Limits on the storage of quantum information in a volume of space

Abstract

We study the fundamental limits on the reliable storage of quantum information in lattices of qubits by deriving tradeoff bounds for approximate quantum error correcting codes. We introduce a notion of local approximate correctability and code distance, and give a number of equivalent formulations thereof, generalizing various exact error-correction criteria. Our tradeoff bounds relate the number of physical qubitsn, the number of encoded qubitsk, the code distanced, the accuracy parameterδthat quantifies how well the erasure channel can be reversed, and the locality parameterℓthat specifies the length scale at which the recovery operation can be done. In a regime where the recovery is successful to accuracyϵthat is exponentially small inℓ, which is the case for perturbations of local commuting projector codes, our bound readskd2D−1≤O(n(log⁡n)2DD−1)for codes onD-dimensional lattices of Euclidean metric. We also find that the code distance of any local approximate code cannot exceedO(ℓn(D−1)/D)ifδ≤O(ℓn−1/D). As a corollary of our formulation of correctability in terms of logical operator avoidance, we show that the code distancedand the sized~of a minimal region that can support all approximate logical operators satisfiesd~d1D−1≤O(nℓDD−1), where the logical operators are accurate up toO((nδ/d)1/2)in operator norm. Finally, we prove that for two-dimensional systems if logical operators can be approximated by operators supported on constant-width flexible strings, then the dimension of the code space must be bounded. This supports one of the assumptions of algebraic anyon theories, that there exist only finitely many anyon types.

Code Properties from Holographic Geometries

“…The quantum Markov condition provides a criterion for local correctability [28]. We say that the state ρ ABC of three disjoint regions A, B, C obeys the quantum Markov condition (also called quantum conditional independence) if 0 ¼ IðA;CjBÞ ¼ SðABÞ þSðBCÞ − SðABCÞ − SðBÞ; ð65Þ which is equivalent to saying that the strong subadditivity inequality is saturated (satisfied as an equality).…”

Tensor chain and constraints in tensor networks

“…S c is called central set. M t is called top tensor chain, which should be prime and satisfy the condition (34). The equivalence between S and S c require κ(M t ) = max M ∈S κ max (M ), as proved in Theorem (8).…”

“…λ c L 2 > 1, the information in the interior of H c can be transmitted to its surface without loss, where we have restored the AdS radius L. While, for a circle H which is larger than H c , its interior information can not be transmitted to its surface without loss. So we can say that H c is the maximal boundary which can holographically store the interior information [34,35]. Thus, for a tensor network which captures the feature of QEC as AdS space, it must not contain circular CP curves, which requires λ c L 2 ≤ 1.…”

Higher-Dimensional Quantum Hypergraph-Product Codes with Finite Rates

“…Essential advantage of topological codes is locality: each stabilizer generator involves only the qubits in the immediate vicinity of each other; it is this feature that makes planar surface codes so practically attractive. However, locality also limits the parameters of topological codes [30][31][32][33]. In particular, for a code of length n with stabilizer generators local in two dimensions, the number of encoded qubits k and the minimal distance d satisfy the inequality [30] kd 2 ≤ O(n).…”