arXiv:2510.08402

Universal Fault Tolerance with Non-Transversal Clifford Gates

Benjamin Anker, Milad Marvian

Published Oct 9, 2025• 0 citations

Abstract

We propose a scheme for the fault-tolerant implementation of arbitrary Clifford circuits. To achieve this, we extend previous work on flag gadgets for syndrome extraction to a general framework that flags any Clifford circuit. This framework opens new pathways toward universal fault tolerance by allowing transversal implementation of $T$ gates alongside fault-tolerant realization of selected non-transversal Clifford gates using flags. The construction we present allows a Clifford circuit consisting of $n$ two-qubit gates and $O(n)$ single-qubit gates acting upon physical qubits in a code of distance $d$ to be made fault tolerant to distance $d$ using $O(d^2 \log(nd^2\log n))$ ancilla qubits and $O(nd^2 \log(nd^2 \log n))$ extra CNOTs. Beyond asymptotic analysis, we demonstrate our construction by implementing the non-transversal logical Hadamard gate for the [[15,1,3]] code, which has transversal T, and compare to alternative approaches for universality using this code. We also apply our construction to magic-state preparation, general state preparation using Clifford circuits, and data-syndrome codes.

DOI: 10.48550/arxiv.2510.08402

Related papers

Flagging the Clifford hierarchy:~Fault-tolerant logical $\fracπ{2^l}$ rotations via measuring circuit gauge operators of non-Cliffords

Shival Dasu, Ben Criger

quant-pharXiv:2603.24573Mar 26, 20260 citations

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

Aranya Chakraborty, Daniel Gottesman

quant-pharXiv:2602.13395PRX QuantumFeb 17, 20260 citations

Magic Quantum Code Surgery

Kathleen Chang, Anasuya Lyons, Yuanjie Ren +5 more

quant-pharXiv:2610.03920Oct 6, 20260 citations

Quantum Codes with Arbitrary Z-Rotation logical Gates and Applications to Fault-Tolerant Code Switching

Reza Dastbasteh, Ruben M. Otxoa, Pedro M. Crespo +1 more

quant-pharXiv:2608.11160Aug 12, 20260 citations

Stabilizer Code-Generic Universal Fault-Tolerant Quantum Computation

Nicholas J. C. Papadopoulos, Ramin Ayanzadeh

quant-pharXiv:2601.10964Jan 19, 20260 citations

Universal Weakly Fault-Tolerant Quantum Computation via Code Switching in the [[8,3,2]] Code

Shixin Wu, Dawei Zhong, Todd A. Brun +1 more

quant-pharXiv:2603.15610Physical Review ResearchMar 17, 20260 citations

arXiv:2510.08402

Universal Fault Tolerance with Non-Transversal Clifford Gates

Benjamin Anker, Milad Marvian

Published Oct 9, 2025• 0 citations

arXiv HTML PDF

Abstract

T

gates alongside fault-tolerant realization of selected non-transversal Clifford gates using flags. The construction we present allows a Clifford circuit consisting of

n

two-qubit gates and

O(n)

single-qubit gates acting upon physical qubits in a code of distance

d

to be made fault tolerant to distance

d

using

O(d^2 \log(nd^2\log n))

ancilla qubits and

O(nd^2 \log(nd^2 \log n))

extra CNOTs. Beyond asymptotic analysis, we demonstrate our construction by implementing the non-transversal logical Hadamard gate for the [[15,1,3]] code, which has transversal T, and compare to alternative approaches for universality using this code. We also apply our construction to magic-state preparation, general state preparation using Clifford circuits, and data-syndrome codes.

DOI: 10.48550/arxiv.2510.08402

Related papers

Flagging the Clifford hierarchy:~Fault-tolerant logical $\fracπ{2^l}$ rotations via measuring circuit gauge operators of non-CliffordsFlagging the Clifford hierarchy:~Fault-tolerant logical $\fracπ{2^l}$ rotations via measuring circuit gauge operators of non-Cliffords

Shival Dasu, Ben Criger

We provide a recursively defined sequence of flag circuits which will detect logical errors induced by non-fault-tolerant $R_{\overline{Z}}(\fracπ{2^l})$ gates on CSS codes with a fault distance of two. As applications, we give a family of circuits with $O(l)$ gates and ancillae which implement fault-tolerant logical $R_{Z}(\fracπ{2^l})$ or $R_{ZZ}(\fracπ{2^l})$ gates on any $[[k + 2, k, 2]]$ iceberg code and fault-tolerant circuits of size $O(l)$ for preparing $|\fracπ{2^l}\rangle$ resource states in the $[[7,1,3]]$ code, which can be used to perform fault-tolerant $R_{\overline{Z}}(\fracπ{2^l})$ rotations via gate teleportation, allowing for implementations of these gates that bypass the high overheads of gate synthesis when $l$ is small relative to the precision required. We show how the circuits above can be generalized to $π( x_0.x_{1}x_{2}\ldots x_{l}) = \sum_{j}^{l} π\frac{x_j}{2^j}$ rotations with identical overheads in $l$ , which could be useful in quantum simulations where time is digitized in binary. Finally, we illustrate two approaches to increase the fault-distance of our construction. We show how to increase the fault distance of a Cliffordized version of the T gate circuit to $3$ in the Steane code and how to increase the fault-distance of the $\fracπ{2}$ iceberg circuit to $4$ through concatenation in two-level iceberg codes. This yields a targeted logical $R_{\overline{Z}}(\fracπ{2})$ gate with fault distance $4$ on any row of logical qubits in an $[[(k_2+2)(k_1+2), k_1k_2, 4]]$ code.We provide a recursively defined sequence of flag circuits which will detect logical errors induced by non-fault-tolerant $R_{\overline{Z}}(\fracπ{2^l})$ gates on CSS codes with a fault distance of two. As applications, we give a family of circuits with $O(l)$ gates and ancillae which implement fault-tolerant logical $R_{Z}(\fracπ{2^l})$ or $R_{ZZ}(\fracπ{2^l})$ gates on any $[[k + 2, k, 2]]$ iceberg code and fault-tolerant circuits of size $O(l)$ for preparing $|\fracπ{2^l}\rangle$ resource states in the $[[7,1,3]]$ code, which can be used to perform fault-tolerant $R_{\overline{Z}}(\fracπ{2^l})$ rotations via gate teleportation, allowing for implementations of these gates that bypass the high overheads of gate synthesis when $l$ is small relative to the precision required. We show how the circuits above can be generalized to $π( x_0.x_{1}x_{2}\ldots x_{l}) = \sum_{j}^{l} π\frac{x_j}{2^j}$ rotations with identical overheads in $l$, which could be useful in quantum simulations where time is digitized in binary. Finally, we illustrate two approaches to increase the fault-distance of our construction. We show how to increase the fault distance of a Cliffordized version of the T gate circuit to $3$ in the Steane code and how to increase the fault-distance of the $\fracπ{2}$ iceberg circuit to $4$ through concatenation in two-level iceberg codes. This yields a targeted logical $R_{\overline{Z}}(\fracπ{2})$ gate with fault distance $4$ on any row of logical qubits in an $[[(k_2+2)(k_1+2), k_1k_2, 4]]$ code.

quant-pharXiv:2603.24573Mar 26, 20260 citations

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

Aranya Chakraborty, Daniel Gottesman

Identifying stabilizer codes that admit fault-tolerant implementations of the full logical Clifford group would significantly advance fault-tolerant quantum computation. 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.Identifying stabilizer codes that admit fault-tolerant implementations of the full logical Clifford group would significantly advance fault-tolerant quantum computation. 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.

quant-pharXiv:2602.13395PRX QuantumFeb 17, 20260 citations

Magic Quantum Code Surgery

Kathleen Chang, Anasuya Lyons, Yuanjie Ren +5 more

Non-Clifford operations required for universal quantum computation can be implemented by measuring logical Clifford operators. We present a generalized framework to fault-tolerantly measure transversal Clifford logical operators in quantum low-density parity check (QLDPC) codes. Our protocol takes any QLDPC code and deforms the code across additional ancillary qubits such that a product of stabilizers measures the desired logical Clifford operator. We prove that the deformed, Clifford-stabilized code remains LDPC and that its distance is preserved up to a constant factor independent of the number of physical qubits. We further prove that the Clifford measurement has a fault distance against data qubit and measurement errors that grows linearly with the distance of the original code. We apply our methods to implement logical non-Clifford gates in algorithmically relevant subroutines and to prepare magic states in quantum stabilizer codes.Non-Clifford operations required for universal quantum computation can be implemented by measuring logical Clifford operators. We present a generalized framework to fault-tolerantly measure transversal Clifford logical operators in quantum low-density parity check (QLDPC) codes. Our protocol takes any QLDPC code and deforms the code across additional ancillary qubits such that a product of stabilizers measures the desired logical Clifford operator. We prove that the deformed, Clifford-stabilized code remains LDPC and that its distance is preserved up to a constant factor independent of the number of physical qubits. We further prove that the Clifford measurement has a fault distance against data qubit and measurement errors that grows linearly with the distance of the original code. We apply our methods to implement logical non-Clifford gates in algorithmically relevant subroutines and to prepare magic states in quantum stabilizer codes.

quant-pharXiv:2610.03920Oct 6, 20260 citations

Quantum Codes with Arbitrary Z-Rotation logical Gates and Applications to Fault-Tolerant Code Switching

Reza Dastbasteh, Ruben M. Otxoa, Pedro M. Crespo +1 more

A technique for realizing a universal set of fault-tolerant quantum operations is the code switching method, which leverages two quantum codes with complementary sets of transversal gates. To date, the application of this technique has been largely limited to families of color codes supporting a logical $T$ gate. No analogous code switching protocols exist for many other prominent families, such as rotated surface codes, or for finer $Z$ -rotation gates. In this work, we first utilize the doubling technique as a unified framework to construct a class of quantum color codes encoding a single logical qubit with an arbitrarily large minimum distance, enabling the transversal realization of arbitrary small logical $Z$ -rotation gates. We investigate the structural properties of this code family, demonstrating that they improve upon the parameters of state-of-the-art triorthogonal codes, achieve lower qubit overhead compared to certain known color codes, and admit single-shot decoding of $Z$ -syndromes via meta-checks. Furthermore, we show that this framework extends beyond color codes; specifically, it enables the generation of $r$ -orthogonal quantum codes, $r \ge 2$ , that inherit the local geometry of rotated surface codes. We then provide an overhead optimization protocol alongside several candidate codes tailored for realizing logical $Z$ -rotation gates within rotated surface codes. Finally, we extend the fault-tolerant code switching protocol based on transversal CNOT gates to incorporate fault-tolerant realization of $Z$ -rotation gates at any level of the Clifford hierarchy for geometries compatible with rotated surface codes. We present the first demonstration of fault-tolerant magic state preparation by means of code switching within a distance-three rotated surface code using a total footprint of only 45 physical qubits, and evaluate its performance through a simulation.A technique for realizing a universal set of fault-tolerant quantum operations is the code switching method, which leverages two quantum codes with complementary sets of transversal gates. To date, the application of this technique has been largely limited to families of color codes supporting a logical $T$ gate. No analogous code switching protocols exist for many other prominent families, such as rotated surface codes, or for finer $Z$-rotation gates. In this work, we first utilize the doubling technique as a unified framework to construct a class of quantum color codes encoding a single logical qubit with an arbitrarily large minimum distance, enabling the transversal realization of arbitrary small logical $Z$-rotation gates. We investigate the structural properties of this code family, demonstrating that they improve upon the parameters of state-of-the-art triorthogonal codes, achieve lower qubit overhead compared to certain known color codes, and admit single-shot decoding of $Z$-syndromes via meta-checks. Furthermore, we show that this framework extends beyond color codes; specifically, it enables the generation of $r$-orthogonal quantum codes, $r \ge 2$, that inherit the local geometry of rotated surface codes. We then provide an overhead optimization protocol alongside several candidate codes tailored for realizing logical $Z$-rotation gates within rotated surface codes. Finally, we extend the fault-tolerant code switching protocol based on transversal CNOT gates to incorporate fault-tolerant realization of $Z$-rotation gates at any level of the Clifford hierarchy for geometries compatible with rotated surface codes. We present the first demonstration of fault-tolerant magic state preparation by means of code switching within a distance-three rotated surface code using a total footprint of only 45 physical qubits, and evaluate its performance through a simulation.

quant-pharXiv:2608.11160Aug 12, 20260 citations

Stabilizer Code-Generic Universal Fault-Tolerant Quantum Computation

Nicholas J. C. Papadopoulos, Ramin Ayanzadeh

Fault-tolerant quantum computation allows quantum computations to be carried out while resisting unwanted noise. Several error-correcting codes have been developed to achieve this task, but none alone are capable of transversal universal quantum computation. This universality is highly desired and often achieved using techniques such as code concatenation, code switching, magic state distillation, or pieceable fault tolerance. Existing constructions can incur substantial overhead or rely on properties specific to particular codes. This work proposes implementing logical Clifford and T gates through novel, ancilla-mediated, pieceable interactions to construct a universal fault-tolerant quantum gate set. Our construction is deterministic, does not modify the underlying data codes or registers, and is generic over all data stabilizer codes. Thus, any single code becomes capable of universal quantum computation by leveraging helper codes in ancilla registers and mid-circuit measurements. Furthermore, since these logical gates are stabilizer code-generic, these implementations enable communication between heterogeneous stabilizer codes.Fault-tolerant quantum computation allows quantum computations to be carried out while resisting unwanted noise. Several error-correcting codes have been developed to achieve this task, but none alone are capable of transversal universal quantum computation. This universality is highly desired and often achieved using techniques such as code concatenation, code switching, magic state distillation, or pieceable fault tolerance. Existing constructions can incur substantial overhead or rely on properties specific to particular codes. This work proposes implementing logical Clifford and T gates through novel, ancilla-mediated, pieceable interactions to construct a universal fault-tolerant quantum gate set. Our construction is deterministic, does not modify the underlying data codes or registers, and is generic over all data stabilizer codes. Thus, any single code becomes capable of universal quantum computation by leveraging helper codes in ancilla registers and mid-circuit measurements. Furthermore, since these logical gates are stabilizer code-generic, these implementations enable communication between heterogeneous stabilizer codes.

quant-pharXiv:2601.10964Jan 19, 20260 citations

Universal Weakly Fault-Tolerant Quantum Computation via Code Switching in the [[8,3,2]] Code

Shixin Wu, Dawei Zhong, Todd A. Brun +1 more

Code-switching offers a route to universal, fault-tolerant quantum computation by circumventing the limitation implied by the Eastin-Knill theorem against a universal transversal gate set within a single quantum code. Here, we present a fault-tolerant code-switching protocol between two versions of the $[[8, 3, 2]]$ code. One version supports weakly fault-tolerant single-qubit Clifford gates, while the other supports a logical $\overline{\mathrm{CCZ}}$ gate via transversal $T/T^\dagger$ together with logical $\overline{\mathrm{CZ}}$ , $\overline{\mathrm{CNOT}}$ , and $\overline{\mathrm{SWAP}}$ gates. Because both codes have distance 2, the protocol operates in a postselected, error-detecting regime: single faults lead to detectable outcomes, and accepted runs exhibit quadratic suppression of logical error rates. This yields a universal scheme for postselected fault-tolerant computation. We validate the protocol numerically through simulations of state preparation, code switching, and a three-logical-qubit implementation of Grover's search.Code-switching offers a route to universal, fault-tolerant quantum computation by circumventing the limitation implied by the Eastin-Knill theorem against a universal transversal gate set within a single quantum code. Here, we present a fault-tolerant code-switching protocol between two versions of the $[[8, 3, 2]]$ code. One version supports weakly fault-tolerant single-qubit Clifford gates, while the other supports a logical $\overline{\mathrm{CCZ}}$ gate via transversal $T/T^\dagger$ together with logical $\overline{\mathrm{CZ}}$, $\overline{\mathrm{CNOT}}$, and $\overline{\mathrm{SWAP}}$ gates. Because both codes have distance 2, the protocol operates in a postselected, error-detecting regime: single faults lead to detectable outcomes, and accepted runs exhibit quadratic suppression of logical error rates. This yields a universal scheme for postselected fault-tolerant computation. We validate the protocol numerically through simulations of state preparation, code switching, and a three-logical-qubit implementation of Grover's search.

quant-pharXiv:2603.15610Physical Review ResearchMar 17, 20260 citations

T

gates alongside fault-tolerant realization of selected non-transversal Clifford gates using flags. The construction we present allows a Clifford circuit consisting of

n

two-qubit gates and

O(n)

single-qubit gates acting upon physical qubits in a code of distance

d

to be made fault tolerant to distance

d

using

O(d^2 \log(nd^2\log n))

ancilla qubits and

O(nd^2 \log(nd^2 \log n))

Flagging the Clifford hierarchy:~Fault-tolerant logical

\fracπ{2^l}

rotations via measuring circuit gauge operators of non-Cliffords

We provide a recursively defined sequence of flag circuits which will detect logical errors induced by non-fault-tolerant

R_{\overline{Z}}(\fracπ{2^l})

gates on CSS codes with a fault distance of two. As applications, we give a family of circuits with

O(l)

gates and ancillae which implement fault-tolerant logical

R_{Z}(\fracπ{2^l})

R_{ZZ}(\fracπ{2^l})

gates on any

[[k + 2, k, 2]]

iceberg code and fault-tolerant circuits of size

O(l)

for preparing

|\fracπ{2^l}\rangle

resource states in the

[[7,1,3]]

code, which can be used to perform fault-tolerant

R_{\overline{Z}}(\fracπ{2^l})

rotations via gate teleportation, allowing for implementations of these gates that bypass the high overheads of gate synthesis when

l

is small relative to the precision required. We show how the circuits above can be generalized to

π( x_0.x_{1}x_{2}\ldots x_{l}) = \sum_{j}^{l} π\frac{x_j}{2^j}

rotations with identical overheads in

l

, which could be useful in quantum simulations where time is digitized in binary. Finally, we illustrate two approaches to increase the fault-distance of our construction. We show how to increase the fault distance of a Cliffordized version of the T gate circuit to

3

in the Steane code and how to increase the fault-distance of the

\fracπ{2}

iceberg circuit to

4

through concatenation in two-level iceberg codes. This yields a targeted logical

R_{\overline{Z}}(\fracπ{2})

gate with fault distance

4

on any row of logical qubits in an

[[(k_2+2)(k_1+2), k_1k_2, 4]]

code.

T

gate. No analogous code switching protocols exist for many other prominent families, such as rotated surface codes, or for finer

Z

-rotation gates. In this work, we first utilize the doubling technique as a unified framework to construct a class of quantum color codes encoding a single logical qubit with an arbitrarily large minimum distance, enabling the transversal realization of arbitrary small logical

Z

-rotation gates. We investigate the structural properties of this code family, demonstrating that they improve upon the parameters of state-of-the-art triorthogonal codes, achieve lower qubit overhead compared to certain known color codes, and admit single-shot decoding of

Z

-syndromes via meta-checks. Furthermore, we show that this framework extends beyond color codes; specifically, it enables the generation of

r

-orthogonal quantum codes,

r \ge 2

, that inherit the local geometry of rotated surface codes. We then provide an overhead optimization protocol alongside several candidate codes tailored for realizing logical

Z

-rotation gates within rotated surface codes. Finally, we extend the fault-tolerant code switching protocol based on transversal CNOT gates to incorporate fault-tolerant realization of

Z

-rotation gates at any level of the Clifford hierarchy for geometries compatible with rotated surface codes. We present the first demonstration of fault-tolerant magic state preparation by means of code switching within a distance-three rotated surface code using a total footprint of only 45 physical qubits, and evaluate its performance through a simulation.

[[8, 3, 2]]

code. One version supports weakly fault-tolerant single-qubit Clifford gates, while the other supports a logical

\overline{\mathrm{CCZ}}

gate via transversal

T/T^\dagger

together with logical

\overline{\mathrm{CZ}}

\overline{\mathrm{CNOT}}

, and

\overline{\mathrm{SWAP}}

gates. Because both codes have distance 2, the protocol operates in a postselected, error-detecting regime: single faults lead to detectable outcomes, and accepted runs exhibit quadratic suppression of logical error rates. This yields a universal scheme for postselected fault-tolerant computation. We validate the protocol numerically through simulations of state preparation, code switching, and a three-logical-qubit implementation of Grover's search.