, and J. Eisert, J. Math. Phys. 48, 052104 (2007), arXiv:quant-ph/​0611002.
https://doi.org/10.1063/1.2716992
arXiv:quant-ph/0611002

[23] F. G. S. L. Brandão, A. W. Harrow, and M. Horodecki, Commun. Math. Phys. 346, 397 (2016).
https://doi.org/10.1007/s00220-016-2706-8

[24] M. J. Bremner, R. Jozsa, and D. J. Shepherd, Proc. Roy. Soc. 467, 2126 (2010), arXiv:1005.1407.
https://doi.org/10.1098/rspa.2010.0301
arXiv:1005.1407

[25] B. M. Terhal and D. P. DiVincenzo, Quant. Inf. Comp. 4, 134 (2004), arXiv:quant-ph/​0205133.
arXiv:quant-ph/0205133

[26] G. Kuperberg, Theory of Computing 11, 183 (2015).
https://doi.org/10.4086/toc.2015.v011a006

[27] K. Fujii and T. Morimae, New J. Phys. 19, 033003 (2017), arXiv:1311.2128.
https://doi.org/10.1088/1367-2630/aa5fdb
arXiv:1311.2128

[28] A. Bouland, B. Fefferman, C. Nirkhe, and U. Vazirani, (2018), arXiv:1803.04402.
arXiv:1803.04402

[29] R. L. Mann and M. J. Bremner, arXiv:1711.00686.
arXiv:1711.00686

[30] S. Aaronson, ``P$\neq$NP?'' in Open problems in mathematics (Springer, 2016).
https://doi.org/10.1007/978-3-319-32162-2

[31] L. Fortnow, in Proceedings of the Thirty-seventh Annual ACM Symposium on Theory of Computing, STOC '05 (ACM, 2005).
https://doi.org/10.1145/1060590.1060609

[32] R. M. Karp and R. J. Lipton, in Proceedings of the Twelfth Annual ACM Symposium on Theory of Computing, STOC '80 (1980).
https://doi.org/10.1145/800141.804678

[33] C. Dankert, R. Cleve, J. Emerson, and E. Livine, Phys. Rev. A 80, 012304 (2009), arXiv:quant-ph/​0606161.
https://doi.org/10.1103/PhysRevA.80.012304
arXiv:quant-ph/0606161

[34] E. Onorati, O. Buerschaper, M. Kliesch, W. Brown, A. H. Werner, and J. Eisert, Commun. Math. Phys. 355, 905 (2017), arXiv:1606.01914.
https://doi.org/10.1007/s00220-017-2950-6
arXiv:1606.01914

[35] A. W. Harrow and R. A. Low, Commun. Math. Phys. 291, 257 (2009), arXiv:0802.1919.
https://doi.org/10.1007/s00220-009-0873-6
arXiv:0802.1919

[36] H. Zhu, R. Kueng, M. Grassl, and D. Gross, (2016), arXiv:1609.08172.
arXiv:1609.08172

[37] J. Emerson, Y. S. Weinstein, M. Saraceno, S. Lloyd, and D. G. Cory, Science 302, 2098 (2003).
https://doi.org/10.1126/science.1090790

[38] W. G. Brown, Y. S. Weinstein, and L. Viola, Phys. Rev. A 77, 040303 (2008), arXiv:0802.2675.
https://doi.org/10.1103/PhysRevA.77.040303
arXiv:0802.2675

[39] C. Neill, P. Roushan, K. Kechedzhi, S. Boixo, S. V. Isakov, V. Smelyanskiy, R. Barends, B. Burkett, Y. Chen, and Z. Chen, (2017), Science 13 360, Issue 6385, (2018).
https://doi.org/10.1126/science.aao4309

[40] P. O. Boykin, T. Mor, M. Pulver, V. Roychowdhury, and F. Vatan, Inf. Proc. Lett. 75, 101 (2000), arXiv:quant-ph/​9906054.
https://doi.org/10.1016/S0020-0190(00)00084-3
arXiv:quant-ph/9906054

[41] A. Kitaev, A. Shen, and M. Vyalyi, Classical and Quantum Computation, Graduate studies in mathematics (American Mathematical Society, 2002).

[42] Y. Shi, Quant. Inf. Comp. 3, 84 (2003), arXiv:quant-ph/​0205115.
arXiv:quant-ph/0205115
http:/​/​dl.acm.org/​citation.cfm?id=2011508.2011515

[43] A. Paetznick and B. W. Reichardt, Phys. Rev. Lett. 111, 090505 (2013), arXiv:1304.3709.
https://doi.org/10.1103/PhysRevLett.111.090505
arXiv:1304.3709

[44] P. W. Shor, in Proc. of 37th Conf. Found. Comp. Sci. (1996) pp. 56-65, arXiv:quant-ph/​9605011.
https://doi.org/10.1109/SFCS.1996.548464
arXiv:quant-ph/9605011

[45] E. Knill, R. Laflamme, and W. Zurek, (1996), arXiv:quant-ph/​9610011.
arXiv:quant-ph/9610011

[46] E. Knill, R. Laflamme, and W. H. Zurek, Proc. Roy. Soc. A 454, 365 (1998).
https://doi.org/10.1098/rspa.1998.0166

[47] D. Gottesman and I. L. Chuang, Nature 402, 390 (1999).
https://doi.org/10.1038/46503

[48] R. Raussendorf, D. E. Browne, and H. J. Briegel, Phys. Rev. A 68, 022312 (2003), arXiv:quant-ph/​0301052.
https://doi.org/10.1103/PhysRevA.68.022312
arXiv:quant-ph/0301052

[49] L. A. Goldberg and H. Guo, (2014), arXiv:1409.5627.
arXiv:1409.5627

[50] A. Y. Kitaev, Russ. Math. Surv. 52, 1191 (1997).
https://doi.org/10.1070/RM1997v052n06ABEH002155

[51] M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information, Cambridge Series on Information and the Natural Sciences (Cambridge University Press, 2000).

[52] C. H. Bennett, E. Bernstein, G. Brassard, and U. Vazirani, SIAM J. Comp. 26, 1510 (1997), arXiv:quant-ph/​9701001.
https://doi.org/10.1137/S0097539796300933
arXiv:quant-ph/9701001

[53] S. Aaronson, Proc. Roy. Soc. A 461, 2063 (2005), arXiv:quant-ph/​0412187.
https://doi.org/10.1098/rspa.2005.1546
arXiv:quant-ph/0412187

[54] H. Dell, T. Husfeldt, D. Marx, N. Taslaman, and M. Wahlén, ACM Trans. Algorithms 10, 21:1 (2014).
https://doi.org/10.1145/2635812

[55] S. X. Cui and Z. Wang, J. Math. Phys. 56, 032202 (2015).
https://doi.org/10.1063/1.4914941

[56] A. Bocharov, M. Roetteler, and K. M. Svore, Phys. Rev. A 91, 052317 (2015), arXiv:1409.3552.
https://doi.org/10.1103/PhysRevA.91.052317
arXiv:1409.3552

[57] R. Cleve, D. Leung, L. Liu, and C. Wang, Quant. Inf. Comp. 16, 0721 (2016), arXiv:1501.04592.
arXiv:1501.04592

[58] R. Koenig and J. A. Smolin, J. Math. Phys. 55, 122202 (2014), arXiv:1406.2170.
https://doi.org/10.1063/1.4903507
arXiv:1406.2170

[59] J. Eisert, M. Friesdorf, and C. Gogolin, Nature Phys 11, 124 (2015), arXiv:1408.5148.
https://doi.org/doi:10.1038/nphys3215
arXiv:1408.5148

[60] A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalattore, Rev. Mod. Phys. 83, 863 (2011), arXiv:1007.5331.
https://doi.org/10.1103/RevModPhys.83.863
arXiv:1007.5331

[61] R. Jozsa, (2006), arXiv:quant-ph/​0603163.
arXiv:quant-ph/0603163

[62] R. Impagliazzo and R. Paturi, in Proc. XIV IEEE Conf. Comp. Compl. (1999) pp. 237-240.
https://doi.org/10.1109/CCC.1999.766282

[63] S. Aaronson and L. Chen, (2016), arXiv:1612.05903.
arXiv:1612.05903

[64] M. Ozols, How to generate a random unitary matrix (Mar, 2009).
http:/​/​home.lu.lv/​~sd20008/​papers/​essays/​Random

[65] F. Mezzadri, (2006), arXiv:math-ph/​0609050.
arXiv:math-ph/0609050

[66] Y. S. Weinstein and C. S. Hellberg, Phys. Rev. A 72, 022331 (2005).
https://doi.org/10.1103/PhysRevA.72.022331

[67] K. Zyczkowski and M. Kus, J. Phys. A 27, 4235 (1994).
https://doi.org/10.1088/0305-4470/27/12/028

[68] M. Pozniak, K. Zyczkowski, and M. Kus, J. Phys. A 31, 1059 (1998), arXiv:chao-dyn/​9707006.
https://doi.org/10.1088/0305-4470/31/3/016
arXiv:chao-dyn/9707006

[69] F. Haake, Quantum signatures of chaos, Springer Series in Synergetics, Vol. 54 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2010).

Cited by

[1] Dominik Hangleiter, Martin Kliesch, Jens Eisert, and Christian Gogolin, "Sample Complexity of Device-Independently Certified “Quantum Supremacy”", arXiv:1812.01023, Physical Review Letters 122 21, 210502 (2019).

[2] Adam Bouland, Bill Fefferman, Chinmay Nirkhe, and Umesh Vazirani, "On the complexity and verification of quantum random circuit sampling", Nature Physics 15 2, 159 (2019).

[3] David T. Stephen, Hendrik Poulsen Nautrup, Juani Bermejo-Vega, Jens Eisert, and Robert Raussendorf, "Subsystem symmetries, quantum cellular automata, and computational phases of quantum matter", Quantum 3, 142 (2019).

[4] Mithuna Yoganathan, Richard Jozsa, and Sergii Strelchuk, "Quantum advantage of unitary Clifford circuits with magic state inputs", Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 475 2225, 20180427 (2019).

[5] Tomoyuki Morimae, "Hardness of classically sampling the one-clean-qubit model with constant total variation distance error", Physical Review A 96 4, 040302 (2017).

[6] Yuki Takeuchi and Tomoyuki Morimae, "Verification of Many-Qubit States", Physical Review X 8 2, 021060 (2018).

[7] Man-Hong Yung and Xun Gao, "Can Chaotic Quantum Circuits Maintain Quantum Supremacy under Noise?", arXiv:1706.08913.

[8] Xun Gao and Luming Duan, "Efficient classical simulation of noisy quantum computation", arXiv:1810.03176.

[9] Adam Bouland, Joseph F. Fitzsimons, and Dax Enshan Koh, "Complexity Classification of Conjugated Clifford Circuits", arXiv:1709.01805.

[10] Ryan L. Mann and Michael J. Bremner, "On the Complexity of Random Quantum Computations and the Jones Polynomial", arXiv:1711.00686.

[11] Adam Bouland, Bill Fefferman, Chinmay Nirkhe, and Umesh Vazirani, "Quantum Supremacy and the Complexity of Random Circuit Sampling", arXiv:1803.04402.

[12] Keisuke Fujii, Hirotada Kobayashi, Tomoyuki Morimae, Harumichi Nishimura, Shuhei Tamate, and Seiichiro Tani, "Impossibility of Classically Simulating One-Clean-Qubit Model with Multiplicative Error", Physical Review Letters 120 20, 200502 (2018).

[13] Rawad Mezher, Joe Ghalbouni, Joseph Dgheim, and Damian Markham, "Efficient approximate unitary t-designs from partially invertible universal sets and their application to quantum speedup", arXiv:1905.01504.

[14] Alexander M. Dalzell, Aram W. Harrow, Dax Enshan Koh, and Rolando L. La Placa, "How many qubits are needed for quantum computational supremacy?", arXiv:1805.05224.

The above citations are from Crossref's cited-by service (last updated 2019-06-17 03:34:48) and SAO/NASA ADS (last updated 2019-06-17 03:34:49). The list may be incomplete as not all publishers provide suitable and complete citation data.