# Quantum supremacy

Quantum supremacy is the potential ability of quantum computing devices to solve problems that classical computers practically cannot.[1] Quantum advantage is the potential to solve problems faster. In computational-complexity-theoretic terms, this generally means providing a superpolynomial speedup over the best known or possible classical algorithm.[2] The term was originally popularized by John Preskill[1] but the concept of a quantum computational advantage, specifically for simulating quantum systems, dates back to Yuri Manin's (1980)[3] and Richard Feynman's (1981) proposals of quantum computing.[4]

Shor's algorithm for factoring integers, which runs in polynomial time on a quantum computer, provides such a superpolynomial speedup over the best known classical algorithm.[5] Although it is yet to be proved, factoring is generally believed to be hard using classical resources. The difficulty of proving what cannot be done with classical computing is a common problem in definitively demonstrating quantum supremacy. It also affects the boson sampling proposal of Aaronson and Arkhipov,[6] D-Wave's specialized frustrated cluster loop problems,[7] and sampling the output of random quantum circuits.[8]

Like factoring integers, sampling the output distributions of random quantum circuits is believed to be hard for classical computers based on reasonable complexity assumptions.[8] Google previously announced plans to demonstrate quantum supremacy before the end of 2017 by solving this problem with an array of 49 superconducting qubits.[9] However, as of early January 2018, only Intel has announced such hardware.[10] In October 2017, IBM demonstrated the simulation of 56 qubits on a conventional supercomputer, increasing the number of qubits needed for quantum supremacy.[11] In November of 2018, Google announced a partnership with NASA that would “analyze results from quantum circuits run on Google quantum processors, and ... provide comparisons with classical simulation to both support Google in validating its hardware and establish a baseline for quantum supremacy.”[12] Theoretical work published in 2018 suggests that quantum supremacy should be possible with a "two-dimensional lattice of 7x7 qubits and around 40 clock cycles" if error rates can be pushed low enough.[13] On June 21, 2019 Scientific American suggested that quantum supremacy could happen in 2019, according to Neven's law.[14]

## Computational complexity

Complexity arguments concern how the amount of some resource needed to solve a problem scales with the size of the input to that problem. As an extension of classical computational complexity theory, quantum complexity theory is about what a working, universal quantum computer could accomplish without necessarily accounting for the difficulty of building one or dealing with decoherence and noise.[15] Since quantum information is a generalization of classical information, it is clear that a quantum computer can efficiently simulate any classical algorithm.[15]

The complexity class of bounded-error quantum polynomial time (BQP) problems is the class of decision problems that can be solved in polynomial time by a universal quantum computer.[16] It is related to important classical complexity classes by the hierarchy${\displaystyle P\subseteq BPP\subseteq BQP\subseteq PSPACE}$ .[17] Whether any of these containments is proper is still an open question.[17]

Contrary to decision problems that require yes or no answers, sampling problems ask for samples from probability distributions.[18] If there is a classical algorithm that can efficiently sample from the output of an arbitrary quantum circuit, the polynomial hierarchy would collapse to the third level, which is considered very unlikely.[8] BosonSampling is a more specific proposal, the classical hardness of which depends upon the intractability of calculating the permanent of a large matrix with complex entries, which is a P#-complete problem.[19] The arguments used to reach this conclusion have also been extended to IQP Sampling,[20] where only the conjecture that the average- and worst-case complexities of the problem are the same is needed.[18]

## Superpolynomial speedups

The following algorithms provide Superpolynomial speedups over the best known classical algorithms:[21]

### Shor's algorithm for factoring integers

This algorithm finds the prime factorization of an n-bit integer in ${\displaystyle {\tilde {O}}(n^{3})}$  time[5] whereas the best known classical algorithm requires ${\displaystyle 2^{O(n^{1/3})}}$ time and the best upper bound for the complexity of this problem is ${\displaystyle O(2^{n/3+o(1)})}$ .[22] It can also provide a speedup for any problem that reduces to integer factoring, including the membership problem for matrix groups over fields of odd order.[23]

This algorithm is important both practically and historically for quantum computing. It was the first polynomial-time quantum algorithm proposed for a problem that is believed to be hard for classical computers.[5] This hardness belief is so strong that the security of today's most common encryption protocol, RSA, is based upon it.[24] A quantum computer successfully and repeatably running this algorithm has the potential to break this encryption system.[25] The need to avoid the imminence of this risk is referred to[by whom?] by the term Y2Q, playing on the name "Y2K" of the Year 2000 problem.

### Boson sampling

This computing paradigm based upon identical photons sent through a linear-optical network can solve certain sampling and search problems that, assuming a few conjectures,[which?] are intractable for classical computers.[6] However, it has been shown that boson sampling in a system with large enough loss and noise can be simulated efficiently.[26]

The largest experimental implementation of boson sampling to date had 6 modes so could handle up to 6 photons at a time.[27] The best proposed classical algorithm for simulating boson sampling runs in time ${\displaystyle O(n2^{n}+mn^{2})}$  for a system with n photons and m output modes.[28][29] BosonSampling is an open-source implementation in R. The algorithm leads to an estimate of 50 photons required to demonstrate quantum supremacy with boson sampling.[28][29]

### Sampling the output distribution of random quantum circuits

The best known algorithm for simulating an arbitrary random quantum circuit requires an amount of time that scales exponentially with the number of qubits, leading one group to estimate that around 50 qubits could be enough to demonstrate quantum supremacy.[13] Google had announced its intention to demonstrate quantum supremacy by the end of 2017 by constructing and running a 49-qubit chip that will be able to sample distributions inaccessible to any current classical computers in a reasonable amount of time.[9] But the largest quantum circuit simulation completed successfully on a supercomputer now contains 56 qubits.[30] This may require increasing the number of qubits to demonstrate quantum supremacy.[11]

## Skepticism

Quantum computers are much more susceptible to errors than classical computers due to decoherence and noise.[31] The threshold theorem states that a noisy quantum computer can use quantum error-correcting codes[32][33] to simulate a noiseless quantum computer assuming the error introduced in each computer cycle is less than some number.[34] Numerical simulations suggest that that number may be as high as 3%.[35]

However, it is not known how the resources needed for error correction will scale with the number of qubits.[36] Skeptics point to the unknown behavior of noise in scaled-up quantum systems as potential roadblocks for successfully implementing quantum computing and demonstrating quantum supremacy.[31][37]

## References

1. ^ a b Preskill, John (2012-03-26). "Quantum computing and the entanglement frontier". arXiv:1203.5813 [quant-ph].
2. ^ Papageorgiou, Anargyros; Traub, Joseph F. (2013-08-12). "Measures of quantum computing speedup". Physical Review A. 88 (2): 022316. arXiv:1307.7488. Bibcode:2013PhRvA..88b2316P. doi:10.1103/PhysRevA.88.022316. ISSN 1050-2947.
3. ^ Manin, Yu. I. (1980). Vychislimoe i nevychislimoe [Computable and Noncomputable] (in Russian). Sov.Radio. pp. 13–15. Archived from the original on 2013-05-10. Retrieved 2013-03-04. Cite uses deprecated parameter |deadurl= (help)
4. ^ Feynman, Richard P. (1982-06-01). "Simulating Physics with Computers". International Journal of Theoretical Physics. 21 (6–7): 467–488. Bibcode:1982IJTP...21..467F. CiteSeerX 10.1.1.45.9310. doi:10.1007/BF02650179. ISSN 0020-7748.
5. ^ a b c Shor, P. (1999-01-01). "Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer". SIAM Review. 41 (2): 303–332. arXiv:quant-ph/9508027. Bibcode:1999SIAMR..41..303S. doi:10.1137/S0036144598347011. ISSN 0036-1445.
6. ^ a b Aaronson, Scott; Arkhipov, Alex (2011). The Computational Complexity of Linear Optics. Proceedings of the Forty-third Annual ACM Symposium on Theory of Computing. STOC '11. New York, NY, USA: ACM. pp. 333–342. arXiv:1011.3245. doi:10.1145/1993636.1993682. ISBN 9781450306911.
7. ^ King, James; Yarkoni, Sheir; Raymond, Jack; Ozfidan, Isil; King, Andrew D.; Nevisi, Mayssam Mohammadi; Hilton, Jeremy P.; McGeoch, Catherine C. (2017-01-17). "Quantum Annealing amid Local Ruggedness and Global Frustration". arXiv:1701.04579 [quant-ph].
8. ^ a b c Aaronson, Scott; Chen, Lijie (2016-12-18). "Complexity-Theoretic Foundations of Quantum Supremacy Experiments". arXiv:1612.05903 [quant-ph].
9. ^ a b "Google Plans to Demonstrate the Supremacy of Quantum Computing". IEEE Spectrum: Technology, Engineering, and Science News. Retrieved 2018-01-11.
10. ^ "CES 2018: Intel's 49-Qubit Chip Shoots for Quantum Supremacy". IEEE Spectrum: Technology, Engineering, and Science News. Retrieved 2017-07-22.
11. ^ a b "Google's quantum computing plans threatened by IBM curveball". October 20, 2017. Retrieved October 22, 2017.
12. ^ Harris, Mark. "Google has enlisted NASA to help it prove quantum supremacy within months". MIT Technology Review. Retrieved 2018-11-30.
13. ^ a b Boixo, Sergio; Isakov, Sergei V.; Smelyanskiy, Vadim N.; Babbush, Ryan; Ding, Nan; Jiang, Zhang; Bremner, Michael J.; Martinis, John M.; Neven, Hartmut (23 April 2018). "Characterizing quantum supremacy in near-term devices". Nature Physics. 14 (6): 595–600. arXiv:1608.00263. doi:10.1038/s41567-018-0124-x.
14. ^ A New "Law" Suggests Quantum Supremacy Could Happen This Year, Scientific American, Daily Digest, June 21, 2019
15. ^ a b Watrous, John (2009). "Quantum Computational Complexity". In Meyers, Robert A. (ed.). Encyclopedia of Complexity and Systems Science. Springer New York. pp. 7174–7201. doi:10.1007/978-0-387-30440-3_428. ISBN 9780387758886.
16. ^ Tereza, Tusarova (2004-09-26). "Quantum Complexity Classes". arXiv:cs/0409051.
17. ^ a b Vazirani, Umesh. "A Survey of Quantum Complexity Theory" (PDF). Proceedings of Symposia in Applied Mathematics.
18. ^ a b Lund, A. P.; Bremner, Michael J.; Ralph, T. C. (2017-04-13). "Quantum sampling problems, BosonSampling and quantum supremacy". Npj Quantum Information. 3 (1): 15. arXiv:1702.03061. Bibcode:2017npjQI...3...15L. doi:10.1038/s41534-017-0018-2. ISSN 2056-6387.
19. ^ Gard, Bryan T.; Motes, Keith R.; Olson, Jonathan P.; Rohde, Peter P.; Dowling, Jonathan P. (August 2015). "An introduction to boson-sampling". From Atomic to Mesoscale: the Role of Quantum Coherence in Systems of Various Complexities. World Scientific. pp. 167–192. arXiv:1406.6767. doi:10.1142/9789814678704_0008. ISBN 978-981-4678-70-4.
20. ^ Bremner, Michael J.; Montanaro, Ashley; Shepherd, Dan J. (2016-08-18). "Average-case complexity versus approximate simulation of commuting quantum computations". Physical Review Letters. 117 (8): 080501. arXiv:1504.07999. Bibcode:2016PhRvL.117h0501B. doi:10.1103/PhysRevLett.117.080501. ISSN 0031-9007. PMID 27588839.
21. ^ Jordan, Stephen. "Quantum Algorithm Zoo". math.nist.gov. Retrieved 2017-07-29.
22. ^ Rubinstein, Michael (2006-10-19). "The distribution of solutions to xy = N mod a with an application to factoring integers". arXiv:math/0610612.
23. ^ Babai, László; Beals, Robert; Seress, Ákos (2009). Polynomial-time Theory of Matrix Groups. Proceedings of the Forty-first Annual ACM Symposium on Theory of Computing. STOC '09. New York, NY, USA: ACM. pp. 55–64. CiteSeerX 10.1.1.674.9429. doi:10.1145/1536414.1536425. ISBN 9781605585062.
24. ^ Rivest, R. L.; Shamir, A.; Adleman, L. (February 1978). "A Method for Obtaining Digital Signatures and Public-key Cryptosystems". Commun. ACM. 21 (2): 120–126. CiteSeerX 10.1.1.607.2677. doi:10.1145/359340.359342. ISSN 0001-0782.
25. ^ Bernstein, Daniel (2009). Post-Quantum Cryptography. Springer. ISBN 9783540887010.
26. ^ Rahimi-Keshari, Saleh; Ralph, Timothy C.; Caves, Carlton M. (2016-06-20). "Sufficient Conditions for Efficient Classical Simulation of Quantum Optics". Physical Review X. 6 (2): 021039. arXiv:1511.06526. Bibcode:2016PhRvX...6b1039R. doi:10.1103/PhysRevX.6.021039.
27. ^ Carolan, Jacques; Harrold, Christopher; Sparrow, Chris; Martín-López, Enrique; Russell, Nicholas J.; Silverstone, Joshua W.; Shadbolt, Peter J.; Matsuda, Nobuyuki; Oguma, Manabu (2015-08-14). "Universal linear optics". Science. 349 (6249): 711–716. arXiv:1505.01182. doi:10.1126/science.aab3642. ISSN 0036-8075. PMID 26160375.
28. ^ a b Clifford, Peter; Clifford, Raphaël (2017-06-05). "The Classical Complexity of Boson Sampling". arXiv:1706.01260 [cs.DS].
29. ^ a b Neville, Alex; Sparrow, Chris; Clifford, Raphaël; Johnston, Eric; Birchall, Patrick M.; Montanaro, Ashley; Laing, Anthony (2017-10-02). "No imminent quantum supremacy by boson sampling". Nature Physics. 13 (12): 1153–1157. arXiv:1705.00686. doi:10.1038/nphys4270. ISSN 1745-2473.
30. ^ Edwin Pednault; John A. Gunnels; Giacomo Nannicini; Lior Horesh; Thomas Magerlein; Edgar Solomonik; Robert Wisnieff (October 2017). "Breaking the 49-Qubit Barrier in the Simulation of Quantum Circuits". arXiv:1710.05867 [quant-ph].
31. ^ a b Kalai, Gil (2011-06-02). "How Quantum Computers Fail: Quantum Codes, Correlations in Physical Systems, and Noise Accumulation". arXiv:1106.0485 [quant-ph].
32. ^ Shor, Peter W. (1995-10-01). "Scheme for reducing decoherence in quantum computer memory". Physical Review A. 52 (4): R2493–R2496. Bibcode:1995PhRvA..52.2493S. doi:10.1103/PhysRevA.52.R2493.
33. ^ Steane, A. M. (1996-07-29). "Error Correcting Codes in Quantum Theory". Physical Review Letters. 77 (5): 793–797. Bibcode:1996PhRvL..77..793S. doi:10.1103/PhysRevLett.77.793. PMID 10062908.
34. ^ Aharonov, Dorit; Ben-Or, Michael (1999-06-30). "Fault-Tolerant Quantum Computation With Constant Error Rate". arXiv:quant-ph/9906129.
35. ^ Knill, E. (2005-03-03). "Quantum computing with realistically noisy devices". Nature. 434 (7029): 39–44. arXiv:quant-ph/0410199. Bibcode:2005Natur.434...39K. doi:10.1038/nature03350. ISSN 0028-0836. PMID 15744292.
36. ^ Kalai, Gil (2016-05-03). "The Quantum Computer Puzzle (Expanded Version)". arXiv:1605.00992 [quant-ph].
37. ^ Dyakonov, M. I. (2007). "Is Fault-Tolerant Quantum Computation Really Possible?". In S. Luryi; J. Xu; A. Zaslavsky (eds.). Future Trends in Microelectronics. Up the Nano Creek. Wiley. pp. 4–18. arXiv:quant-ph/0610117. Bibcode:2006quant.ph.10117D.