The surface code is a promising candidate for fault-tolerant quantum computation and has been implemented in many quantum hardware platforms. In this work, we propose a new non-local unitary circuit to encode surface code states based on a code conversion between rotated and regular surface codes, which halves the circuit depth of the fastest unitary encoder known previously. While the unitary encoders can be used to increase the code size, the fault-distance remains fixed. Nonetheless, they can be used for space-time efficient preparation of eigenstates of the logical operators that cannot be easily accessed transversally such as the Pauli Y -eignestate and Clifford eigenstates. It may be expected that error propagation in the non-local circuit will make decoding more challenging compared to local ones. However, we find that conventional matching decoders can still be effectively used. Furthermore, we perform numerical simulations to benchmark the performance of our encoder against a local unitary encoder and the conventional stabilizer-measurement based approach for preparing the Pauli Y -eigenstate. We find that our encoder can outperform other approaches in certain experimentally relevant noise regimes. Therefore, our encoder provides practical advantage in platforms where non-local interactions are available such as neutral atoms and trapped ions.
The surface code is a promising candidate for fault-tolerant quantum computation and has been implemented in many quantum hardware platforms. In this work, we propose a new non-local unitary circuit to encode surface code states based on a code conversion between rotated and regular surface codes, which halves the circuit depth of the fastest unitary encoder known previously. While the unitary encoders can be used to increase the code size, the fault-distance remains fixed. Nonetheless, they can be used for space-time efficient preparation of eigenstates of the logical operators that cannot be easily accessed transversally such as the Pauli Y -eignestate and Clifford eigenstates. It may be expected that error propagation in the non-local circuit will make decoding more challenging compared to local ones. However, we find that conventional matching decoders can still be effectively used. Furthermore, we perform numerical simulations to benchmark the performance of our encoder against a local unitary encoder and the conventional stabilizer-measurement based approach for preparing the Pauli Y -eigenstate. We find that our encoder can outperform other approaches in certain experimentally relevant noise regimes. Therefore, our encoder provides practical advantage in platforms where non-local interactions are available such as neutral atoms and trapped ions.
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 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.
Preparing logical states is a constantly recurring primitive at the start of any surface-code-based quantum computation. The standard measurement-based protocol is expensive on neutral atoms, since measurements are orders of magnitude slower than gates and, on single-zone architectures, additionally require shuttling the atoms to a readout zone. In this work, we present a measurement-free, unitary preparation of surface-code Pauli eigenstates tailored to neutral atoms. A bidirectional stabilizer-expanding CNOT cascade grows the patch from its middle line outward in depth (d+3)/2, while retaining fault distance d in the protected error direction. Each layer maps to one collective atom move followed by a single global Rydberg pulse, with about one row pickup per two layers. We compile and simulate the resulting schedules with the open-source bloqade toolchain under a hardware-calibrated, circuit-level neutral-atom noise model. The bidirectional cascade achieves the lowest logical error rate at every distance, and is the only construction tested that operates below threshold for both logical states at the considered device-level noise.
The surface code is a leading candidate quantum error correcting code, owing to its high threshold, and compatibility with existing experimental architectures. Bravyi et al. (2006) showed that encoding a state in the surface code using local unitary operations requires time at least linear in the lattice size L , however the most efficient known method for encoding an unknown state, introduced by Dennis et al. (2002), has O(L2) time complexity. Here, we present an optimal local unitary encoding circuit for the planar surface code that uses exactly 2L time steps to encode an unknown state in a distance L planar code. We further show how an O(L) complexity local unitary encoder for the toric code can be found by enforcing locality in the O(logL) -depth non-local renormalisation encoder. We relate these techniques by providing an O(L) local unitary circuit to convert between a toric code and a planar code, and also provide optimal encoders for the rectangular, rotated and 3D surface codes. Furthermore, we show how our encoding circuit for the planar code can be used to prepare fermionic states in the compact mapping, a recently introduced fermion to qubit mapping that has a stabiliser structure similar to that of the surface code and is particularly efficient for simulating the Fermi-Hubbard model.
The surface code is a leading candidate quantum error correcting code, owing to its high threshold, and compatibility with existing experimental architectures. Bravyi {et al.} [7] showed that encoding a state in the surface code using local unitary operations requires time at least linear in the lattice size L, however the most efficient known method for encoding an unknown state, introduced by Dennis {et al.} [18], has O(L^{2}) time complexity. Here, we present an optimal local unitary encoding circuit for the planar surface code that uses exactly 2L time steps to encode an unknown state in a distance L planar code. We further show how an O(L) complexity local unitary encoder for the toric code can be found by enforcing locality in the O(log L) -depth non-local renormalisation encoder. We relate these techniques by providing an O(L) local unitary circuit to convert between a toric code and a planar code, and also provide optimal encoders for the rectangular, rotated and 3D surface codes. Furthermore, we show how our encoding circuit for the planar code can be used to prepare fermionic states in the compact mapping, a recently introduced fermion to qubit mapping that has a stabiliser structure similar to that of the surface code and is particularly efficient for simulating the Fermi-Hubbard model.
Quantum error correction represents a significant milestone in large-scale quantum computing, with the surface code being a prominent strategy due to its high error threshold and experimental feasibility. However, it is challenging to implement non-Clifford logical gates in a fault-tolerant way with low overhead, through the conventional magic-state distillation technique. Here, we enhance the performance of the conventional surface code by incorporating non-Pauli stabilizers and introduce two innovative initialization protocols. Our approach enhances the fidelity of the initialization of the non-Clifford logical state by avoiding unprotected operations before the encoding process. This improved fidelity of the initialization of non-Clifford logical states reduces the necessary number of logical qubits for precise state distillation, ultimately decreasing the overall resource overhead. Furthermore, we demonstrate the ability to entangle logical qubits in non-Pauli and Pauli bases via the lattice surgery technique. This integration enables the use of Pauli-based surface codes for computation while non-Pauli codes are employed for auxiliary qubit initialization, thus making them compatible with the conventional wisdom of logical Clifford operation based on the Pauli-based surface code.
Stabilizer quantum error correction (QEC) codes, in particular topological surface codes, are prime candidates to enable practical quantum computing. While it is widely believed that strictly fault-tolerant protocols can only be implemented using single- and two-qubit gates, several quantum computing platforms, including trapped ions, neutral atoms, and superconducting qubits, support native multi-qubit operations. In this Letter, we show that stabilizer measurement circuits for unrotated surface codes can be fault tolerant using single auxiliary qubits and three-qubit gates. These gates enable lower-depth circuits with fewer fault locations and potentially shorter QEC cycle times. We find that in an optimistic parameter regime where fidelities of three-qubit gates are the same as those of two-qubit gates, the logical error rate can be up to one order of magnitude lower and the threshold significantly higher, increasing from ≈0.63% to ≈0.83%. Our results, applicable to a wide range of platforms, motivate further investigation into multi-qubit gates for fault-tolerant QEC as they can offer substantial time and physical qubit resource advantages to reach a given target logical error rate.