Letters to the Universe

Letters to the Universe

Issue 1 | On “Error-Resilience Phase Transitions in Encoding-Decoding Quantum Circuits”

THE WHY

My first iteration of Universum Labs was half a year ago and felt like an attempt to fantasize my way out of the career I had and not at all like a career I would actively pursue. As I refined the details of it over time, I realized that it wasn’t at all an escape from reality, but perhaps the reality in which the wave function collapses on its existence. Though much is yet to come, at the heart of Universum Labs, and by extension The Daily Qubit, is my passion for physics and the need to make it more accessible to the global community. As a professor once told me, “Physics is hard to understand, and even harder to teach.” This will be increasingly applicable as we enter the infantile stages of the quantum era as quantum theory itself moves from an isolated, eyebrow-raising antithesis of classical mechanics into the wider public eye.

Quantum computing is currently a mix of public market frenzy and honorable academics detailing their hard-won findings in peer-reviewed journals. Now, these journals can understandably seem like dense conglomerations of important but hard to extract information followed by seemingly unending paper trails of past discoveries. But, science requires such things and we must be able to sift through them to reach a deep level of comprehension. Quantum computing is rooted deep in quantum theory and it is difficult enough to understand the state of progress let alone contemplate viable solutions without understanding the science behind the “magic”. I’m sure few would argue that magic and quantum mechanics seem interchangeable at times.

While The Daily Qubit provides your daily hit of quick facts curated for importance and presented in a way to show you I understand that time is in fact money and both seem in short demand these days, this series of weekly letters will serve as an expanded analysis on the research or news that resonated the most. And, as a twist that stems from my deep love of physics, I must impart the wonder I see in it along the way.

These analyses will favor comprehensive explanations that isolate why this is applicable instead of how it was performed. While mathematical proof has its place, this is all about the “why”. While I’ll often focus on research submissions, the mathematical formulation will be scant with an emphasis on retaining the key vocabulary. With each word or phrase that is new to you, I encourage you to seek out sources for further explanation. This will not only accustom you to research as a necessary skill – we’ll need this for later – but also solidify your comprehension.

FOCUS & SUMMARY

Let’s break this down further.

ENCODING & DECODING CIRCUITS

Encoding circuits are used to encode logical qubits into a larger number of physical qubits, distributing the quantum information so that it becomes resilient to certain types of errors. Said another way, this process involves entangling the logical qubits with additional qubits so that the original information can be recovered even if some of the qubits are affected by errors.

Decoding circuits are used to disentangle the information from the error-protective state back into a smaller number of qubits or to interpret the error syndromes without necessarily disentangling the qubits. This process often involves measuring some of the qubits to identify errors and applying corrections based on the outcomes of these measurements. The design and implementation of these circuits are tailored to the specific quantum error correction code being used.

Encoding/decoding circuits are one of many types of circuits that can be found within quantum computing devices. These specific circuits are most relevant in quantum error correction (QEC) and quantum communication protocols, such as quantum key distribution and quantum teleportation. These communication protocols require the encoding and decoding of quantum information to ensure secure communication channels over potentially noisy environments.

ERROR-PROTECTING & ERROR-VULNERABLE STATES

In an error-protecting state, a quantum system has the ability to effectively prevent, correct, or minimize the impact of errors on the quantum information it holds. This is critical for fault-tolerant quantum computing.

An effective QEC implementation maintains the system in an error-protecting phase so that computations may proceed reliably.

In an error-vulnerable state, a quantum system is susceptible to errors which significantly impact the integrity of the quantum information it holds. In this state, the errors can multiply or become too complex for the QEC protocols to correct which leads to a breakdown in accuracy and reliability of the computation. In other words, this is the state that can prevent a quantum computer from reaching or even maintaining quantum advantage.

SIGNIFICANCE OF THE PHASE TRANSITION

This returns us to the purpose of the research itself. If the overall goal is to create a quantum system with an infallible QEC protocol, it’s first crucial to understand what quantum processes are at work that would cause a quantum computer to move between error-protecting to error-vulnerable states. The phase transition itself is of general interest to physics, because it provides insights into the behavior of quantum many-body systems.

If you’re new to the world of physics, your exposure to many-body systems may be limited to that in the media (Netflix’s recent adaptation of “The Three Body-Problem”). It’s important to note that there is a difference between classical many-body systems and quantum many-body systems. The three-body-problem is a classical example of a system that is challenging to predict long-term. This is due to the complex and unpredictable dynamics influenced by the gravitational forces acting between them.

A quantum many-body system can be simply stated as a large number of particles governed by the principles of quantum mechanics. These systems are even more challenging to predict due to the combinatorial nature of the particles' ability to exist in multiple states simultaneously (superposition). Plus, entanglement adds another layer of complication due to non-local interactions that cannot be described by classical mechanics.

Studying phase transitions in quantum many-body systems, such as within encoding and decoding circuits, can provide new insights into the behavior of these systems which has applications not just in quantum computing, but in other areas such as condensed matter physics and statistical mechanics.

RÉNYI ENTROPY TRANSITIONS

Before we add in our friends Rényi and Shannon, we need to be clear on entropy.

Our reality is guided by the laws of physics. Along with many other physicists, I take great comfort in knowing that we are not just abstract beings inhabiting an abstract plane, but rather the result of traceable processes we’ve worked hard to define since we could question our existence. This thought staves off the existential musings, no?

Enter entropy. Entropy is randomness, disorder, uncertainty, and unpredictability. And no, you can’t ignore it because the second law of thermodynamics tells us that the total entropy of an isolated system can never decrease over time AND that systems naturally evolve towards a state of maximum disorder.

While entropy is understandably a lesser-enjoyed topic within physics and statistics, the concept has undeniable implications for our natural world and the understanding of society’s historical progress. Consider the pre-industrial era with its simple (albeit, shorter) lifestyle which persisted for many centuries. The dawn of the industrial revolution was marked by a significant acceleration in technological progress and as a byproduct, societal complexity. Within 200 years, we had nuclear energy followed by the personal computer. Within 20 years, the internet. And within a decade, AI and further exploration into quantum computing. This gentle cruise followed by exponential increase (and eventually plateau) is best modeled by the sigmoid curve, or S-curve. The S-curve is a potent symbol for many phenomena in our world, capturing the trend of growth, saturation, and eventual stabilization. Consider the following:

• Population growth

• Biological processes such as growth of bacteria

• The economic growth of nations, the boom and fizzle of empires

All this to say, embrace the disorder as a lens through which to recognize patterns and navigate the phenomena of our world.

As we dial back the musings and return to encoding/decoding quantum circuits and their phase transitions, let’s consider two different applications of entropy to informational systems (of which quantum circuits are).

Shannon entropy is a cornerstone of information theory. It quantifies the amount of uncertainty in a variable or message.

Rényi entropy, of which we are particularly interested in, adds to Shannon entropy a parameter of order. This allows for a deeper exploration of a system’s randomness. As it relates to quantum computing, Rényi entropy provides insights into the complexity and error tendencies of quantum states. Such insights are important for understanding the error resilience of these states.

The research in question considers how quantum circuits transition between states that are resistant to errors and states that are susceptible to errors. Rényi entropy helps to unveil how these transitions show a shift in the system’s informational structure and its proneness to errors. By characterizing these transitions, we can refine QEC protocols and get one step closer to realized fault-tolerance. (Nobel prize for anyone that does so, I’m sure.)

MULTIFRACTALITY

For some reason of which I’m not sure and try not to apply too much meaning to, I remember clearly the day a high school math teacher brought my attention to fractals. I had never paid attention to the concept before and I was shocked how beautiful a geometric concept in mathematics could be. Up until that point, I received off and on B’s in mathematics which cultivated in me a bitterness towards the subject. But fractals fascinated me in that they provided this bridge from the sterile, coldness of the black and white pages of Times New Roman mathematical theory and numerical equations in which I encapsulated math to the wonder and warmth of the natural world. I could walk outside and easily see evidence of fractals within the branching of trees, Romanesco cauliflower, the phyllotaxy of a plant, honeycomb, the crest of a wave.

This is so distinct because it was my first realization that the world is not just chaos (thermodynamics was in the future), but instead a collection of intentional and repeatable occurrences. I saw math for what it was: the language in which reality communicates with us.

Quantum can be fractal, and unsurprisingly, quantum fractality is not simple. It is actually multifractal. Unlike a simple fractal pattern, a multifractal occurrence is a layered pattern of fractal behavior across a spectrum. If we apply this to the context of quantum circuits, it refers to the demonstrated pattern of probability distributions and correlations that emerge from the interplay between the coherent and incoherent errors, as evidenced by the overlaid sigmoid curves from the numeric analysis of the research. See how they elevate a once black and white (though terribly interesting) research paper into art?

FULL CIRCLE

And with that, I believe we’re ready to once again return to the brevity of the thing.

My goal is that this now feels more approachable and it can be seen for what it is — an incredible contribution to quantum computing knowledge as well as the physics community at large.

Error-Resilience Phase Transitions in Encoding-Decoding Quantum Circuits reveals a phase transition in encoding-decoding quantum circuits from error-protecting to error-vulnerable states, marked by Rényi entropy transitions and multifractality. From this, we may extract insights into phase transitions within many-body quantum systems as well as insights into the refinement of QEC, and ultimately move one step closer to widespread quantum advantage.