Optimized Compilation of Logical Clifford Circuits

TL;DR

The paper addresses the high resource cost of gate-by-gate logical Clifford compilation in fault-tolerant quantum computing by introducing a peephole optimization framework that mines small instances of Clifford circuits on the $[[n,n-2,2]]$ code to derive depth-optimized primitives and closed-form strategies. It validates these primitives through noisy simulations, showing reductions in depth and higher logical success rates, and introduces three size-invariant strategies that adapt to sparse, moderate, and dense Hadamard placements, with edge regimes demonstrating the largest gains. The combination of LCS/SAS-derived primitives with scalable templates provides a compact, extensible tool for peephole-based Clifford compilation that scales to larger systems and other code families. While the method focuses on Clifford subcircuits within the $[[n,n-2,2]]$ code family and does not enforce full fault tolerance, its modular design invites extensions to higher-distance codes, broader Clifford subroutines, and limited non-Clifford integration.

Abstract

Fault-tolerant quantum computing hinges on efficient logical compilation, in particular, translating high-level circuits into code-compatible implementations. Gate-by-gate compilation often yields deep circuits, requiring significant overhead to ensure fault-tolerance. As an alternative, we investigate the compilation of primitives from quantum simulation as single blocks. We focus our study on the [[n,n-2,2]] code family, which allows for the exhaustive comparison of potential compilation primitives on small circuit instances. Based upon that, we then introduce a methodology that lifts these primitives into size-invariant, depth-efficient compilation strategies. This recovers known methods for circuits with moderate Hadamard counts and yields improved realizations for sparse and dense placements. Simulations show significant error-rate reductions in the compiled circuits. We envision the approach as a core component of peephole-based compilers. Its flexibility and low hand-crafting burden make it readily extensible to other circuit structures and code families.

Figures (7)

• Figure 1: Clifford quantum simulation kernel (C-QSK) on three qubits with a Hadamard gate only on the second qubit, i.e. $I_h = \left\{ 2 \right\}$ and $\left| I_h \right| = 1$.
• Figure 2: Full sequencing rule obtained by the $\mathcal{C}^{H}_{mid}$ compilation strategy, which is equivalent to the from chen2024tailoring (up to re-formulations with different notation). Operations highlighted in gray occur only for odd Hadamard configurations. The embedding function is given by $\mathcal{E}: [k] \rightarrow \{2,\ldots,n-1\}$ with $\mathcal{E}(i) = i+1$.
• Figure 3: Comparison of depth for logical realizations of on varying system sizes as a function of the Hadamard count $h$. We plot the four position-invariant solutions of for circuits on $k=20$ qubits. While solution $\mathcal{S}^{H}_{\text{mid}}$ produces particularly shallow circuits in the regime of about $5$ to $15$ Hadamard gates, $\mathcal{S}^{H}_{\text{low}}$ is clearly superior in case of sparse and $\mathcal{S}^{H}_{\text{high}}$ for dense Hadamard placement.
• Figure 4: Building blocks for logical realizations of with even Hadamard count: (1) CNOT-entanglement layer, (2) IQP-like structure, (3) Z-diagonal layer. While shown for $k=4$, the same structure is observed for all system sizes. The concrete instantiation of the blocks depends on the Hadamard count, position, and the choice of strategy $\mathcal{S}$.
• Figure 5: Exemplary instances of the three blocks from \ref{['fig:building_blocks']}, corresponding to solution $\mathcal{S}^{H}_{\text{mid}}$. The underlying circuits are of size $k=4$ and contain an even number of Hadamard gates. Based on just these few instances, it is possible to compactly formulate the compilation strategy in \ref{['eq:sequencing_rule_1', 'eq:sequencing_rule_2', 'eq:sequencing_rule_3']}. The full compilation strategy $\mathcal{C}^{H}_{\text{mid}}$ just sequentially concatenates the three compiled blocks.

Theorems & Definitions (6)

• Theorem 1
• Theorem 2
• Theorem
• proof
• Theorem
• proof