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Adaptive Loss-tolerant Syndrome Measurements

Yuanjia Wang, Todd A. Brun

Abstract

In the presence of qubit losses, the building blocks of fault-tolerant error correction (FTEC) must be revisited. Existing loss-tolerant approaches are mainly architecture-specific, and little attention has been given to optimizing the syndrome measurement sequences under loss. Schemes designed for the standard Pauli error model are not directly applicable because the syndrome patterns differ when both Pauli errors and erasures can occur. Based on recent advances in loss detection units and loss-tolerant syndrome extraction gadgets, we extend the study of adaptive Shor-style measurement sequences to the mixed error model. We begin by discussing how to adaptively convert correctable erasures into located errors. The minimal overhead is quantified by the number of stabilizer measurements, which can be reduced to a subgroup dimension problem for erasures arising in any FTEC circuit for qubits and prime-dimensional qudits. As a byproduct, we provide the construction of the canonical generating set with respect to a given bipartite partition for a stabilizer group on qudits of composite dimension. We then generalize both the weak and strong FTEC conditions. Finally, we present adaptive syndrome-measurement protocols for the mixed error model, generalizing the adaptive protocols for the standard Pauli error model.

Adaptive Loss-tolerant Syndrome Measurements

Abstract

In the presence of qubit losses, the building blocks of fault-tolerant error correction (FTEC) must be revisited. Existing loss-tolerant approaches are mainly architecture-specific, and little attention has been given to optimizing the syndrome measurement sequences under loss. Schemes designed for the standard Pauli error model are not directly applicable because the syndrome patterns differ when both Pauli errors and erasures can occur. Based on recent advances in loss detection units and loss-tolerant syndrome extraction gadgets, we extend the study of adaptive Shor-style measurement sequences to the mixed error model. We begin by discussing how to adaptively convert correctable erasures into located errors. The minimal overhead is quantified by the number of stabilizer measurements, which can be reduced to a subgroup dimension problem for erasures arising in any FTEC circuit for qubits and prime-dimensional qudits. As a byproduct, we provide the construction of the canonical generating set with respect to a given bipartite partition for a stabilizer group on qudits of composite dimension. We then generalize both the weak and strong FTEC conditions. Finally, we present adaptive syndrome-measurement protocols for the mixed error model, generalizing the adaptive protocols for the standard Pauli error model.
Paper Structure (45 sections, 11 theorems, 62 equations, 3 figures, 1 algorithm)

This paper contains 45 sections, 11 theorems, 62 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related work
  3. Adaptive erasure error correction.
  4. Mixed error correctability.
  5. Shor-style sequences.
  6. Preliminaries
  7. Stabilizer states and stabilizer codes
  8. Quantum error correction and fault tolerance
  9. Losses, erasures and loss detection
  10. Standard LDUs.
  11. Teleportation-based LDUs.
  12. Adaptive erasure error "correction"
  13. Reduction to a subgroup-dimension problem for prime qudits
  14. Canonical form of bipartite stabilizer states for qudits of composite dimension
  15. Step 1. Construct the nonlocal pairs from $\mathcal{S}|_A$.
  16. ...and 30 more sections

Key Result

Lemma 3

Let $S=\{s_0,s_1,\dots,s_{k-1}\}\subseteq\overline{\mathcal{P}}_n$ and let $A_{ij}=\llbracket s_i,s_j\rrbracket_d$ define a matrix $A\in\mathbb{Z}_d^{k\times k}$. The group $G:=\langle S\rangle$ has a Gram-Schmidt generating set $S_1\cup S_2\cup U$ where $|S_1|=|S_2|=\Theta(M_A)/2$, $M_A$ is the sub

Figures (3)

  • Figure 1: The circuit representation for the standard loss detection unit and the teleportation-based loss detection unit perrin2025quantumerrorcorrectionresilient.
  • Figure 2: The circuit representation for the qudit teleportation-based loss detection unit.
  • Figure 3: Three possible loss scenarios in a teleportation-based LDU (TLDU). The red asterisks indicate qudit losses. All subsequent operations acting on a lost qudit, including single- and two-qudit gates, have no effect until the qudit is replaced by a fresh ancilla. A measurement performed on a lost qudit returns a loss symbol rather than a classical bit, and has no effect on the lost qudit. In Case $2$, by incorrectly measuring the older data qudit, a random and useless outcome $a$ is extracted.

Theorems & Definitions (33)

  • Definition 1
  • Definition 2: The canonical form of generators for a stabilizer group with respect to a given bipartition fattal2004entanglement
  • Lemma 3: Lemma 6.11 in sarkar2024qudit
  • Lemma 4: The cleaning lemma bravyi2009no
  • Definition 5
  • Definition 6: Strong FTEC conditions tansuwannont2023adaptive
  • Definition 7: Weak conditions for fault-tolerant error correction tansuwannont2023adaptive
  • Definition 8
  • Theorem 9: The equivalence between the Pauli error correctability and the mixed error correctability haselgrove2006trade
  • Theorem 10
  • ...and 23 more