Three-dimensional self-correcting quantum memory

Problem

Does there exist a passive self-correcting quantum memory in three spatial dimensions? Consider finite-dimensional spins on a three-dimensional lattice $\Lambda_L$ of linear size $L$ and a Hamiltonian $H_L=\sum_{X\subseteq\Lambda_L}h_X$ whose interaction range, local strength $\lVert h_X\rVert$, and number of terms incident on each spin are bounded independently of $L$. Let its ground space $\mathcal C_L$ encode a logical space $\mathcal Q_L$ with $\dim\mathcal Q_L\ge2$, and let $\mathcal V_L:\mathcal S(\mathcal Q_L)\to\mathcal S(\mathcal C_L)$ be the channel induced by an isometric encoding into $\mathcal C_L$.

During storage no active control or error correction is permitted. The encoded state evolves under a local thermal channel $\mathcal E_{t,\beta}^{(L)}$ at inverse temperature $\beta$, such as a Davies semigroup, followed only at readout by a decoder $\mathcal D_L$. For a fixed $0<\varepsilon<1$, define the worst-case storage time by

\begin{equation} \tau_L(\beta,\varepsilon) :=\sup\!\left\{t\ge0: \sup_{0\le s\le t}\ \sup_{\rho\in\mathcal S(\mathcal Q_L)} \left\| \bigl(\mathcal D_L\circ\mathcal E_{s,\beta}^{(L)} \circ\mathcal V_L\bigr)(\rho)-\rho \right\|_1 \le\varepsilon\right\}, \tag{1} \end{equation}

where $\mathcal S(\mathcal Q_L)$ is the set of logical states. The problem is to construct such a Hamiltonian family with efficient decoders and a finite critical inverse temperature $\beta_c$ for which

\begin{equation} \beta>\beta_c \quad\Longrightarrow\quad \lim_{L\to\infty}\tau_L(\beta,\varepsilon)=\infty. \tag{2} \end{equation}

Equation (2) requires the lifetime in Eq. (1) to diverge at fixed nonzero temperature as $L\to\infty$; growth that terminates at a temperature-dependent system size is only partial self-correction.

Source

Brown et al. explicitly identify the construction of a self-correcting quantum memory in three spatial dimensions as an outstanding open problem [BLP+16].

Progress

Self-correcting quantum memories are known from four-dimensional local models. The absence of a three-dimensional construction, and the standard thermal-memory framework against which candidates were assessed, were summarized by Brown et al. [BLP+16].
Haah’s three-dimensional cubic code has no string-like logical operators and possesses a logarithmic energy barrier. Under Davies dynamics, Bravyi and Haah proved a low-temperature lifetime bound of the form

\begin{equation} T_{\mathrm{mem}}\ge L^{c\beta} \quad\text{for some $c>0$, but only when}\quad L\le L^\star\sim e^{\beta/3}. \tag{3} \end{equation}

Because the admissible size in Eq. (3) is bounded at fixed $\beta$, the cubic code is partially self-correcting but does not satisfy Eq. (2) [BH13].
Recent three-dimensional layer codes have polynomial energy barriers and provably exhibit partial self-correction. For suitable families, their memory time grows exponentially with $L$ only up to a temperature-dependent cutoff that is exponential in $\beta$ [Wil25]. Baspin then proved that these codes have a constant free-energy barrier and at most polynomial thermalization time under Davies dynamics, ruling them out as strict self-correcting memories [Bas25].
A 2026 preprint by Balasubramanian, Davydova, and Lin constructs bounded-density, geometrically local three-dimensional CSS stabilizer Hamiltonians encoding one qubit, gives an explicit renormalization decoder, and proves a low-temperature lifetime $T_{\mathrm{mem}}\ge\exp(\Theta(n^\eta))$ for some $\eta>0$ under local Gibbs-stationary Lindbladian dynamics. Its theorem uses the mixed thermal encoding $\rho_L\otimes\rho_{\beta,\mathrm{syndrome}}$, however, rather than the isometric ground-space encoding $\mathcal V_L$ imposed in this problem. It therefore establishes self-correction in the paper’s thermal-encoding model but does not directly prove Eq. (2) as formulated here [BDL26].

Comment

The 2026 construction resolves the long-standing question under a thermal syndrome encoding and provides substantially stronger progress than earlier finite-size candidates. Under the deliberately stricter formulation in this section, the remaining issue is to establish the same thermodynamic lifetime starting from the isometric ground-space encoding $\mathcal V_L$ in Eq. (1), or to prove that the two initialization models are equivalent for this construction.

References

[BLP+16]: B. J. Brown, D. Loss, J. K. Pachos, C. N. Self, and J. R. Wootton, “Quantum Memories at Finite Temperature,” Reviews of Modern Physics 88, 045005 (2016).DOI arXiv
[BH13]: S. Bravyi and J. Haah, “Quantum Self-Correction in the 3D Cubic Code Model,” Physical Review Letters 111, 200501 (2013).DOI arXiv
[Wil25]: D. J. Williamson, “Partial Self-Correction in Layer Codes,” arXiv:2510.09218 (2025).arXiv
[Bas25]: N. Baspin, “The Free Energy Barrier: An Eyring–Polanyi Bound for Stabilizer Hamiltonians, with Applications to Quantum Error Correction,” arXiv preprint (2025).arXiv
[BDL26]: S. Balasubramanian, M. Davydova, and T.-C. Lin, “A Passive Self-Correcting Quantum Memory in Three Dimensions,” arXiv preprint (2026).arXiv

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