bibkey: basslerheinrichkliesch2024gzz
authors: Pascal Baßler; Markus Heinrich; Martin Kliesch
year: 2024
title: "Time-optimal multi-qubit gates: Complexity, efficient heuristic and gate-time bounds"
doi: 10.22331/q-2024-03-13-1279
url: https://arxiv.org/abs/2307.11160v2
claim: "The paper casts the time-optimal synthesis of global ZZ gates as a linear program over the cut-polytope vertices m m^T, proves gate-time bounds and a tightness theorem for M = -C m m^T, and conjectures that the optimal gate time is at most the largest coupling times n (odd n) or n-1 (even n)."
strata_touched:
  - D5/S3/Quantum/Optimization/GlobalZZGateTimeBound
license: citation-only
triage: anchor

Time-optimal multi-qubit gates: complexity, efficient heuristic and gate-time bounds

P. Baßler, M. Heinrich and M. Kliesch, arXiv:2307.11160v2 [quant-ph]; Quantum 8, 1279 (2024). Quotations are from the v2 TeX source (\HTL is \mathrm{Sym}_0, \linfnorm{M} is $\Vert M{\Vert }_{{\ell }_{\infty }}$).

The linear program:

Our objective is to minimize the total gate time and the amount of $\vec{m}$’s needed to express the matrix $M\in \text{HTL}({\text{RR}}^n)$. To this end we formulate the following \acf{LP}: minimize ${\vec{1}}^T\vec{\lambda }$ subject to $V\vec{\lambda }=\vec{v}(M)$, $\vec{\lambda }\in {\mathbb{R}}_{\ge 0}^{2^{n-1}}$

(the optimization display is written inline here).

The conjecture (Conjecture conj:gate-time):

The optimal gate time of $\text{Gate}GZZ(A)$ with $A\in \text{HTL}({\text{RR}}^n)$ is tightly upper bounded by ${\vec{1}}^T{\vec{\lambda }}^{\ast }\le \text{linfnorm}M\cdot n$ for odd $n$, $n-1$ for even $n$.

(the cases display is written inline here).

The status:

Unfortunately, we were not able to prove this result and state the challenges in \cref{app:proof_problems}.

Verified locator

Source: https://arxiv.org/abs/2307.11160v2

• URL: https://arxiv.org/abs/2307.11160v2 (source J_shaping.tex, md5 3c751c31a7a40505b6013390b841878a):
  • ${\mathrm{Sym}}_0$, l. 240;
  • the linear program, l. 270–290;
  • the norms, l. 448;
  • the conjecture, l. 919–927;
  • the tightness theorem, l. 1121–1130;
  • the status sentence, l. 349.
• DOI: 10.22331/q-2024-03-13-1279 (https://doi.org/10.22331/q-2024-03-13-1279; Crossref lists Quantum, volume 8, article 1279, published 13 March 2024).