bibkey: wei2024magicnoise authors: Fuchuan Wei and Zi-Wen Liu year: 2024 title: Noise robustness and threshold of many-body quantum magic doi: null url: https://arxiv.org/abs/2410.21215v1 claim: The local-depolarizing CCZ magic-capacity threshold is conjectured to be 1/3; stabilizer normal forms and robustness faithfulness specify its meaning. strata_touched:

• D5/S3/Quantum/Information/CCZMagicCapacityThreshold license: citation-only triage: anchor

Noise robustness and the CCZ capacity conjecture

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DOI: null

Source: https://arxiv.org/abs/2410.21215v1

The Crossref title query for “Noise robustness and threshold of many-body quantum magic” returns no exact title match. The locator is the arXiv v1 manuscript.

Definitions and source statements

On printed page 2:

Let ${\mathrm{C}}^{n-1}Z=\mathrm{diag}(1,\cdots ,1,-1)$ denote the multi-controlled-Z gate on $n$-qubits, with ${\mathrm{C}}^{0}Z=Z$.

Let ${\mathrm{STAB}}_n$ denote the set of all $n$-qubit stabilizer states, namely the convex hull of all pure stabilizer states.

For a $n$-qubit state $\rho$, its robustness of magic (RoM), denoted by $\mathcal{R}(\rho )$, is defined as $\mathcal{R}(\rho )={\min }_{\sigma ,\tau \in {\mathrm{STAB}}_n}\{2a+1\mid \rho =(a+1)\sigma -a\tau ,a\ge 0\}$.

The source’s property i) on page 2 states that “$\mathcal{R}(\cdot )$ is faithful, i.e., $\mathcal{R}(\sigma )=1$ iff $\sigma \in \mathrm{STAB}$”. Faithfulness makes capacity one equivalent to every permitted stabilizer input having stabilizer-mixture output. The Lean claim uses that membership formulation; it does not define or optimize RoM.

On printed page 2:

As a standard noise model, we primarily consider the independent depolarizing noise ${\mathcal{E}}_{\lambda }^{\otimes n}$ acting on the $n$-body quantum system, which leaves the qubits it acts on unchanged with probability $1-\lambda$, and replaces them with ${\mathbb{I}}_2/2$ with probability $\lambda$.

The source writes $\mathcal{G}(\sigma )=\mathrm{Tr}(\sigma ){\mathbb{I}}_2/2$ and ${\mathcal{E}}_{\lambda }=(1-\lambda )\mathcal{I}+\lambda \mathcal{G}$. Thus depol acts on arbitrary single-qubit matrices, and depolA applies it to coordinates 0, 1 and 2 of the six-qubit space.

On printed page 3, following Seddon–Campbell [31]:

Define the magic capacity [31] of the ${\mathrm{C}}^{n-1}Z$ gate as $\mathcal{C}({\mathrm{C}}^{n-1}Z)={\max }_{|s⟩}\mathcal{R}({\mathrm{C}}^{n-1}Z\otimes {\mathbb{I}}_{2^n}|s⟩)$, where the maximum is taken over all $2n$-qubit pure stabilizer states $|s⟩$.

On printed page 11, Appendix D, Eq. (D2):

Any pure $n$-qubit stabilizer state has the form $|\mathcal{K},q,\mathbf{b}⟩:=\frac{1}{\sqrt{|\mathcal{K}|}}{\sum }_{x\in \mathcal{K}}{i}^{\mathbf{b}\cdot x}(-1{)}^{q(x)}|x⟩$, where $\mathcal{K}\subset {\mathbb{F}}_2^n$ is an affine subspace, $\mathbf{b}\in {\mathbb{F}}_2^n$, $q$ is a quadratic form, and $i=\sqrt{-1}$.

The coefficient encoding uses Fin n → Bool for computational labels and Fin n → ZMod 2 for the affine support. The binary dot product in the exponent of $i$ uses integer lifts before summation. Quadratic polynomials use upper-triangular binary coefficients, including diagonal terms. Nonempty support is required. These conventions are fixed in the source-to-Lean statement.

On printed page 6, Section VII:

We conjecture that the magic capacity threshold for $\mathrm{CC}Z$ under local depolarizing noise is $1/3$.

The conjecture concerns a gate with a reference system. The paper’s proved $1/3$ threshold for the three-qubit state $\mathrm{CC}Z|+++⟩$ is a different statement.

Use and boundary

The settling module refutes the membership claim at $\lambda =1/2$ with the six-qubit stabilizer input $\Omega =8^{-1/2}{\sum }_x|x,x⟩$. A separating matrix has nonnegative trace pairing with every element of STAB 6 and trace pairing $-7/1024$ with this noisy output. The diagonal restriction of the stabilizer normal form uses the quadratic carry term ${\sum }_i{b}_A(i)b_B(i)x_i^2$ together with the binary phase vector $b_A+b_B$.

The exact capacity threshold, the general reference advantage for ${\mathrm{C}}^{n-1}Z$, and the source’s numerically reported dephasing threshold near $0.645$ remain open. The source’s state-threshold theorem and its noiseless diagonal-gate capacity bound are not refuted by this example.