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  • Open Access

No-Go Theorem on Fault Tolerant Gadgets for Multiple Logical Qubits

Aranya Chakraborty1,* and Daniel Gottesman2,†

  • *Contact author: aranyac@umd.edu
  • †Contact author: dgottesm@umd.edu
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PRX Quantum 7, 033027 – Published 13 August, 2026

DOI: https://doi.org/10.1103/y14y-7kp3

Abstract

Identifying stabilizer codes that admit fault tolerant implementations of the full logical Clifford group would significantly advance fault tolerant quantum computation, as such implementations prevent uncontrolled error propagation. Motivated by this goal, we study several classes of fault tolerant gadget constructions consisting of Clifford gates acting on the physical qubits, including transversal gadgets, code automorphisms, and fold-transversal gadgets. While stabilizer codes encoding a single logical qubit—most notably the [[7,1,3]] Steane code—are known to admit transversal implementations of the full logical Clifford group, no analogous examples are known for codes encoding multiple logical qubits. In this work, we prove a no-go theorem establishing that no stabilizer code admits a fully transversal implementation of the Clifford group on more than one logical qubit. We further strengthen this result by showing that fold-transversal implementations of the full logical Clifford group are impossible for stabilizer codes encoding more than two logical qubits. More generally, we introduce the notion of k-fold transversal gadgets and prove that implementing the full Clifford group on k logical qubits requires at least k-fold transversal gadgets at the physical level. In addition, we analyze code-automorphism-based constructions and demonstrate that they also fail to realize the full Clifford group on multiple logical qubits for any stabilizer code. Together, these results place fundamental constraints on fault tolerant Clifford gadget design and show that stabilizer codes supporting the full logical Clifford group on multiple logical qubits via these architectures do not exist. Since the Clifford group is a core component of universal gate sets, our findings imply that quantum computing with codes encoding multiple logical qubits within a single code block necessarily entails more complex constructions for fault tolerance.

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To build a working quantum computer, we must protect quantum information from errors while still enabling reliable computation. A central tool for this is stabilizer codes, which encode logical qubits into highly entangled collections of physical qubits. However, performing operations on these logical qubits requires carefully designed fault tolerant gadgets that prevent errors from spreading across many qubits. In this work, we study several important classes of fault tolerant constructions, including transversal gates, fold-transversal gates, and code automorphisms. We show that these approaches are fundamentally limited in their ability to realize the full Clifford group on logical qubits, a key ingredient for universal quantum computation. In particular, for stabilizer codes encoding multiple logical qubits within a single block, the full Clifford group cannot be implemented using any of these gadget classes. Achieving universal fault tolerant computation in such settings therefore requires more sophisticated constructions.

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PRX Quantum 7, 033027– Published 13 August, 2026
  • Received 27 February 2026
  • Revised 29 April 2026
  • Accepted 23 June 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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