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Adam Zalcman
Adam Zalcman
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TL;DR: 1. The code is unique under the stated equivalence. 2. No.

Weight of a logical operator is ill-defined

The crux of the issue is that the weight of a logical operator is ill-defined. Logical operator is not a single Pauli operator in the $n$-qubit Pauli group $\mathcal{P}_n$, but a coset $X_L\mathcal{S}$ of the stabilizer group $\mathcal{S}$ where $X_L$ is some representative of the logical operator.

For example, in the $[\![5,1,3]\!]$ code folks typically choose $X_L=XXXXX$, but $g_1=XZZXI\in\mathcal{S}$, so $X_Lg_1\equiv IYYIX\in X_L\mathcal{S}$ is another valid representative of the logical operator $X_L\mathcal{S}$. Clearly, the two representatives have different weight.

A more dramatic example occurs in an odd distance $d$ planar surface code where people typically represent $X_L$ as a weight $d$ chain of physical $X$ operators along a straight line connecting two boundaries. However, an alternative representative is the operator that applies $X$ to all $d^2$ data qubits of the code.

Weights in a tableau are not fixed under choice of generators

It is true that weights of the generators listed in a tableau remain unchanged under local Cliffords and qubit permutations. However, they do not remain fixed under multiplication of generators. Consequently, it is in fact not true that two different tableaus can only represent equivalent codes if all the weights of their Pauli strings are the same.

As a simple counterexample, consider the following two tableaus representing the group stabilizing $\mathrm{span}(|000\rangle,|001\rangle)$ $$ \begin{array}{ccc|ccc} ZII&III\\ IZI&III \end{array}\quad\quad\quad\quad\quad \begin{array}{cc} ZZI&III\\ IZI&III \end{array} $$$$ \begin{array}{ccc} ZII\\ IZI \end{array}\quad\quad\quad \begin{array}{ccc} ZZI\\ IZI \end{array} $$ The issue is that if $h,g_1,g_2,\dots,g_k$ are independent generators of a group $\mathcal{G}$ then so are $h,hg_1,hg_2,\dots,hg_k$ and the weights of $g_i$ and $hg_i$ are generally not the same.

TL;DR: 1. The code is unique under the stated equivalence. 2. No.

Weight of a logical operator is ill-defined

The crux of the issue is that the weight of a logical operator is ill-defined. Logical operator is not a single Pauli operator in the $n$-qubit Pauli group $\mathcal{P}_n$, but a coset $X_L\mathcal{S}$ of the stabilizer group $\mathcal{S}$ where $X_L$ is some representative of the logical operator.

For example, in the $[\![5,1,3]\!]$ code folks typically choose $X_L=XXXXX$, but $g_1=XZZXI\in\mathcal{S}$, so $X_Lg_1\equiv IYYIX\in X_L\mathcal{S}$ is another valid representative of the logical operator $X_L\mathcal{S}$. Clearly, the two representatives have different weight.

A more dramatic example occurs in an odd distance $d$ planar surface code where people typically represent $X_L$ as a weight $d$ chain of physical $X$ operators along a straight line connecting two boundaries. However, an alternative representative is the operator that applies $X$ to all $d^2$ data qubits of the code.

Weights in a tableau are not fixed under choice of generators

It is true that weights of the generators listed in a tableau remain unchanged under local Cliffords and qubit permutations. However, they do not remain fixed under multiplication of generators. Consequently, it is in fact not true that two different tableaus can only represent equivalent codes if all the weights of their Pauli strings are the same.

As a simple counterexample, consider the following two tableaus representing the group stabilizing $\mathrm{span}(|000\rangle,|001\rangle)$ $$ \begin{array}{ccc|ccc} ZII&III\\ IZI&III \end{array}\quad\quad\quad\quad\quad \begin{array}{cc} ZZI&III\\ IZI&III \end{array} $$ The issue is that if $h,g_1,g_2,\dots,g_k$ are independent generators of a group $\mathcal{G}$ then so are $h,hg_1,hg_2,\dots,hg_k$ and the weights of $g_i$ and $hg_i$ are generally not the same.

TL;DR: 1. The code is unique under the stated equivalence. 2. No.

Weight of a logical operator is ill-defined

The crux of the issue is that the weight of a logical operator is ill-defined. Logical operator is not a single Pauli operator in the $n$-qubit Pauli group $\mathcal{P}_n$, but a coset $X_L\mathcal{S}$ of the stabilizer group $\mathcal{S}$ where $X_L$ is some representative of the logical operator.

For example, in the $[\![5,1,3]\!]$ code folks typically choose $X_L=XXXXX$, but $g_1=XZZXI\in\mathcal{S}$, so $X_Lg_1\equiv IYYIX\in X_L\mathcal{S}$ is another valid representative of the logical operator $X_L\mathcal{S}$. Clearly, the two representatives have different weight.

A more dramatic example occurs in an odd distance $d$ planar surface code where people typically represent $X_L$ as a weight $d$ chain of physical $X$ operators along a straight line connecting two boundaries. However, an alternative representative is the operator that applies $X$ to all $d^2$ data qubits of the code.

Weights in a tableau are not fixed under choice of generators

It is true that weights of the generators listed in a tableau remain unchanged under local Cliffords and qubit permutations. However, they do not remain fixed under multiplication of generators. Consequently, it is in fact not true that two different tableaus can only represent equivalent codes if all the weights of their Pauli strings are the same.

As a simple counterexample, consider the following two tableaus representing the group stabilizing $\mathrm{span}(|000\rangle,|001\rangle)$ $$ \begin{array}{ccc} ZII\\ IZI \end{array}\quad\quad\quad \begin{array}{ccc} ZZI\\ IZI \end{array} $$ The issue is that if $h,g_1,g_2,\dots,g_k$ are independent generators of a group $\mathcal{G}$ then so are $h,hg_1,hg_2,\dots,hg_k$ and the weights of $g_i$ and $hg_i$ are generally not the same.

Source Link
Adam Zalcman
Adam Zalcman
  • 24.8k
  • 3
  • 38
  • 92

TL;DR: 1. The code is unique under the stated equivalence. 2. No.

Weight of a logical operator is ill-defined

The crux of the issue is that the weight of a logical operator is ill-defined. Logical operator is not a single Pauli operator in the $n$-qubit Pauli group $\mathcal{P}_n$, but a coset $X_L\mathcal{S}$ of the stabilizer group $\mathcal{S}$ where $X_L$ is some representative of the logical operator.

For example, in the $[\![5,1,3]\!]$ code folks typically choose $X_L=XXXXX$, but $g_1=XZZXI\in\mathcal{S}$, so $X_Lg_1\equiv IYYIX\in X_L\mathcal{S}$ is another valid representative of the logical operator $X_L\mathcal{S}$. Clearly, the two representatives have different weight.

A more dramatic example occurs in an odd distance $d$ planar surface code where people typically represent $X_L$ as a weight $d$ chain of physical $X$ operators along a straight line connecting two boundaries. However, an alternative representative is the operator that applies $X$ to all $d^2$ data qubits of the code.

Weights in a tableau are not fixed under choice of generators

It is true that weights of the generators listed in a tableau remain unchanged under local Cliffords and qubit permutations. However, they do not remain fixed under multiplication of generators. Consequently, it is in fact not true that two different tableaus can only represent equivalent codes if all the weights of their Pauli strings are the same.

As a simple counterexample, consider the following two tableaus representing the group stabilizing $\mathrm{span}(|000\rangle,|001\rangle)$ $$ \begin{array}{ccc|ccc} ZII&III\\ IZI&III \end{array}\quad\quad\quad\quad\quad \begin{array}{cc} ZZI&III\\ IZI&III \end{array} $$ The issue is that if $h,g_1,g_2,\dots,g_k$ are independent generators of a group $\mathcal{G}$ then so are $h,hg_1,hg_2,\dots,hg_k$ and the weights of $g_i$ and $hg_i$ are generally not the same.