Hierarchical quantum decoders

TL;DR

This work proposes a family of hierarchical quantum decoders with a tunable trade-off between speed and accuracy while retaining guarantees of optimality, and uses the Lasserre Sum-of-Squares hierarchy from optimization theory to relax the decoding problem.

Abstract

Decoders are a critical component of fault-tolerant quantum computing. They must identify errors based on syndrome measurements to correct quantum states. While finding the optimal correction is NP-hard and thus extremely difficult, approximate decoders with faster runtime often rely on uncontrolled heuristics. In this work, we propose a family of hierarchical quantum decoders with a tunable trade-off between speed and accuracy while retaining guarantees of optimality. We use the Lasserre Sum-of-Squares (SOS) hierarchy from optimization theory to relax the decoding problem. This approach creates a sequence of Semidefinite Programs (SDPs). Lower levels of the hierarchy are faster but approximate, while higher levels are slower but more accurate. We demonstrate that even low levels of this hierarchy significantly outperform standard Linear Programming relaxations. Our results on rotated surface codes and honeycomb color codes show that the SOS decoder approaches the performance of exact decoding. We find that Levels 2 and 3 of our hierarchy perform nearly as well as the exact solver. We analyze the convergence using rank-loop criteria and compare the method against other relaxation schemes. This work bridges the gap between fast heuristics and rigorous optimal decoding.

Figures (5)

• Figure 1: Comparison of different SOS levels for decoding of surface code. We show logical error $p_\text{L}$ against physical error $p$ for rotated surface code for different types of decoders, where we fix distance $d=7$. We show no decoding at all, SOS decoder with level $\ell=1,\dots,4$ and the MIP decoder.
• Figure 2: Decoding rotated surface code. We show logical error $p_\text{L}$ against physical error $p$ for rotated surface code with distance $d\in\{2,5,7,9\}$. We show SOS hierarchy a) level 1, b) level 2, c) level 3 and d) MIP decoder. Dots are simulated data, while curves are fit with \ref{['eq:fit_thr']}. Vertical dashed line shows the numerically fit of threshold $p_\text{th}=\{\text{N/A}, 0.083, 0.085, 0.093\}$.
• Figure 3: Decoding of color code. We show logical error $p_\text{L}$ against physical error $p$ for different code distances $d$. We show SOS hierarchy a) level 1, b) level 2, c) level 3 and d) MIP decoder. Vertical dashed line shows the numerically fit of threshold $p_\text{th}=\{\text{N/A}, 0.095, 0.098, 0.103\}$.
• Figure 4: Comparison of different SOS levels for decoding of color code. We show logical error $p_\text{L}$ against physical error $p$ for rotated surface code for different types of decoders, where we fix distance $d=7$. We show no decoding at all (blue curve), SOS decoder with level $\ell=1,\dots,3$ (orange, green, yellow, respectively) and the MIP decoder (red).
• Figure 5: Relationship of $\ell$-th level of $SA^{(\ell)}$ (Sheralli-Adams), $\ LS^{(\ell)}$, $\ LS_+^{(\ell)}$ (Lovasz-Schrijver), and $\ Las^{(\ell)}$ (Lasserre) relaxations, where the direction of arrows denotes tighter relaxations chlamtac2012convex.

Theorems & Definitions (2)

• Definition 1: Lasserre hierarchy
• Definition 2: Protection matrix