Fourier samples are uniform in the annihilator

Given: The exact product QFT, projective measurement in its character-label basis, and discarded-output state from One finite abelian hidden-subgroup Fourier sample and Discarding the hiding register yields a coset-state mixture.

Put $N:=\mathrm{\mid }G\mathrm{\mid }$, $M:=\mathrm{\mid }H\mathrm{\mid }$, and $Q:=\mathrm{\mid }{H}^{\perp }\mathrm{\mid }$, and write $N_{\mathbb{R}}:={\iota }_{\mathbb{R}}(N)$, $M_{\mathbb{R}}:={\iota }_{\mathbb{R}}(M)$, and $Q_{\mathbb{R}}:={\iota }_{\mathbb{R}}(Q)$ for their positive canonical real images, viewed also in $\mathbb{C}$. A normalized uniform coset state transforms to amplitude $Q_{\mathbb{R}}^{-1\mathrm{/}2}\overline{\chi (g)}$ on $H^{\perp }$ and zero outside. In the probability ratios in the Statement and below, $1\mathrm{/}\mathrm{\mid }{H}^{\perp }\mathrm{\mid }$ means $1\mathrm{/}{Q}_{\mathbb{R}}$ and $\mathrm{\mid }H\mathrm{\mid }\mathrm{/}\mathrm{\mid }G\mathrm{\mid }$ means $M_{\mathbb{R}}\mathrm{/}{N}_{\mathbb{R}}$. (Fourier transform of a subgroup coset indicator)

Every value of an additive character of a finite abelian group has modulus one. (Values of additive characters of finite abelian groups have modulus one)

The discarded-output input is the uniform mixture over distinct coset states. (Discarding the hiding register yields a coset-state mixture)

$\mathrm{\mid }{H}^{\perp }\mathrm{\mid }=\mathrm{\mid }G\mathrm{\mid }\mathrm{/}\mathrm{\mid }H\mathrm{\mid }$. (Subgroup-annihilator cardinality)

The abstract normalized Fourier coefficient matrix is unitary, with its column orthogonality supplied by the dual-index character sum. This is a matrix identity for the abstract group and character index spaces; it does not establish orthonormality of encoded register states. (Finite abelian Fourier transform is unitary, Dual-index character orthogonality)

A density operator is a nonnegative endomorphism of trace one. (Density operator)

The endomorphism trace is the sum of diagonal entries in an ordered basis. (The basis-independent trace of an endomorphism of a finite-dimensional vector space)

The rank-one operator $\mathrm{\mid }x⟩⟨y\mathrm{\mid }$ sends $v$ to $⟨v,y{⟩}_{\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{t}}x$ by definition, and its coordinate matrix in ordered orthonormal bases is $[x][y{]}^{†}$. (Dirac Notation Conversion, Dirac Notation Coordinate Identity)

A projective-measurement outcome has weight $\mathrm{tr}(\rho P_a)$. (Finite projective measurements)

In a fixed ordered basis, the coordinate matrix of a finite sum of endomorphisms is the sum of their coordinate matrices; the definition of endomorphism trace and matrix-trace linearity therefore give trace linearity for the finite sums used here. (The basis-independent trace of an endomorphism of a finite-dimensional vector space, Trace is a linear functional on $M_n(F)$)

$H^{\perp }=\{\chi \in \widehat{G}:\chi (h)=1\text{ for every }h\in H\}$. (Character annihilator of a subgroup)

For $N\ge 2$, the sample procedure's supplied $Q_G$ is a unitary on the full $b$-qubit register and its exact action on each valid group code is the prescribed Fourier superposition of valid character codes; the procedure measures the full $b$-bit computational basis. The $N=1$ branch instead returns the unique trivial character without a query. (One finite abelian hidden-subgroup Fourier sample)

The problem definition makes the group code injective and uses the same least-residue, fixed-width encoding for character labels, which is injective coordinatewise. An $n$-qubit register is an ordered tensor product of $Q_2$ factors (Finite registers and their ordered computational bases), with the declared bit order (Big-endian encoding of ordered bit strings). The factor bases are orthonormal (The computational basis is orthonormal), and the tensor inner product is the product of the factor inner products (Ordered tensor registers, their inner product, and wire permutations); hence the factor-basis theorem makes the lexicographic product computational basis orthonormal (The product basis is orthonormal and tensor dimensions multiply). Thus distinct valid group or character codes give orthonormal register states. (Finite abelian hidden-subgroup promise and output encoding)