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Quantum

Designing fault-tolerant circuits using detector error models

Peter-Jan H. S. Derks, Alex Townsend-Teague, Ansgar G. Burchards, Jens Eisert

Published Nov 6, 2025• 11 citations
Publisher

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.

DOI
10.22331/q-2025-11-06-1905

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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.
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.
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.
In fault-tolerant quantum computing, quantum algorithms are implemented through quantum circuits capable of error correction. These circuits are typically constructed based on specific quantum error correction codes, with consideration given to the characteristics of the underlying physical platforms. Optimizing these circuits within the constraints of today's quantum computing technologies, particularly in terms of error rates, qubit counts, and network topologies, holds substantial implications for the feasibility of quantum applications in the near future. This paper presents a toolkit for designing and analyzing fault-tolerant quantum circuits. We introduce a framework for representing stabilizer circuits using classical low-density parity-check (LDPC) codes. Each codeword in the representation corresponds to a quantum-mechanical equation regarding the circuit, formalizing the correlations utilized in parity checks and delineating logical operations within the circuit. Consequently, the LDPC code provides a means of quantifying fault tolerance and verifying logical operations. We outline the procedure for generating LDPC codes from circuits using the Tanner graph notation, alongside proposing graph-theory tools for constructing fault-tolerant quantum circuits from classical LDPC codes. These findings offer a systematic approach to applying classical error correction techniques in optimizing existing fault-tolerant protocols and developing new ones. As an example, we develop a resource-efficient scheme for universal fault-tolerant quantum computing on hypergraph product codes based on the LDPC representation.
Stabilizer channels are stabilizer circuits that implement logical operations while mapping from an input stabilizer code to an output stabilizer code. They are widely used to implement fault tolerant error correction and logical operations in stabilizer codes such as surface codes and LDPC codes, and more broadly in subsystem, Floquet and space-time codes. We introduce a rigorous and general formalism to analyze the fault tolerance properties of any stabilizer channel under a broad class of noise models. This includes rigorous but easy-to-work-with definitions and algorithms for the fault distance and hook faults for stabilizer channels. The generalized notion of hook faults which we introduce, defined with respect to an arbitrary subset of a circuit's faults rather than a fixed phenomenological noise model, can be leveraged for fault-tolerant circuit design. Additionally, we establish necessary conditions such that channel composition preserves the fault distance. We apply our framework to design and analyze fault tolerant stabilizer channels for surface codes, revealing novel aspects of fault tolerant circuits.
Quantum computation promises asymptotic speedups over classical algorithms, but realizing these advantages requires overcoming the noisiness of quantum hardware. Although fault-tolerant error correction can allow for reliable quantum computation even using unreliable components, the resource overheads required can substantially erode the asymptotic performance gains. This dissertation focuses on constructing and optimizing fault-tolerant procedures with lower overhead than previous methods were capable of. We present new frameworks for fault-tolerant syndrome extraction using flag gadgets with exponentially reduced ancilla requirements, explicit measurement schedules that achieve asymptotically fewer measurements than stabilizer generators, and a general method for making arbitrary Clifford circuits fault tolerant. In addition, we optimize surface code measurement schedules in the presence of constraints imposed by fabrication defects to reduce logical error rate. Together, these results advance low-overhead fault tolerance and improve performance in resource-constrained and near-term quantum computing settings.
Floquet quantum error-correcting codes provide an operationally economical route to fault tolerance by dynamically generating stabilizer structures using only two-body Pauli measurements. But while it is well established that stabilizer codes in higher spatial dimensions gain additional levels of intrinsic robustness, higher-dimensional Floquet codes have hitherto been explored only in limited scope. Here we introduce a 3d generalization of a Floquet code whose instantaneous stabilizer group realizes a 3d fermionic toric code, while crucially preserving all three logical qubits throughout the entire measurement sequence. One central ingredient is the identification of a 3d lattice geometry that generalizes the features of the Kekulé lattice underlying the 2d Hastings-Haah code - specifically, a structure where deleting any one edge color yields a two-color subgraph that decomposes into short, closed loops rather than homologically nontrivial chains. This loop property avoids the collapse of logical information that plagues naive sequential two-color measurement schedules on many 3d lattices. Although, for our lattice geometry, a simple 3-round cycle that sequentially measures the three types of parity checks does not expose the full error syndrome set, we show that one can append a measurement sequence to extract the missing syndromes without disturbing the logical subspace. Beyond code design, 3d tricoordinated lattice geometries define a family of 3d monitored Kitaev models, in which random measurements of the non-commuting parity checks give rise to dynamically created entangled phases with nontrivial topology. In discussing the general structure of their underlying phase diagrams and, in particular, the existence of certain quantum critical points, we again make a connection to the general preservation of logical information in time-ordered Floquet protocols.