Permutation-invariant qudit codes from polynomials

Abstract

A permutation-invariant quantum code on $N$ qudits is any subspace stabilized by the matrix representation of the symmetric group $S_N$ as permutation matrices that permute the underlying $N$ subsystems. When each subsystem is a complex Euclidean space of dimension $q\ge 2$, any permutation-invariant code is a subspace of the symmetric subspace of $({\mathbb{C}}^q{)}^N.$ We give an algebraic construction of new families of of $d$-dimensional permutation-invariant codes on at least $(2t+1{)}^2(d-1)$ qudits that can also correct $t$ errors for $d\ge 2$. The construction of our codes relies on a real polynomial with multiple roots at the roots of unity, and a sequence of $q-1$ real polynomials that satisfy some combinatorial constraints. When $N>(2t+1{)}^2(d-1)$, we prove constructively that an uncountable number of such codes exist.

14 pages. Minor corrections made, to appear in Linear Algebra and its Applications

References in corpus (2)

• New class of quantum error-correcting codes for a bosonic mode
• Permutation-invariant codes encoding more than one qubit

Cited by in corpus (25)

• Performance and structure of single-mode bosonic codes
• Quantum information processing with bosonic qubits in circuit QED
• Permutation-invariant constant-excitation quantum codes for amplitude damping
• Preparing Dicke states in a spin ensemble using phase estimation
• Quantum storage in quantum ferromagnets
• Avoiding coherent errors with rotated concatenated stabilizer codes
• Permutation-invariant quantum coding for quantum deletion channels
• Permutation-Invariant Quantum Codes for Deletion Errors
• Computing spectral bounds of the Heisenberg ferromagnet from geometric considerations
• Trade-offs on number and phase shift resilience in bosonic quantum codes
• Symmetric quantum states: a review of recent progress
• Robust quantum metrology with explicit symmetric states
• Linear programming bounds for quantum channels acting on quantum error-correcting codes
• Class of codes correcting absorptions and emissions
• Constructing quantum codes from any classical code and their embedding in ground space of local Hamiltonians
• Degenerate quantum erasure decoding
• Robust projective measurements through measuring code-inspired observables
• Measurement-free code-switching for low overhead quantum computation using permutation invariant codes
• Optimizing quantum codes with an application to the loss channel with partial erasure information
• The equivalence between correctability of deletions and insertions of separable states in quantum codes
• Quantum error correction beyond $SU(2)$: spin, bosonic, and permutation-invariant codes from convex geometry
• Describing quantum metrology with erasure errors using weight distributions of classical codes
• Linear Optical Schemes to Postselect High-Dimensional Dicke States
• Fault-Tolerant Encoding of Logical Qudits in Spin Systems
• Generalized Conditional Displacement