Researchers at the International Quantum Academy and the National University of Singapore have constructed a 48-qubit code supporting fully addressable transversal T gates, a critical component for building practical, error-corrected quantum computers. The work improves existing guarantees for controlling quantum phase, using a novel cyclic encoding that separates logical phase control from the design of code rate and distance.

This approach yielded a performance improvement by a factor of approximately 2.86, and establishes an upper bound only 1.28 times greater than the lower bound achieved by this construction. These results provide concrete targets for advancing efficient quantum code designs.

Cyclic Encoding Enables Addressable Transversal Phase Gates

A deterministic construction now establishes a lower bound of 52/(675h) for both rate and relative distance in quantum codes, advancing the field beyond previous limitations. The new encoding allows for two distinct construction approaches, each with differing computational demands, offering flexibility in implementation. As a demonstration of this work, the team constructed quantum codes with 48 and 90 physical qubits, both supporting fully addressable transversal T gates, essential components for fault-tolerant quantum computation.

The 8-to-10 inner encoding preserves logical phase action, using registers represented by h qubits, and relies on the invertibility of matrix B to ensure an injective map. The paper states that “the code defined by E admits fully addressable transversal Z(h) gates,” detailing the addressing matrix F ∈ R_h^(K× hM) where 1 ≤ j ≤ M and 0 ≤ b < h. Supplemental material provides proofs and construction details, including derivations of error-compression identities and the polynomial-time algebraic-geometric construction.

Fully Addressable T Gates with 48 and 90 Qubits

These codes, constructed through a novel cyclic encoding beyond standard binary CSS frameworks, enable precise control over quantum phase, a critical component for scalable fault-tolerant computation. The construction process involved sampling candidate matrices and applying exact rejection tests to verify performance, ensuring detection of Pauli errors up to a defined weight.

Rate and Distance Limits of Existing Quantum Codes

This code, alongside a 90-qubit counterpart, was realized through a cyclic encoding scheme operating outside the conventional binary CSS framework, representing each coordinate with a register of physical qubits. However, an exponential-time construction, incorporating exact verification through sampling and rejection tests, improves upon this bound by approximately 2.86, pushing the performance of fully addressable transversal phase gates further than previously achieved. The paper states, “If η > $c_{u p}$ , these classical codes eventually have rate and relative distance greater than t,” outlining the contradiction.

Polynomial Construction Yields 2.57% Rate and Distance

A deterministic, polynomial-time algebraic-geometric construction now yields a rate of 2.57% alongside a corresponding relative distance, representing an improvement in the design of quantum error-correcting codes. The work details how this approach separates the control of logical phase from the simultaneous design of code rate and distance, enabling the creation of two constructions with differing computational demands. This comparison highlights the increasing efficiency of these codes and how close current designs are to theoretical limits of addressability.

Numerical values were rounded to the displayed precision, with the resulting parameters detailed in a table within the paper. The evaluation matrices used in the construction are generated deterministically in polynomial time, with binary expansion, basis completion, and ring-matrix inversion also completed within the same timeframe.

Section S6 of the supplemental material verifies the addressable logical gates and exact distances of these finite examples, and describes an ancilla-free encoder for the 48-qubit code. The team’s analysis of Garcia-Stichtenoth towers over F_(256) yields AG code families with positive asymptotic rate and relative minimum distance, where degrees satisfy 2g_i-2 < a_(0,i) < a_(1,i) < n_i for sufficiently large ‘i’. Substituting into Eq. (S13), with N_i = 10hn_i, yields finite-length guarantees of K_i/N_i ≥ 52/675h-4/5hn_i and D_i/N_i ≥ 52/675h, demonstrating the practical implications of these theoretical advancements.

Exponential Construction Improves Rate to 7.34%

An exponential-time construction gives an exact verification, improving a lower bound of 52/(675h) on both rate and relative distance by a factor of approximately 2.86. The researchers also establish a universal upper bound for codes supporting fully addressable transversal single-qubit phase gates. Within their cyclic encoding framework, they obtain a stronger upper bound that is approximately 1.28 times the lower bound achieved by their exponential-time construction.

Scaling of Rate and Distance with Clifford Hierarchy Level

The resulting approach yields two distinct construction methods, differing in computational cost but both aimed at enhancing fully addressable transversal phase gates at each fixed level h ≥ 3 of the Clifford hierarchy. This bound effectively doubles the certified physical-qubit distance at a 25% increase in blocklength, while maintaining the number of logical qubits.

For the T gate specifically, this construction initially provides a lower bound of approximately 2.57% on both rate and distance; however, an exponential-time construction using a counting argument over encoding matrices and quadratic phases elevates this guarantee to approximately 7.34%. The researchers detail how, at m = 8 and *κ* = 1/15, balancing rate and distance necessitates specific parameter choices, a_0,i = and a_1,i =, to satisfy genus bounds and rounding conditions.

These calculations ensure that for every i ≥ 1, the inequalities 2*g*_i-2 < a_0,i < a_1,i < n_i hold true. The work establishes a connection between the encoding and classical binary codes, extracting a classical [ M_i, K_i, ≥ D_i ] code with a lower-limit rate of hR_∞ and a lower-limit relative distance of at least hd_∞, where the positive rate and Plotkin bound ensure hd_∞ < 1/2. Theorem S1, as presented in the paper, formalizes this relationship, linking quantum code parameters to the characteristics of the extracted classical code.

Separating Addressability from Rate and Distance Design

allows for the recovery of binary logical labels as linear functions of coordinates within the encoding matrix. This decoupling replaces previous constraints on code construction, specifically, higher-order overlap and divisibility conditions, with a simpler linear coordinate identity, streamlining the process of building error-correcting codes for quantum computers. The polynomial-time construction detailed in the research begins with nested algebraic-geometric (AG) codes over the finite field F₂₅₆.

Universal Upper Bound for Transversal Z(h) Gate Codes

These codes are built using a cyclic encoding framework, moving beyond the limitations of traditional binary CSS approaches. This separation is key to achieving the improved bounds and demonstrates a pathway toward more flexible code designs. A deterministic polynomial-time algebraic-geometric construction gives a lower bound of 52/(675h) on both quantities.

👉 More information
🗞 Substantially Improved Rate and Distance for Addressable Transversal Phase Gates
✍️ Bujiao Wu and Tongyin Lin (International Quantum Academy, Shenzhen, 518048, China); Bin Cheng (Centre for Quantum Technologies, National University of Singapore, Singapore 117543)
🧠 ArXiv: https://arxiv.org/abs/2611.04964

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