Unitary fault-tolerant encoding of Pauli states in surface codes

TL;DR

This work introduces a unitary, distance-preserving encoding scheme for preparing Pauli eigenstates in surface codes that uses only geometrically local gates and achieves a circuit depth of $O(d)$. By generalizing RL-discovered strategies to arbitrary code distance and both rotated and unrotated surface codes, the authors provide two architectures (with and without ancilla qubits) for stabilizer-expanding circuits that avoid measurements. Numerical simulations under depolarizing noise show that the unitary encoding without ancillas can outperform standard stabilizer-measurement-based encoding, reducing logical error rates by up to an order of magnitude in some regimes and reducing measurement overhead on hardware platforms with costly measurements. The scheme thus bridges unitary and measurement-based encodings, offering a distance-preserving pathway for preparing high-quality Pauli states in fault-tolerant quantum computation, with particular relevance for trapped ions and neutral-atom platforms.

Abstract

In fault-tolerant quantum computation, the preparation of logical states is a ubiquitous subroutine, yet significant challenges persist even for the simplest states required. In the present work, we present a unitary, scalable, distance-preserving encoding scheme for preparing Pauli eigenstates in surface codes. Unlike previous unitary approaches whose fault-distance remains constant with increasing code distance, our scheme ensures that the protection offered by the code is preserved during state preparation. Building on strategies discovered by reinforcement learning for the surface-17 code, we generalize the construction to arbitrary code distances and both rotated and unrotated surface codes. The proposed encoding relies only on geometrically local gates, and is therefore fully compatible with planar 2D qubit connectivity, and it achieves circuit depth scaling as $\mathcal{O}(d)$, consistent with fundamental entanglement-generation bounds. We design explicit stabilizer-expanding circuits with and without ancilla-mediated connectivity and analyze their error-propagation behavior. Numerical simulations under depolarizing noise show that our unitary encoding without ancillas outperforms standard stabilizer-measurement-based schemes, reducing logical error rates by up to an order of magnitude. These results make the scheme particularly relevant for platforms such as trapped ions and neutral atoms, where measurements are costly relative to gates and idling noise is considerably weaker than gate noise. Our work bridges the gap between measurement-based and unitary encodings of surface-code states and opens new directions for distance-preserving state preparation in fault-tolerant quantum computation.

Figures (10)

• Figure 1: a) and c) Rotated surface code of code distance $d=3$. b) and d) Unrotated surface code of distance $d=3$. Red and blue denote $X$ and $Z$ stabilizers respectively. Logical operators $Z_L$ and $X_L$ are defined along the vertical and horizontal boundary, respectively. Yellow and dark blue dots represent data and ancilla qubits respectively. The black lines represent the qubit connectivity.
• Figure 2: Circuits for expanding the $X$ stabilizers. a) and b) Circuit for coherently initializing an $X$ plaquette with and without ancilla in the middle, respectively. Each circuit assumes that the pivot qubit (positioned at the top of each diagram) is initialized in the state $|+\rangle$ (shown as a Hadamard gate in this depiction). The circuit then transforms the pivot qubit into an $X$-type stabilizer. This procedure is general and applies to stabilizers of any weight, although only weight-two and weight-four examples are shown. Notably, the construction requires at most one ancilla qubit—if any—independent of the stabilizer weight. Numbers indicate the order of the gates to avoid proliferation of dangerous hook errors. c) and d) show the potential hook errors for the circuit realization with and without ancilla, respectively. In addition, several potentially dangerous hook $Z$ errors are discussed in detail in Appendix \ref{['appendix:ft_both']}.
• Figure 3: Unitary FT encoding for the $d=5$ rotated surface code using ancillas as bridge qubits. In this layout, blue (red) denotes the $Z$ ($X$) stabilizers and the logical operators $X_L$ and $Z_L$ lie along the horizontal and vertical directions, respectively. Step 1 initializes the data and ancilla qubits in the states $|0\rangle$ except the pivot qubits that are initialized in the state $|+\rangle$. The blue irregular regions at this stage denote possible groupings of $|0\rangle$ qubits that will form the standard $Z$ stabilizers at the end of the encoding procedure. Steps 2 through 6 prepare the $X$-type stabilizers on the topmost row. Each plaquette within the row can be prepared in parallel. Steps 7 through 11 perform the preparation of $X$ stabilizers on the second row. Steps 12 through 16 prepare the $X$ stabilizers on the third row. Here, the location of the weight-two $X$ stabilizer alternates between the left and right boundaries, such that the gate scheduling appears the same only every second row. Finally, steps 17 through 21 complete the preparation of the $X$ stabilizers on the bottom row. Step 22 depicts the whole set of stabilizers after the last gate is applied. At the end of the circuit, only correctable weight-two $X$ errors remain. However, certain hook $Z$ errors may still be potentially harmful; these are discussed in detail in Appendix \ref{['appendix:ft_both']}.
• Figure 4: Unitary FT encoding circuit for the $d=5$ rotated surface code with full intra-plaquette connectivity. This encoding circuit assumes full connectivity among data qubits within each plaquette, eliminating the need for ancilla-mediated interactions. As a result, the total circuit depth is reduced to 13 time steps, compared to the 21 steps required in the ancilla-based version shown in Fig. \ref{['fig:circuit_ancilla']}. This reduction is due to the simplification of the stabilizer-expanding circuits, which require two fewer gates per stabilizer, as illustrated in Fig. \ref{['fig:stab_circuits']}.
• Figure 5: Logical error rate $p_L$ for preparing $|0\rangle_L$ for the unitary encoding schemes with, panels a) and b), and without, panels c) and d), ancilla for the rotated and unrotated surface code for $d=3,5,7,9,11$. $p$ is the physical error rate of gates and resets. Dashed lines are the measurement encoding (ME) while solid lines show different unitary encodings. For both types of codes, the encodings without ancillas perform better than the respective measurement encoding.