The Exercise

Take a register of $n$ qubits and a binary linear code $C_2$ of length $n$, a subspace of the bit vectors $\mathbb{Z}_2^n$. Its dual $C_2^\perp$ is the code of the bit vectors $z$ with $z \cdot y = 0$ for every codeword $y \in C_2$, where $z \cdot y$ counts, mod 2, the positions at which both carry a one.^✦

A coset $a + C_2$ of the code gives the equal superposition of the computational basis states $|a + y\rangle$ over the codewords $y$, and a bit vector $z$ puts the sign $(-1)^{z \cdot y}$ on each term. With one such state for each coset of $C_2$ and each coset of $C_2^\perp$, the exercise asks to show that these states resolve the identity on the register.

A Register of Qubits

A quantum system $S$ is a physical system whose state space is a finite dimensional complex Hilbert space.^✦ A pure state $\psi$ of $S$ is a vector of unit norm in the state space of $S$, $\lVert \psi \rVert = 1$, determined physically up to a global phase.^✦

For a finite group $G$, the group system $\mathbb{C}[G]$ is the quantum system whose state space is the space of complex functions on $G$.^✦ The bit register of $n$ qubits, one for each of the $n$ positions of a bit vector, is the group system $\mathbb{C}[\mathbb{Z}_2^n]$ of the bit vectors under addition mod 2.^✦ The computational basis $\{|w\rangle\}$ of the bit register $\mathbb{C}[\mathbb{Z}_2^n]$ is the orthonormal basis of its state space with one vector $|w\rangle$ for each bit vector $w$, the function equal to $1$ at $w$ and to $0$ elsewhere.^✦

Coset States

For bit vectors $u$, $v$ and $x$, the twisted coset state $|x + C_2\rangle_{u,v}$ of $C_2$ is the pure state of the register $$|x + C_2\rangle_{u,v} = \frac{1}{\sqrt{|C_2|}} \sum_{y \in C_2} (-1)^{u \cdot y}\, |x + y + v\rangle .$$(1)^✦

The phase coset state $|\xi_{z,a}\rangle$ of $C_2$, for bit vectors $z$ and $a$, is the twisted coset state of $a$ with its signs set by $z$ and no shift: $$|\xi_{z,a}\rangle = |a + C_2\rangle_{z,0} = \frac{1}{\sqrt{|C_2|}} \sum_{y \in C_2} (-1)^{z \cdot y}\, |a + y\rangle .$$(2)^✦

The coset index set $\mathcal{I}$ of $C_2$ is the set of pairs of a coset of $C_2$ and a coset of $C_2^\perp$ in $\mathbb{Z}_2^n$: $$\mathcal{I} = \mathbb{Z}_2^n / C_2 \times \mathbb{Z}_2^n / C_2^\perp .$$(3)^✦

The phase coset family of $C_2$ assigns to each index $p = (a + C_2,\, z + C_2^\perp)$ the phase coset state built from a fixed representative $a$ of the first coset and a fixed representative $z$ of the second: $$|\xi_p\rangle = |\xi_{z,a}\rangle .$$(4)^✦

The Claim

What Is Checked

The claim is stated over the phase coset family (4) on the index set (3), as vectors in the state space of the bit register.

About these constructions the library proves that each twisted coset state (1) has unit norm.