[Paper Review] Transversal switching between generic stabilizer codes

Cupjin Huang, Michael Newman|arXiv (Cornell University)|Sep 26, 2017

Quantum Computing Algorithms and Architecture14 references3 citations

TL;DR

This paper introduces a randomized stabilizer rewiring algorithm (rSRA) that constructs transversal circuits for fault-tolerant code switching between any two stabilizer codes while preserving code distance throughout intermediate steps. It proves that such a path exists with at most linear overhead in distance, using O(d log(n/d) + log(1/ε)) ancilla qubits with high probability, enabling universal, distance-preserving code transformations critical for fault-tolerant quantum computation.

ABSTRACT

We propose a randomized variant of the stabilizer rewiring algorithm (SRA), a method for constructing a transversal circuit mapping between any pair of stabilizer codes. As gates along this circuit are applied, the initial code is deformed through a series of intermediate codes before reaching the final code. With this randomized variant, we show that there always exists a path of deformations which preserves the code distance throughout the circuit, while using at most linear overhead in the distance. Furthermore, we show that a random path will almost always suffice, and discuss prospects for implementing general fault-tolerant code switching circuits.

Motivation & Objective

• To address the challenge of fault-tolerantly transforming logical information between arbitrary stabilizer codes without exposing it to errors during conversion.
• To overcome the limitation of prior stabilizer rewiring algorithms (SRA), which may produce intermediate codes with low distance, compromising fault tolerance.
• To demonstrate that a randomized variant of SRA (rSRA) can reliably generate distance-preserving paths between codes with high probability.
• To provide a universal upper bound on ancilla qubit overhead for transversal, distance-preserving code transformations.
• To lay the groundwork for practical implementation of general fault-tolerant code switching circuits in quantum computing.

Proposed method

• Proposes a randomized variant of the stabilizer rewiring algorithm (rSRA) that selects random intermediate stabilizer generators to deform one code into another via transversal gates.
• Uses a probabilistic construction to ensure that intermediate codes maintain distance at least d with high probability, leveraging concentration bounds on Pauli weight distributions.
• Employs a commutativity matrix H derived from generator matrices to verify that intermediate codes remain valid stabilizer codes with full rank.
• Applies KL-divergence analysis to bound failure probability and derive overhead scaling: O(d log(n/d) + log(1/ε)) ancilla qubits.
• Introduces a framework for constructing transversal circuits that deform the initial code through a sequence of intermediate codes, each with distance ≥ d.
• Uses Shor-style measurement circuits for fault-tolerant verification of intermediate code states in the implementation schematic.

Experimental results

Research questions

• RQ1 Can a transversal circuit be constructed between any two stabilizer codes such that all intermediate codes preserve the minimum distance d?
• RQ2 What is the minimal overhead (in terms of ancilla qubits) required to ensure such a distance-preserving path exists with high probability?
• RQ3 Is there a randomized algorithm that can efficiently find such a path without relying on specific code presentations?
• RQ4 Can the rSRA be used as a general schema to search for fault-tolerant code switching circuits for small codes?
• RQ5 What is the relationship between code distance preservation and fault tolerance in transversal code switching?

Key findings

• The rSRA guarantees the existence of a transversal circuit mapping between any two [[n,k,d]] stabilizer codes, where all intermediate codes have distance at least d.
• With high probability (1−ε), the rSRA constructs such a path using O(d log(n/d) + log(1/ε)) ancilla qubits.
• The algorithm ensures that the number of intermediate codes is bounded and the transformation remains transversal throughout, preserving logical information.
• The method provides a universal upper bound on ancilla overhead for distance-preserving code switching, independent of code structure.
• The construction is robust: even when the initial SRA fails due to low-distance intermediates, the randomized variant succeeds with high probability.
• Small examples demonstrate the method's feasibility, including a path between the [[5,1,3]] and [[7,1,3]] codes that protects against erasure with no overhead.