This paper presents conditions for constructing permutation-invariant quantum codes for deletion errors and provides a method for constructing them. Our codes give the first example of quantum codes that can correct two or more deletion errors. Also, our codes give the first example of quantum codes that can correct both multiple-qubit errors and multiple-deletion errors. We also discuss a generalization of the construction of our codes at the end.
Quantum deletions, which are harder to correct than erasure errors, occur in many realistic settings. It is therefore pertinent to develop quantum coding schemes for quantum deletion channels. To date, not much is known about which explicit quantum error correction codes can combat quantum deletions. We note that {\em any} permutation-invariant quantum code that has a distance of $t+1$ can correct $t$ quantum deletions for any positive integer $t$ in both the qubit and the qudit setting. Leveraging on coding properties of permutation-invariant quantum codes under erasure errors, we derive corr…
Permutation-invariant codes: a numerical study and qudit constructions
QuSoft, Amsterdam（QuSoft）；Institute of Physics, University of Amsterdam（阿姆斯特丹大学物理研究所）；Institute for Logic, Language and Computation, University of Amsterdam（阿姆斯特丹大学逻辑、语言与计算研究所）；Korteweg-de Vries Institute for Mathematics, University of Amsterdam（阿姆斯特丹大学科特韦格-德弗里斯数学研究所）；Centre for the Mathematics of Quantum Theory, University of Copenhagen（哥本哈根大学量子理论数学中心）
A permutation-invariant code on m qubits is a subspace of the symmetric subspace of the m qubits. We derive permutation-invariant codes that can encode an increasing amount of quantum information while suppressing leading order spontaneous decay errors. To prove the result, we use elementary number theory with prior theory on permutation invariant codes and quantum error correction.
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 …
The study of entanglement properties of multi-qubit states that are invariant under permutations of qubits is motivated by potential applications in quantum computing, quantum communication, and quantum metrology. In this work, we generalize the notions of symmetrization, Dicke states, and the Majorana representation to the alternating, cyclic, and dihedral subgroups of the full group of permutations. We use these tools to characterize states that are invariant under these subgroups and analyze their entanglement properties.
We describe the implications of permutation symmetry for the state space and dynamics of quantum mechanical systems of matrices of general size $N$. We solve the general 11- parameter permutation invariant quantum matrix harmonic oscillator Hamiltonian and calculate the canonical partition function. The permutation invariant sector of the Hilbert space, for general Hamiltonians, can be described using partition algebra diagrams forming the bases of a tower of partition algebras $P_k(N)$. The integer $k$ is interpreted as the degree of matrix oscillator polynomials in the quantum mechanics. Fam…
