Quantum Error-Control Codes

Information is physical. It is sensible to use quantum mechanics as a basis of computation and information processing [19]. Here at the intersection of information theory, computing, and physics, mathematicians and computer scientists must think in terms of the quantum physical realizations of messages. The often philosophical debates among physicists over the nature and interpretations of quantum mechanics shift to harnessing its power for information processing and testing the theory for completeness.

One cannot directly access information stored and processed in massively entangled quantum systems without destroying the content. Turning large-scale quantum computing into practical reality is massively challenging. To start with, it requires techniques for error control that are much more complex than those implemented effectively in classical systems. As a quantum system grows in size and circuit depth, error control becomes ever more important.

Quantum error-control is a set of methods to protect quantum information from unwanted environmental interactions, known as decoherence. Classically, one encodes information-carrying vectors into a larger space to allow for sufficient redundancy for error detection and correction. In the quantum setup, information is stored in a subspace embedded in a larger Hilbert space, which is a finite dimensional, normed, vector space over the field of complex numbers $\mathbb{C}$. Codewords are quantum states and errors are operators.

The good news is that noise, if it can be kept below a certain level, is not an obstacle to resilient quantum computation. This crucial insight is arrived at based on seminal results that form the so-called treshold theorems. Theoretical references include the exposition of Knill et al.in [34], the work of Preskill in [42], the results shown by Steane in [49], and the paper of Ahoronov and Ben-Or [1]. A comprehensive review on related experiments is available in [9].

The possibility of correcting errors in quantum systems was shown, e.g., in the early works of Shor [46], Steane [47] and Laflamme et al.[35]. While the quantum codes that these pioneers proposed may nowadays seem to be rather trivial in performance, their construction highlighted both the main obstacles and their respective workarounds. Measurement collapses the information contained in the state into something useless. One should measure the error, not the data. Since repetition is ruled out due to the no-cloning theorem [53], we use redundancy from spreading the states to avoid repetition. There are multiple types of errors, as we will soon see. The key is to start by correcting the phase errors and, then, use the Hadamard transform to exchange the bit flips and the phase errors. Quantum errors are continuous. Controlling them seemed to be too daunting a task. It turned out that handling a set of discrete error operators, represented by tensor product of Pauli matrices, allows for the control of every $\mathbb{C}$-linear combination of these operators.

Advances continue to be made as effort intensifies to scale quantum computing up. Research in quantum error-correcting codes (QECs) has attracted the sustained attention of established researchers and students alike. Several excellent online lecture notes, surveys, and books are available. Developments up to 2011 have been well-documented in [36]. It is impossible to describe the technical details of every important research direction in QECs. We focus on quantum stabilizer codes and their variants. The decidedly biased take here is for the audience with more applied mathematics background, including coding theorist, information theorist, researchers in discrete mathematics and finite geometries. No knowledge of quantum mechanics is required beyond the very basic. This chapter is meant to serve as an entry point for those who want to understand and get involved in building upon this foundational aspect of quantum information processing and computation, which have been tipped to be indispensable in future technologies.

A quantum stabilizer code is designed so that errors with high probability of occuring transform information-carrying states to an error space which is orthogonal to the original space. The beauty lies in how natural the determination of the location and type of each error in the system is. Correction becomes a simple application of the type of error at the very location.

Consider the field extensions $\mathbb{F}_{p}$ to $\mathbb{F}_{q=p^{r}}$ to $\mathbb{F}_{q^{m}=p^{rm}}$, for positive integers $r$ and $m$. For $\alpha\in\mathbb{F}_{q^{m}}$, the trace mapping from $\mathbb{F}_{q^{m}}$ to $\mathbb{F}_{q}$ is given by

The trace of $\alpha$ is the sum of its conjugates. If the extension $\mathbb{F}_{q}$ of $\mathbb{F}_{p}$ is contextually clear, the notation $\Tr$ is sufficient. Properties of the trace map can be found in standard textbooks, e.g., [37, Chapter 2]. The key idea is that $\Tr_{\mathbb{F}_{q^{m}}/\mathbb{F}_{q}}$ serves as a description for all linear transformations from $\mathbb{F}_{q^{m}}$ to $\mathbb{F}_{q}$.

Let $G$ be a finite abelian group, written multiplicatively, with the identity $1_{G}$. Let $U$ be the multiplicative group of complex numbers of modulus $1$, i.e., the unit circle of radius $1$ on the complex plane $\mathbb{C}$. A character $\chi:G\mapsto U$ is a homomorphism. For any $g\in G$, the images of $\chi$ are $\absolutevalue{G}$-th roots of unity since $\left(\chi(g)\right)^{\absolutevalue{G}}=\chi\left(g^{\absolutevalue{G}}\right)=1$. Let $\overline{c}$ denote the complex conjugate of $c$. Then $\chi(g^{-1})=\left(\chi(g)\right)^{-1}=\overline{\chi(g)}$. The only trivial character is $\chi_{0}:g\mapsto 1$ for all $g\in G$. One can associate $\chi$ and $\overline{\chi}$ by using $\overline{\chi}(g)=\overline{\chi(g)}$. The set of all characters of $G$ forms, under composition $\circ$, an abelian group $\widehat{G}$.

For $g,h\in G$ and $\chi,\Psi\in\widehat{G}$ we have two orthogonality relations

The additive character $\chi_{1}:=c\mapsto e^{\frac{2\pi}{p}\Tr(c)}$, for all $c\in\mathbb{F}_{q}$, is called the canonical character. For a chosen $b\in\mathbb{F}_{q}$ and for all $c\in\mathbb{F}_{q}$,

is a character of $(\mathbb{F}_{q},+)$. Every character of $(\mathbb{F}_{q},+)$ can, in fact, be expressed in this manner. The extension to $(\mathbb{F}_{q}^{n},+)$ is straightforward.

A qubit, a term coined by Schumacher in [45], is the canonical quantum system consisting of two distinct levels. The states of a qubit live in $\mathbb{C}^{2}$ and are defined by their continuous amplitudes. A qudit refers to a system of $q\geq 3$ distinct levels, with a qutrit commonly used when $q=3$. Physicists prefer the “bra” $\ket{\cdot}$ and “ket” $\bra{\cdot}$ notation to describe quantum systems. A $\ket{\varphi}$ is a (column) vector while $\bra{\psi}$ is the vector dual of $\ket{\psi}$.

The inner product of $\ket{\psi}:=\sum_{{\mathbf{a}}\in\mathbb{F}_{2}^{n}}c_{{\mathbf{a}}}\ket{{\mathbf{a}}}$ and $\ket{\varphi}:=\sum_{{\mathbf{a}}\in\mathbb{F}_{2}^{n}}b_{{\mathbf{a}}}\ket{{\mathbf{a}}}$ is

Their (Kronecker) tensor product is written as $\ket{\varphi}\otimes\ket{\psi}$ and is often abbreviated to $\ket{\varphi\psi}$. The states $\ket{\psi}$ and $\ket{\varphi}$ are orthogonal or distinguishable if $\innerproduct{\psi}{\varphi}=0$. Let $A$ be a $2^{n}\times 2^{n}$ complex unitary matrix with conjugate transpose $A^{\dagger}$. The (Hermitian) inner product of $\ket{\varphi}$ and $A\ket{\psi}$ is equal to that of $A^{\dagger}\ket{\varphi}$ and $\ket{\psi}$. Henceforth, $i:=\sqrt{-1}$.

The actions of the error operators on a qubit $\ket{v}=\alpha\ket{0}+\beta\ket{1}\in\mathbb{C}^{2}$ can be considered based on their types. The trivial operator $I_{2}$ leaves the qubit unchanged. The bit-flip error $\sigma_{x}$ flips the probabilities

The phase-flip error $\sigma_{z}$ modifies the angular measures

The combination error $\sigma_{y}$ contains both bit-flip and phase-flip, implying

It is immediate to confirm that $\sigma_{x}^{2}=\sigma_{y}^{2}=\sigma_{z}^{2}=I_{2}$ and $\sigma_{x}\sigma_{z}=-\sigma_{z}\sigma_{x}$. The Pauli matrices generate a group of order $16$. Each of its elements can be uniquely represented as $i^{\lambda}w$, with $\lambda\in\{0,1,2,3\}$ and $w\in\{I_{2},\sigma_{x},\sigma_{z},\sigma_{y}\}$.

The most common route from classical coding theory to QEC is via the stabilizer formalism, from which numerous specific constructions emerge. Classical codes can not be used as quantum codes but can model the error operators in some quantum channels. The capabilities of a QEC can then be inferred from the properties of the corresponding classical codes. The main tools come from character theory and symplectic geometry over finite fields. Our focus is on the qubit setup since it is the most deployment-feasible and because the general qudit case naturally follows from it.

Let ${\mathbf{a}}=(a_{1},a_{2},\ldots,a_{n})\in\mathbb{F}_{2}^{n}$, $\lambda\in\{0,1,2,3\}$, and $w_{j}\in\{I_{2},\sigma_{x},\sigma_{z},\sigma_{y}\}$. A quantum error operator on $\mathbb{C}^{2^{n}}$ is of the form ${\rm E}:=i^{\lambda}w_{1}\otimes w_{2}\otimes\ldots\otimes w_{n}$. It is a $\mathbb{C}$-linear unitary operator acting on a $\mathbb{C}^{2^{n}}$-basis $\{\ket{{\mathbf{a}}}=\ket{a_{1}}\otimes\ket{a_{2}}\otimes\ldots\otimes\ket{a_{n}}\}$ by ${\rm E}\ket{{\mathbf{a}}}:=i^{\lambda}\left(w_{1}\ket{a_{1}}\otimes w_{2}\ket{a_{2}}\otimes\ldots\otimes w_{n}\ket{a_{n}}\right)$. The set of error operators

is a non-abelian group of cardinality $4^{n+1}$. Given ${\rm E}:=i^{\lambda}w_{1}\otimes w_{2}\otimes\ldots\otimes w_{n}$ and ${\rm E}^{\prime}:=i^{\lambda^{\prime}}w_{1}^{\prime}\otimes w_{2}^{\prime}\otimes\ldots\otimes w_{n}^{\prime}$ in ${\mathcal{E}}_{n}$, we have

Expanding ${\rm E}^{\prime}\,{\rm E}$ makes it clear that ${\rm E}\,{\rm E}^{\prime}=\pm 1~{}{\rm E}^{\prime}\,{\rm E}$.

The center of ${\mathcal{E}}_{n}$ is $\mathcal{C}({\mathcal{E}}_{n}):=\{i^{\lambda}I_{2}\otimes I_{2}\otimes\ldots\otimes I_{2}\}$. Let $\overline{{\mathcal{E}}_{n}}$ denote the quotient group ${\mathcal{E}}_{n}/\mathcal{C}({\mathcal{E}}_{n})$ of cardinality $\absolutevalue{\overline{{\mathcal{E}}_{n}}}=4^{n}$. This group is an abelian $2$-elementary group $\cong(\mathbb{F}_{2}^{2n},+)$, since $\overline{{\rm E}}^{2}=I_{2}\otimes\cdots\otimes I_{2}=I_{2^{n}}$ for any $\overline{{\rm E}}\in\overline{{\mathcal{E}}_{n}}$.

We switch notation to define the product of error operators in terms of an inner product of their vector representatives. We write ${\rm E}=i^{\lambda}w_{1}\otimes w_{2}\otimes\ldots\otimes w_{n}$ as ${\rm E}=i^{\lambda+\epsilon}X({\mathbf{a}})Z({\mathbf{b}})$, where ${\mathbf{a}}=(a_{1},\ldots,a_{n}),{\mathbf{b}}=(b_{1},\ldots,b_{n})\in\mathbb{F}_{2}^{n}$ and $\epsilon:=\absolutevalue{\{1\leq i\leq n\}:w_{i}=\sigma_{y}}$, by replacing $(a_{i},b_{i})$ with $(0,0)$ if $w_{i}=I_{2}$, by $(1,0)$ if $w_{i}=\sigma_{x}$, by $(0,1)$ if $w_{i}=\sigma_{z}$, and by $(1,1)$ if $w_{i}=\sigma_{y}$.

The respective actions of $X({\mathbf{a}})$ and $Z({\mathbf{b}})$ on any vector $\ket{{\mathbf{v}}}\in\mathbb{C}^{2^{n}}$, for ${\mathbf{v}}\in\mathbb{F}_{2}^{n}$, are $X({\mathbf{a}})\ket{{\mathbf{v}}}=\ket{{\mathbf{a}}+{\mathbf{v}}}$ and $Z({\mathbf{b}})\ket{{\mathbf{v}}}=(-1)^{{\mathbf{b}}\cdot{\mathbf{v}}}\ket{{\mathbf{v}}}$. The matrix for $X({\mathbf{a}})$ is a symmetric $\{0,1\}$ matrix. It represents a permutation consisting of $2^{n-1}$ transpositions. The matrix for $Z({\mathbf{b}})$ is diagonal with diagonal entries $\pm 1$. Hence, writing the operators in ${\mathcal{E}}_{n}$ as ${\rm E}:=i^{\lambda}X({\mathbf{a}})Z({\mathbf{b}})$ and ${\rm E}^{\prime}:=i^{\lambda^{\prime}}X({\mathbf{a}}^{\prime})Z({\mathbf{b}}^{\prime})$, one gets ${\rm E}\,{\rm E}^{\prime}=(-1)^{{\mathbf{a}}\cdot{\mathbf{b}}^{\prime}+{\mathbf{a}}^{\prime}\cdot{\mathbf{b}}}~{}{\rm E}^{\prime}\,{\rm E}$. The symplectic inner product of $({\mathbf{a}}|{\mathbf{b}})$ and $({\mathbf{a}}^{\prime}|{\mathbf{b}}^{\prime})$ in $\mathbb{F}_{2}^{2n}$ is

or, in matrix form,

The symplectic dual of $C\subseteq\mathbb{F}_{2}^{2n}$ is $C^{\perp_{s}}=\{{\mathbf{u}}\in\mathbb{F}_{2}^{2n}:\langle{\mathbf{u}},{\mathbf{c}}\rangle_{s}=0\,\forall\,{\mathbf{c}}\in C\}$. Thus, a subgroup $G$ of ${\mathcal{E}}_{n}$ is abelian if and only if $\overline{G}$ is a symplectic self-orthogonal subspace of $\overline{{\mathcal{E}}_{n}}\cong\mathbb{F}_{2}^{2n}$.

The quantum weight of an error operator ${\rm E}=i^{\lambda}X({\mathbf{a}})Z({\mathbf{b}})\in{\mathcal{E}}_{n}$ is

By definition, $\operatorname{w_{Q}}({\rm E}\,{\rm E}^{\prime})\leq\operatorname{w_{Q}}({\rm E})+\operatorname{w_{Q}}({\rm E}^{\prime})$, for any ${\rm E},{\rm E}^{\prime}\in{\mathcal{E}}_{n}$. We can define the set of all error operators of weight at most $\delta$ in ${\mathcal{E}}_{n}$ and determine its cardinality. Let

Then $\absolutevalue{{\mathcal{E}}_{n}(\delta)}=4\sum_{j=0}^{\delta}3^{j}\binom{n}{j}$ and $\absolutevalue{\overline{{\mathcal{E}}_{n}}(\delta)}=\sum_{j=0}^{\delta}3^{j}\binom{n}{j}$.

In the classical setup, both errors and codewords are vectors over the same field. In the quantum setup, errors are linear combinations of the tensor products of Pauli matrices. A qubit code $Q\subseteq\mathbb{C}^{2^{n}}$ has three parameters: its length $n$, dimension $K$ over $\mathbb{C}$, and minimum distance $d=d(Q)$. We use

to signify that $Q$ describes the encoding of $k$ logical qubits as $n$ physical qubits, with $d$ being the smallest number of simultaneous errors that can transform a valid codeword into another.

Given an $((n,K,d))$-qubit code $Q$ and an ${\rm E}\in{\mathcal{E}}_{n}$, ${\rm E}\,Q$ is a subspace of $\mathbb{C}^{2^{n}}$. The fact that $Q$ corrects errors of weight up to $\ell=\lfloor\frac{d-1}{2}\rfloor$ does not imply that the subspaces $\{\overline{{\rm E}}Q:\overline{{\rm E}}\in\overline{{\mathcal{E}}_{n}}(\ell)\}$ are orthogonal to each other. It is possible that a codeword $\ket{{\mathbf{v}}}$ is fixed by some ${\rm E}\neq I_{2^{n}}$, say, when $\ket{{\mathbf{v}}}$ is an eigenvector of ${\rm E}$ satisfying ${\rm E}\ket{{\mathbf{v}}}=\alpha\ket{{\mathbf{v}}}$ for some nonzero $\alpha\in\mathbb{C}$. If the subspaces $\{\overline{{\rm E}}\,Q:\overline{{\rm E}}\in\overline{{\mathcal{E}}_{n}}(\ell)\}$ are orthogonal to each other, then $Q$ is said to be pure. Otherwise, the code is degenerate or impure.

To formally define a qubit stabilizer code, we choose an abelian group $G$, which is a subgroup of ${\mathcal{E}}_{n}$, and associate $G$ with a classical code $C\subset\mathbb{F}_{2}^{2n}$, which is self-orthogonal under the symplectic inner product. The action of $G$ partitions $\mathbb{C}^{2^{n}}$ into a direct sum of $\chi$-eigenspaces $Q(\chi)$ with $\chi\in\widehat{G}$. The properties of $Q:=Q(\chi)$ follow from the properties of $C$ and $C^{\perp_{s}}$. The stabilizer formalism, first introduced by Gottesman in his thesis [23] and described in the language of group algebra by Calderbank et al.in [8], remains the most widely-studied approach to control quantum errors. Ketkar et al.generalized the formalism to qudit codes derived from classical codes over $\mathbb{F}_{q^{2}}$ in [31].

Let $G$ be a finite abelian group acting on a finite dimensional $\mathbb{C}$-vector space $V$. Each $g\in G$ is a Hermitian operator of $V$ and, for any $g,g^{\prime}\in G$ and for all $\ket{{\mathbf{v}}}\in V$, $(gg^{\prime})\ket{{\mathbf{v}}}=g(g^{\prime}(\ket{{\mathbf{v}}}))\mbox{ and }gg^{-1}(\ket{{\mathbf{v}}})=\ket{{\mathbf{v}}}$. Let $\widehat{G}$ be the character group of $G$. For any $\chi\in\widehat{G}$, the map $L_{\chi}:=\frac{1}{\absolutevalue{G}}\sum_{g\in G}\overline{\chi}(g)\,g$ is a linear operator over $V$. The set $\{L_{\chi}:\chi\in\widehat{G}\}$ is the system of orthogonal primitive idempotent operators.

It is also a well-known fact that $V(\chi)$ and $V(\chi^{\prime})$ are Hermitian orthogonal for all $\chi\neq\chi^{\prime}\in\widehat{G}$ when all $g\in G$ are unitary linear operators on $V$.

All the tools to connect qubit stabilizer codes to classical codes are now in place. We choose $G:=\left\langle g_{1},g_{2},\ldots,g_{k}\right\rangle$ to be an abelian subgroup of ${\mathcal{E}}_{n}$ with $g_{j}:=i^{\lambda_{j}}~{}X({\mathbf{a}}_{j})~{}Z({\mathbf{b}}_{j})$ for $1\leq j\leq k$, where ${\mathbf{a}}_{j},{\mathbf{b}}_{j}\in\mathbb{F}_{2}^{n}$ and $\lambda_{j}\equiv{\mathbf{a}}_{j}\cdot{\mathbf{b}}_{j}\pmod{2}$. Since $\sigma_{x}$ and $\sigma_{z}$ are Hermitian unitary matrices, $X({\mathbf{a}}_{j})$ and $Z({\mathbf{b}}_{j})$ are also Hermitian matrices. The basis element $g_{j}$ is Hermitian since