Holographic Quantum Error Correcting Codes

Exploring the intersection of quantum gravity in the theory of quantum error correction.

Quantum error-correction is necessary if we want to have fault-tolerant quantum computers that possess the massive computational capability to solve some exponentially hard problems. Qubit systems become more and more susceptible to noise as they scale up in number. Error-correcting codes that can fix bit-flip errors are important in outputting accurate results from our quantum computers.

Holographic quantum error-correcting codes are a new method to approach this problem and they were introduced by the famous physicist John Preskill. To understand how they work we need to dive deeper into the theoretical side of quantum gravity and the holographic principle.

Quantum Mechanics + General Relativity = ?

We suck at explaining how the universe works as whole, there are direct distinctions between the laws of small things (atoms) versus big things (planets).

The biggest challenge for theoretical physicists over the past couple decades has been to find a unifying theory between general relativity and quantum mechanics. The most incompatible theories which work separately but never together! We need both classical and quantum laws to describe nature. A single theory that encompasses all fundamental laws of the universe is commonly termed The Theory of Everything.

The Differences:

Quantum mechanics are the rules that govern how microscopic particles interact, think the size of electrons for scale. Undoubtedly the smaller you get the weirder reality seems to be because nature now allows for these smaller particles to behave in ways that simply don’t make sense on larger scale. One specific phenomena is quantum entanglement, the idea of how two particles can have a special bond between them that will not break no matter how far apart these particles are. Entanglement is fundamentally a very abstract concept that spooked even Einstein! There are more weird phenomenons such as superposition and tunnelling but I won’t go deep into what they are in this article, you can view my past articles for more background info (I will link them down below).

General relativity is what describes the force of gravity. The theory is built on the framework that our universe has a dynamic geometry defined by curvature of spacetime. The universe can be perceived as a lot of empty space with matter and time which is a linear. Spacetime creates a 4 dimensional continuum of the universe with 3D space and one dimension of time. We can roughly picture our universe geometrically as 4d landscape of spacetime however curvature is introduced into the equation due to gravity. Without gravity we would be experiencing what you call Minkowski Space (virtually the same as our universe expect spacetime is flat). However we have gravitational force acting upon us thus Einstein’s theory of general relativity provided a solution that included spacetime and how matter interacts while having gravity as a present force. So spacetime is curved. Large cosmological bodies curve the geometry of spacetime based on parameters such as mass and this in turn determines how other objects will move around it.

A Model of Spacetime Curvature

Both these theories are a lot more complex but the gist is quantum mechanics explain all the weird phenomenons of really small things and general relativity explains the interaction of very larger things through spacetime. BUT just like oil and water, these theories don’t mix yet.

This happens because our understanding of gravity in quantum mechanical equations breakdown. The wave function or the famous Schrodinger’s equation gives us the probability of where an electron is at a given point in time. This fundamentally describes the superposition states of quantum systems however these calculations only work in Minkowski space where spacetime doesn’t have a gravity tensor so no curvature of spacetime is considered. This is a problem because we know large bodies such as black holes or even us create a curvature in spacetime. Subatomic particles are representations of almost flat-spacetime where our quantum calculations work but there is no way to define quantum systems where gravity is a present force.

That is why many smart people are trying to find ways to conceptualize a single theory called Quantum Gravity.

Quantum gravity strives to find ways to incorporate gravitational effects in quantum mechanics. We don’t have a complete theory however there are many clues which have been uncovered by smart physicists that bring us a lot closer to having a complete picture of the dynamic of our universe.

AdS/CFT Correspondence

One really important physicist in this field is Juan Maldacena. He theorized a special duality of spacetimes using Anti de-Sitter (AdS) and conformal field theory (CFT). AdS is representation of our universe but with negative curvature of spacetime, often referred to as the bulk. It obeys our laws of physics but space itself neither expands nor contracts. The other key component to Juan’s research was the boundary theory which obeyed the laws of a quantum system called Conformal Field Theory. It’s a model with less dimensions than AdS and without gravity. The unique insight was combining these two models together to get AdS/CFT correspondence. It creates a bottled structure where AdS is the volume inside and CFT describes the outside membrane which mathematically is always an infinite distance away from any point in the bulk.

AdS/CFT Correspondence

It can be hard to imagine what this space looks like so a simple analogy is to think of the bulk as the volume of air in a balloon and casing of the same balloon as the boundary. The correspondence would be knowing information about the air inside the balloon by examining the surface area geometry of the casing of balloon.

Okay but why does this absurd and abstract representation of the universe matter?

Introducing The Holographic Principle:

The AdS/CFT correspondence describe a special duality which wreaks a sense of havoc on our understanding of the physical world, its called the holographic principle. To describe this lets examine blackholes, the theory explains that the information stored in a black hole is not calculated based on its volume but the surface area of the event horizon. Which means the information represented inside of a black hole is a hologram of the information stored on the 2d surface of the event horizon. This holds true for AdS/CFT correspondence as well because the boundary of CFT can actually reveal information about the bulk AdS much like a hologram. The craziest part is that information of degrees of freedom/qubits are proportional to the boundary of spacetime itself. This duality emerges because of the entangled geometry of CFT boundary. To understand this at a deeper level I would recommend diving deep into understanding black holes because they hold a lot of significance in revealing many counter-intuitive parts of nature.

So the holographic principle is believed to be a part of quantum gravity. Much more can be revealed from black holes as well since they are objects with massive amounts of gravity which curves spacetime but stored in an extremely small space called the singularity. It’s absolutely fascinating how singularities bend spacetime but are also so small that quantum effects should be taken account and the only way to have a complete understanding is by having a theory for quantum gravity.

Black Hole Geometry

A really interesting implication was derived by physicist Mark Van Raamsdonk who proposed a theory of quantum gravity by researching on how the spacetime geometry is derived from quantum entanglement. This is a a whole new topic which is equally as mind-blowing.

So now we understand briefly what the holographic principle is and how it is represented in AdS/CFT correspondence and black holes. Next step is to take this theory and apply it as quantum error-correction codes. But first lets understand what quantum error-correction is.

Quantum Error-Correction:

In classical error-correction information is copied and stored multiple times in case there is an error then output will still give correct answers based on the majority of information all copied bits carry. But this is impossible in quantum computing due to the no-cloning theorm where it states quantum information cannot be copied so error-correction must be done in a new way. Peter Shor introduced a new way of quantum error-correction. He spread out the quantum information of one qubit into a high entangled state of many other qubits. We can call these logical and physical qubits. The information is protected from the environment such that local errors will not affect the entangled states. If there are errors in the system (bit-flips, phase errors, etc.) we cannot measure it without destroying the superposition state. Which is why error syndrome(stabilizer codes) measurements are applied to retrieve data only on the error and not the actual encoded information. Once information on the error is known locally, specific gates can be applied to fix the error without disturbing the superposition state.

With this intuition its easy to understand how quantum error-correction works and now how it can be done using holography. We know that there is bulk/boundary duality where information about the bulk is related to entanglement structure on the boundary.

Holographic codes create the same duality by embedding the bulk Hilbert space to the boundary subset of Hilbert space. The logical qubits would be within the bulk and the physical qubits reside on the boundary. The codes use a perfect tensor network to create the bulk/boundary geometry.

An example holographic code

Each pentagon is a tensor with 6 indices (the red dot refers to an index going out of the page). It creates a symmetrical network of tensors such that if a cut is made on the network, tensors will retain a maximally entangled state. Then you can map an error-correcting code onto the tensors, in this example it will be 1 qubit mapped onto 5 indices which will correct 2 errors by erasure.

The tensor above has maximally entangled qubits. The R is a reference qubit entangled with our encoded qubit. Subsystem E are physical qubits with errors which are erased while their counterpart qubits are preserved. Due to their entanglement the reference qubit and Ec can decode the information about the logical qubit after erasing qubit E by using a unitary operator. And viola you have error-correction by kicking out only the affected qubits.

Writing a Error-Correcting Code in Q#:

Checking for errors with destroying quantum information

Quantum computing is going to become a technology which is going to allow us to reach a new wave of computation and enable us to learn more about nature itself. Surely it will unlock many secrets about how our universe works but only if we can make robust machines that aren’t affected by noise. This will be done by improving our hardware and our error-correction protocols. That is why continuous research into new methods of quantum error-correction like using holography is so important.

Next Steps:

My goal is to continue researching about how to leverage quantum computers to solve really important problems in our world today. We start by tackling all the issues we are facing right now, one of them being error-correction. I will continue building more projects to improve my skillsets and knowledge base. Check out my last articles to learn the very basics of what quantum computing is :)

Past quantum computing articles:

A Brief Introduction to Quantum Computing_Quantum physics to me was always this mystifying property which give unreal capabilities to subatomic particles. It was…_hackernoon.com

Decoherence: Quantum Computer’s Greatest Obstacle_Okay, so I have explained the basic theory behind how quantum computers will work in my previous article (A Brief…_hackernoon.com

I am still improving my knowledge and skills so feel free to reach out to me for anything, I am always happy to talk to people interested in quantum computing!

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