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’s Theorem 1 gives, for lattice sizes encoding two qubits and every fixed \(0<a<1\), a lifetime bound of the form

\begin{equation} T_{\mathrm{mem}}\ge C_{\varepsilon}L^{ac\beta-3} \qquad\text{when}\qquad L\le e^{(1-a)\beta/3}, \tag{3} \end{equation}

where \(c>0\) and \(C_{\varepsilon}>0\) are independent of \(L\) and \(\beta\). The bound in Eq. (3) establishes partial self-correction at low temperature. Its admissible size is bounded at fixed \(\beta\), so this bound does not establish the thermodynamic divergence in 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’s Theorem 3 bounds the Davies mixing time by \(O(\beta\lVert H\rVert_{\infty}n e^{\beta\bar f})\), where \(n\) is the number of spins and \(\bar f\) is the free-energy barrier. For bounded local terms with \(\lVert H\rVert_{\infty}=O(n)\) and the constant barrier established for layer codes in Section V, this is \(O(\beta n^2e^{\beta\bar f})\). It excludes superpolynomial storage times at fixed temperature, but does not by itself exclude the merely diverging lifetime required by Eq. (2) [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. Theorem 1.1 and Eq. (49) use 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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