Quantum supremacy

This algorithm finds the prime factorization of an n-bit integer in ${\displaystyle {\tilde {O}}(n^{3})}$ time[31] 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 ${\displaystyle O(2^{n/3+o(1)})}$.[32] 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.[33]

This algorithm is important both practically and historically for quantum computing. It was the first polynomial-time quantum algorithm proposed for a real-world problem that is believed to be hard for classical computers.[31] Namely, it gives a superpolynomial speedup under the reasonable assumption that RSA, today's most common encryption protocol, is secure.[34]

Factoring has some benefit over other supremacy proposals because factoring can be checked quickly with a classical computer just by multiplying integers, even for large instances where factoring algorithms are intractably slow. However, implementing Shor's algorithm for large numbers is infeasible with current technology,[35][36] so it is not being pursued as a strategy for demonstrating supremacy.

This computing paradigm based upon identical photons sent through a linear-optical network can solve certain sampling and search problems that, assuming a few complexity-theoretical conjectures (that calculating the permanent of Gaussian matrices is #P-Hard and that the polynomial hierarchy does not collapse) are intractable for classical computers.[7] However, it has been shown that boson sampling in a system with large enough loss and noise can be simulated efficiently.[37]

The largest experimental implementation of boson sampling to date had 6 modes so could handle up to 6 photons at a time.[38] 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.[39][40] 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.[39][40]

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.[14] 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.[10] But the largest quantum circuit simulation completed successfully on a supercomputer now contains 56 qubits.[41] This may require increasing the number of qubits to demonstrate quantum supremacy.[12] On October 23, 2019, Google published the results of this quantum supremacy experiment in the Nature article, “Quantum Supremacy Using a Programmable Superconducting Processor” in which they developed a new 53-qubit processor, named “Sycamore”, that is made of fast, high-fidelity quantum logic gates, in order to perform the benchmark testing. Google claims that their machine performed the target computation in 200 seconds, and estimated that their classical algorithm would take 10,000 years in the world’s fastest supercomputer to solve the same problem.[42] IBM disputed this claim, saying that an improved classical algorithm should be able to solve that problem in two and a half days in that same supercomputer. [43]

Some researchers have suggested that the term "quantum supremacy" should not be used, arguing that the word "supremacy" evokes distasteful comparisons to the racist belief of white supremacy. A controversial[52][53] Nature commentary signed by thirteen researchers asserts that the alternative phrase "quantum advantage" should be used instead.[54] John Preskill, the professor of theoretical physics at the California Institute of Technology who coined the term, clarified: "I proposed the term 'quantum supremacy' to describe the point where quantum computers can do things that classical computers can’t, regardless of whether those tasks are useful. With that new term, I wanted to emphasize that this is a privileged time in the history of our planet, when information technologies based on principles of quantum physics are ascendant."[55] He further explained: "I considered but rejected several other possibilities, deciding that quantum supremacy best captured the point I wanted to convey. One alternative is 'quantum advantage,' which is also now widely used. But to me, 'advantage' lacks the punch of 'supremacy.' In a race, a horse has an advantage if it wins by a nose. In contrast, the speed of a quantum computer vastly exceeds that of classical computers, for certain tasks."[56]