Synthesis of Fault-tolerant State Preparation Circuits using Steane-type Error Detection

TL;DR

This work tackles fault-tolerant state preparation for Steane-type error correction by introducing a general, automated synthesis framework that does not rely on code symmetries. It combines Gaussian-elimination-based circuit construction with fault-set guided synthesis to produce four $t$-distinct FTSP circuits for arbitrary CSS codes, demonstrated up to distance $d\le 7$ with detailed simulations. The approach achieves low-depth, constant-overhead preparation while maintaining strong fault-tolerance guarantees, at the cost of reduced acceptance rates due to post-selection. The methodology and open-source tooling advance the practical realization of high-fidelity ancilla states for near-term fault-tolerant quantum computing.

Abstract

Fault-tolerant state preparation is essential for reliable quantum error correction, particularly in Steane-type error correction, which relies on robust ancilla states for syndrome readout. One method of fault-tolerant state preparation is to initialize multiple ancilla states and check them against each other to detect problematic errors. In the worst case, the number of states required for successful initialization grows polynomially with the code distance, but it has been shown that this can be reduced to constant ancilla overhead-in the best case, only four states are required. However, existing techniques for finding low-overhead initialization schemes are limited to codes with large symmetry groups, such as the Golay code. In this work, we propose a general, automated synthesis methodology for Steane-type fault-tolerant state preparation circuits that applies to arbitrary Calderbank-Shor-Steane (CSS) codes and does not rely on code symmetries. We apply the proposed methods to various CSS codes up to a distance of seven and simulate the successful fault-tolerant initialization of logical basis states under circuit-level depolarizing noise. The circuits synthesized using the proposed methodology provide an important step towards experimental realizations of high-fidelity ancilla states for near-term demonstration of fault-tolerant quantum computation.

Figures (5)

• Figure 1: Non-deterministic Steane-type state preparation. (a) The state is prepared by non-fault-tolerantly preparing multiple states and copying errors on data qubits over to ancilla qubits, which are measured out in the respective basis for error detection. This process is repeated until no errors are detected on any of the states. This process requires $O(nd^2)$ physical qubits and measurements if the states are prepared using the same circuit, since the ancilla need to be recursively verified using additional ancilla. (b) Four states are prepared using different state preparation circuits. This construction is fault-tolerant if the sets of propagated errors of all circuits are sufficiently different.
• Figure 2: Block I and II are state preparation circuits for the logical zero state of the code. Block I represents the data structure, and Block II is the ancilla structure. Block III is the connection of both state preparation circuits through transversal CNOT gates. Together, all three Blocks form the base structure of Steane-type error correction.
• Figure 3: State preparation circuits for the $\ket{0}_L$ of the $\llbracket19,1,5\rrbracket$ and $\llbracket17,1,5\rrbracket$ color codes bombinTopologicalQuantumDistillation2006. Black vertices represent qubits prepared in $\ket{+}$, and white vertices represent qubits prepared in $\ket{0}$. The CNOT order is indicated by the edge colors (orange, magenta, yellow, blue, and purple).
• Figure 4: Fault-set–guided synthesis of a state-preparation circuit. Given the check matrix $H_X$ and a reference fault set $\mathcal{E}_{\text{ref}}$, the algorithm constructs a circuit whose propagated fault set is $t$-distinct from $\mathcal{E}_{\text{ref}}$. Panels (a–c) illustrate three different scenarios in the search process: the evolving check matrix (top), the partially constructed circuit assembled from right to left (middle), and the remaining candidate CNOTs (bottom). Blue CNOTs satisfy the $t$-distinctness constraint, while red CNOTs violate it and are therefore pruned. (a) A valid CNOT is selected according to the cost function, and the fault set is updated. (b) A candidate CNOT violates $t$-distinctness and is discarded in favor of an alternative choice. (c) All candidates violate $t$-distinctness, forcing the algorithm to backtrack by removing previously placed CNOTs.
• Figure 5: Logical error and acceptance rates for Steane-type fault-tolerant state preparation circuits for the logical $\ket{0}_L$ state constructed with our methods for various $d = 5$, $d=7$ and $d = 9$ CSS codes under circuit-level noise using a LUT decoder.

Theorems & Definitions (5)

• Definition 1
• Definition 2
• Example 1
• Example 2
• Definition 3