Designing the Quantum Channels Induced by Diagonal Gates

Let $\mathbb{F}_{2}=\{0,1\}$ denote the binary field. Let $m\geq 1$, and let $x_{1}$, $x_{2}$, $\dots$, $x_{m}$ be binary variables (monomials of degree $1$). Monomials of degree $r$ can be written as $x_{i_{1}}x_{i_{2}}\cdots x_{i_{r}}$ where $i_{j}\in\{1,2,\dots,m\}$ are distinct. A boolean function with degree $r$ is a binary linear combination of monomials with degrees at most $r$. There is a one-to-one correspondence between boolean functions $h$ and evaluation vectors $\bm{h}=[h(x_{1},x_{2},\cdots,x_{m})]_{(x_{1},x_{2},\dots,x_{m})\in\mathbb{F}_{2}^{m}}$. The degree $0$ boolean function corresponds to the constant evaluation vector $\bm{1}\in\mathbb{F}_{2}^{2^{m}}$.

For $0\leq r\leq m$, the Reed-Muller code RM$(r,m)$ is the set of all evaluation vectors $\bm{h}$ associated with boolean functions $h(x_{1},x_{2},\cdots,x_{m})$ of degree at most $r$, $\mathrm{RM}(r,m)\coloneqq\{\bm{h}\in\mathbb{F}_{2}^{2^{m}}\mid h\in\mathbb{F}_{2}[x_{1},x_{2},\cdots,x_{m}],$ $\deg(h)\leq r\}.$ The length of the RM$(r,m)$ code is $2^{m}$, the dimension is given by $k=\sum_{j=0}^{r}\binom{m}{j}$, and the minimal distance is $2^{m-r}$. The dual of RM$(r,m)$ is RM$(m-r-1,m)$, and we can construct the RM codes by a recursively observing RM$(r,m+1)=\{(\bm{u},\bm{u}+\bm{v})\mid\bm{u}\in\mathrm{RM}(r,m),\bm{v}\in\mathrm{RM}(r-1,m)\}$ [28]. Note that all weights in RM($r,m$) are multiples of $2^{\left\lfloor(m-1)/r\right\rfloor}$ [3, 29, 28], and the highest power of $2$ that divides all weights of codewords in RM($r,m$) is exactly $2^{\left\lfloor(m-1)/r\right\rfloor}$ [6].

Let $\imath\coloneqq\sqrt{-1}$ be the imaginary unit. We denote the Hamming weight of a binary vector $\bm{v}$ by $w_{H}(\bm{v})$. The weight enumerator of a binary linear code $\mathcal{C}\subset\mathbb{F}_{2}^{m}$ is the polynomial

$$P_{\mathcal{C}}(x,y)=\sum_{\bm{v}\in\mathcal{C}}x^{m-w_{H}\left(\bm{v}\right)}y^{w_{H}\left(\bm{v}\right)}.$$

(1)

The MacWilliams Identities [27] relate the weight enumerator of a code $\mathcal{C}$ to that of the dual code $\mathcal{C}^{\perp}$, and are given by

$$P_{\mathcal{C}}(x,y)=\frac{1}{|\mathcal{C}^{\perp}|}P_{\mathcal{C}^{\perp}}(x+y,x-y).$$

(2)

Given an angle $\theta\in(0,2\pi)$, we make the substitution $x=\cos\frac{\theta}{2}$ and $y=-\imath\sin\frac{\theta}{2}$, and define

	$\displaystyle P_{\theta}[\mathcal{C}]$	$\displaystyle\coloneqq P_{\mathcal{C}}\left(\cos\frac{\theta}{2},-\imath\sin\frac{\theta}{2}\right)$		(3)
		$\displaystyle=\sum_{\bm{v}\in\mathcal{C}}\left(\cos\frac{\theta}{2}\right)^{m-w_{H}(\bm{v})}\left(-\imath\sin\frac{\theta}{2}\right)^{w_{H}(\bm{v})}.$		(4)

Any $2\times 2$ Hermitian matrix can be uniquely expressed as a real linear combination of the four single qubit Pauli matrices/operators

$\displaystyle I_{2}\coloneqq\begin{bmatrix}1&0\\ 0&1\end{bmatrix},\leavevmode\nobreak\ X\coloneqq\begin{bmatrix}0&1\\ 1&0\end{bmatrix},\leavevmode\nobreak\ Z\coloneqq\begin{bmatrix}1&0\\ 0&-1\end{bmatrix},$

(5)

and $Y\coloneqq\imath XZ$. The operators satisfy $X^{2}=Y^{2}=Z^{2}=I_{2},\leavevmode\nobreak\ XY=-YX,\leavevmode\nobreak\ XZ=-ZX,\leavevmode\nobreak\ \text{ and }YZ=-ZY.$

Let $A\otimes B$ denote the Kronecker product (tensor product) of two matrices $A$ and $B$. Let $n\geq 1$ and $N=2^{n}$. Given binary vectors $\bm{a}=[a_{1},a_{2},\dots,a_{n}]$ and $\bm{b}=[b_{1},b_{2},\dots,b_{n}]$ with $a_{i},b_{j}=0$ or $1$, we define the operators

	$\displaystyle D(\bm{a},\bm{b})$	$\displaystyle\coloneqq X^{a_{1}}Z^{b_{1}}\otimes\cdots\otimes X^{a_{n}}Z^{b_{n}},$		(6)
	$\displaystyle E(\bm{a},\bm{b})$	$\displaystyle\coloneqq\imath^{\bm{a}\bm{b}^{T}\bmod 4}D(\bm{a},\bm{b}).$		(7)

We often abuse notation and write $\bm{a},\bm{b}\in\mathbb{F}_{2}^{n}$, though entries of vectors are sometimes interpreted in $\mathbb{Z}_{4}=\{0,1,2,3\}$. Note that $D(\bm{a},\bm{b})$ can have order $1,2$ or $4$, but $E(\bm{a},\bm{b})^{2}=\imath^{2\bm{a}\bm{b}^{T}}D(\bm{a},\bm{b})^{2}=\imath^{2ab^{T}}(\imath^{2\bm{a}\bm{b}^{T}}I_{N})=I_{N}$. The $n$-qubit Pauli group is defined as

$$\mathcal{HW}_{N}\coloneqq\{\imath^{\kappa}D(\bm{a},\bm{b}):\bm{a},\bm{b}\in\mathbb{F}_{2}^{n},\kappa\in\mathbb{Z}_{4}\},$$

(8)

where $\mathbb{Z}_{2^{l}}=\{0,1,\dots,2^{l}-1\}$. The $n$-qubit Pauli matrices form an orthonormal basis for the vector space of $N\times N$ complex matrices ($\mathbb{C}^{N\times N}$) under the normalized Hilbert-Schmidt inner product $\langle A,B\rangle\coloneqq\mathrm{Tr}(A^{\dagger}B)/N$ [18].

We use the Dirac notation, $|\cdot\rangle$ to represent the basis states of a single qubit in $\mathbb{C}^{2}$. For any $\bm{v}=[v_{1},v_{2},\cdots,v_{n}]\in\mathbb{F}_{2}^{n}$, we define $|\bm{v}\rangle=|v_{1}\rangle\otimes|v_{2}\rangle\otimes\cdots\otimes|v_{n}\rangle$, the standard basis vector in $\mathbb{C}^{N}$ with $1$ in the position indexed by $\bm{v}$ and $0$ elsewhere. We write the Hermitian transpose of $|\bm{v}\rangle$ as $\langle\bm{v}|=|\bm{v}\rangle^{\dagger}$. We may write an arbitrary $n$-qubit quantum state as $|\psi\rangle=\sum_{\bm{v}\in\mathbb{F}_{2}^{n}}\alpha_{\bm{v}}|\bm{v}\rangle\in\mathbb{C}^{N}$, where $\alpha_{\bm{v}}\in\mathbb{C}$ and $\sum_{\bm{v}\in\mathbb{F}_{2}^{n}}|\alpha_{\bm{v}}|^{2}=1$. The Pauli matrices act on a single qubit as $X|{0}\rangle=|{1}\rangle,X|{1}\rangle=|{0}\rangle,Z|{0}\rangle=|{0}\rangle,\text{ and }Z|{1}\rangle=-|{1}\rangle.$

The symplectic inner product is $\langle[\bm{a},\bm{b}],[\bm{c},\bm{d}]\rangle_{S}=\bm{a}\bm{d}^{T}+\bm{b}\bm{c}^{T}\bmod 2$. Since $XZ=-ZX$, we have

$$E(\bm{a},\bm{b})E(\bm{c},\bm{d})=(-1)^{\langle[\bm{a},\bm{b}],[\bm{c},\bm{d}]\rangle_{S}}E(\bm{c},\bm{d})E(\bm{a},\bm{b}).$$

(9)

The Clifford hierarchy of unitary operators was introduced in [20]. The first level of the hierarchy is defined to be the Pauli group $\mathcal{C}^{(1)}=\mathcal{HW}_{N}$. For $l\geq 2$, the levels $l$ are defined recursively as

$$\mathcal{C}^{(l)}:=\{U\in\mathbb{U}_{N}:U\mathcal{HW}_{N}U^{\dagger}\subset\mathcal{C}^{(l-1)}\},$$

(10)

where $\mathbb{U}_{N}$ is the group of $N\times N$ unitary matrices. The second level is the Clifford Group, $\mathcal{C}^{(2)}$, which can be generated (up to overall phases) using the “elementary" unitaries Hadamard, Phase, and either of Controlled-NOT (C$X$) or Controlled-$Z$ (C$Z$) defined respectively as

$$H\coloneqq\frac{1}{\sqrt{2}}\begin{bmatrix}1&1\\ 1&-1\end{bmatrix},P\coloneqq\begin{bmatrix}1&0\\ 0&\imath\end{bmatrix},$$

(11)

	$\displaystyle\text{C}Z_{ab}$	$\displaystyle\coloneqq|{0}\rangle\langle{0}|_{a}\otimes(I_{2})_{b}+|{1}\rangle\langle{1}|_{a}\otimes Z_{b},$		(12)
	$\displaystyle\leavevmode\nobreak\ \text{C}X_{a\rightarrow b}$	$\displaystyle\coloneqq|{0}\rangle\langle{0}|_{a}\otimes(I_{2})_{b}+|{1}\rangle\langle{1}|_{a}\otimes X_{b}.$		(13)

Note that Clifford unitaries in combination with any unitary from a higher level can be used to approximate any unitary operator arbitrarily well [7]. Hence, they form a universal set for quantum computation. A widely used choice for the non-Clifford unitary is the $T$ gate in the third level defined by

$$T:=\begin{bmatrix}1&0\\ 0&e^{\frac{\imath\pi}{4}}\end{bmatrix}=Z^{\frac{1}{4}}\equiv\begin{bmatrix}e^{-\frac{\imath\pi}{8}}&0\\ 0&e^{\frac{\imath\pi}{8}}\end{bmatrix}=e^{-\frac{\imath\pi}{8}Z}.$$

(14)

We define a stabilizer group $\mathcal{S}$ to be a commutative subgroup of the Pauli group $\mathcal{HW}_{N}$, where every group element is Hermitian and no group element is $-I_{N}$. We say $\mathcal{S}$ has dimension $r$ if it can be generated by $r$ independent elements as $\mathcal{S}=\langle\nu_{i}E(\bm{c_{i}},\bm{d_{i}}):i=1,2,\dots,r\rangle$, where $\nu_{i}\in\{\pm 1\}$ and $\bm{c_{i}},\bm{d_{i}}\in\mathbb{F}_{2}^{n}$. Since $\mathcal{S}$ is commutative, we must have $\langle[\bm{c_{i}},\bm{d_{i}}],[\bm{c_{j}},\bm{d_{j}}]\rangle_{S}=\bm{c_{i}}\bm{d_{j}}^{T}+\bm{d_{i}}\bm{c_{j}}^{T}=0\bmod 2$.

Given a stabilizer group $\mathcal{S}$, the corresponding stabilizer code is the fixed subspace $\mathcal{V}(\mathcal{S)}:=\{|\psi\rangle\in\mathbb{C}^{N}:g|\psi\rangle=|\psi\rangle\text{ for all }g\in\mathcal{S}\}$. We refer to the subspace $\mathcal{V}(\mathcal{S})$ as an $\left[\left[n,k,d\right]\right]$ stabilizer code because it encodes $k:=n-r$ logical qubits into $n$ physical qubits. The minimum distance $d$ is defined to be the minimum weight of any operator in $\mathcal{N}_{\mathcal{HW}_{N}}\left(\mathcal{S}\right)\setminus\mathcal{S}$. Here, the weight of a Pauli operator is the number of qubits on which it acts non-trivially (i.e., as $X,\leavevmode\nobreak\ Y$ or $Z$), and $\mathcal{N}_{\mathcal{HW}_{N}}\left(\mathcal{S}\right)$ denotes the normalizer of $\mathcal{S}$ in $\mathcal{HW}_{N}$ defined by

	$\displaystyle\mathcal{N}_{\mathcal{HW}_{N}}\left(\mathcal{S}\right)$	$\displaystyle\coloneqq\{\imath^{\kappa}E\left(\bm{a},\bm{b}\right)\in\mathcal{HW}_{N}:$
		$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ E\left(\bm{a},\bm{b}\right)\mathcal{S}E\left(\bm{a},\bm{b}\right)=\mathcal{S},\kappa\in\mathbb{Z}_{4}\}$
		$\displaystyle=\{\imath^{\kappa}E\left(\bm{a},\bm{b}\right)\in\mathcal{HW}_{N}:$
		$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ E\left(\bm{a},\bm{b}\right)E\left(\bm{c},\bm{d}\right)E\left(\bm{a},\bm{b}\right)=E\left(\bm{c},\bm{d}\right)$
		$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \text{ for all }E\left(\bm{c},\bm{d}\right)\in\mathcal{S},\kappa\in\mathbb{Z}_{4}\}.$		(15)

Note that the second equality defines the centralizer of $\mathcal{S}$ in $\mathcal{HW}_{N}$, and it follows from the first since Pauli matrices commute or anti-commute.

For any Hermitian Pauli matrix $E\left(\bm{c},\bm{d}\right)$ and $\nu\in\{\pm 1\}$, the operator $\frac{I_{N}+\nu E\left(\bm{c},\bm{d}\right)}{2}$ projects onto the $\nu$-eigenspace of $E\left(\bm{c},\bm{d}\right)$. Thus, the projector onto the codespace $\mathcal{V}(\mathcal{S})$ of the stabilizer code defined by $\mathcal{S}=\langle\nu_{i}E\left(\bm{c_{i}},\bm{d_{i}}\right):i=1,2,\dots,r\rangle$ is

	$\displaystyle\Pi_{\mathcal{S}}$	$\displaystyle=\prod_{i=1}^{r}\frac{\left(I_{N}+\nu_{i}E\left(\bm{c_{i}},\bm{d_{i}}\right)\right)}{2}$
		$\displaystyle=\frac{1}{2^{r}}\sum_{j=1}^{2^{r}}\epsilon_{j}E\left(\bm{a_{j}},\bm{b_{j}}\right),$		(16)

where $\epsilon_{j}\in\{\pm 1\}$ is a character of the group $\mathcal{S}$, and is determined by the signs of the generators that produce $E(\bm{a_{j}},\bm{b_{j}})$: $\epsilon_{j}E\left(\bm{a_{j}},\bm{b_{j}}\right)=\prod_{t\in J\subset\{1,2,\dots,r\}}\nu_{t}E\left(\bm{c_{t}},\bm{d_{t}}\right)$ for a unique $J$.

Let $|{\bm{\alpha}}\rangle_{L}$, $\bm{\alpha}\in\mathbb{F}_{2}^{k}$ be the protected logical state. We define the generating set $\{X^{L}_{j},Z^{L}_{j}\in\mathcal{HW}_{2^{k}}:j=1,\dots k=k_{1}-k_{2}\}$ for the logical Pauli operators by the actions

$\displaystyle X_{j}^{L}|{\bm{\alpha}}\rangle_{L}=|{\bm{\alpha^{\prime}}}\rangle_{L},$

(17)

where

$\displaystyle\alpha^{\prime}_{i}=\left\{\begin{array}[]{lc}\alpha_{i},&\text{ if }i\neq j,\\ \alpha_{i}\oplus 1,&\text{ if }i=j,\end{array}\right.$

(20)

and $Z_{j}^{L}|{\bm{\alpha}}\rangle_{L}=(-1)^{\alpha_{j}}|{\bm{\alpha}}\rangle_{L}.$ Let $\bar{X}_{j},\bar{Z}_{j}$ be the $n$-qubit operators which are physical representatives of $X_{j}^{L},Z^{L}_{j}$ for $j=1,\dots,k$. Then $\bar{X}_{j},\bar{Z}_{j}$ commute with the stabilizer group $S$ and satisfy

$\displaystyle\bar{X}_{i}\bar{Z}_{j}=\left\{\begin{array}[]{lc}\bar{Z}_{j}\bar{X}_{i},&\text{ if }i\neq j,\\ -\bar{Z}_{j}\bar{X}_{i},&\text{ if }i=j.\end{array}\right.$

(23)

A CSS (Calderbank-Shor-Steane) code is a particular type of stabilizer code with generators that can be separated into strictly $X$-type and strictly $Z$-type operators. Consider two classical binary codes $\mathcal{C}_{1},\mathcal{C}_{2}$ such that $\mathcal{C}_{2}\subset\mathcal{C}_{1}$, and let $\mathcal{C}_{1}^{\perp}$, $\mathcal{C}_{2}^{\perp}$ denote the dual codes. Note that $\mathcal{C}_{1}^{\perp}\subset\mathcal{C}_{2}^{\perp}$. Suppose that $\mathcal{C}_{2}=\langle\bm{c_{1}},\bm{c_{2}},\dots,\bm{c_{k_{2}}}\rangle$ is an $[n,k_{2}]$ code and $\mathcal{C}_{1}^{\perp}=\langle\bm{d_{1}},\bm{d_{2}}\dots,\bm{d_{n-k_{1}}}\rangle$ is an $[n,n-k_{1}]$ code. Then, the corresponding CSS code has the stabilizer group

	$\displaystyle\mathcal{S}$	$\displaystyle=\langle\nu_{(\bm{c_{i}},\bm{0})}E\left(\bm{c_{i}},\bm{0}\right),\nu_{(\bm{0},\bm{d_{j}})}E\left(\bm{0},\bm{d_{j}}\right)\rangle_{i=1;\leavevmode\nobreak\ j=1}^{i=k_{2};\leavevmode\nobreak\ j=n-k_{1}}$
		$\displaystyle=\{\epsilon_{(\bm{a},\bm{0})}\epsilon_{(\bm{0},\bm{b})}E\left(\bm{a},\bm{0}\right)E\left(\bm{0},\bm{b}\right):\bm{a}\in\mathcal{C}_{2},\bm{b}\in\mathcal{C}_{1}^{\perp}\},$

where $\nu_{(\bm{c_{i}},\bm{0})},\nu_{(\bm{0},\bm{d_{j}})},\epsilon_{(\bm{a},\bm{0})},\epsilon_{(\bm{0},\bm{b})}\in\{\pm 1\}$. The CSS code projector can be written as the product:

$\displaystyle\Pi_{\mathcal{S}}=\Pi_{\mathcal{S}_{X}}\Pi_{\mathcal{S}_{Z}},$

(24)

where

	$\displaystyle\Pi_{\mathcal{S}_{X}}$	$\displaystyle\coloneqq\prod_{i=1}^{k_{2}}\frac{(I_{N}+\nu_{(\bm{c_{i}},\bm{0})}E(\bm{c_{i}},\bm{0}))}{2}$
		$\displaystyle=\frac{\sum_{\bm{a}\in\mathcal{C}_{2}}\epsilon_{(\bm{a},\bm{0})}E(\bm{a},\bm{0})}{|\mathcal{C}_{2}|},$		(25)

and

	$\displaystyle\Pi_{\mathcal{S}_{Z}}$	$\displaystyle\coloneqq\prod_{j=1}^{n-k_{1}}\frac{(I_{N}+\nu_{(\bm{0},\bm{d_{j}})}E(\bm{0},\bm{d_{j}}))}{2}$
		$\displaystyle=\frac{\sum_{\bm{b}\in\mathcal{C}_{1}^{\perp}}\epsilon_{(\bm{0},\bm{b})}E(\bm{0},\bm{b})}{|\mathcal{C}_{1}^{\perp}|}.$		(26)

Each projector defines a resolution of the identity, and we focus on $\Pi_{\mathcal{S}_{X}}$ since we consider diagonal gates. Note that any $n$-qubit Pauli $Z$ operator can be expressed as $E(\bm{0},\bm{b})E(\bm{0},\bm{\gamma})E(\bm{0},\bm{\mu})$ for a $Z$-stabilizer representation $\bm{b}\in\mathcal{C}_{1}^{\perp}$, a $Z$-logical representation $\bm{\gamma}\in\mathcal{C}_{2}^{\perp}/\mathcal{C}_{1}^{\perp}$, and a $X$-syndrome representation $\bm{\mu}\in\mathbb{F}_{2}^{n}/\mathcal{C}_{2}^{\perp}$. For $\bm{\mu}\in\mathbb{F}_{2}^{n}/\mathcal{C}_{2}^{\perp}$, we define

	$\displaystyle\mathcal{S}_{X}(\bm{\mu})$	$\displaystyle\coloneqq\left\{(-1)^{\bm{a}\bm{\mu}^{T}}\epsilon_{(\bm{a},0)}E(\bm{a},\bm{0}):\bm{a}\in\mathcal{C}_{2}\right\},$		(27)
	$\displaystyle\Pi_{\mathcal{S}_{X}(\bm{\mu})}$	$\displaystyle\coloneqq\frac{1}{|\mathcal{C}_{2}|}\sum_{\bm{a}\in\mathcal{C}_{2}}(-1)^{\bm{a}\bm{\mu}^{T}}\epsilon_{(\bm{a},\bm{0})}E(\bm{a},\bm{0}).$		(28)

Then, we have

	$\displaystyle\Pi_{\mathcal{S}_{X}(\bm{\mu})}\Pi_{\mathcal{S}_{X}(\bm{\mu}^{\prime})}=\left\{\begin{array}[]{lc}\Pi_{\mathcal{S}_{X}(\bm{\mu})},&\text{if}\ \bm{\mu}=\bm{\mu}^{\prime},\\ 0,&\text{if}\ \bm{\mu}\neq\bm{\mu}^{\prime},\end{array}\right.$		(31)
	$\displaystyle\text{and }\sum_{\bm{\mu}\in\mathbb{F}_{2}^{n}/\mathcal{C}_{2}^{\perp}}\Pi_{S_{X}(\bm{\mu})}=I_{2^{n}}.$		(32)

If $\mathcal{C}_{1}$ and $\mathcal{C}_{2}^{\perp}$ can correct up to $t$ errors, then $S$ defines an $\left[\left[n,k_{1}-k_{2},d\right]\right]$ CSS code with $d\geq 2t+1$, which we will represent as CSS($X,\mathcal{C}_{2};Z,\mathcal{C}_{1}^{\perp}$). If $G_{2}$ and $G_{1}^{\perp}$ are the generator matrices for $\mathcal{C}_{2}$ and $\mathcal{C}_{1}^{\perp}$ respectively, then the $(n-k_{1}+k_{2})\times(2n)$ matrix

$$G_{\mathcal{S}}=\left[\begin{array}[]{c|c}G_{2}&\\ \hline\cr&G_{1}^{\perp}\end{array}\right]$$

(33)

generates $\mathcal{S}$.

Given an $[\![n,k,d]\!]$ CSS($X,\mathcal{C}_{2};Z,\mathcal{C}_{1}^{\perp}$) code with all positive signs, let $G_{\mathcal{C}_{1}/\mathcal{C}_{2}}$ be the generator matrix for all coset representatives for $\mathcal{C}_{2}$ in $\mathcal{C}_{1}$ (note that the choice of coset representatives is not unique). The canonical encoding map $e:\mathbb{F}_{2}^{k}\to\mathcal{V}(\mathcal{S})$ is given by $e(|{\bm{\alpha}}\rangle_{L})\coloneqq\frac{1}{\sqrt{|\mathcal{C}_{2}|}}\sum_{\bm{x}\in\mathcal{C}_{2}}|{\bm{\alpha}G_{\mathcal{C}_{1}/\mathcal{C}_{2}}\oplus\bm{x}}\rangle$. Note that the signs of stabilizers change the fixed subspace by changing the eigenspaces that enter into the intersection. Thus, the encoding map needs to include information about nontrivial signs.

We capture sign information through character vectors $\bm{y}\in\mathbb{F}_{2}^{n}/\mathcal{C}_{1},\bm{r}\in\mathbb{F}_{2}^{n}/\mathcal{C}_{2}^{\perp}$ (note that the choice of coset representatives is not unique) defined for $Z$-stabilizers and $X$-stabilizers respectively by

$$\mathcal{B}=\mathcal{C}_{1}^{\perp}\cap\bm{y}^{\perp},\text{equivalently, }\mathcal{B}^{\perp}=\langle\mathcal{C}_{1},\bm{y}\rangle,$$

(34)

and

$$\mathcal{D}=\mathcal{C}_{2}\cap\bm{r}^{\perp},\text{equivalently, }D^{\perp}=\langle\mathcal{C}_{2}^{\perp},\bm{r}\rangle.$$

(35)

Then, for $\epsilon_{(\bm{a},\bm{0})}\epsilon_{(\bm{0},\bm{b})}E\left(\bm{a},\bm{0}\right)E\left(\bm{0},\bm{b}\right)\in S$, we have $\epsilon_{(\bm{a},\bm{0})}=(-1)^{\bm{a}\bm{r}^{T}}$ and $\epsilon_{(\bm{0},\bm{b})}=(-1)^{\bm{b}\bm{y}^{T}}$. In Example 3, we may choose the character vectors $\bm{r}=\bm{0}$ (character vector of $X$-stabilizers) and $\bm{y}=[1,0,0]$ (character vector of $Z$-stabilizers).

The generalized encoding map $g_{e}:|{\bm{\alpha}}\rangle_{L}\in\mathbb{F}_{2}^{k}\to|{\overline{\bm{\alpha}}}\rangle\in\mathcal{V}(\mathcal{S})$ is defined by

$$|{\overline{\bm{\alpha}}}\rangle\coloneqq\frac{1}{\sqrt{|\mathcal{C}_{2}|}}\sum_{\bm{x}\in\mathcal{C}_{2}}(-1)^{\bm{x}\bm{r}^{T}}|{\bm{\alpha}G_{\mathcal{C}_{1}/\mathcal{C}_{2}}\oplus\bm{x}\oplus\bm{y}}\rangle.$$

(36)

To verify that the image of the general encoding map $g_{e}$ is in $\mathcal{V}(\mathcal{S})$, we show that for $\epsilon_{(\bm{a},\bm{0})}\epsilon_{(\bm{0},\bm{b})}E\left(\bm{a},\bm{0}\right)E\left(\bm{0},\bm{b}\right)\in\mathcal{S}$ (that is $\bm{a}\in\mathcal{C}_{2}$, $\epsilon_{(\bm{a},\bm{0})}=(-1)^{\bm{a}\bm{r}^{T}}$, $\bm{b}\in\mathcal{C}_{1}^{\perp}$, and $\epsilon_{(\bm{0},\bm{b})}=(-1)^{\bm{b}\bm{y}^{T}}$),

	$\displaystyle\epsilon_{(\bm{a},\bm{0})}\epsilon_{(\bm{0},\bm{b})}E\left(\bm{a},\bm{0}\right)E\left(\bm{0},\bm{b}\right)|{\overline{\bm{\alpha}}}\rangle$
	$\displaystyle=\frac{1}{\sqrt{|\mathcal{C}_{2}|}}\sum_{\bm{x}\in\mathcal{C}_{2}}\Big{(}\epsilon_{(\bm{a},\bm{0})}(-1)^{\bm{x}\bm{r}^{T}}\epsilon_{(\bm{0},\bm{b})}(-1)^{\bm{b}(\bm{\alpha}G_{\mathcal{C}_{1}/\mathcal{C}_{2}}\oplus\bm{x}\oplus\bm{y})^{T}}|{\bm{\alpha}G_{\mathcal{C}_{1}/\mathcal{C}_{2}}\oplus\bm{a}\oplus\bm{x}\oplus\bm{y}}\rangle\Big{)}$
	$\displaystyle=\frac{1}{\sqrt{|\mathcal{C}_{2}|}}\sum_{\bm{x}\in\mathcal{C}_{2}}(-1)^{(\bm{a}\oplus\bm{x})\bm{r}^{T}}|{\bm{\alpha}G_{\mathcal{C}_{1}/\mathcal{C}_{2}}\oplus\bm{a}\oplus\bm{x}\oplus\bm{y}}\rangle$
	$\displaystyle=|{\overline{\bm{\alpha}}}\rangle.$		(37)

Given the choice of $G_{\mathcal{C}_{1}/\mathcal{C}_{2}}$, there exists a unique set of vectors $\{\bm{\gamma_{1}},\cdots,\bm{\gamma_{k}}\in\mathcal{C}_{2}^{\perp}:G_{\mathcal{C}_{1}/\mathcal{C}_{2}}\bm{\gamma_{i}}=\bm{e_{i}}\text{ for all }i=1,\dots,k\}$, where $\{\bm{e_{i}}\}_{i=1,\dots,k}$ is the standard basis of $\mathbb{F}_{2}^{k}$. If $\bm{\gamma_{i}}$ is the $i$-the row of generator matrix $G_{\mathcal{C}_{2}^{\perp}/\mathcal{C}_{1}^{\perp}}$, then

$$G_{\mathcal{C}_{1}/\mathcal{C}_{2}}G_{\mathcal{C}_{2}^{\perp}/\mathcal{C}_{1}^{\perp}}^{T}=I_{k}.$$

(38)

Assume we have

$\displaystyle G_{\mathcal{C}_{1}/\mathcal{C}_{2}}=\left[\begin{array}[]{c}\bm{w_{1}}\\ \bm{w_{2}}\\ \vdots\\ \bm{w_{k}}\end{array}\right],\leavevmode\nobreak\ G_{\mathcal{C}_{2}^{\perp}/\mathcal{C}_{1}^{\perp}}=\left[\begin{array}[]{c}\bm{\gamma_{1}}\\ \bm{\gamma_{2}}\\ \vdots\\ \bm{\gamma_{k}}\end{array}\right].$

(47)

Thus, we have for $i=1,\dots,k$

	$\displaystyle E(\bm{w_{i}},\bm{0})|{\overline{\bm{\alpha}}}\rangle$
	$\displaystyle=\frac{1}{\sqrt{|\mathcal{C}_{2}|}}\sum_{\bm{x}\in\mathcal{C}_{2}}(-1)^{\bm{x}\bm{r}^{T}}|{\bm{\alpha}G_{\mathcal{C}_{1}/\mathcal{C}_{2}}\oplus\bm{w_{i}}\oplus\bm{x}\oplus\bm{y}}\rangle$
	$\displaystyle=\frac{1}{\sqrt{|\mathcal{C}_{2}|}}\sum_{\bm{x}\in\mathcal{C}_{2}}(-1)^{\bm{x}\bm{r}^{T}}|{(X_{i}^{L}\bm{\alpha})G_{\mathcal{C}_{1}/\mathcal{C}_{2}}\oplus\bm{x}\oplus\bm{y}}\rangle$
	$\displaystyle=\bar{X}_{i}|{\overline{\bm{\alpha}}}\rangle,$		(48)

and

	$\displaystyle(-1)^{\bm{\gamma_{i}}\bm{y}^{T}}E(\bm{0},\bm{\gamma_{i}})|{\overline{\bm{\alpha}}}\rangle$
	$\displaystyle=\frac{1}{\sqrt{|\mathcal{C}_{2}|}}\Big{(}\sum_{\bm{x}\in\mathcal{C}_{2}}(-1)^{{\bm{x}\bm{r}^{T}}\oplus{\bm{\gamma_{i}}\bm{y}^{T}}\oplus{\bm{\gamma_{i}}(\bm{\alpha}G_{\mathcal{C}_{1}/\mathcal{C}_{2}}\oplus\bm{x}\oplus\bm{y})^{T}}}$
	$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ |{\bm{\alpha}G_{\mathcal{C}_{1}/\mathcal{C}_{2}}\oplus\bm{x}\oplus\bm{y}}\rangle\Big{)}$
	$\displaystyle=\frac{1}{\sqrt{|\mathcal{C}_{2}|}}\sum_{\bm{x}\in\mathcal{C}_{2}}(-1)^{\bm{x}\bm{r}^{T}}(-1)^{\bm{\alpha}\bm{e_{i}}^{T}}|{\bm{v}G_{\mathcal{C}_{1}/\mathcal{C}_{2}}\oplus\bm{x}\oplus\bm{y}}\rangle$
	$\displaystyle=\bar{Z}_{i}|{\overline{\bm{\alpha}}}\rangle,$		(49)

where the second to last step follows from (38). Thus we can choose

$$\bar{X}_{i}=E(\bm{w_{i}},\bm{0})\text{ and }\bar{Z}_{i}=\epsilon_{(\bm{0},\bm{\gamma_{i}})}E(\bm{0},\bm{\gamma_{i}}),$$

(50)

where $\bm{w_{i}},\bm{\gamma_{i}}$ are the $i$-th rows of the above coset generator matrices $G_{\mathcal{C}_{1}/\mathcal{C}_{2}}$, $G_{\mathcal{C}_{2}^{\perp}/\mathcal{C}_{1}^{\perp}}$ respectively.