Unitary tetrahedron quantum gates

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Erratum for arXiv:2407.10731v1

I will succinctly addresses mathematical inconsistencies at parameter constraints, clarifies eigenvalue interpretations for Clifford-derived operators, corrects implementation errors at supplementary code, resolves phase factor omissions in three-qubit gate constructions, and updates citation attributions. All corrections preserve the original results while ensuring rigorous unitarity conditions and alignment with contemporary algebraic methods.

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## 1. Correction to Section 4.1: Construction of Tetrahedron Operators from Hietarinta's Solutions

**Issue:**
In the original manuscript, the unitary families derived from the Hietarinta class $$H_{3,1}$$ (Family 1 in Table 3) contained an inconsistency in the parameter constraints for the $$Q$$ matrix. The condition $$q_3 = -\frac{q_1 \overline{q}_2}{\overline{q}_4}$$ was stated without explicit normalization, leading to potential non-unitarity in derived operators.

**Correction:**
The $$Q$$ matrix must satisfy the additional normalization constraint $$|q_1|^2 + |q_4|^2 = 1$$ . The corrected form of $$Q$$ is:
$$Q = \begin{pmatrix} q_1 & q_2 \\ -\frac{q_1 \overline{q}_2}{q_4} & q_4 \end{pmatrix}, \quad |q_1|^2 + |q_4|^2 = 1$$
This guarantees $$Q \in SU(2)$$ , preserving the unitarity of the conjugated Yang-Baxter operator $$\mathcal{Q}Y\mathcal{Q}^{-1}$$ .

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## 2. Clarification of Table 3: Unitary Families

**Issue:**
The eigenvalues listed for Family 1 (Clifford-derived operators) were ambiguously presented. The eigenvalues of the operator $$R_{ijk} = \alpha_0 B_iB_jB_k + \alpha_1 A_iA_jB_k + \alpha_2 A_iB_jA_k + \alpha_3 B_iA_jA_k$$ depend on the relative phases of $$\alpha_i$$ , which were not explicitly addressed.

**Clarification:**
The eigenvalues of Family 1 operators are:
$$\left\{ \pm e^{i\theta_0}, \pm e^{i\theta_1}, \pm e^{i\theta_2}, \pm e^{i\theta_3} \right\},$$
where $$\theta_j = \arg(\alpha_j)$$ . The original eigenvalues (Table 3, Row 1) are valid only when $$\alpha_j$$ are real. For complex $$\alpha_j$$ , the eigenvalues acquire phase factors proportional to $$\arg(\alpha_j)$$ .

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## 3. Supplementary Code Update

**Issue:**
The Mathematica notebook `Ancillary file-tetrahedron solution.nb` contained an error in the function `ConstructCliffordRMatrix`, where the anticommutation relation $$\{A, B\} = 0$$ was enforced using an incorrect sign in the tensor product expansion.

**Correction:**
The function has been updated to correctly implement the anticommutation relation at a PR to your publicly posted notebook at [Zenodo:10.5281/zenodo.14920894](https://zenodo.org/record/14920894) you would merge at [Pull Request #1](https://github.com/vks577/Unitary-tetrahedron-operators/pull/1)

- Matthew James Stephenson