## Description

A block quantum code that forms the ground-state subspace 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 original criteria for a self-correcting quantum memory, informally known as the Caltech rules [3,4], also required finite-spin Hamiltonians.

The effect of a Markovian thermal environment consists of a Lindbladian in Davies form admitting a Gibbs steady state at some temperature \(T\) [5]. To test whether a system is self-correcting, an initial codeword \(\rho(0)\) is evolved under the Davies Lindbladian and the code Hamiltonian (or, if we are to allow extra passive protection, the 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\} \tag*{(1)}\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.

## Protection

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 [5,6]. 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 [7–10] 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. There is a general upper bound on the relaxation rate of a qubit stabilizer or qubit subsystem stabilizer quantum memory interacting with a Markovian environment [11].

An \(n\)-dependent energy barrier to creating all logical errors is likely necessary for a thermally stable memory, having been shown as such for a large class of 2D topological phases [12–15]. Two-dimensional stabilizer codes [16] and encodings of frustration-free code Hamiltonians [17] admit only constant-energy excitations, and so do not have admit such a barrier. No-go theorems for 3D models are much more restrictive [18], e.g., a 3D lattice stabilizer code with a locality-preserving non-Clifford gate cannot have a microscopic energy barrier [19]. There exist several candidates for self-correction as well as several partially self-correcting memories (see cousins below).

## Notes

## Parent

- Symmetry-protected self-correcting quantum code — A self-correcting quantum memory does not require symmetry for self correction.

## Child

- Loop toric code — The 4D loop toric code is a self-correcting quantum memory [1,2].

## Cousins

- 3D lattice stabilizer code — 3D translationally-invariant qubit stabilizer code families with constant \(k\) support logical string operators and thus cannot be self-correcting [21]. For non-constant \(k\), such families can support at most a logarithmic energy barrier [22].
- 3D surface code — The 3D welded surface code is partially self-correcting with a power-law energy barrier [23]. The 3D toric code is a classical self-correcting memory, whose protected bit admits a membrane-like logical operator [4], but it is not a quantum self-correcting memory [24].
- Color code — The 6D color code is a self-correcting quantum memory and admits fault-tolerant universal gate set in 7D [25].
- Haah cubic code (CC) — Cubic code 1 is partially self-correcting with a logarithmic energy barrier [26].
- Quantum repetition code — The bit-flip repetition code associated with the 2D classical Ising model is a self-correcting classical memory [5; Sec. V.A].
- Repetition code — The repetition code associated with the 2D classical Ising model is a self-correcting classical memory [27][5; Sec. V.A].
- Bacon-Shor code — 3D Bacon-Shor codes were conjectured to be self-correcting [28], but there remain issues to be resolved in order to validate this conjecture (see [5; Sec. IX.B]).
- Expander code — Constant-rate random (quantum) expander codes are self-correcting (quantum) memories, but have no thermodynamic phase transitions [29].
- Quantum expander code — Constant-rate random (quantum) expander codes are self-correcting (quantum) memories, but have no thermodynamic phase transitions [29].
- Concatenated cat code — A concatenation of the repetition code with the two-component cat code is a candidate for a memory that may be self-correcting, but only in the limit of infinite energy per mode [30].
- Matrix-model code — Matrix-model codes are similar to self-correcting memories in the sense that memory time becomes infinite in the thermodynamic limit, but with corrections being polynomial in \(N\).
- Single-shot code — The presence of an energy barrier (i.e., confinement) is sufficient for a code to be single shot, and is also conjectured to be necessary for a code to be a self-correcting memory.
- Quantum locally testable code (QLTC) — The notion of an energy barrier in a self-correcting memory is intimately related to the soundness of a QLTC.
- Layer code — The energy barrier for layer-code excitations for codes constructed using asymptotically good QLDPC codes scales as \(\Theta{n^{1/3}}\).
- Cubic theory code — A family of five-dimensional cubic theory codes with non-Abelian excitations is argued to be self-correcting below a critical temperature via a Peierls argument [31].
- Kitaev surface code — Various candidates for self-correcting quantum memories have been constructed by coupling neighboring anyons so as to prevent them from spreading [32–36]
- Fractal surface code — The classical codes underlying the fractal product code form classical self-correcting memories [37–39].
- 3D subsystem surface code — The 3D subsystem surface code is not a self-correcting quantum memory despite being a single-shot code [24].

## References

- [1]
- E. Dennis et al., “Topological quantum memory”, Journal of Mathematical Physics 43, 4452 (2002) arXiv:quant-ph/0110143 DOI
- [2]
- R. Alicki et al., “On thermal stability of topological qubit in Kitaev’s 4D model”, (2008) arXiv:0811.0033
- [3]
- C. G. Brell, “A proposal for self-correcting stabilizer quantum memories in 3 dimensions (or slightly less)”, New Journal of Physics 18, 013050 (2016) arXiv:1411.7046 DOI
- [4]
- O. Landon-Cardinal et al., “Perturbative instability of quantum memory based on effective long-range interactions”, Physical Review A 91, (2015) arXiv:1501.04112 DOI
- [5]
- B. J. Brown et al., “Quantum memories at finite temperature”, Reviews of Modern Physics 88, (2016) arXiv:1411.6643 DOI
- [6]
- R. Peierls, “On Ising’s model of ferromagnetism”, Mathematical Proceedings of the Cambridge Philosophical Society 32, 477 (1936) DOI
- [7]
- R. Alicki, M. Fannes, and M. Horodecki, “A statistical mechanics view on Kitaev’s proposal for quantum memories”, Journal of Physics A: Mathematical and Theoretical 40, 6451 (2007) arXiv:quant-ph/0702102 DOI
- [8]
- Z. Nussinov and G. Ortiz, “Autocorrelations and thermal fragility of anyonic loops in topologically quantum ordered systems”, Physical Review B 77, (2008) arXiv:0709.2717 DOI
- [9]
- R. Alicki, M. Fannes, and M. Horodecki, “On thermalization in Kitaev’s 2D model”, Journal of Physics A: Mathematical and Theoretical 42, 065303 (2009) arXiv:0810.4584 DOI
- [10]
- F. Pastawski et al., “Limitations of Passive Protection of Quantum Information”, (2009) arXiv:0911.3843
- [11]
- S. Chesi et al., “Thermodynamic stability criteria for a quantum memory based on stabilizer and subsystem codes”, New Journal of Physics 12, 025013 (2010) arXiv:0907.2807 DOI
- [12]
- A. Kay and R. Colbeck, “Quantum Self-Correcting Stabilizer Codes”, (2008) arXiv:0810.3557
- [13]
- K. Temme, “Thermalization time bounds for Pauli stabilizer Hamiltonians”, (2016) arXiv:1412.2858
- [14]
- A. Kómár, O. Landon-Cardinal, and K. Temme, “Necessity of an energy barrier for self-correction of Abelian quantum doubles”, Physical Review A 93, (2016) arXiv:1601.01324 DOI
- [15]
- A. Lucia, D. Pérez-García, and A. Pérez-Hernández, “Thermalization in Kitaev’s quantum double models via tensor network techniques”, Forum of Mathematics, Sigma 11, (2023) arXiv:2107.01628 DOI
- [16]
- S. Bravyi and B. Terhal, “A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes”, New Journal of Physics 11, 043029 (2009) arXiv:0810.1983 DOI
- [17]
- O. Landon-Cardinal and D. Poulin, “Local Topological Order Inhibits Thermal Stability in 2D”, Physical Review Letters 110, (2013) arXiv:1209.5750 DOI
- [18]
- S. Bravyi and J. Haah, “Energy Landscape of 3D Spin Hamiltonians with Topological Order”, Physical Review Letters 107, (2011) arXiv:1105.4159 DOI
- [19]
- F. Pastawski and B. Yoshida, “Fault-tolerant logical gates in quantum error-correcting codes”, Physical Review A 91, (2015) arXiv:1408.1720 DOI
- [20]
- J. R. Wootton, “Quantum memories and error correction”, Journal of Modern Optics 59, 1717 (2012) arXiv:1210.3207 DOI
- [21]
- B. Yoshida, “Feasibility of self-correcting quantum memory and thermal stability of topological order”, Annals of Physics 326, 2566 (2011) arXiv:1103.1885 DOI
- [22]
- J. Haah, “Local stabilizer codes in three dimensions without string logical operators”, Physical Review A 83, (2011) arXiv:1101.1962 DOI
- [23]
- K. P. Michnicki, “3D Topological Quantum Memory with a Power-Law Energy Barrier”, Physical Review Letters 113, (2014) arXiv:1406.4227 DOI
- [24]
- Y. Li et al., “Phase diagram of the three-dimensional subsystem toric code”, (2023) arXiv:2305.06389
- [25]
- H. Bombin et al., “Self-Correcting Quantum Computers”, (2012) arXiv:0907.5228
- [26]
- S. Bravyi and J. Haah, “Quantum Self-Correction in the 3D Cubic Code Model”, Physical Review Letters 111, (2013) arXiv:1112.3252 DOI
- [27]
- L. E. Thomas, “Bound on the mass gap for finite volume stochastic ising models at low temperature”, Communications in Mathematical Physics 126, 1 (1989) DOI
- [28]
- D. Bacon, “Operator quantum error-correcting subsystems for self-correcting quantum memories”, Physical Review A 73, (2006) arXiv:quant-ph/0506023 DOI
- [29]
- Y. Hong, J. Guo, and A. Lucas, “Quantum memory at nonzero temperature in a thermodynamically trivial system”, (2024) arXiv:2403.10599
- [30]
- S. Lieu, Y.-J. Liu, and A. V. Gorshkov, “Candidate for a passively protected quantum memory in two dimensions”, (2023) arXiv:2205.09767
- [31]
- P.-S. Hsin, R. Kobayashi, and G. Zhu, “Non-Abelian Self-Correcting Quantum Memory”, (2024) arXiv:2405.11719
- [32]
- A. Hamma, C. Castelnovo, and C. Chamon, “Toric-boson model: Toward a topological quantum memory at finite temperature”, Physical Review B 79, (2009) arXiv:0812.4622 DOI
- [33]
- S. Chesi, B. Röthlisberger, and D. Loss, “Self-correcting quantum memory in a thermal environment”, Physical Review A 82, (2010) arXiv:0908.4264 DOI
- [34]
- C. Stark et al., “Localization of Toric Code Defects”, Physical Review Letters 107, (2011) arXiv:1101.6028 DOI
- [35]
- M. Herold et al., “Cellular-automaton decoders for topological quantum memories”, npj Quantum Information 1, (2015) arXiv:1406.2338 DOI
- [36]
- C.-E. Bardyn and T. Karzig, “Exponential lifetime improvement in topological quantum memories”, Physical Review B 94, (2016) arXiv:1512.04528 DOI
- [37]
- A. Vezzani, “Spontaneous magnetization of the Ising model on the Sierpinski carpet fractal, a rigorous result”, Journal of Physics A: Mathematical and General 36, 1593 (2003) arXiv:cond-mat/0212497 DOI
- [38]
- R. Campari and D. Cassi, “Generalization of the Peierls-Griffiths theorem for the Ising model on graphs”, Physical Review E 81, (2010) arXiv:1002.1227 DOI
- [39]
- M. Shinoda, “Existence of phase transition of percolation on Sierpiński carpet lattices”, Journal of Applied Probability 39, 1 (2002) DOI

## Page edit log

- Victor V. Albert (2022-09-28) — most recent
- Victor V. Albert (2022-05-18)
- Yi-Ting (Rick) Tu (2022-05-02)

## Cite as:

“Self-correcting quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/self_correct