# How space and time can be a quantum error correction code

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### The same codes required to prevent errors in quantum computers can give space-time tissue its inherent strength

*In the toy “holographic” universes (even if not in our present) the space-time fabric arises on the basis of a network of quantum particles. Physicists have discovered that this works on the principle of quantum error correction.*

In 1994, a mathematician from an AT&T research unit named Peter Shore instantly glorified quantum computers (QC), discovering that these hypothetical devices could quickly factor out large numbers - thereby breaking most of modern cryptography. But the fundamental problem stood in the way of the real creation of spacecraft: the natural instability of their physical components.

Unlike the binary bits of information of conventional computers, qubits are composed of quantum particles, which have the probability of being in one of two states, denoted | 0> and | 1>, at the same time. When qubits interact, their possible states become interdependent, and the chances of finding themselves in the states | 0> and | 1> depend on each other. The proportional probabilities grow the more, the stronger after each operation, the qubits become entangled with each other. Supporting and managing this exponentially growing number of simultaneous capabilities makes CCs theoretically powerful.

However, qubits are insanely error prone. The weakest magnetic field or random microwave pulse causes them to “throw bits”, changing their chances to be equal to | 0> or | 1> in relation to other qubits, or to “turn phases”, which reverses the mathematical relation of their two states. In order for the spacecraft to work, scientists need to find ways to protect information even when individual qubits are damaged. Moreover, these methods should detect and correct errors without measuring the qubits themselves directly, since measurement leads to a collapse of the coexisting capabilities of the qubit into a certain reality - and the good old 0 and 1 cannot support quantum computing.

In 1995, Shore, following his decomposition algorithm, published another astounding proof., about the existence of "codes correcting quantum errors." Computer scientists Dorit Aaronova and Michael Ben-Ohr (as well as other independent researchers) a year later proved that these codes could theoretically bring the number of errors to almost zero. “It was a major discovery in the 90s that convinced people that scalable quantum computing was possible in principle,” said Scott Aaronson , a leading quantum computer scientist at the University of Texas. “And that it’s just the hardest engineering challenge.”

*Peter Shore, Dorit Aaronova and Michael Ben-Or*

Today, although small QCs materialize in laboratories around the world, useful QCs that can outperform ordinary QCs have to wait a few more years or even decades. Much more effective error correction codes are required in order to deal with the discouragingly large number of errors generated by real qubits. Attempts to develop improved codes are “one of the most important issues in the field,” Aaronson said, along with improving iron.

However, as part of the persistent search for these codes over the past 25 years, something funny happened in 2014 - physicists have found evidence of the deep connections between quantum error correction and the nature of space, time and gravity. In the general theory of relativity by Albert Einstein, gravity is defined as the curvature of the fabric of space and time - or "space-time" - around massive objects. The ball thrown into the air moves in a straight line in space-time, and it is already bending towards the Earth. However, despite the full power of Einstein's theory, physicists believe that gravity should have a deeper, quantum origin, from which something similar to the fabric of space-time somehow emerges.

In 2014, three young researchers of quantum gravity came to an incredible result. They worked in a field beloved by theorists: the toy universe , known as the " anti-desitter space, " which works like a hologram. The curved fabric of space-time inside the universe arises as a projection of entangled quantum particles existing on its outer boundary. Ahmed Almeyri , Sea Don and Daniel Harlow performed calculations, from which it follows that this holographic “manifestation” of space-time works exactly like a quantum error correction code. In a journal of high energy physics, they published the assumptionthat spacetime itself is a code - at least in antidesitter space. The work generated a wave of activity in the community of quantum gravity researchers, and new quantum error correction codes were discovered, which included more space-time properties.

John Preskil , a theoretical physicist at the California Institute of Technology, says quantum error correction explains the reliability of space-time, despite being woven from brittle quantum matter. “We don’t have to deal with this idea too carefully so as not to ruin the geometry,” Preskil said. “I think this connection with quantum error correction is the deepest explanation of all that we have.”

The quantum error correction language is also beginning to allow researchers to study the mysteries of black holes: spherical sections in which space-time is so bent to the center that even light cannot escape from it. “All traces lead to black holes,” said Almeyri, currently working at the Princeton Institute for Advanced Studies. In these places filled with paradoxes, gravity reaches its peak, and Einstein's general theory of relativity ceases to work. “There are some signs that if we understand which code uses space-time,” he said, “it can help us understand the internal structure of black holes.”

As a bonus, the researchers hope that the holographic space-time can also suggest a way to scale the spacecraft, realizing the old dream of Shor and others. “Space-time is smarter than us,” said Almeyri. “A very effective quantum error correction code is built into these designs.”

*Ahmed Almeiri, Sea Don and Daniel Harlow*

How do quantum error correction codes work? The secret to storing information in troubled qubits is to store it not in separate qubits, but in a system of many confusing qubits.

As a simple example, consider the code for three qubits: they use three “physical” qubits to protect one “logical” qubit of information from flipping bits. Such a code is not particularly useful for quantum error correction, since it does not protect against phase reversal, but performs an explanatory function. The state of the logical qubit | 0> corresponds to the finding of all three physical qubits in the state | 0>, and the state | 1> corresponds to the fact that all three are in the state | 1>. The system is in a superposition of these states, which is written as | 000> + | 111>. But, let's say, one of the qubits beat a bit. How to detect and correct an error without measuring qubits directly?

A qubit can be entered into a quantum circuit through two gateways. One checks the "parity" of the first and second physical qubits - they are the same or different - and the other checks the parity of the first and third. When there are no errors (that is, the qubits are in the state | 000> + | 111>), the parity gateways determine that the first and second, as well as the first and third qubits are identical. However, if a bit is accidentally thrown at the first qubit, this leads to the state | 100> + | 011>, and the gateway determines the difference in both pairs. A bit transfer in the second qubit gives | 010> + | 101>, the gateways determine that the first and second qubits are different, and the second and third coincide; in the case of a transfer of the third qubit, the result will be “coincide; differ. " These unique results show which corrective surgery is needed, and whether it is necessary at all - that is, the operation of flipping back the first, second or third physical qubit, which does not lead to a collapse of the logical qubit. “The quantum error correction seems to me magic,” said Almeyri.

Better patch codes can usually recover all encoded information based on slightly more physical qubits than half, even if all the others are damaged. This is what prompted Almeyri, Don, and Harlow in 204 to think about the possible connection between quantum error correction and how antidesitter space (AdS) arises from quantum entanglement.

It is important to note that AdS space is different from the space-time geometry of our desitter space. Our Universe is saturated with positive vacuum energy, which forces it to expand without boundaries, while in AdS space the vacuum energy is negative, because of which it acquires a hyperbolic geometry similar to M. K. Escher's “Limit - Circle”. Escher's mosaic creatures are getting smaller, spreading from the center of the circle, eventually disappearing on the perimeter. In the same way, the spatial dimension emanating from the center of the AdS of space is constantly compressed and eventually disappears, indicating the outer boundary of the universe. AdS space became popular with theoretical physicists studying quantum gravity in 1997, after the famous physicist Juan Maldacena discovered,

*Hyperbolic geometry on Escher's 1959 engraving “Limit - Circle III” depicts AdS space*

Studying how duality works like hundreds of other physicists in the last couple of decades, Almeyri and colleagues noticed that any point inside AdS space can be created on the basis of a little more than half boundaries are just like the optimal quantum error correction code.

In their work, which suggests that holographic space-time and quantum error correction are the same thing, they described how even the simplest code can be represented as a two-dimensional hologram. It consists of three coutrites"- particles that exist in one of three states - located at equal distances from each other in a circle. An entangled trio of coutrites encodes one logical coutrite, corresponding to one point in space-time in the center of the circle. The code protects the point from erasing any of the three coutrites.

Of course one point -. so imagine the universe In 2015, Harlow, Presque Fernando Pastavsky and Beni Yoshida discovered another holographic code named HappY, to adopt more properties of the space of the ads code divides the space into five-sided tiles. - "as a small Circuits Design Tinkertoy », said Patrick Haydenfrom Stanford University, research leader. Each detail represents one point in space-time. “These tiles would play the role of fish in Escher's mosaic,” said Hayden.

In HaPPY code and other open holographic error correction schemes, everything inside a space-time region called an entanglement wedge can be recreated from qubits in a region adjacent to the border. Confluent wedges will be superimposed on overlapping regions on the border, Hayden said, just as a logical qubit in a spacecraft can be recreated from many different subsets of physical qubits. “And here the bug fix feature comes into play.”

“Quantum error correction gives us a general idea of geometry in this code language,” Preskil said. The same language, he said, “can probably be applied to more generalized situations,” in particular to a desitter universe like ours. But it turned out that the Desitter space, which has no boundaries, is much more difficult to imagine in the form of a hologram.

So far, researchers such as Almeyri, Harlow, and Hayden have been working with the AdS space, since it has many similarities with the Desitter world, but it is easier to study. The space-time geometries of both spaces obey Einstein's theory; they simply bend in different directions. And which is probably much more important, there are black holes in the universes of both types. “The most fundamental property of gravity is the presence of black holes,” said Harlow, now an associate professor of physics at MIT. “This is what distinguishes gravity from all other interactions. Therefore, with quantum gravity is so hard. "

The quantum error correction language has provided a new way of describing black holes. The presence of a black hole is defined as a “failure to fix”, said Hayden: “When you have so many errors that you can no longer track what is happening in a lot of space-time, you come to a black hole. It’s something like a drain for ignorance. ”

Ignorance inevitably accumulates in relation to the insides of black holes. Stephen Hawking’s revelation of 1974 that BHs radiate heat and, therefore, evaporate sooner or later, gave rise to the notorious “black hole information paradox” that asks what happens with all the BH information swallowed. This problem may be related to cosmology and the birth of the Universe, since the expansion of the Big Bang singularity is very similar to the gravitational collapse of BH in reverse reproduction.

AdS space simplifies the issue of information. Since the boundary of the ADS of the universe is holographically dual to everything that it contains - to all BHs and other things - the information falling into the BHs is not guaranteed to be lost; it will always be holographically encoded at the boundary of the universe. Calculations show that in order to reconstruct information about the interiors of a black hole based on qubits at the border, you will need access to tangled qubits at about three quarters of the border. “A little more than half is not enough,” Almeyri said. He added that the demand for three quarters should report something important about quantum gravity, but the question of why such a fraction is obtained "still remains open."

In Almeyri’s first application of 2012, the tall thin physicist from the UAE and his three associates deepenedinformation paradox. Their reasoning suggested that the information might not get inside the BH at all, since the “firewall” on the horizon of the BH events would stop it.

Like most physicists, Almeyri does not believe that BH firewalls really exist, but it turned out to be difficult to bypass this concept. Now he believes that the formation of firewalls is hindered by quantum error correction, which protects information even after crossing the BH horizon. In his last independent workthat appeared in October, he said that quantum error correction is “necessary to maintain the smoothness of space-time on the horizon” of a black hole of two components, known as a wormhole. He suggests that quantum error correction not only prevents firewalls, but also allows qubits to escape from the black hole after falling into it, since the entanglement threads between the inner and outer parts of the black hole are themselves similar to miniature wormholes. That would explain Hawking's paradox.

This year, the Department of Defense allocated funding for the study of holographic space-time, in particular because advances in this area could lead to the emergence of more efficient error correction codes for quantum computers.

Physics will still have to figure out whether it is possible to describe the Desitter universe, for example, ours, using a hologram and in terms of qubits and codes. “All this connection is known to a world that is obviously not ours,” Aaronson said. In a work from last year, Don, now working at the University of California at Santa Barbara, and his co-authors Eva Silverstein and Gonzalo Torroba, took a step in the desitter direction, trying to create a primitive holographic description. Researchers are still studying this proposal, but Preskil believes that the language of quantum error correction will still be transferred to real space-time.

“In fact, space is held in by intricacy,” he said. - If you want to sew space-time from small pieces, they must be properly confused. And it will be right to do this by creating a quantum error correction code. "