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Quantum error correction beyond $SU(2)$: spin, bosonic, and permutation-invariant codes from convex geometry

Arda Aydin, Victor V. Albert, Alexander Barg

Abstract

We develop a framework for constructing quantum error-correcting codes and logical gates for three types of spaces -- composite permutation-invariant spaces of many qubits or qudits, composite constant-excitation Fock-state spaces of many bosonic modes, and monolithic nuclear state spaces of atoms, ions, and molecules. By identifying all three spaces with discrete simplices and representations of the Lie group $SU(q)$, we prove that many codes and their gates in $SU(q)$ can be inter-converted between the three state spaces. We construct new code instances for all three spaces using classical $\ell_1$ codes and Tverberg's theorem, a classic result from convex geometry. We obtain new families of quantum codes with distance that scales almost linearly with the code length $N$ by constructing $\ell_1$ codes based on combinatorial patterns called Sidon sets and utilizing their Tverberg partitions. This improves upon the existing designs for all the state spaces. We present explicit constructions of codes with shorter length or lower total spin/excitation than known codes with similar parameters, new bosonic codes with exotic Gaussian gates, as well as examples of short codes with distance larger than the known constructions.

Quantum error correction beyond $SU(2)$: spin, bosonic, and permutation-invariant codes from convex geometry

Abstract

We develop a framework for constructing quantum error-correcting codes and logical gates for three types of spaces -- composite permutation-invariant spaces of many qubits or qudits, composite constant-excitation Fock-state spaces of many bosonic modes, and monolithic nuclear state spaces of atoms, ions, and molecules. By identifying all three spaces with discrete simplices and representations of the Lie group , we prove that many codes and their gates in can be inter-converted between the three state spaces. We construct new code instances for all three spaces using classical codes and Tverberg's theorem, a classic result from convex geometry. We obtain new families of quantum codes with distance that scales almost linearly with the code length by constructing codes based on combinatorial patterns called Sidon sets and utilizing their Tverberg partitions. This improves upon the existing designs for all the state spaces. We present explicit constructions of codes with shorter length or lower total spin/excitation than known codes with similar parameters, new bosonic codes with exotic Gaussian gates, as well as examples of short codes with distance larger than the known constructions.

Paper Structure

This paper contains 30 sections, 24 theorems, 135 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Summary of technical results
  3. Preliminaries
  4. PI codes
  5. Quantum deletion channel
  6. Quantum deletions acting on Dicke states
  7. Error correction conditions for PI codes
  8. Fock state codes
  9. Amplitude damping errors
  10. Error correction conditions
  11. Equivalence of Fock state codes and PI codes
  12. Spin codes
  13. and the Jordan-Schwinger map
  14. Error Correction Conditions
  15. Equivalence of spin codes and PI codes
  16. ...and 15 more sections

Key Result

Lemma 4.1

Let $q\geq 2$ and $N\ge 0$ be integers. For a Dicke state $\ket{D_{\underline n}},{\underline n} \in {\EuScript S}_{q,N}$, all $i\in\{1,2,\ldots,N\}$ and $j\in \{0,1,\ldots,q-1\}$, where ${\boldsymbol u}_j\in U_q$.

Figures (5)

  • Figure 1: (a) Angular momentum ladders of spin-$N/2$ irreps of $SU(2)$ are in one-to-one correspondence with points of the discrete simplex ${\EuScript S}_{2,N}$. This correspondence can be extended to one between completely symmetric $SU(q)$-irreps and higher-dimensional discrete simplices ${\EuScript S}_{q,N}$ (see \ref{['subsec:js']}). (b) Simplices can be mapped into permutation-invariant (PI), bosonic, and spin spaces by associating simplex points with $N$-$q$udit Dicke states, $N$-excitation $q$-mode Fock states, and $SU(q)$ "spin" states, respectively. (c) This gives rise to new qudit PI, Fock state, and spin codes via classical $\ell_1$ codes and results from convex geometry (see \ref{['sec: classical']}).
  • Figure 2: Relations between $SU(q)$ code families. The solid arrows apply to all $q\ge 2$ and the dashed arrows apply only to $q=2$. The transformation Spin-to-PI in \ref{['lemma:ericandian']} is from IanEricDihedralIEEE; the other transitions form new results.
  • Figure 3: Graph of the discrete simplex ${\EuScript S}_{3,3}$, where the vertices $u,v$ are connected iff $d_1(u,v)=1$; see \ref{['eq: d1']}. This space is used to define qudit PI codes and Fock state codes (as the index set of basis coefficients), as well as classical $\ell_1$ codes of \ref{['sec: classical']} (as a metric space).
  • Figure 4: Conditions for construction and error correction with qudit PI codes. Eqns. (C3), (C4) represent the KL conditions. We relate similar conditions for spin and Fock state codes to these using multinomial coefficient identities.
  • Figure 5: The simplex ${\EuScript S}_{4,4}$. The colored set of 11 vertices represents a $3$-dimensional code $B$ with $\ell_1$ distance 2 in the simplex. Each color shows a subset of code points ${\boldsymbol x}$ of fixed composition $\mathop{\mathrm{\sf C}}\nolimits({\boldsymbol x})$. There are 3 such subsets, $B_0,B_1,B_2$, which form a Tverberg partition ${\mathscr I}_{11,3}$ of the codeword set with respect to the set of points/equations of the form \ref{['eq: K system']} with coefficients given in \ref{['eq: coefficients']}.

Theorems & Definitions (67)

  • Remark 1: Related results
  • Definition 3.1
  • Definition 3.2
  • Definition 4.1
  • Definition 4.2
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • ...and 57 more