Stabilizer Code-Generic Universal Fault-Tolerant Quantum Computation

TL;DR

This work presents a stabilizer-code-generic framework for universal fault-tolerant quantum computation that uses ancilla-mediated protocols with the Generalized Shor Code ($GSC$) and its Hadamard dual ($GSCH$) to implement a Clifford+$T$ gate set across arbitrary stabilizer codes without altering the data codes. The approach enables deterministic, code-agnostic universal quantum computation and cross-code communication by mediating gates through ancillary $GSC$ registers and mid-circuit measurements, avoiding resource-intensive methods like magic state distillation. The paper provides detailed constructions for stabilizer-generic Hadamard, controlled-flip, and $Z$-rotation (including $T$) gates, analyzes resource overheads, and validates the framework through simulations showing correct logical transformations and error-rate scaling that respect the constituent codes’ distances. The results suggest a scalable, architecture-independent path toward heterogeneous quantum computing, with potential applications in distributed quantum systems and modular quantum architectures, while offering a comprehensive overhead framework through Supplementary Notes. Overall, the work broadens the landscape of universal fault-tolerant QC by decoupling gate implementations from specific codes and enabling interoperability between diverse stabilizer codes.

Abstract

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 universal quantum computation. This universality is highly desired and often achieved using additional techniques such as code concatenation, code switching, or magic state distillation, which can be costly and only work for specific codes. This work implements logical Clifford and T gates through novel ancilla-mediated protocols to construct a universal fault-tolerant quantum gate set. Unlike traditional techniques, our implementation is deterministic, does not consume ancilla registers, does not modify the underlying data codes or registers, and is generic over all 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. These features collectively open the door to countless possibilities for existing and undiscovered codes as well as their scalable, heterogeneous coexistence.

Figures (6)

• Figure 1: $GSC_{3,3}$ in a grid structure. Data qubits are represented by black circles with center labels. The $i$th subregister, $si$, is labeled and circled in green. Red operators indicate a collection of X gates, while blue operators indicate a collection of Z gates. Hence, the qubits used for the $\bar{X}$ logical gate are shown intersecting the horizontal, solid blue line, and those used for the $\bar{Z}$ logical gate are shown intersecting the vertical, solid red line. Those used for the X-stabilizers are shown intersecting the dotted, red squares, and those used for the Z-stabilizers are shown intersecting the dotted, blue lines.
• Figure 2: The transformation of states via the modified $GSCH_{3,b}$ X-stabilizers for the first iteration of $\bar{X}_{GSCH_{3,b}, C1}$, which performs $\bar{X}_{s1, C1}$, where the $GSCH_{3,b}$ register encodes $\ket{0}_{GSCH_{3,b}}$. The arrows indicate which pairs of states are transformed into each other given a stabilizer. $\ket{a..b..c..}$ is shorthand for $\bigotimes_{i \in \{a,b,c\}} \ket{i}^{\otimes b}$. Colors are used as a visual aid to distinguish different encodings.
• Figure 3: Diagram of the fault-tolerant process to perform logical flip gate $\bar{O}_{GSC_{3,b},C1}$. $(\ket{0..} + \ket{1..})$ is shorthand for $(\ket{0}^{\otimes b} + \ket{1}^{\otimes b})$, and $\ket{a..b..c..}$ is shorthand for $\bigotimes_{i \in \{a,b,c\}} \ket{i}^{\otimes b}$. Colors are used as a visual aid to distinguish different encodings.
• Figure 4: Circuit representation of fault-intolerant stabilizer-generic (a) $\bar{H}_{C1}$ and (b) $\bar{O}_{C1,C2}$ gates. The $A$ subscript denotes a single-qubit ancilla register, and $C1$ denotes an arbitrary target code.
• Figure 5: Circuit representation of fault-tolerant stabilizer-generic (a) $\bar{H}_{C1}$, (b) $\bar{O}_{C1,C2}$, and (c) $\bar{T}_{C1}$ gates. Gates controlled by a $GSC$ register use Lemma \ref{['lem:hcxh']}, and the double-qubit gate in (c) uses Theorem \ref{['thm:flip']}. The measurement in (c) is done in the X-basis. The $RC$ subscript denotes a code that had a fault-tolerant implementation of a desired logical $\bar{Z}^p$ gate.

Theorems & Definitions (5)

• Lemma 2.1
• Lemma 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5