Partial Self-Correction in Layer Codes

TL;DR

This work shows that a family of Layer Codes, based on good Quantum Tanner Codes, exhibit partial self-correction, and positions Layer Codes as the leading candidate for a partially self-correcting memory in three dimensions.

Abstract

The storage of large-scale quantum information at finite temperature requires an autonomous and reliable quantum hard drive, also known as a self-correcting quantum memory. It is a long-standing open problem to find a self-correcting quantum memory in three dimensions. The recently introduced Layer Codes achieve the best possible scaling of code parameters and logical energy barrier in three dimensions, these are tantalizing features for the purposes of self-correction. In this work we show that a family of Layer Codes, based on good Quantum Tanner Codes, exhibit partial self-correction. Their memory time grows exponentially with linear system size, up to a length scale that is exponential in the inverse temperature. At this length scale, the memory time scales as a double exponential of inverse temperature. To establish this result we introduce a concatenated matching decoder that combines three rounds of parallelized minimum-weight perfect-matching with a decoder for good Quantum Tanner Codes. We show that our decoder corrects errors up to a constant fraction of the energy barrier, and a constant fraction of the code distance, for a family of Layer Codes. Our results position Layer Codes as the leading candidate for a partially self-correcting memory in three dimensions. While they fall short of achieving strict self-correction in the thermodynamic limit, our work highlights the potential of these local codes in three dimensions, with fast distance and logical qubit growth, fast decoders, and a long memory time over a wide range of parameters.

Figures (6)

• Figure 1: The Layer Code based on the [[4,2,2]] input code. The grey $xz$-layers, blue $xy$-layer, and red $yz$-layer, depict surface codes that correspond to physical qubits, the $XXXX$ check, and the $ZZZZ$ check, of the input code, respectively. The blue $x$-oriented, red $z$-oriented, and green $y$-oriented, junction lines correspond to nontrivial topological defects, see Fig. \ref{['fig:Defects']}. The intersection of two layers without a junction line indicates a simple decoupled crossing. The grey and red surface codes have rough e-condensing boundaries on the top and bottom, while the grey and blue surface codes have smooth m-condensing boundaries on the front and back. The dashed line indicates an $X$-type logical operator.
• Figure 2: The line defects in a Layer Code. Here, we depict generators for the syndromes that can be created or destroyed (condensed) locally at the defect. This is dual to a stabilizer description, which can be found in Ref. Williamson2023. The 3-partite line defects where a red (blue) layer ends can be resolved into the above 4-partite red (blue) defect and a rough e-condensing (smooth m-condensing) boundary, see Fig. \ref{['fig:Boundaries']}.
• Figure 3: The 3-partite line defects where a red $yz$ (blue $xy$) layer ends can be resolved into a 4-partite defect from Fig. \ref{['fig:Defects']} and a smooth e-condensing (rough m-condensing) boundary.
• Figure 4: A depiction of an $X$-type error (dashed lines) and a correction operator (dotted lines) on the [[4,2,2]] Layer Code.
• Figure 5: A depiction of the multistage decoder applied to the $X$-type error (dashed lines) on the [[4,2,2]] Layer code in Fig. \ref{['fig:422error']}. The errors and corrections (dotted lines) applied at each step can create further syndromes in the subseqeunt steps due to the properties of the defects in Fig. \ref{['fig:Defects']}. In the first step, the syndromes (endpoints of dashed lines) on the blue layer are matched using rough boundary conditions on all sides. At the second step, the syndromes in the grey layers (including any from the first correction) are matched (blue dotted lines) using standard surface code boundary conditions. In the third step the parity of syndromes (including any from previous corrections) on the red layer is computed and the input decoder is run on this effective syndrome. The correction returned by the input decoder is applied by flipping the logical sector of the matching on the layers that correspond to qubits flipped by the correction operator. In this example the red layer has negative parity, and so a correction operator $X_0$ is returned. This flips the matching on the leftmost layer (purple dotted lines) to restore even parity on the red layer. In the fourth, final, step, the syndromes on the red layer (including those from previous corrections) are matched assuming smooth boundary conditions (dotted lines).

Theorems & Definitions (7)

• Theorem 1: Partial Self-Correction in Layer Codes
• proof
• Corollary 1: Layer Code memory time
• Lemma 1: Concatenated matching decoder energy barrier
• proof
• Lemma 2: Concatenated matching decoder distance
• proof