diff --git a/src/thesis/chapters/3_fault_tolerant_qec.tex b/src/thesis/chapters/3_fault_tolerant_qec.tex index 4e356e9..2728f22 100644 --- a/src/thesis/chapters/3_fault_tolerant_qec.tex +++ b/src/thesis/chapters/3_fault_tolerant_qec.tex @@ -1,18 +1,42 @@ % TODO: Make all [H] -> [t] -\chapter{Fault Tolerant QEC} +\chapter{Fault-Tolerant Quantum Error Correction} % Intro -An important challenge of \ac{qec} that was recognized early on is -the fact that the error correction machinery itself may introduce new -errors \cite[Sec.~III]{shor_scheme_1995}. -Specifically for stabilizer codes, errors may happen during the -syndrome extraction process, since it is implemented in quantum hardware itself. -We call the errors the \ac{qec} procedure is supposed to correct -\emph{input errors} and the errors introduced by the procedure itself -\emph{internal errors}. -In order to be \emph{fault-tolerant}, the procedure must be able to -address both types of errors. +In the previous chapter, we introduced the fundamental concepts of +classical and quantum error correction. +In this chapter, we shift our focus to some practical aspects of the +implementation of \ac{qec} systems. + +% Fault tolerance as a general notion + +The reason we turned to \ac{qec} in the first place is to allow us to +perform quantum computing despite the errors inherent to operations +using qubits. +While the use of error correcting codes may facilitate this, it also +introduces two new challenges \cite[Sec.~4]{gottesman_introduction_2009}: +\begin{itemize} + \item We must be able to perform operations on the encoded state + in such a way that we do not lose the protection against errors. + \item \ac{qec} systems are themselves partially implemented in + quantum hardware. In addition to the errors we have + originally introduced them for, these systems must + be able to acount for the fact they are implemented on noisy + hardware themselves. +\end{itemize} +In the literature, both of these points are viewed under the umbrella +of \emph{fault tolerance}. +We focus only on the second aspect in this work. + +It was recognized early on as a challenge of \ac{qec} that the correction +machinery itself may introduce new faults \cite[Sec.~III]{shor_scheme_1995}. +Specifically for stabilizer codes, this can happen during the +syndrome extraction process. +We distinguish between the errors the QEC procedure is supposed to +correct, which we call \emph{input errors}, and +those introduced by the procedure itself, which we call \emph{internal errors}. +In order to be fault-tolerant, the procedure must be able to +address both. % Definition of fault tolerance @@ -33,7 +57,7 @@ indicating which errors occurred, with \end{align*} \autoref{fig:fault_tolerance_overview} illustrates the flow of errors. Specifically for \ac{css} codes, a \ac{qec} procedure is deemed -\emph{fault-tolerant}, if \cite[Def.~4.2]{derks_designing_2025} +fault-tolerant, if \cite[Def.~4.2]{derks_designing_2025} \begin{gather*} \lVert \bm{e}_{\text{output},X} \rVert \le t \hspace{5mm} \forall\, \bm{e}_\text{input}, \bm{e}_\text{internal} \in \{0,1\}^N: @@ -51,9 +75,9 @@ errors the code is able to correct. The vectors $\bm{e}_{\text{output},X}$ and $\bm{e}_{\text{output},Z}$ denote only $X$ and $Z$ errors respectively. -In order to deal with errors during the syndrome extraction that flip -single syndrome bits, multiple rounds of syndrome measurements must -be performed. +% TODO: Properly introduce d_min for QEC, specifically for CSS codes +In order to deal with internal errors that flip syndrome bits, +multiple rounds of syndrome measurements must be performed. Typically, the number of syndrome extraction rounds is chosen as $d_\text{min}$. % % This is the definition of a fault-tolerant QEC gadget @@ -147,7 +171,9 @@ opposed to the system-level view introduced in We visualize the different types of noise models in \autoref{fig:noise_model_types}. -% Bit-flip noise +%%%%%%%%%%%%%%%% +\subsection{Bit-Flip Noise} +\label{subsec:Bit-Flip Noise} The simplest type of noise model is \emph{bit-flip} noise. This corresponds to the classical \ac{bsc}, i.e., only $X$ errors on the @@ -156,7 +182,9 @@ Note that we cannot use bit-flip noise to develop fault-tolerant systems, as it doesnt't account for errors during the syndrome extraction. This type of noise model is shown in \autoref{subfig:bit_flip}. -% Depolarizing channel +%%%%%%%%%%%%%%%% +\subsection{Depolarizing Channel} +\label{subsec:Depolarizing Channel} Extending bit-flip noise to consider $X,Z$ or $Y$ instead of just $X$ errors, we obtain the \emph{depolarizing channel} @@ -168,7 +196,13 @@ decoherence. Bit-flip noise and the depolarizing channel are sometimes referred to as \emph{code capacity noise models}. -% Phenomenological noise +While the depolarizing channel is still not suited for the design and +simulation of fault-tolerant systems, it is already complex enough to +be used to gauge the suitability of a code for the \ac{qec} problem. + +%%%%%%%%%%%%%%%% +\subsection{Phenomenological Noise} +\label{subsec:Phenomenological Noise} The \emph{phenomenological noise model} is the first type of noise model we examine that accounts for faults during the syndrome extraction. @@ -179,12 +213,18 @@ locations right before each measurement \cite[Appendix~A]{gidney_new_2023}. Note that it is enough to only consider $X$ errors at this point, since that is the only type of error directly affecting the measurement outcomes. -While not fully capturing all possible error mechanisms, -phenomenological noise is already expressive enough to be used for -the design of fault-tolerant circuitry \cite[Sec.~4.2]{derks_designing_2025}. This model is depicted in \autoref{subfig:phenomenological}. -% Circuit-level noise +While not fully capturing all possible error mechanisms, +phenomenological noise is already \ldots . +Additionally, there are applications were the consideration of +phenomenological noise is enough. +It can, for example, be used for \ldots \red{the design of +fault-tolerant circuitry} \cite[Sec.~4.2]{derks_designing_2025}. + +%%%%%%%%%%%%%%%% +\subsection{Circuit-Level Noise} +\label{subsec:Circuit-Level Noise} The most general type of noise model is \emph{circuit-level noise}. Here we not only consider noise inbetween syndrome extraction rounds @@ -193,30 +233,11 @@ Specifically, we allow arbitrary $n$-qubit Pauli errors after each $n$-qubit gate \cite[Def.~2.5]{derks_designing_2025}. An $n$-qubit Pauli error is simply a series of correlated Pauli errors on each related individual qubit. +This type of noise model is shown in \autoref{subfig:circuit_level}. + While phenomenological noise is useful for some design aspects of fault tolerant circuitry, for simulations, circuit-level noise should always be used \cite[Sec.~4.2]{derks_designing_2025}. -This type of noise model is shown in \autoref{subfig:circuit_level}. - -% Practical simulation aspects - -While these types of noise models give us some constraints on the -types and locations of errors, the question of how exactly to choose -their probabilities remains. -When performing simulations, we typically sweep a range of values of -the \emph{physical error rate} $p$, and there are different ways of -obtaining the noise model $\bm{p}$ from this physical error rate. -For the depolarizing channel in particular, we usually choose the -same error rate $p/3$ for $X$, $Z$, and $Y$ errors. -For circuit-level noise, various options exist, such as the \emph{SI1000} -(superconducting inspired) or the \emph{EM3} (entangling -measurements) models \cite[Sec.~2.1]{gidney_fault-tolerant_2021}. -These differ in the way they compute individual error probabilities -from the physical error rate. -In this work we only consider \emph{standard circuit-based depolarizing -noise}, as this is the standard approach in the literature. -We thus set the error probabilities of all error locations in the -circuit-level noise model to the same value, the physical error rate $p$. \begin{figure}[t] \centering @@ -383,10 +404,13 @@ fault-tolerant quantum computing schemes \cite[Sec.~1]{derks_designing_2025}. % benefit from the fact that \content{Benefits of this approach \cite[Sec.~4.2]{derks_designing_2025}} +\content{Where they were introduced originally} + % Core idea -The goal we strive to achieve is to consider the possible error -locations in the syndrome extraction circuitry during decoding. +To achieve fault tolerance, the goal we strive towards is to +consider the internal errors in addition to the input errors during +the decoding process. The core idea behind detector error models is to do this by defining a new \emph{circuit code} that describes the circuit. Each \ac{vn} of this new code corresponds to an error location in the @@ -665,7 +689,7 @@ may wish to combine them in some way. We call such combinations \emph{detectors}. Formally, a detector is a parity constraint on a set of measurement outcomes \cite[Def.~2.1]{derks_designing_2025}. -Chaning the perspective in this way does not change the theoretical +Changing the perspective in this way does not alter the theoretical error correcting capabilities of the circuit, but it may change the decoding performance when using a practical decoder. \red{[Possibly a few more words on this (maybe a mathematical @@ -829,6 +853,9 @@ identical to obtain this structure. \subsection{Detector Error Models} \label{subsec:Detector Error Models} +A \emph{detector error model} is the combination of the detector +error matric $\bm{H}$ and the noise model $\bm{p}$. + \content{Combination of detector error matrix and noise model} \content{Contains all information necessary for decoding \cite[Intro.]{derks_designing_2025}} @@ -838,7 +865,25 @@ identical to obtain this structure. \section{Practical Considerations} \label{sec:Practical Considerations} -% Intro +% Practical simulation aspects + +While these types of noise models give us some constraints on the +types and locations of errors, the question of how exactly to choose +their probabilities remains. +When performing simulations, we typically sweep a range of values of +the \emph{physical error rate} $p$, and there are different ways of +obtaining the noise model $\bm{p}$ from this physical error rate. +For the depolarizing channel in particular, we usually choose the +same error rate $p/3$ for $X$, $Z$, and $Y$ errors. +For circuit-level noise, various options exist, such as the \emph{SI1000} +(superconducting inspired) or the \emph{EM3} (entangling +measurements) models \cite[Sec.~2.1]{gidney_fault-tolerant_2021}. +These differ in the way they compute individual error probabilities +from the physical error rate. +In this work we only consider \emph{standard circuit-based depolarizing +noise}, as this is the standard approach in the literature. +We thus set the error probabilities of all error locations in the +circuit-level noise model to the same value, the physical error rate $p$. \content{Intro}