Borrowed Identities: Malleable Distillation Factories and a Unified Numerical Search

Technical Summary: Borrowed Identities and Unified Numerical Search for Magic-State Distillation

Problem Statement
Magic-state distillation (MSD) is a primary overhead in fault-tolerant quantum computation (FTQC), required to implement non-Clifford gates via gate teleportation. Existing methods for designing distillation factories rely on the "Heisenberg picture," demanding that a transversal gate act correctly on the entire codespace. This constraint, often formalized as tri-orthogonality or quasi-transversality, limits search generality and efficiency, turning the design problem into non-linear matrix equations. Furthermore, current frameworks typically treat factories for different magic states (e.g., $|T\rangle$, $|CS\rangle$, $|CCZ\rangle$) as distinct problems, preventing a unified search across the Clifford hierarchy.

Methodology: The Borrowed-Identity Framework
The authors introduce a strictly weaker condition called the "borrowed identity," formulated in the Schrödinger picture. Instead of requiring a circuit to act as the identity on the full codespace, the framework only requires that a circuit $C$ act as the identity (up to a global phase) on a specific input state:
$$C |+\rangle^{\otimes n} = e^{i\phi} |+\rangle^{\otimes n}$$
The "identity" is effectively borrowed from the input state. A distillation factory is constructed by taking such a borrowed identity circuit on $n$ qubits and removing all gates that act solely on a subset of $k$ output qubits. The remaining $n-k$ qubits are measured in the $X$ basis for error detection, while the $k$ output qubits are left in a distilled magic state $|\theta\rangle$.

The framework utilizes parity phase gates $Z^{\otimes w}(\theta)$ where $\theta = \pi/2^l$, corresponding to the $l$-th level of the Clifford hierarchy. The authors derive algebraic conditions for these circuits to function as factories, parameterized by $\theta$, which hold uniformly across all levels $l$ without modification.

The search is conducted in three settings:

1. Symmetric Circuits: Invariant under the full symmetric group $S_n$. This yields closed-form master equations (Theorem 3) that recover known families like the quantum Reed–Muller codes.
2. Two-Group Symmetric Circuits: Qubits are partitioned into $k$ output and $n-k$ check qubits, with symmetry within each group ($S_k \times S_{n-k}$). This allows for a brute-force search where the validity of sign assignments can be verified in polynomial time $O(k(n-k))$ via a linear condition over $\mathbb{Z}_{2^{l+1}}$ (Theorem 5).
3. Malleable Circuits: A sequential construction where a single parent borrowed identity encodes multiple factories. By removing gates sequentially, the output magic state type (e.g., $|T\rangle$, $|CS\rangle$, $|CCZ\rangle$) can be chosen at compile time rather than being fixed by the hardware design.

Key Contributions and Results

• Unified Search Space: The framework unifies the search for factories distilling different magic states (unentangled $T$, entangled $CS$, and $CCZ$) within a single numerical search. It recovers known factories from disparate approaches, including the Bravyi–Haah, H-code, generalized triorthogonal, and synthillation families.
• Recovery of Known and New Factories: A brute-force search over two-group asymmetric circuits (on a standard laptop) recovered all known distance-2 factories within the swept range ($l \in \{2,3,4\}$, $k \le 7$, $n \le 11$) in approximately 9 seconds. This includes:
  • The $[[2^{l+1}-1, 1, 3]]$ quantum Reed–Muller family (distance 3).
  • The $[[2^l, l, 2]]$ hypercube quantum code family.
  • The $[[12, 2, 2]]$ T-to-CS factory and $[[24, 6, 2]]$ T-to-CCZ factory, which match constructions derived from algebraic-geometry (AG) codes.
  • Previously unconnected families like the Campbell–Howard synthillation circuits, which break two-group symmetry.
• Malleability: The authors demonstrate that a single parent circuit can encode multiple distinct factories. For example, a four-qubit parent circuit can yield $[[8, 3, 2]]$ ($|CCZ\rangle$), $[[12, 2, 2]]$ ($|CS\rangle$), and $[[14, 1, 2]]$ ($|T\rangle$) simply by removing different sets of gates. This allows the output type to be deferred to compilation time.
• Extension Beyond CSS: The framework naturally extends to non-CSS constructions, including catalytic factories (e.g., $[[11, 1, 2]]$ and $[[10, 2, 2]]$) that utilize Clifford gates in addition to the borrowed identity components.

Significance
The paper claims that the borrowed-identity condition provides a "strictly weaker" and more general alternative to transversal gate constraints, enabling a unified, efficient, and exhaustive search for distillation factories. By shifting from the Heisenberg picture (tracking operator transversality) to the Schrödinger picture (tracking state evolution), the authors achieve a linear, level-uniform algebraic structure. This unification reveals that many previously distinct factory constructions are endpoints of a single parent circuit, offering a new degree of freedom for fault-tolerant compilation: the ability to select the specific magic state resource at compile time rather than hard-coding it into the circuit design. The work establishes a single algebraic foundation for distillation across all Clifford levels and output types, bridging the gap between code-based constructions and circuit-based searches.