Quantum Self-Correction in the 3D Cubic Code Model

A big open question in the quantum information theory concerns the feasibility of a self-correcting quantum memory. A quantum state recorded in such memory can be stored reliably for a macroscopic time without need for active error correction, if the memory is in contact with a cold enough thermal bath. Here we report analytic and numerical evidence for self-correcting behavior in the quantum spin lattice model known as the 3D cubic code. We prove that its memory time is at least $L^{c\beta }$, where $L$ is the lattice size, $\beta$ is the inverse temperature of the bath, and $c>0$ is a constant coefficient. However, this bound applies only if the lattice size $L$ does not exceed a critical value which grows exponentially with $\beta$. In that sense, the model can be called a partially self-correcting memory. We also report a Monte Carlo simulation indicating that our analytic bounds on the memory time are tight up to constant coefficients. To model the readout step we introduce a new decoding algorithm, which can be implemented efficiently for any topological stabilizer code. A longer version of this work can be found in Bravyi and Haah, arXiv:1112.3252.

Figure 1

Stabilizer generators of the 3D cubic code. Here $X\equiv {\sigma }^x$ and $Z\equiv {\sigma }^x$ represent single-qubit Pauli operators, while $I$ is the identity operator. Double-letter indices represent two-qubit Pauli operators, for example, $IZ\equiv I\otimes Z$, $ZZ\equiv Z\otimes Z$, $II\equiv I\otimes I$, etc.

Figure 2

The memory time $T_{\mathrm{mem}}$ versus the system size $L$. The data for $\beta =4.3$, 4.5, 4.7, 4.9, 5.1, 5.25 are shown (starting from the bottom). The left bottom plot shows the exponent of the power law fit of $T_{\mathrm{mem}}$ for the first few system sizes. It indicates that $T_{\mathrm{mem}}\propto L^{2.93\beta -10.5}$ for $L<L^{\star }$, where $L^{\star }$ is the optimal system size where $T_{\mathrm{mem}}$ reaches maximum.

Figure 3

The maximum memory time $T_{\mathrm{mem}}$ versus the inverse temperature $\beta$. The memory time is maximized with respect to the system size. The logarithm of $T_{\mathrm{mem}}$ clearly follows a quadratic relation with $\beta$ as opposed to a linear one (inset).