SciPost Submission Page
Unitary tetrahedron quantum gates
by Vivek Kumar Singh, Akash Sinha, Pramod Padmanabhan, Vladimir Korepin
Submission summary
| Authors (as registered SciPost users): | Vivek Singh |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2407.10731v1 (pdf) |
| Code repository: | https://github.com/vks577/Unitary-tetrahedron-operators |
| Date submitted: | 2024-07-17 12:31 |
| Submitted by: | Singh, Vivek |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
Quantum simulations of many-body systems using 2-qubit Yang-Baxter gates offer a benchmark for quantum hardware. This can be extended to the higher dimensional case with $n$-qubit generalisations of Yang-Baxter gates called $n$-simplex operators. Such multi-qubit gates potentially lead to shallower and more efficient quantum circuits as well. Finding them amounts to identifying unitary solutions of the $n$-simplex equations, the building blocks of higher dimensional integrable systems. These are a set of highly non-linear and over determined system of equations making it notoriously hard to solve even when the local Hilbert spaces are spanned by qubits. We systematically overcome this for higher simplex operators constructed using two methods: from Clifford algebras and by lifting Yang-Baxter operators. The $n=3$ or the tetrahedron case is analyzed in detail. For the qubit case our methods produce 13 inequivalent families of unitary tetrahedron operators. 12 of these families are obtained by appending the 5 unitary families of 4 by 4 constant Yang-Baxter operators of Dye-Hietarinta, with a single qubit operator. As applications, universal sets of single, two and three qubit gates are realized using such unitary tetrahedron operators. The ideas presented in this work can be naturally extended to the higher simplex cases.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Reports on this Submission
Strengths
1. The presentation is very clear.
2. The main results are summarized in tables, making them readily accessible for the reference.
Weaknesses
1. The constructions are all compositions of lower-ranked operators and heavily depend on [63], making the solutions less original.
2. With the present title, I believe the paper should elaborate further on the "quantum gates" aspect of the work rather than limiting it to a brief discussion section.
Report
This paper identifies a non-exhaustive list of unitary solutions to the constant tetrahedron equation, a generalization of the Yang-Baxter equation. The first class of solutions is constructed solely from Clifford operators. The second class generalizes known Yang-Baxter operators, such as Hietarinta’s solutions or Clifford solutions. After identifying equivalence classes, the authors summarize 13 families of unitary tetrahedron operators in Table 3. Additionally, the paper argues that these operators can be utilized to construct universal quantum gates, with results summarized in Tables 4 and 5.
I believe the paper meets the journal’s acceptance criteria provided the authors address the following concerns. Otherwise, I recommend publication in SciPost Physics Core.
Requested changes
1. Regarding Weakness 2: The current discussion focuses on interpreting universal gates as tetrahedron operators. To make it reasonable, the authors should address at least one of the following questions:
(a) Does this framework enable shallower or more efficient quantum circuits?
(b) Do these unitary tetrahedron operators admit simpler physical realizations (e.g., in spin systems)?
2. Beyond the equivalence classes discussed in Section 4, the manuscript does not sufficiently justify why the remaining families are inequivalent. A clearer argument should be provided.
3. The footnote explaining "nilpotent" should be relocated to its first occurrence (page 9, line 6) rather than its second (page 11).
4. The prerequisites for propositions (e.g., unitarity in Proposition 2.1 or off-diagonal terms in Proposition 2.4) should be explicitly stated within the propositions themselves, not merely in preceding text.
5. Several typos need correction:
Page 4, line 8: Replace "the set second" with "the second set."
Page 5, line 7: Clarify that the tetrahedron equation (not the operators) satisfies the symmetry conditions.
Page 14, line 1: Correct the misprinted equation for $A_1$.
Page 23, line 22: Replace "other words in" with "in other words."
Recommendation
Ask for minor revision
Matthew James Stephenson on 2025-02-25 [id 5242]
Errors in user-supplied markup (flagged; corrections coming soon)
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.
---
## 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 $$\mathcal{Q}Y\mathcal{Q}^{-1}$$.
---
## 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 $$\arg(\alpha_j)$$.
---
## 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