CE-Preserving Logical CNOT Gates in FTQEC

• The paper introduces a fault-tolerant framework for constant-excitation CSS codes, featuring CE-preserving logical CNOT gates that maintain the excitation subspace against collective noise.
• It details a novel gate design interleaving transversal CNOTs with corrective layers or zero-controlled-NOTs to ensure constant weight is preserved during quantum operations.
• Simulations and minimal code constructions, like the [[12,1,3]] code, demonstrate practical error thresholds and scalability for quantum processors facing global coherent errors.

A CE-preserving logical CNOT gate is a form of two-qubit logical gate designed for use with constant-excitation (CE) codes, in which the encoded state space consists solely of fixed–excitation–number basis states. Such codes are of central interest when addressing collective coherent noise—namely, errors that act as global rotations, such as $U = \exp(-i\theta\sum_j Z_j)$, affecting all qubits identically. Standard transversal CNOT gates break the constant-excitation constraint and are thus incompatible with CE codes. Recent work has established the first complete fault-tolerant framework for constant-excitation CSS codes, introducing CE-preserving logical CNOT gates, modified syndrome extraction circuits, and tailored simulation techniques that together enable robust fault-tolerant quantum error correction (FTQEC) against both stochastic and collective coherent noise (Lai et al., 14 Jul 2025).

1. Constant-Excitation Codes and the Constraints on Logical Gates

Constant-excitation (CE) codes, also called fixed-weight codes, encode quantum information in the subspace of all computational basis states with a fixed Hamming weight $w$. That is,

$$\ket{\psi} = \sum_{|\mathbf{x}|=w} \alpha_{\mathbf{x}}\, \ket{\mathbf{x}},$$

where $\mathrm{wt}(\mathbf{x}) = w$ for all basis states in the codeword. CE codes are naturally robust against collective coherent (CC) noise of the form

$U = \exp\!\left(-i\theta \sum_j Z_j\right)$

because all physical codewords are eigenstates of the total excitation number operator, and such collective errors induce only a global phase within the code space.

However, standard transversal logical CNOT operations do not generally preserve the set of CE codewords. Applying each CNOT between corresponding qubits in two code blocks changes the weight of computational basis states, potentially mapping the code out of the constant-excitation subspace. This fundamental incompatibility requires the design of alternative CNOT constructions that preserve excitation number—so-called “CE-preserving logical CNOT gates.”

2. Structure and Implementation of the CE-Preserving Logical CNOT Gate

The CE-preserving logical CNOT is achieved by modifying the standard transversal logic. The key idea is to interleave transversal CNOTs with additional correction layers so the excitation number constraint holds after the operation. The central result is that a transversal CNOT, followed by a product of Pauli-X operators on all target qubits, implements a logical CNOT in the constant-excitation code subspace: $$\mathrm{CE\!-\!CNOT} = \left( \prod_{j=1}^n X_{2j} \right) \circ \mathrm{CNOT}_{1\to2}$$ where the subscripts denote the logical blocks.

Alternatively, the same logic can be implemented by interleaving transversal CNOTs with zero-controlled-NOT (ZCNOT) gates—these are gates that flip the target qubit if and only if the control is in $|0\rangle$, as opposed to the usual CNOT which acts when the control is $|1\rangle$. ZCNOT gates can be built from a standard CNOT and an $X$ on the target. By carefully arranging standard and zero-controlled gates, one ensures that excitation number is always conserved throughout the logical operation, and the computation remains entirely within the CE subspace [(Lai et al., 14 Jul 2025), Lemma "prop:CNOT"].

Moreover, the propagation of errors, including how collective rotations and $X$ errors interact with CNOT layers, has precise algebraic characterization: $$e^{-i\theta Z} X = X\, e^{i2\theta Z} e^{-i\theta Z}$$

$$CX_{1,2}\, e^{-i\theta Z_2} = e^{-i\theta Z_1 Z_2}\, CX_{1,2}$$

These identities elucidate how logical and coherent errors pass through CE-preserving CNOT constructions and motivate the alternation of corrective $X$ layers and CNOTs.

3. Fault-Tolerant Syndrome Extraction Compatible with Constant-Excitation Codes

Because CE-preserving logical gate constructions are fundamentally different from standard Clifford-based stabilizer codes, fault-tolerant syndrome extraction circuits must also be adapted for CE constraints. The framework introduces modified Shor- and Steane-type protocols:

• Modified Shor extraction replaces conventional ancilla cat states with “CE cat states,” defined as

$$\ket{\mathrm{cat}^{\mathrm{CE}}(w)} = \frac{1}{\sqrt{2}}\Bigl( \ket{01}^{\otimes w} + \ket{10}^{\otimes w} \Bigr)$$

and all controlled operations are replaced by their CE-compatible versions (using zero-controlled-based constructions as needed).

• Modified Steane extraction similarly adapts ancilla preparation and logical gate sequences to utilize only CE-preserving gates, ensuring that both data and ancilla remain in the constant-excitation subspace throughout.

Key to these protocols is the use of code-specific ancilla states whose preparation and measurement procedures are themselves CE-preserving, such that error information is extracted without ever leaving the code space.

4. Simulation and Error Propagation under Coherent and Stochastic Noise

Given that collective coherent errors (CC noise) do not commute with all standard Clifford circuits and lie outside the Clifford group, their simulation cannot rely on Clifford-only techniques. An extended stabilizer simulation algorithm is introduced, representing coherent errors as pairs $(\theta, c)$, where $\theta$ is a rotation phase and $c$ labels qubit support. For example, after a coherent layer: $$e^{-i\theta Z_j} X_j = X_j\, e^{i 2\theta Z_j} e^{-i\theta Z_j}$$ which means $X$ errors commute but pick up an additional phase that is carefully tracked in the error record. When measurements (e.g., in the $X$ basis) occur, the simulation probabilistically projects the coherent errors onto appropriate Pauli errors.

This simulator tracks Pauli errors and coherent errors in parallel, updating both as each logical CE-preserving circuit layer is executed. The total computational resource is efficient in the code size, allowing for practical simulation and threshold estimation of the CE FTQEC protocol even with collective errors [(Lai et al., 14 Jul 2025), Algorithm 2].

5. Explicit Code Constructions and Error Correction Performance

Minimal CE CSS codes built by dual-rail concatenation illustrate the feasibility and performance of CE-preserving CNOTs within larger FTQEC protocols. The $[[12,1,3]]$ code is the smallest distance-3 CE CSS code constructed in the framework, and the $[[14,3,3]]$ code encodes three logical qubits at distance 3.

Benchmarked under both circuit-level depolarizing noise and collective coherent noise, the $[[12,1,3]]$ code achieves a logical error threshold near $10^{-4}$. The code, combined with the CE-preserving CNOT and compatible syndrome extraction, maintains logical states within the CE subspace at every stage, thus realizing high error suppression and compatibility with the CE code constraints.

The framework generalizes to larger and higher-distance CE CSS codes, showing that fault tolerance and performance benefits extend as the code grows [(Lai et al., 14 Jul 2025), Theorem 1 & simulations].

6. Implications for Hardware and Quantum Architectures

The CE-preserving logical CNOT framework is particularly well suited to quantum processors dominated by collective coherent noise, such as quantum-optical systems with global fluctuations or architectures targeting low-decoherence through passive encoding constraints. Because CE codes and their CNOT gates never leave the excitation subspace, systems with excitation-number conservation laws—e.g., certain photonic, atom, or cavity QED hardware—can implement robust FTQEC without violating physical constraints.

In contrast with standard stabilizer protocols, which require codewords and logical operations that do not conserve excitation number, CE-preserving architectures provide strong passive protection against a dominant class of noise channels (global phase rotations) while supporting high-threshold, practical quantum error correction in realistic device contexts.

7. Summary

CE-preserving logical CNOT gates form the cornerstone of a modern FTQEC framework for constant-excitation codes. They achieve this by using modified physical gate sequences (interleaving CNOT and zero-controlled-NOT gates, or appending collective $X$ flips) that ensure the code remains entirely within the constant-excitation subspace, even in the presence of Pauli or coherent errors. Modified syndrome extraction schemes deploy CE-compatible ancillas and gate layers, and a new simulation methodology enables performance prediction and threshold estimation under both stochastic and coherent noise. Minimal code implementations (such as the $[[12,1,3]]$ code) highlight the feasibility and robustness of the approach for scalable quantum processing, prominently in architectures subject to collective coherent noise (Lai et al., 14 Jul 2025).