Description
Block quantum code encoding one subsystem into \(n\) subsystems whose encoding isometry is a perfect tensor. This code stems from an AME\((n,q)\) AME state, or equivalently, a \(((n+1,1,\lfloor (n+1)/2 \rfloor + 1))_{\mathbb{Z}_q}\) code.
Absolutely maximally entangled (AME) state: A state on \(n\) subsystems is \(d\)-uniform [1,2] (a.k.a. \(d\)-undetermined [3] or \(d\)-maximally mixed [4]) if all reduced density matrices on up to \(d\) subsystems are maximally mixed. A \(K\)-dimensional subspace of \(d-1\)-uniform states of \(q\)-dimensional subsystems is equivalent to a pure \(((n,K,d))_q\) code [5,6]. An AME state (a.k.a. maximally multi-partite entangled state [7,8]) is a \(\lfloor n/2 \rfloor\)-uniform state, corresponding to a pure \(((n,1,\lfloor n/2 \rfloor + 1))_{\mathbb{Z}_q}\) code. The rank-\(n\) tensor formed by the encoding isometry of such codes is a perfect tensor (a.k.a. multi-unitary tensor), meaning that it is proportional to an isometry for any bipartition of its indices into a set \(A\) and a complementary set \(A^{\perp}\) such that \(|A|\leq|A^{\perp}|\).
Stabilizer Galois-qudit perfect-tensor codes can be converted to AME states via established shortening/lengthening procedures [10][9; Table 1]. For example, an \([[n,0,d]]\) AME state can be converted into an \([[n-1,1,d-1]]\) perfect-tensor code by tracing over one qubit [11; Sec. 3.5].
Notes
Parent
- Planar-perfect-tensor code — Planar-perfect tensors are automatically planar-perfect tensors.
Children
- \(((3,6,2))_{\mathbb{Z}_6}\) Euler code — The \(((3,6,2))_{\mathbb{Z}_6}\) Euler code is an example of a non-stabilizer perfect-tensor code [13].
- \([[5,1,3]]_{\mathbb{Z}_q}\) modular-qudit code — The \([[5,1,3]]_{\mathbb{Z}_q}\) code is a perfect-tensor code because it stems from the \([[6,0,4]]_{\mathbb{Z}_q}\) AME state [14; Thm. 13].
- Three-qutrit code — The three-qutrit code stems from the \([[4,0,3]]_3\) AME state [15–17].
Cousins
- Quantum maximum-distance-separable (MDS) code — AME states for even \(n\) are examples of quantum MDS codes with no logical qubits [1,18]. A family of conjectured perfect-tensor codes is quantum MDS [19].
- Maximum distance separable (MDS) code — MDS codes can be used to obtain perfect-tensor codes with minimal support [16,18,20].
- Combinatorial design — Combinatorial designs and \(d\)-uniform quantum states are related [16].
- Orthogonal array (OA) — Orthogonal arrays and \(d\)-uniform quantum states are related [2,21].
- Modular-qudit cluster-state code — Since any modular-qubit stabilizer code is equivalent to a graph quantum code for \(G=\mathbb{Z}_q\) via a single-modular-qudit Clifford circuit [22] (see also [23,24]), stabilizer AME states can be understood as modular-qudit cluster states [15].
- Galois-qudit GRS code — GRS codes can yield perfect tensors via a generalized Hermitian construction [25,26].
- Hermitian qubit code — The sole codeword of some \([[n,0,d]]\) Hermitian codes is an AME state [17].
- Pastawski-Yoshida-Harlow-Preskill (HaPPY) code — The encoding of a HaPPy code is a holographic tensor network consisting of pentagon and hexagon perfect tensors.
References
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Page edit log
- Victor V. Albert (2024-07-01) — most recent
Cite as:
“Perfect-tensor code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/ame