It’s one of the buzz of the moment, the Canadian start-up Xanadufounded in 2016 in Toronto, has announced that it has achieved quantum supremacy — or what can be said to be very similar — with its programmable quantum computer at photons.

Remember that we tend to define quantum supremacy as being able to perform an algorithm on a machine using the laws of Quantum mechanicsessentially those of the superposition of states and theentanglementto perform a calculation very quickly, say in seconds, whereas the best algorithm that could do the same calculation on a conventional supercomputer would take such a long time that a human lifetime might not be enough to wait for the result.

There is not yet a precise and unanimous formulation on what should be understood by quantum supremacyor the quantum advantage which means almost the same thing, if not a general idea close to the explanation which has just been given.

A number of big announcements today from the incredible team @xanaduai!

1. Xanadu has achieved quantum computational advantage (supremacy) on our new photonic quantum computer called Borealis and is peer reviewed in @nature (https://t.co/KlpY7i8S6V)

— Christian Weedbrook (@_cweedbrook) June 1, 2022

Concretely, in the case of Xanadu, and as its members explain in an open access article published in the journal *Nature*their quantum machine baptized Borealis performed in about 36 microseconds a calculation that would take a conventional supercomputer about 9,000 years, a dizzying speed factor.

A presentation of Borealis from Xanadu. To obtain a fairly accurate French translation, click on the white rectangle at the bottom right. The English subtitles should then appear. Then click on the nut to the right of the rectangle, then on “Subtitles” and finally on “Translate automatically”. Choose “French”. © Xanadu

## From Planck to Feynman with quantum mechanics

However, it is customary to take tweezers with this kind of result because it has sometimes happened that the discovery of a new classic algorithm makes it possible to obtain the same result, or even faster. A calculator, or a universally programmable quantum computer, does not systematically achieve quantum supremacy either.

Finally, let us recall that quantum calculations are physically affected on physical systems by very significant disturbances which, at the very least, produce many errors or make it impossible to complete the execution of an algorithm all the more quickly as it requires a large number of qubitsthe quantum equivalents classic bits of information. It is the famous decoherence problem which has, in fact, made many specialists skeptical about the real performance that could have quantum computers universally programmable. Although in recent years many have become more optimistic about the performance of machines that can be called computers, or quantum simulators — because they specialize in the execution of certain algorithms –, we may still be at the threshold of a quantum revolution.

That would at least prove right once again the visionary genius of Richard Feynmana pioneer in quantum computer theory, who had proposed using the physical laws and systems of quantum mechanics to quickly simulate the behavior of other quantum systems that were intractable with classical computers or at least very difficult, such as the properties of molecules in quantum chemistry or the behavior of quarks and gluons in the hadrons.

Discover in animation-video the history of quantum physics: from the ultraviolet catastrophe to the promises of the quantum computer, passing through the first and the second quantum revolution with the ideas of Feynman and Peter Shor. A video-animation co-produced with *The Sorcerer’s Spirit*. © CEA Research

## Qubits with photonic compressed states

But back to Xanadu and Borealis. The machine operates with photons and it does not need to be cooled almost to the absolute zero unlike many other quantum circuits that protect themselves from thermal noise that quickly degrades quantum calculations.

Quantum calculations with photons are often done with two photon polarization states allowing to have a quantum qubit with a superposition of these two states equivalent to a quantum superposition of 1 and 0. But the case of Borealis, these are quantum states of electromagnetic field from light different ones that are used, examples of what are called compressed states or *squeezed-states* in English.

Technically, an example of these states is realized in quantum mechanics with a harmonic oscillator, that is to say essentially the equivalent of an oscillating weight at the end of a spring. We can consider an electromagnetic field as a set of such springs since charged particles in an oscillating field would behave like these harmonic oscillators. A compressed state would be that, or in accordance with Heisenberg’s relations, the product of “uncertainties” about the position and the amount of movement of the particle at the end of an oscillating spring would have the minimum possible value and therefore would characterize in a way the least blurry and quantum noisy state for an oscillator — it’s a little different in the case of states compressed with photons but the purpose and the idea are very similar.

Borealis therefore exploits a quantum superposition of compressed states of photons and in this case 216 qubits with these states.

A description of the machine can be found on the Xanadu site and in the accompanying videos.

Basically, it can be thought of as a black box with photons entering through several channels and exiting through others with detectors to count the number of those leaving through each channel in each experiment carried out.

Inside the box is a set of devices ( mirrors and blades separating light rays in particular) affecting the incoming light beams, first producing compressed states, then separating them into other beams, changing what is called their phase and producing entanglements between the photons present as well as interference

All of these devices behave like portes elementary logics of classical computers that can be combined to form many operations and therefore carry out a large number of different algorithms that one may wish to program. But these are quantum logic gates like the ones discussed in the video below.

A further presentation of the concept of computer and quantum algorithm. © CEA Research

## Graph theory quantum calculations

Technically again, Borealis behaves mathematically like a matrix, an array of numbers, allowing to produce from a column of numbers in input, a column of numbers in output (the channels of photons with a count of those which arrive for each channel). This matrix encodes a programmable algorithm, as many matrices, as many algorithms.

Ultimately, Borealis performs what is also called Gaussian bosonic sampling (photons are bosons) still called *Gaussian boson sampling* (GBS) in English. In each experiment, the photons entering through the channels follow different paths in the machine and exit according to a different distribution in the final channels.

But the average distribution of photons for each detector at the channel output must behave in accordance with a probability distribution which is proportional to a quantity which is calculated with the matrix of the algorithm implemented in Borealis, a quantity which mathematicians call the Hafnian of a matrix (not to be confused with what is called the determinant). The Hafnian becomes more and more difficult to compute classically with the number of inputs and outputs — which are equal, the matrix being square as we say.

But as we said, Borealis performed a Hafnian calculation in 36 microseconds instead of 9,000 years and this, 50 million times faster than with previous quantum calculators with photons. Cerise on top of that, Xanadu has given free online cloud access to its machine so that, in principle, anyone can try programming an algorithm on Borealis.

Xanadu has uploaded tutorials to program in python with some ” softwares ” as PennyLane et Strawberry Fields.

A presentation of the Xanadu photonic chip. To obtain a fairly accurate French translation, click on the white rectangle at the bottom right. The English subtitles should then appear. Then click on the nut to the right of the rectangle, then on “Subtitles” and finally on “Translate automatically”. Choose “French”. © Xanadu

Among the perspectives opened up by Xanadu, there is the *Machine Learning* quantum and the fact that the technology behind Borealis makes it possible in theory to easily increase the size of the photonic quantum computer. The members of Xanadu even think that they can reach the million qubits!

Achieving supremacy or what is called the quantum advantage is one thing but prodigiously speeding up the execution of an algorithm only has an impact if the algorithm in question has the potential to concretely solve important practical problems. . So what’s the point to be able to quickly calculate the Hafnian of large matrices?

It turns out that the Hafnian, for GBS determinations, is involved in calculations of graph theory to do optimization, *Machine Learning* and in quantum chemistry when trying to determine spectres of molecules which vibrate. In the latter case, it makes it possible to predict and study how materials absorb light at different frequencies. The spectres d’absorption studied can be useful for optimizing the yield of solar cells or in the development of pharmaceutical products, as explained on one of the pages of Xanadu whose comments we include in this paragraph and the following.

Finally, let us recall that the graph theory can be used to model financial markets, biological networks and social networks. The graphs are composed of a set of knots interconnected and one of the problems that we often try to solve with them is to find clusters, that is to say regions with a high level of connectivity. They can correspond to communities in social networks, correlated assets in a market or proteins mutually influential in a biological network.

A quantum computer with a photonic chip like Borealis’ should be able to work with 1 million Qubits. To obtain a fairly accurate French translation, click on the white rectangle at the bottom right. The English subtitles should then appear. Then click on the nut to the right of the rectangle, then on “Subtitles” and finally on “Translate automatically”. Choose “French”. © Xanadu

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