Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2509.09779

Lower-depth local encoding circuits for the surface code

Jahan Claes

Published Sep 11, 2025• 0 citations
arXivHTMLPDF

Abstract

The surface code is the most studied error-correcting code thanks to its high threshold, simple decoding, and locality in two dimensions (2D). A key component of any code is its encoding circuit, which maps an unencoded state to the corresponding encoded state. The best previous surface code encoding circuit compatible with 2D local connectivity requires depth $2d$ to encode distance-$d$ surface codes. This paper presents depth $d$ encoding circuits for the rotated surface code. Our circuit is constructed inductively from circuits that grow the code from $d$ to $(d+2)$. We prove that depth $d+O(1)$ is optimal for inductively constructed circuits.

DOI
10.48550/arxiv.2509.09779

Related papers

Optimal local unitary encoding circuits for the surface code

Oscar Higgott, Wilson, M, Hefford, J +4 more

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.

•Jan 1, 2021•2 citations

Unitary fault-tolerant encoding of Pauli states in surface codes

Luis Colmenarez, Remmy Zen, Jan Olle +2 more

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.

quant-pharXiv:2601.05113•npj Quantum Information•Jan 9, 2026•0 citations

Optimal local unitary encoding circuits for the surface code

Oscar Higgott, Matthew Wilson, James Hefford +4 more

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(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.

arXiv:2002.00362•Quantum•Aug 5, 2021•17 citations

A unitary encoder for surface codes

Pei-Kai Tsai, Shruti Puri

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.

arXiv:2506.04084•npj Quantum Information•Aug 4, 2026•0 citations

Local tensor-network codes

Terry Farrelly, David K. Tuckett, Thomas M. Stace

Abstract Tensor-network codes enable the construction of large stabilizer codes out of tensors describing smaller stabilizer codes. An application of tensor-network codes was an efficient and exact decoder for holographic codes. Here, we show how to write some topological codes, including the surface code and colour code, as simple tensor-network codes. We also show how to calculate distances of stabilizer codes by contracting a tensor network. The algorithm actually gives more information, including a histogram of all logical coset weights. We prove that this method is efficient in the case of stabilizer codes encoded via local log-depth circuits in one dimension and holographic codes. Using our tensor-network distance calculator, we find a modification of the rotated surface code that has the same distance but fewer minimum-weight logical operators by ‘doping’ the tensor network, i.e., we break the homogeneity of the tensor network by locally replacing tensors. For this example, this corresponds to an improvement in successful error correction of almost 2% against depolarizing noise (in the perfect-measurement setting), but comes at the cost of introducing three higher-weight stabilizers. Our general construction lets us pick a network geometry (e.g., a Euclidean lattice in the case of the surface code), and, using only a small set of seed codes (constituent tensors), build extensive codes with the potential for optimisation.

arXiv:2109.11996•New Journal of Physics•Mar 16, 2022•11 citations

Multi-path Summation for Decoding 2D Topological Codes

Ben Criger, Imran Ashraf

Fault tolerance is a prerequisite for scalable quantum computing. Architectures based on 2D topological codes are effective for near-term implementations of fault tolerance. To obtain high performance with these architectures, we require a decoder which can adapt to the wide variety of error models present in experiments. The typical approach to the problem of decoding the surface code is to reduce it to minimum-weight perfect matching in a way that provides a suboptimal threshold error rate, and is specialized to correct a specific error model. Recently, optimal threshold error rates for a variety of error models have been obtained by methods which do not use minimum-weight perfect matching, showing that such thresholds can be achieved in polynomial time. It is an open question whether these results can also be achieved by minimum-weight perfect matching. In this work, we use belief propagation and a novel algorithm for producing edge weights to increase the utility of minimum-weight perfect matching for decoding surface codes. This allows us to correct depolarizing errors using the rotated surface code, obtaining a threshold of 17.76 ± 0.02 % . This is larger than the threshold achieved by previous matching-based decoders ( 14.88 ± 0.02 % ), though still below the known upper bound of ∼ 18.9 % .

arXiv:1709.02154•Quantum•Oct 19, 2018•46 citations
The surface code is the most studied error-correcting code thanks to its high threshold, simple decoding, and locality in two dimensions (2D). A key component of any code is its encoding circuit, which maps an unencoded state to the corresponding encoded state. The best previous surface code encoding circuit compatible with 2D local connectivity requires depth 2d2d2d to encode distance-ddd surface codes. This paper presents depth ddd encoding circuits for the rotated surface code. Our circuit is constructed inductively from circuits that grow the code from ddd to (d+2)(d+2)(d+2). We prove that depth d+O(1)d+O(1)d+O(1) is optimal for inductively constructed circuits.
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.
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)\mathcal{O}(d)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.
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(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.
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.
Abstract Tensor-network codes enable the construction of large stabilizer codes out of tensors describing smaller stabilizer codes. An application of tensor-network codes was an efficient and exact decoder for holographic codes. Here, we show how to write some topological codes, including the surface code and colour code, as simple tensor-network codes. We also show how to calculate distances of stabilizer codes by contracting a tensor network. The algorithm actually gives more information, including a histogram of all logical coset weights. We prove that this method is efficient in the case of stabilizer codes encoded via local log-depth circuits in one dimension and holographic codes. Using our tensor-network distance calculator, we find a modification of the rotated surface code that has the same distance but fewer minimum-weight logical operators by ‘doping’ the tensor network, i.e., we break the homogeneity of the tensor network by locally replacing tensors. For this example, this corresponds to an improvement in successful error correction of almost 2% against depolarizing noise (in the perfect-measurement setting), but comes at the cost of introducing three higher-weight stabilizers. Our general construction lets us pick a network geometry (e.g., a Euclidean lattice in the case of the surface code), and, using only a small set of seed codes (constituent tensors), build extensive codes with the potential for optimisation.
Fault tolerance is a prerequisite for scalable quantum computing. Architectures based on 2D topological codes are effective for near-term implementations of fault tolerance. To obtain high performance with these architectures, we require a decoder which can adapt to the wide variety of error models present in experiments. The typical approach to the problem of decoding the surface code is to reduce it to minimum-weight perfect matching in a way that provides a suboptimal threshold error rate, and is specialized to correct a specific error model. Recently, optimal threshold error rates for a variety of error models have been obtained by methods which do not use minimum-weight perfect matching, showing that such thresholds can be achieved in polynomial time. It is an open question whether these results can also be achieved by minimum-weight perfect matching. In this work, we use belief propagation and a novel algorithm for producing edge weights to increase the utility of minimum-weight perfect matching for decoding surface codes. This allows us to correct depolarizing errors using the rotated surface code, obtaining a threshold of 17.76 ± 0.02 % . This is larger than the threshold achieved by previous matching-based decoders ( 14.88 ± 0.02 % ), though still below the known upper bound of ∼ 18.9 % .