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arXiv:1610.06169 · Quantum

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

Steven T. Flammia, Jeongwan Haah, Michael J. Kastoryano, Isaac H. Kim

Published Apr 25, 2017• 36 citations

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 qubits n , the number of encoded qubits k , the code distance d , the accuracy parameter δ that quantifies how well the erasure channel can be reversed, and the locality parameter ℓ that specifies the length scale at which the recovery operation can be done. In a regime where the recovery is successful to accuracy δ that is exponentially small in ℓ , which is the case for perturbations of local commuting projector codes, our bound reads kd2D−1≤O(n(log⁡n)2DD−1) for codes on D -dimensional lattices of Euclidean metric. We also find that the code distance of any local approximate code cannot exceed O(ℓn(D−1)/D) if δ≤O(ℓn−1/D) . As a corollary of our formulation of correctability in terms of logical operator avoidance, we show that the code distance d and the size d~ of a minimal region that can support all approximate logical operators satisfies d~d1D−1≤O(nℓDD−1) , where the logical operators are accurate up to O((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.

DOI: 10.22331/q-2017-04-25-4

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We ask whether there are fundamental limits on storing quantum information reliably in a bounded volume of space. To investigate this question, we study quantum error correcting codes specified by geometrically local commuting constraints on a 2D lattice of finite-dimensional quantum particles. For these 2D systems, we derive a tradeoff between the number of encoded qubits k, the distance of the code d, and the number of particles n. It is shown that kd{2}=O(n) where the coefficient in O(n) depends only on the locality of the constraints and dimension of the Hilbert spaces describing individual particles. The analogous tradeoff for the classical information storage is k sqrt[d]=O(n).

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We study the structure of logical operators in local

D

-dimensional quantum codes, considering both subsystem codes with geometrically local gauge generators and codes defined by geometrically local commuting projectors. We show that if the code distance is

d

, then any logical operator can be supported on a set of specified geometry containing $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{d}$ qubits, where $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{d}{d}^{1/(D\ensuremath{-}1)}=O(n)$ and

n

is the code length. Our results place limitations on partially self-correcting quantum memories, in which at least some logical operators are protected by energy barriers that grow with system size. We also show that for any two-dimensional local commuting projector code there is a nontrivial logical ``string'' operator supported on a narrow strip, where the operator is only slightly entangling across any cut through the strip.

The following open problems, which concern a fundamental limit on coding properties of quantum codes with realistic physical constraints, are analyzed and partially answered here: (a) the upper bound on code distances of quantum error-correcting codes with geometrically local generators, (b) the feasibility of a self-correcting quantum memory. To investigate these problems, we study stabilizer codes supported by local interaction terms with translation and scale symmetries on a

D

-dimensional lattice. Our analysis uses the notion of topology emerging in geometric shapes of logical operators, which sheds a surprising new light on theory of quantum codes with physical constraints.

Topologically ordered phases naturally realize quantum error correction through nonlocal encoding of quantum information. More recently, conformal field theories have been shown to realize approximate quantum error-correcting codes, but such constructions generally require fine tuning to criticality. Here we introduce a family of approximate quantum error-correcting codes realized by the chiral edges of two-dimensional topologically ordered phases. The proposed encoding combines the robustness of a gapped topological bulk with the flexibility of gapless edge conformal field theories. To characterize its robustness, we study coherent-information loss under local erasure. We derive an exact expression relating coherent-information loss to relative entropy, reducing the recoverability problem to universal properties of the edge theory. This leads to power-law scaling of coherent-information loss with the size of the erased region. We further show that, for geometrically local erasures near one edge, the two-dimensional chiral edge code is at least as robust as the dimensionally reduced CFT code, and is strictly more robust in several representative examples. For Abelian code subspaces, we further construct a power-law-range recovery map supported on the erased region together with a power-law-range buffer; this recovery map depends only on the code subspace, not on the unknown encoded state. We provide numerical calculations for lattice realizations of compact free boson and Ising CFT examples that support the theoretical predictions of the power-law exponents.