Designing fault-tolerant circuits using detector error models

TL;DR

The paper introduces detector error models as a comprehensive framework to certify and design fault-tolerant Clifford circuits across three abstraction levels: syndrome extraction, measurement scheduling, and fault-tolerant logical measurements. It formalizes detectors, detector matrices $D$, measurement syndrome matrices $\boldsymbol{\Omega}$, error vectors $\mathbf{e}$, and observables, and shows how the detector error matrix $H = D\boldsymbol{\Omega}$ encodes the fault-tolerance properties via Tanner graphs. Practical contributions include (i) syndrome extraction circuits for the surface code with enhanced resilience to measurement errors, (ii) systematic color-code measurement schedules including distance-3 and distance-5 constructions, and (iii) a fault-tolerant logical-measurement protocol based on the AGP scheme and a QED-enhanced variant to reduce overhead. The results demonstrate that detector error models enable targeted circuit design under realistic noise (e.g., measurement bias) and offer a path toward more efficient fault-tolerant quantum architectures with improved decoding and reduced resource overhead.

Abstract

Quantum error-correcting codes, such as subspace, subsystem, and Floquet codes, are typically constructed within the stabilizer formalism, which does not fully capture the idea of fault-tolerance needed for practical quantum computing applications. In this work, we explore the remarkably powerful formalism of detector error models, which fully captures fault-tolerance at the circuit level. We introduce the detector error model formalism in a pedagogical manner and provide several examples. Additionally, we apply the formalism to three different levels of abstraction in the engineering cycle of fault-tolerant circuit designs: finding robust syndrome extraction circuits, identifying efficient measurement schedules, and constructing fault-tolerant procedures. We enhance the surface code's resistance to measurement errors, devise short measurement schedules for color codes, and implement a more efficient fault-tolerant method for measuring logical operators.

Figures (6)

• Figure 1: Fault-tolerant quantum computation can be performed using fault-tolerant gadgets, which contain measurement schedules, which contain syndrome extraction circuits. A detector error model contains all information necessary to verify the error-correcting capabilities of Clifford circuits subject to a noise model at these three levels.
• Figure 2: Input errors occur before the QEC procedure and internal errors occur during the procedure. Together they lead to an output error.
• Figure 3: The CSS QEC procedure is split up into two parts, one for correcting $X$ errors and one for correcting $Z$ errors. Inside each of these parts is a measurement schedule, whose measurement outcomes are passed to a decoder, which outputs a correction. a) Measurements are performed using fault-tolerant syndrome extraction circuits. b) Errors happening during the syndrome extraction circuit are equivalent to errors that can happen in the phenomenological noise model.
• Figure 4: Teraquop plots obtained using memory experiments with the SDMB noise model for measurement biases $\eta = 1, 5, 10, 20$.
• Figure 5: Teraquop plots obtained using stability experiments with the SDMB noise model for measurement biases $\eta = 1, 5, 10, 20$.

Theorems & Definitions (54)

• Definition 2.1: Detector
• Example 1: Finding detectors
• Definition 2.2: Detector matrix
• Example 2: Constructing a detector matrix
• Definition 2.3: Errors
• Definition 2.4: Noise model
• Example 3: A circuit's noise model
• Definition 2.5: Circuit-level noise
• Definition 2.6: Circuit error vector
• Definition 2.7: Measurement syndrome matrix