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Optimal local unitary encoding circuits for the surface code

Oscar Higgott, Wilson, M, Hefford, J, James Dborin, Hanif, F, Burton, S, Dan E. Browne

Published Jan 1, 2021• 2 citations
OpenAlex

Abstract

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.

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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.
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.
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 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 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.
Abstract. In recent years, surface codes have become the preferred method for quantum error correction in large scale computational and communications architectures. Their comparatively high fault-tolerant thresholds and their natural 2-dimensional nearest neighbour (2DNN) structure make them an obvious choice for large scale designs in experimentally realistic systems. While fundamentally based on the toric code of Kitaev, there are many variants, two of which are the planar-and defect- based codes. Planar codes require fewer qubits to implement (for the same strength of error correction), but are restricted to encoding a single qubit of information. Interactions between encoded qubits are achieved via transversal operations, thus destroying the inherent 2DNN nature of the code. In this paper we introduce a new technique enabling the coupling of two planar codes without transversal operations, maintaining the 2DNN of the encoded computer. Our lattice surgery technique comprises splitting and merging planar code surfaces, and enables us to perform universal quantum computation (including magic state injection) while removing the need for braided logic in a strictly 2DNN design, and hence reduces the overall qubit resources for logic operations. We show how lattice surgery allows us to distribute encoded GHZ states in a more direct (and overhead friendly) manner, and how a demonstration of an encoded CNOT between two distance 3 logical states is possible with 53 physical qubits, half of that required in any other known construction in 2D. ar
We propose a systematic and efficient quantum circuit composed solely of Clifford gates for simulating the ground state of the surface code model. This approach yields the ground state of the toric code in ⌈2L+2+log2(d)+L2d⌉ time steps, where L refers to the system size and d represents the maximum distance to constrain the application of the CNOT gates. Our algorithm reformulates the problem into a purely geometric one, facilitating its extension to attain the ground state of certain 3D topological phases, such as the 3D toric model in 3L+8 steps and the X-cube fracton model in 12L+11 steps. Furthermore, we introduce a gluing method involving measurements, enabling our technique to attain the ground state of the 2D toric code on an arbitrary planar lattice and paving the way to more intricate 3D topological phases.