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

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


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.

[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.

[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.

[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.

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

[6] M. J. Bremner, A. Montanaro, and D. J. Shepherd, Phys. Rev. Lett. 117, 080501 (2016), arXiv:1504.07999.

[7] M. J. Bremner, A. Montanaro, and D. J. Shepherd, Quantum 1, 8 (2017).

[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.

[9] X. Gao, S.-T. Wang, and L.-M. Duan, Phys. Rev. Lett. 118, 040502 (2017), 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.

[11] T. Morimae, Phys. Rev. A 96, 040302 (2017), arXiv:1704.03640.

[12] J. Miller, S. Sanders, and A. Miyake, Phys. Rev. A 96, 062320 (2017), arXiv:1703.11002.

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

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

[15] D. Hangleiter, M. Kliesch, M. Schwarz, and J. Eisert, Quantum Sci. Technol. 2, 015004 (2017), arXiv:1602.00703.

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

[17] L. Stockmeyer, Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing, STOC '83, 118 (1983).

[18] A. P. Lund, M. J. Bremner, and T. C. Ralph, npj Quant. Inf. 3, 15 (2017), arXiv:1702.03061.

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

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

[21] Y. Nakata, M. Koashi, and M. Murao, New J. Phys. 16, 053043 (2014), arXiv:1311.1128.

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

[23] F. G. S. L. Brandão, A. W. Harrow, and M. Horodecki, Commun. Math. Phys. 346, 397 (2016).

[24] M. J. Bremner, R. Jozsa, and D. J. Shepherd, Proc. Roy. Soc. 467, 2126 (2010), arXiv:1005.1407.

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

[26] G. Kuperberg, Theory of Computing 11, 183 (2015).

[27] K. Fujii and T. Morimae, New J. Phys. 19, 033003 (2017), arXiv:1311.2128.

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

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

[30] S. Aaronson, ``P$\neq$NP?'' in Open problems in mathematics (Springer, 2016).

[31] L. Fortnow, in Proceedings of the Thirty-seventh Annual ACM Symposium on Theory of Computing, STOC '05 (ACM, 2005).

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

[33] C. Dankert, R. Cleve, J. Emerson, and E. Livine, Phys. Rev. A 80, 012304 (2009), 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.

[35] A. W. Harrow and R. A. Low, Commun. Math. Phys. 291, 257 (2009), arXiv:0802.1919.

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

[37] J. Emerson, Y. S. Weinstein, M. Saraceno, S. Lloyd, and D. G. Cory, Science 302, 2098 (2003).

[38] W. G. Brown, Y. S. Weinstein, and L. Viola, Phys. Rev. A 77, 040303 (2008), 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).

[40] P. O. Boykin, T. Mor, M. Pulver, V. Roychowdhury, and F. Vatan, Inf. Proc. Lett. 75, 101 (2000), 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.

[43] A. Paetznick and B. W. Reichardt, Phys. Rev. Lett. 111, 090505 (2013), arXiv:1304.3709.

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

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

[46] E. Knill, R. Laflamme, and W. H. Zurek, Proc. Roy. Soc. A 454, 365 (1998).

[47] D. Gottesman and I. L. Chuang, Nature 402, 390 (1999).

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

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

[50] A. Y. Kitaev, Russ. Math. Surv. 52, 1191 (1997).

[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.

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

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

[55] S. X. Cui and Z. Wang, J. Math. Phys. 56, 032202 (2015).

[56] A. Bocharov, M. Roetteler, and K. M. Svore, Phys. Rev. A 91, 052317 (2015), arXiv:1409.3552.

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

[58] R. Koenig and J. A. Smolin, J. Math. Phys. 55, 122202 (2014), arXiv:1406.2170.

[59] J. Eisert, M. Friesdorf, and C. Gogolin, Nature Phys 11, 124 (2015), arXiv:1408.5148.

[60] A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalattore, Rev. Mod. Phys. 83, 863 (2011), arXiv:1007.5331.

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

[62] R. Impagliazzo and R. Paturi, in Proc. XIV IEEE Conf. Comp. Compl. (1999) pp. 237–240.

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

[64] M. Ozols, How to generate a random unitary matrix (Mar, 2009).

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

[66] Y. S. Weinstein and C. S. Hellberg, Phys. Rev. A 72, 022331 (2005).

[67] K. Zyczkowski and M. Kus, J. Phys. A 27, 4235 (1994).

[68] M. Pozniak, K. Zyczkowski, and M. Kus, J. Phys. A 31, 1059 (1998), 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] Daniel Stilck França and Raul Garcia-Patron, "A game of quantum advantage: linking verification and simulation", Quantum 6, 753 (2022).

[2] J. Haferkamp, D. Hangleiter, A. Bouland, B. Fefferman, J. Eisert, and J. Bermejo-Vega, "Closing Gaps of a Quantum Advantage with Short-Time Hamiltonian Dynamics", Physical Review Letters 125 25, 250501 (2020).

[3] Sachin Kasture, Oleksandr Kyriienko, and Vincent E. Elfving, "Protocols for classically training quantum generative models on probability distributions", Physical Review A 108 4, 042406 (2023).

[4] Aram W. Harrow and Saeed Mehraban, "Approximate Unitary t-Designs by Short Random Quantum Circuits Using Nearest-Neighbor and Long-Range Gates", Communications in Mathematical Physics 401 2, 1531 (2023).

[5] Alexander M. Dalzell, Aram W. Harrow, Dax Enshan Koh, and Rolando L. La Placa, "How many qubits are needed for quantum computational supremacy?", Quantum 4, 264 (2020).

[6] Abhinav Deshpande, Arthur Mehta, Trevor Vincent, Nicolás Quesada, Marcel Hinsche, Marios Ioannou, Lars Madsen, Jonathan Lavoie, Haoyu Qi, Jens Eisert, Dominik Hangleiter, Bill Fefferman, and Ish Dhand, "Quantum computational advantage via high-dimensional Gaussian boson sampling", Science Advances 8 1, eabi7894 (2022).

[7] 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).

[8] Michał Oszmaniec, Ninnat Dangniam, Mauro E.S. Morales, and Zoltán Zimborás, "Fermion Sampling: A Robust Quantum Computational Advantage Scheme Using Fermionic Linear Optics and Magic Input States", PRX Quantum 3 2, 020328 (2022).

[9] Ramis Movassagh, "The hardness of random quantum circuits", Nature Physics 19 11, 1719 (2023).

[10] Antonio Anna Mele, "Introduction to Haar Measure Tools in Quantum Information: A Beginner's Tutorial", Quantum 8, 1340 (2024).

[11] Lucas Kocia and Peter Love, "Stationary Phase Method in Discrete Wigner Functions and Classical Simulation of Quantum Circuits", Quantum 5, 494 (2021).

[12] 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).

[13] 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).

[14] Abhinav Deshpande, Pradeep Niroula, Oles Shtanko, Alexey V. Gorshkov, Bill Fefferman, and Michael J. Gullans, "Tight Bounds on the Convergence of Noisy Random Circuits to the Uniform Distribution", PRX Quantum 3 4, 040329 (2022).

[15] Rawad Mezher, Joe Ghalbouni, Joseph Dgheim, and Damian Markham, "Fault-tolerant quantum speedup from constant depth quantum circuits", Physical Review Research 2 3, 033444 (2020).

[16] Nishad Maskara, Abhinav Deshpande, Adam Ehrenberg, Minh C. Tran, Bill Fefferman, and Alexey V. Gorshkov, "Complexity Phase Diagram for Interacting and Long-Range Bosonic Hamiltonians", Physical Review Letters 129 15, 150604 (2022).

[17] Dolev Bluvstein, Simon J. Evered, Alexandra A. Geim, Sophie H. Li, Hengyun Zhou, Tom Manovitz, Sepehr Ebadi, Madelyn Cain, Marcin Kalinowski, Dominik Hangleiter, J. Pablo Bonilla Ataides, Nishad Maskara, Iris Cong, Xun Gao, Pedro Sales Rodriguez, Thomas Karolyshyn, Giulia Semeghini, Michael J. Gullans, Markus Greiner, Vladan Vuletić, and Mikhail D. Lukin, "Logical quantum processor based on reconfigurable atom arrays", Nature 626 7997, 58 (2024).

[18] Dominik Hangleiter and Jens Eisert, "Computational advantage of quantum random sampling", Reviews of Modern Physics 95 3, 035001 (2023).

[19] Michal Oszmaniec, Adam Sawicki, and Michal Horodecki, "Epsilon-Nets, Unitary Designs, and Random Quantum Circuits", IEEE Transactions on Information Theory 68 2, 989 (2022).

[20] Supanut Thanasilp, Jirawat Tangpanitanon, Marc-Antoine Lemonde, Ninnat Dangniam, and Dimitris G. Angelakis, "Quantum supremacy and quantum phase transitions", Physical Review B 103 16, 165132 (2021).

[21] Leonardo Novo, Juani Bermejo-Vega, and Raúl García-Patrón, "Quantum advantage from energy measurements of many-body quantum systems", Quantum 5, 465 (2021).

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

[23] Martin Kliesch and Ingo Roth, "Theory of Quantum System Certification", PRX Quantum 2 1, 010201 (2021).

[24] Keerthi Kumaran, Manas Sajjan, Sangchul Oh, and Sabre Kais, "Random projection using random quantum circuits", Physical Review Research 6 1, 013010 (2024).

[25] Oussama Houhou, Darren W. Moore, Sougato Bose, and Alessandro Ferraro, "Unconditional measurement-based quantum computation with optomechanical continuous variables", Physical Review A 105 1, 012610 (2022).

[26] Byeongseon Go, Changhun Oh, Liang Jiang, and Hyunseok Jeong, "Exploring shallow-depth boson sampling: Toward a scalable quantum advantage", Physical Review A 109 5, 052613 (2024).

[27] Kaifeng Bu and Dax Enshan Koh, "Efficient Classical Simulation of Clifford Circuits with Nonstabilizer Input States", Physical Review Letters 123 17, 170502 (2019).

[28] Meng Zhang, Chao Wang, and Yongjian Han, "Noisy Random Quantum Circuit Sampling and its Classical Simulation", Advanced Quantum Technologies 6 7, 2300030 (2023).

[29] Yuki Takeuchi, Yasuhiro Takahashi, and Seiichiro Tani, "Hardness of efficiently generating ground states in postselected quantum computation", Physical Review Research 3 1, 013213 (2021).

[30] Jirawat Tangpanitanon, Supanut Thanasilp, Marc-Antoine Lemonde, Ninnat Dangniam, and Dimitris G Angelakis, "Signatures of a sampling quantum advantage in driven quantum many-body systems", Quantum Science and Technology 8 2, 025019 (2023).

[31] Jirawat Tangpanitanon, Supanut Thanasilp, Ninnat Dangniam, Marc-Antoine Lemonde, and Dimitris G. Angelakis, "Expressibility and trainability of parametrized analog quantum systems for machine learning applications", Physical Review Research 2 4, 043364 (2020).

[32] Brian Barch, Razieh Mohseninia, and Daniel Lidar, "Low overhead universality and quantum supremacy using only Z control", Physical Review Research 3 3, 033207 (2021).

[33] Alexander M. Dalzell, Nicholas Hunter-Jones, and Fernando G. S. L. Brandão, "Random Quantum Circuits Anticoncentrate in Log Depth", PRX Quantum 3 1, 010333 (2022).

[34] Xun Gao and Luming Duan, "Efficient classical simulation of noisy quantum computation", arXiv:1810.03176, (2018).

[35] Adam Bouland, Bill Fefferman, Chinmay Nirkhe, and Umesh Vazirani, "Quantum Supremacy and the Complexity of Random Circuit Sampling", arXiv:1803.04402, (2018).

[36] 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).

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

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

[39] Alexander M. Dalzell, Nicholas Hunter-Jones, and Fernando G. S. L. Brandão, "Random quantum circuits anti-concentrate in log depth", arXiv:2011.12277, (2020).

[40] Man-Hong Yung and Xun Gao, "Can Chaotic Quantum Circuits Maintain Quantum Supremacy under Noise?", arXiv:1706.08913, (2017).

[41] 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, (2019).

[42] Martin Kliesch and Ingo Roth, "Theory of quantum system certification: a tutorial", arXiv:2010.05925, (2020).

[43] Ryan L. Mann and Michael J. Bremner, "On the Complexity of Random Quantum Computations and the Jones Polynomial", arXiv:1711.00686, (2017).

[44] Adam Bouland, Joseph F. Fitzsimons, and Dax Enshan Koh, "Complexity Classification of Conjugated Clifford Circuits", arXiv:1709.01805, (2017).

The above citations are from Crossref's cited-by service (last updated successfully 2024-06-22 09:29:32) and SAO/NASA ADS (last updated successfully 2024-06-22 09:29:32). The list may be incomplete as not all publishers provide suitable and complete citation data.