1

Is there a finite field equivalent to the Euler decomposition of symplectic matrices?

Over $\Bbb R$: Finding Euler decomposition of a symplectic matrix

Rodrigo de Azevedo
  • 23,223
unknown
  • 1,050

1 Answers1

1

Over the field $\mathbb{F}_2$, here a possible decomposition (by the way, this question is very relevant to quantum computing and quantum error correction):

A matrix $F\in\operatorname{Sp}(2m,\mathbb{F}_2)$ can be decomposed as $$ F=A_{Q_1}\Omega T_{R_1}G_k T_{R_2} A_{Q_2} , $$ for some $k\leq m$, and some $m\times m$ matrices $Q_1,Q_2,R_1,R_2$, where

\begin{aligned} \Omega &= \begin{bmatrix}0&I_m\\I_m&0\end{bmatrix}, \quad A_Q = \begin{bmatrix}Q&0\\0&Q^{-T}\end{bmatrix}, \quad T_R = \begin{bmatrix}I_m&R\\0&I_m\end{bmatrix} \text{ with } R=R^T, \\[6pt] G_t &= \begin{bmatrix}L_{m-t}&U_t\\U_t&L_{m-t}\end{bmatrix} \text{ with } U_t = \operatorname{diag}(I_t,0_{m-t}), L_{m-t} = \operatorname{diag}(0_t,I_{m-t}) \end{aligned} (all matrix elements above are in $\mathbb{F}_2$).

See appendix B here, or chapter 5.3.2 here for a proof and detailed explanations.

Ooooscar
  • 105
  • I think this is also closely related to "Bruhat decomposition" and I found that angle of looking at the problem in that setting helpful. Thanks for the answer. – unknown Sep 11 '23 at 14:25
  • 1
    By the way, I recently found another quantum computing paper using the bruhat decomposition: https://arxiv.org/pdf/2003.09412.pdf. Their results are even implemented into qiskit (a python library for quantum computing) already. – Ooooscar Feb 09 '24 at 03:40
  • 1
    Thanks for the reference; Bravyi's papers are usually pretty good. I also found another implementation in GAP a while ago : https://github.com/danielrademacher/BruhatDecomposition . It works pretty well and is useful to check some results on small circuits. It is fairly slow though; so the largest order of the symplectic group I managed is <10. I'll see if the python version can handle larger values. – unknown Feb 09 '24 at 15:27
  • 1
    Just for reference, here's another master's thesis on the topic: https://uwspace.uwaterloo.ca/bitstream/handle/10012/14622/Huq_Jaron.pdf – Ooooscar Feb 09 '24 at 18:45