Self-correcting quantum code[1][2]


Also called a self-correcting quantum memory or thermally stable encoding. A ground-state encoding of an \(n\)-body geometrically local Hamiltonian whose logical information is recoverable for arbitrary long times in the \(n\to\infty\) limit after interaction with a sufficiently cold thermal environment. Typically, one also requires a decoder whose decoding time scales polynomially with \(n\) and a finite energy density.

The effect of a Markovian thermal environment consists of a Lindbladian in Davies form admitting a Gibbs steady state at some temperature \(T\) [3]. To test whether a system is self-correcting, an initial codeword \(\rho(0)\) is evolved under the Davies Lindbladian and the code Hamiltonian (or code Lindbladian) to the state \(\rho(t)\) at time \(t\), after which it is decoded via decoding map \(\cal{D}\). The memory time \(\tau\) is defined to be \begin{align} \tau=\sup\left\{ t>0\,|\left\Vert {\cal D}(\rho(t))-\rho(0)\right\Vert _{1}<\epsilon\right\} \end{align} for some fixed \(\epsilon\). For a self-correcting memory, there exists a critical temperature \(T_\star>0\) such that \(\tau\to\infty\) (typically, exponentially with \(n\)) as \(n\to\infty\) for any temperature \(T<T_{\star}\) and any codeword \(\rho(0)\). A memory is partially self-correcting if \(\tau\) scales polynomially with \(n\) up to some cutoff \(n_{max}\). A self-correcting memory is typically associated with a (stable) phase of quantum matter.


Self-correcting classical memories exist in two and higher dimensions, with the canonical example being the classical Ising model. In that model, a classical bit is stored in the overall magnetization. The magnetization is thermally stable due to the fact that there is an \(n\)-dependent (i.e., macroscopic) energy cost of flipping a contiguous region of physical bits [4][3]. This cost scales with the surface area of the region, and the surface area is \(n\)-dependent for dimensions greater than one.

Self-correcting quantum memories currently exist in four and higher dimensions, with their existence in three dimensions being an open question. For similar reasons as the classical Ising model, the four-dimensional toric code is a self-correcting quantum memory due to an order \(O(n)\) energy cost of creating a logical error [1][2]. On the other hand, the 2D toric code is not thermally stable because its string-like logical operators anti-commite with stabilizer generators supported only at their ends, and thus have a constant energy cost of creation.

An \(n\)-dependent energy barrier to all logical errors is likely necessary for a thermally stable memory, having been shown as such for a large class of 2D topological phases [5][6][7]. Two-dimensional stabilizer codes [8] and encodings of frustration-free code Hamiltonians [9] admit only constant-energy excitations, and so do not have admit such a barrier. No-go theorems for 3D models are much more restrictive, and there exist several candidates for self-correction as well as several partially self-correcting memories (see cousins below).




  • Translationally-invariant stabilizer code — 3D translationally-invariant qubit stabilizer code families with constant \(k\) support logical string operators and thus cannot be self-correcting [10]. For non-constant \(k\), such families can support at most a logarithmic energy barrier [11].
  • Higher-dimensional surface code — The 4D toric code is a self-correcting quantum memory [1][2].
  • Solid code — The 3D welded solid code is partially self-correcting with a power-law energy barrier [12].
  • Color code — The 6D color code is a self-correcting quantum memory [13].
  • Haah cubic code — Cubic code 1 is partially self-correcting with a logarithmic energy barrier [14].
  • Quantum repetition code — The bit-flip repetition code associated with the 2D classical Ising model is a self-correcting classical memory.

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Cite as:
“Self-correcting quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
@incollection{eczoo_self_correct, title={Self-correcting quantum code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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Angelo Lucia, David Pérez-García, and Antonio Pérez-Hernández, “Thermalization in Kitaev's quantum double models via Tensor Network techniques”. 2107.01628
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B. Yoshida, “Feasibility of self-correcting quantum memory and thermal stability of topological order”, Annals of Physics 326, 2566 (2011). DOI; 1103.1885
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H. Bombin et al., “Self-Correcting Quantum Computers”. 0907.5228
S. Bravyi and J. Haah, “Quantum Self-Correction in the 3D Cubic Code Model”, Physical Review Letters 111, (2013). DOI; 1112.3252

Cite as:

“Self-correcting quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.