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Exhaustive Optimisation of Automorphism Groups for Stabiliser Codes

Aisling Mac Aree, Mark Howard

Abstract

An important measure of utility for a quantum code is the identification of which logical operations can be implemented fault-tolerantly on its codespace. We introduce a framework which leverages the automorphism groups of associated classical codes, the choice of logical basis and exploitation of code equivalence to construct all distinct implementable realisations of each valid logical operation for a given $[[n,k,d]]$ code. We establish conjugacy classes and group transversals (unrelated to transversality) as key explanatory concepts. We subsequently motivate and calculate two figures-of-merit that can be optimised with this framework. Our results yield a table of optimal logical operations and their corresponding physical circuits for all small stabiliser codes with $n \leq 7$ and $k \leq 2$, drawn from quantum databases. This exhaustive table of results provides the optimal physical implementations of logical operations which may be advantageous for both magic state cultivation and experimental purposes.

Exhaustive Optimisation of Automorphism Groups for Stabiliser Codes

Abstract

An important measure of utility for a quantum code is the identification of which logical operations can be implemented fault-tolerantly on its codespace. We introduce a framework which leverages the automorphism groups of associated classical codes, the choice of logical basis and exploitation of code equivalence to construct all distinct implementable realisations of each valid logical operation for a given code. We establish conjugacy classes and group transversals (unrelated to transversality) as key explanatory concepts. We subsequently motivate and calculate two figures-of-merit that can be optimised with this framework. Our results yield a table of optimal logical operations and their corresponding physical circuits for all small stabiliser codes with and , drawn from quantum databases. This exhaustive table of results provides the optimal physical implementations of logical operations which may be advantageous for both magic state cultivation and experimental purposes.

Paper Structure

This paper contains 23 sections, 5 theorems, 88 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background of algebraic framework
  3. Logical Operations and Corresponding Physical Circuits
  4. Logical action
  5. Conjugacy classes of the symplectic group $Sp(2k,2)$
  6. Choice of logical basis
  7. Code equivalence
  8. Worked Example
  9. Results
  10. Discussion
  11. Acknowledgements
  12. Proofs of Propositions
  13. Proof of Proposition \ref{['eqn logical action']}
  14. Proofs of Proposition \ref{['eqn L=ALAinv']} and \ref{['prop code equivalence no effect']}
  15. Proof of Proposition \ref{['eqn L=ALAinv']}
  16. ...and 8 more sections

Key Result

Proposition 1

The logical action of the circuit $\pi$ acting on a basis $\mathcal{B}$ is given by the trace inner product of each row of the dual basis $\mathcal{D}_{\mathcal{B}}$ with the transformed basis $\pi(\mathcal{B})^{T}$, $\blacktriangleleft$$\blacktriangleleft$

Figures (3)

  • Figure 1: Figure showing the elements of the automorphism group of $\mathcal{C}$ associated with the $[[4,2,2]]$ code. Each dot represents a distinct logical operation. There are $4$ different automorphisms which implement each logical operation. Elements $\pi^{i} \in \Gamma(\mathcal{C})$ composed in the same colour implement logical operations belonging to the same conjugacy class. These conjugacy classes are also given in Table \ref{['table of class reps of SP4,2']}. The first wheel (left) has explicit index labels $\pi^{i} \in \Gamma(\mathcal{C})$, $i \in \{1, \dots, 144\}$, showing the partitioning of the automorphism group based on the logical operations they implement. The second wheel (right) is the same figure shown without the indexing.
  • Figure 2: Figure showing the conjugacy classes implemented by the automorphism groups of the $216$ distinct equivalent versions of the $[[4,2,2]]$ code. This figure highlights how the conjugacy classes of the automorphism-induced logical actions are unaffected by code equivalence. Taken together, the $144$ automorphisms for each of the $216$ versions of $[[4,2,2]]$ exhaust the Hamming group of $\mathcal{C}$.
  • Figure 3: Relating to $[[4,2,2]]$ example, showing that the best implementation of $\mathcal{L}$ is given by $\pi'^{30}$. $\Gamma(\mathcal{C'})$ denotes the automorphism group obtained by using (in our case), the $6^{th}$ element of the transversal group $\tau^{6} \in \mathcal{T}$ (the circuit diagram of this transversal $\tau$ is given explicitly in the example).

Theorems & Definitions (5)

  • Proposition 1
  • Proposition 2
  • Theorem 1
  • Proposition 3
  • Theorem 2