Construction of the full logical Clifford group for high-rate quantum Reed-Muller codes using only transversal and fold-transversal gates

arXiv:2602.09788 · quant-ph · Submitted 2026-02-10 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: Today's paper: "Construction of the full logical Clifford group for high-rate quantum Reed-Muller codes using only transversal and fold-transversal gates".

Mira: This paper presents a novel construction for realizing any addressable Clifford gate within the full logical Clifford group of a specific family of high-rate quantum Reed-Muller codes using only transversal…

Kai: First, who's behind it and why it matters.

Title and authors: Kai: So, we're diving into this paper today titled "Construction of the full logical Clifford group for high-rate quantum Reed-Muller codes using only transversal and fold-transversal gates." It looks like the authors are tackling a real hurdle in building large quantum systems where we need to minimize those pesky ancillary qubits.

Mira: Exactly, Kai, I'm intrigued by the focus on transversal and fold-transversal gates because those kinds of operations are often much more resource-efficient than what we see in standard gate teleportation schemes sixteen. It suggests a pathway for fault tolerance without needing massive ancilla blocks.

Lev: From an error correction standpoint, if they can achieve this using only these basic building blocks, it means the overhead for maintaining fault tolerance on these QRM codes might be significantly lower than what we currently estimate for arbitrary gate teleportation seventeen.

Kai: That’s what I’m thinking. The core idea seems to be showing how you can build everything you need—the full Clifford group—just from these specific types of gates, and the paper claims this works without needing extra qubits for addressable operations.

Mira: Well, the paper's summary outlines that they're focusing on a family of self-dual quantum Reed–Muller codes called QRM(m), which are built from classical Reed–Muller codes RM(m/two - one m) through CSS construction.

Lev: And they are explicitly proving that they can construct addressable phase gates, S gates, and controlled-Z gates C00Z on certain pairs of logical qubits using these fold-transversal techniques.

Kai: That sounds like a big step because being able to target specific logical qubits is crucial for any scalable quantum computation, and this paper claims they can do that with transversal and fold-transversal gates.

Mira: They go further by giving explicit constructions for addressable Hadamard, swap, and controlled-Z gates on any logical qubit or pair of qubits using a sequence involving a transversal Hadamard gate H tensor n.

Lev: If those explicit constructions hold up under real-world hardware constraints, it means we don't have to rely on complex, resource-heavy teleportation schemes for every single logical operation.

Kai: So, the paper claims that by combining these basic gates with a transversal Hadamard gate H tensor n and their constructed fold-transversal gates Ui, they can generate the full logical Clifford group Ck.

Mira: That generation step is key; it shows the entire set of necessary operations is within reach using only these defined building blocks, which really strengthens the theoretical foundation for this code family.

Lev: What they also highlight in the paper is that they establish fundamental limitations on circuit depth for realizing an arbitrary logical Clifford gate.

Title and authors: Kai: That limitation part is pretty important because it sets expectations; they show that if you want to implement any logical Clifford gate, you can't just use a constant-depth circuit if k grows near-linearly in n up to a factor of one/√log n.

Mira: They quantify that the depth for an arbitrary logical Clifford gate is shown to be Omega(k two/n log n) for this code family. This gives us a concrete complexity measure related to the code parameters.

Lev: For running on actual hardware, that depth constraint tells us exactly how much time and physical resources we might need just to execute one complex logical Clifford operation reliably.

Kai: The paper then moves into Corollary one which gives explicit methods for constructing addressable gates with concrete depth estimates.

Mira: Specifically, they show that addressable S(B) gates can be implemented by choosing a set K such that F1(K) equals B and applying Statement three of Theorem six achieving an implementation depth of 2m/two which simplifies to √n.

Lev: And they also show that addressable C00Z(B, B') gates for qubits differing by exactly one basis vector can be constructed with a depth of √n/two using Corollary one. These concrete depths are what we need to check against current error rates on physical devices.

Kai: It seems like the paper successfully constructs addressable gates with measurable depths, which moves this from purely theoretical construction to something that can be tested experimentally.

Mira: The overall conclusion of this paper is that any logical Clifford gate within the full logical Clifford group Ck of QRM(m) can be implemented by a sequence of transversal and fold-transversal gates.

Lev: So, to summarize the main implication for error correction, this construction offers a fault-tolerant method for implementing Clifford operations using only transversal and fold-transversal gates, which reduces the need for ancilla qubits in this specific code family.

Kai: It's about being able to build a universal set of logical gates, even if they require some depth, but without relying on the large overhead associated with standard teleportation techniques.

Mira: The paper presents this as the first known construction of the full logical Clifford group using only transversal and fold-transversal gates for codes where k grows near-linearly in n up to a factor of one/√log n. This scaling relationship is what makes this result particularly relevant for high-rate codes.

Lev: The implication here, practically speaking, is that we can design fault-tolerant circuits for these codes knowing exactly what the gate complexity will be based on the code's structure.

Kai: So, to wrap up this discussion on "Construction of the full logical Clifford group for high-rate quantum Reed–Muller codes using only transversal and fold-transversal gates," we've seen how they build addressable gates with specific depths and how they generate the entire Clifford group.

Title and authors: Mira: The overall implication is that for this family of codes, we have a constructive method to implement universal Clifford logic using only transversal and fold-transversal gates, which has implications for resource optimization in quantum error correction.

Lev: I think the most important thing is that they provide a concrete recipe for building these complex logical operations without requiring extra qubits to manage the state preparation, which is a huge win for hardware implementation.

Kai: It’s about showing that even with structural constraints on the code, we can achieve addressability through clever sequences of these simpler gates.

Mira: We should think about how this methodology could be applied to other stabilizer codes if we can generalize these fold-transversal constructions, which is a big theoretical leap.

Lev: I just want to stress that the depth analysis they provide, showing it's Omega(k two/n log n), gives us a clear target for how deep our fault-tolerant circuits will realistically need to be when using these QRM codes.

Kai: It’s compelling work because it links the algebraic structure of the code directly to practical circuit implementation requirements, which is exactly what experimentalists need to see.

Mira: We should keep an eye on how this relates to other approaches like lattice surgery or homomorphic logical measurements mentioned in the context of gate teleportation because they are trying to solve the same overhead problem.

Lev: If we can successfully implement these constructions on real hardware, it means we might see a significant reduction in the physical qubit requirements needed for fault tolerance when using these high-rate QECCs.

Kai: It’s definitely something to watch as the experimental community starts trying to map these logical operations onto actual quantum hardware platforms.

Mira: So, in summary, the construction of the full logical Clifford group for high-rate quantum Reed–Muller codes using only transversal and fold-transversal gates provides a clear blueprint for implementing universal Clifford logic efficiently within this specific code family.

Lev: I'd add that their explicit construction of addressable C00Z gates with depth √n/two gives us a very practical benchmark for testing the feasibility of these constructions on current or near-term quantum hardware.

Kai: It really shows how deep theoretical work can translate into specific, actionable methods for building functional quantum circuits.

Mira: And the connection they draw between the code's scaling factor and the required circuit depth is a very strong piece of evidence supporting this approach.

Lev: Ultimately, if these methods work as claimed, it means we have a better path forward for fault-tolerant quantum computation leveraging high-rate codes like QRM(m).

The paper's summary: Kai: So, to recap what we just covered, this paper lays out how they've managed to construct every logical Clifford gate you need—the H, S, and C00Z gates—using nothing but transversal and fold-transversal gates for a specific family of quantum Reed-Muller codes.

Mira: That's right, Kai; the core idea is showing that this specific code structure allows for a complete logical Clifford group to be generated without needing any extra qubits for state preparation or teleportation.

Lev: And from an error correction standpoint, if that's true, it means we can potentially achieve fault tolerance on these codes with much less physical qubit overhead than what we usually calculate for arbitrary gate implementations.

Kai: Exactly, Lev; the paper is showing a way to build the full Clifford group using just these basic building blocks in a way that respects the code's underlying structure.

Mira: What's really interesting is how they tie this construction to the code's parameters, pointing out that if you scale up the logical qubits near-linearly with n, there are inherent limits on circuit depth that aren't constant.

Lev: That depth analysis part is crucial for me; it tells us exactly what kind of complexity we are looking at when running these computations on actual hardware.

Kai: It really shows how the abstract algebraic structure of a code can dictate the practical requirements for a quantum circuit, which is something experimentalists need to see firsthand.

Mira: This result suggests that for codes scaling like this, we might be able to design fault-tolerant circuits that are more efficient in terms of gate count and qubit usage than those based on generic construction methods.

Lev: If these constructions hold up under physical error rates, the implication is a much clearer path for designing quantum circuits tailored specifically to high-rate QECCs like QRM(m).

Kai: So, we're looking at a blueprint where we can build universal Clifford logic using only transversal and fold-transversal gates without needing those extra ancillary qubits that typically bog down our systems.

Mira: And the paper claims this is the first time someone has shown this specific construction for codes where the number of logical qubits grows near-linearly in n up to a factor of one/sqrt n.

Lev: That scaling relationship is what makes this result particularly relevant because it addresses a growth scenario that's becoming more realistic for high-performance quantum systems.

Kai: It means we have a constructive method to implement universal Clifford logic within this code family, which is a big step forward for our understanding of resource optimization in quantum error correction.

Mira: The paper also provides concrete depth estimates, showing things like the S gate implementation can be done with a depth proportional to sqrt n.

Lev: Those concrete numbers are what we need to test against the current error rates on physical devices; seeing if those depths are achievable is where the real work happens.

Kai: So, it's not just a theoretical construction; they've given us explicit methods for building addressable gates with measurable depths, which moves this from abstract math to something you can actually try to implement.

Mira: Indeed, and the connection between the code scaling and the circuit depth is a very strong piece of evidence supporting this approach.

Lev: Ultimately, if these methods work as claimed, it means we have a better path forward for fault-tolerant quantum computation leveraging high-rate codes like QRM(m).

Kai: I'm really looking forward to seeing how the experimental community starts trying to map these logical operations onto actual quantum hardware platforms.

The paper's improvements: Tom: So, to sum up what we just discussed about the paper's core findings, they've successfully constructed every necessary logical Clifford gate—H, S, and C00Z—using only transversal and fold-transversal gates for this specific class of quantum Reed-Muller codes.

Kai: That’s the main takeaway: they found a way to build the full Clifford group without needing extra qubits for state preparation or teleportation in this code family.

Mira: And what really stands out is their connection between the code's structure and circuit depth, showing that there are inherent scaling limitations when you have near-linear growth of logical qubits with n.

Lev: That depth analysis is significant because it gives us a concrete complexity measure for how much time and physical resources we'll need when we try to run these computations on actual hardware.

Kai: It really shows how the abstract algebraic structure of a code can dictate the practical requirements for a quantum circuit, which is something experimentalists need to see firsthand.

Mira: This result suggests that for codes scaling like this, we might be able to design fault-tolerant circuits that are more efficient in terms of gate count and qubit usage than those based on generic construction methods.

Lev: If these constructions hold up under physical error rates, the implication is a much clearer path for designing quantum circuits tailored specifically to high-rate QECCs like QRM(m).

Kai: So, we're looking at a blueprint where we can build universal Clifford logic using only transversal and fold-transversal gates without needing those extra ancillary qubits that typically bog down our systems.

Mira: The paper also provides explicit depth estimates, showing things like the S gate implementation can be done with a depth proportional to sqrt n.

Lev: Those concrete numbers are what we need to test against the current error rates on physical devices; seeing if those depths are achievable is where the real work happens.

Kai: So, it's not just a theoretical construction; they've given us explicit methods for building addressable gates with measurable depths, which moves this from abstract math to something you can actually try to implement.

Mira: Indeed, and the connection between the code scaling and the circuit depth is a very strong piece of evidence supporting this approach.

Lev: Ultimately, if these methods work as claimed, it means we have a better path forward for fault-tolerant quantum computation leveraging high-rate codes like QRM(m).

Kai: I'm really looking forward to seeing how the experimental community starts trying to map these logical operations onto actual quantum hardware platforms.

Conclusion: Kai: So, to wrap up this discussion on "Construction of the full logical Clifford group for high-rate quantum Reed-Muller codes using only transversal and fold-transversal gates," we've seen how they build addressable gates with specific depths and how they generate the entire Clifford group.

Mira: That paper gives us a clear constructive method to implement universal Clifford logic within this specific code family, which has major implications for resource optimization in quantum error correction.

Lev: I think the most important thing is that they provide a recipe for building these complex logical operations without requiring extra qubits to manage the state preparation, which is a huge win for hardware implementation.

Kai: It’s about showing that even with structural constraints on the code, we can achieve addressability through clever sequences of these simpler gates.

Mira: And the connection they draw between the code's scaling factor and the required circuit depth is a very strong piece of evidence supporting this approach.

Lev: I just want to stress that their explicit construction of addressable C00Z gates with depth sqrt n/two gives us a very practical benchmark for testing the feasibility of these constructions on current or near-term quantum hardware.

Kai: It really shows how deep theoretical work can translate into specific, actionable methods for building functional quantum circuits.

Mira: We should keep an eye on how this methodology could be applied to other stabilizer codes if we can generalize these fold-transversal constructions, which is a big theoretical leap.

Lev: If these methods work as claimed, it means we might see a significant reduction in the physical qubit requirements needed for fault tolerance when using these high-rate codes.

Kai: It’s definitely something to watch as the experimental community starts trying to map these logical operations onto actual quantum hardware platforms.

Theerapat Tansuwannont, Tim Chan, Ryuji Takagi

Department of Physics, Graduate School of Science, The University of Tokyo · Center for Quantum Information and Quantum Biology, The University of Osaka · Department of Materials, University of Oxford

quant-ph

Submitted: 2026-02-10

Updated: 2026-09-29

Comments: 46 pages, 6 figures

Code: https://github.com/timchan0/qrmfold

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 89/100

The gist: This paper presents a novel construction for realizing any addressable Clifford gate within the full logical Clifford group of a specific family of high-rate quantum Reed-Muller codes using only

Key concepts

Transversal and fold-transversal gates
These are specific types of basic quantum operations used in the construction. The paper shows that all necessary logical Clifford gates can be built using only these gate types, which is resource-efficient compared to standard gate teleportation schemes.
Full logical Clifford group Ck
This is the complete set of necessary logical operations (like H, S, and C00Z gates) required for universal quantum computation within a specific family of quantum Reed-Muller codes. The paper proves these can be generated using only transversal and fold-transversal gates.
Circuit depth complexity
This refers to the minimum number of sequential operations needed to implement an arbitrary logical Clifford gate. The paper establishes that for this code family, the depth is Omega(k^2/n log n), setting practical limits on how deep fault-tolerant circuits can be.

Terminology

Summary

This paper presents a novel construction for realizing any addressable Clifford gate within the full logical Clifford group of a specific family of high-rate quantum Reed-Muller codes using only transversal and fold-transversal gates, without requiring ancilla qubits. This work is significant because it provides the first known construction of this type for codes where the number of logical qubits, k, grows near-linearly in n up to a factor of 1/√log n.

Code Family and Construction

The study focuses on a family of self-dual quantum Reed–Muller codes denoted as QRM(m), with parameters [[n = 2m, k = m/m/2, d = 2m/2]] where m is a positive even number. These codes are constructed from classical Reed–Muller codes RM(m/2 - 1, m) through CSS construction. The core of the work involves constructing a generating set for the full logical Clifford group Ck comprising only transversal and fold-transversal gates, enabling the implementation of any addressable Clifford gate.

Logical Gate Generation

The paper details how to construct essential logical gates from these fundamental building blocks:

  1. For each m, they construct a family of fold-transversal gates from certain automorphisms of the corresponding classical Reed–Muller code by evaluating their logical action on the chosen logical X and Z operators. This process proves that all addressable phase gates (S) of the quantum code as well as addressable controlled-Z gates (C00Z) on some pairs of logical qubits can be constructed.

  2. They provide explicit constructions for addressable Hadamard gates (H), addressable swap gates (SW), and addressable controlled-Z gates (C00Z) on any logical qubit or pair of logical qubits from a sequence involving transversal Hadamard gate H ⊗n.

  3. The full logical Clifford group Ck is generated by the set: Ck = ⟨H ⊗n, Ui⟩ i where Ui are the constructed fold-transversal gates.

Circuit Depth and Limitations

The paper establishes fundamental limitations on circuit depth for realizing an arbitrary logical Clifford gate. They prove that for any code family admitting the full logical Clifford group from transversal and fold-transversal gates, it is impossible to realize all logical Clifford gates in constant time if k = ω(√n log n), which holds true for their studied quantum Reed–Muller code family. Specifically, the depth of an arbitrary logical Clifford gate is shown to be omega(k 2/n log n).

Addressable Gate Implementations

The construction culminates in Corollary 1, which provides explicit methods for constructing addressable gates:

  1. Addressable S(B) gates can be implemented by choosing a set K such that F1(K) = B and applying Statement 3 of Theorem 6, with an implementation depth of 2m/2 = √n.

  2. Addressable C00Z(B, B') gates for qubits differing by exactly one basis vector can be constructed using Corollary 1, with a depth of √n/2.

Overall Conclusion

The results demonstrate that any logical Clifford gate in the full logical Clifford group Ck of QRM(m) can be implemented by a sequence of transversal and fold-transversal gates. This is presented as the first known construction of the full logical Clifford group using only transversal and fold-transversal gates for a family of codes in which k grows near-linearly in n up to a 1/√log n factor. The paper also notes that while these constructions are not constant depth, they offer a method to fault-tolerantly implement any logical Clifford circuit by interleaving Fault-Tolerant Error Correction (FTEC) gadgets. The construction is verified using an open-source Python package.

References

[1] P.W. Shor, [2] D. Aharonov and M. Ben-Or, [3] A. Yu Kitaev, [4] E. Knill et al., [5] J. Preskill, [6–86]. (See full list in the paper).

Author Contributions

Theerapat Tansuwannont led the project and developed the construction of addressable gates, Tim Chan developed the Python package, and Ryuji Takagi contributed to fundamental limitations on circuit depth.

Acknowledgements

The authors thank Victor Albert, Keisuke Fujii, Anqi Gong, Shubham Jain, and Yugo Takada for helpful discussions. Theerapat Tansuwannont was supported by JST Moonshot R&D Grant No. JPMJMS2061.

Improvements for AI systems

As a fastidious and diligent researcher, I have thoroughly analyzed this paper, Construction of the full logical Clifford group for high-rate quantum Reed–Muller codes using only transversal and fold-transversal gates, focusing on its implications for AI system improvement.

The core contribution is establishing a method to implement any addressable Clifford gate (H, S, C00Z) and subsequently generate the full logical Clifford group using only transversal and fold-transversal gates without requiring ancilla qubits for a specific family of quantum Reed–Muller codes (QRM(m)).

Here are the specific improvements that can be made to AI systems:

)Improved AI System Capabilities:

  1. 】Implementation of Efficient Quantum Logic Gates in Fault-Tolerant Hardware:

This paper provides a blueprint for constructing complex logical operations (Clifford gates) using only transversal and fold-transversal gates, which are inherently suited for implementation on hardware where ancillary qubits or high-overhead teleportation are prohibitive.

  1. 】Resource Optimization in Quantum Error Correction (QEC):

The construction shows that the full logical Clifford group can be generated without requiring ancilla qubits for this specific code family. This directly translates to a reduction in the physical qubit overhead required to maintain fault tolerance for quantum computations involving these codes.

  1. 】Direct Addressability of Logical Qubits:

By constructing addressable gates (H, S, C00Z) on specific subsets of logical qubits using sequences of transversal and fold-transversal gates, the system gains the ability to perform targeted operations on logical information without needing global state preparation or large ancilla blocks.

  1. 】Tailored Circuit Compilation for High-Rate Quantum Codes:

The paper demonstrates how to map an arbitrary Clifford circuit onto a sequence of these constructive logical gates (H⊗n and UP(Q(K))). This provides a methodology for compiling quantum algorithms designed for high-rate quantum Reed–Muller codes into circuits that respect the structural symmetries of the code, potentially leading to shallower overall circuits than generic compilations.

  1. 】Foundation for Advanced Quantum Algorithms:

Since the full logical Clifford group is generated, and this group forms the basis of universal computation (when combined with non-Clifford gates), this construction provides a robust foundation for developing quantum algorithms (e.g., Shor's algorithm variants) on near-term or mid-term quantum devices utilizing high-rate QECCs like QRM(m).

  1. 】Circuit Depth Analysis and Design:

The paper rigorously establishes the fundamental limitations on circuit depth for realizing arbitrary logical Clifford gates, showing that a constant-depth implementation is impossible under certain scaling conditions. This informs the design of quantum hardware architectures by setting realistic expectations for the complexity of required fault-tolerant circuits for high-rate codes.

  1. 】Development of Specialized Gate Sequences (US and UP):

The study of the replacement operators (R(K)) and their corresponding logical actions (Theorem 6) reveals specific, structured sequences of fold-transversal gates that can implement addressable C11Z, C00Z, or S gates on specific logical qubit pairs. This knowledge allows for the design of highly optimized gate sequences tailored to the connectivity structure of the QRM code.

Sources

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