Define H-count, computational nullity, OCE, and new ensembles

[ChanceSiyuan]

Step 3a: Define new monotones and ensembles

Goal: Formally define the coherence-framework analogues of all quantities used in the Clifford+T theorem.

Definitions needed

Clifford+T framework	{CX,CCX,S}+H framework
T-count $\tau(U)$	H-count $\eta(U)$: min # of $H$ gates in $\mathcal{C}(\mathcal{B}_{\text{comp}})$+$H$ decomposition
Unitary nullity $\nu(U)$ (Pauli stabilizers)	Computational nullity $\nu_{\text{comp}}(U)$: based on $\mathcal{P}(\mathcal{B}_{\text{comp}})$ stabilizers
OSE $M^{(\alpha)}(O_U)$ (Pauli basis)	Operator coherence entropy (OCE) $M^{(\alpha)}_{\text{comp}}(O_U)$: entropy of $O_U$ in computational basis
$T$-doped ensemble $\mu_\tau$	$H$-doped ensemble $\mu_\eta$: random $\mathcal{C}(\mathcal{B}_{\text{comp}})$ circuits interspersed with $\eta$ single-qubit $H$ gates
$\nu$-compressible ensemble $\mu_\nu$	$\nu_{\text{comp}}$-compressible ensemble $\mu_{\nu_{\text{comp}}}$: $U = G_0(V_\ell \otimes \mathbb{I})G_1$ with $G_i \in \mathcal{C}(\mathcal{B}_{\text{comp}})$

Subtasks

• Formal definition of H-count $\eta(U)$ with proof that it is well-defined
• Formal definition of computational nullity $\nu_{\text{comp}}(U)$ based on $\mathcal{P}(\mathcal{B}_{\text{comp}})$ stabilizers
• Formal definition of operator coherence entropy (OCE) $M^{(\alpha)}_{\text{comp}}(O_U)$
• Formal definition of $H$-doped ensemble $\mu_\eta$
• Formal definition of $\nu_{\text{comp}}$-compressible ensemble $\mu_{\nu_{\text{comp}}}$

Notes

• Choice of initial operator $O$: In the Clifford+T case, the initial operator is a Pauli string $O \in \mathcal{P}N \setminus {\mathbb{I}}$. The analogous "free" operators for $\mathcal{C}(\mathcal{B}{\text{comp}})$ are computational-basis elements $|j\rangle\langle k|$. Need to verify that the 4-fold average simplifies analogously when $X = O^{\otimes 4}$ for such $O$.

Files

• docs/operator_magic.qmd — existing magic-coherence duality discussion