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arXiv:2511.22290

Untangling Surface Codes: Bridging Braids and Lattice Surgery

Alexandru Paler

quant-phSubmitted Nov 27, 2025• Announced Nov 28, 2025• 0 citations
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Abstract

We present a systematic method for translating fault-tolerant quantum circuits between their braiding and lattice surgery (LS) representations within the surface code. Our approach employs the ZX calculus to establish an equivalence between these two paradigms, enabling verified, bidirectional conversion of arbitrary surface-code-level circuits. We show that both braiding and LS operations can be uniformly expressed as compositions of multibody measurements and demonstrate that the Raussendorf compression rule encompasses all known braid and bridge optimizations. We also introduce a novel CNOT circuit with LS. Our framework provides a foundation for the automated verification, compilation, and benchmarking of large-scale surface code computations, advancing toward a unified formal language for topological quantum computation.

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The traditional method for computation in either the surface code or in the Raussendorf model is the creation of holes or “defects” within the encoded lattice of qubits that are manipulated via topological braiding to enact logic gates. However, this is not the only way to achieve universal, fault-tolerant computation. In this work, we focus on the lattice surgery representation, which realizes transversal logic operations without destroying the intrinsic 2D nearest-neighbor properties of the braid-based surface code and achieves universality without defects and braid-based logic. For both techniques there are open questions regarding the compilation and resource optimization of quantum circuits. Optimization in braid-based logic is proving to be difficult and the classical complexity associated with this problem has yet to be determined. In the context of lattice-surgery-based logic, we can introduce an optimality condition, which corresponds to a circuit with the lowest resource requirements in terms of physical qubits and computational time, and prove that the complexity of optimizing a quantum circuit in the lattice surgery model is NP-hard.

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String Diagrams for Defect-Based Surface Code Computing

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Surface codes are a popular choice for implementing fault-tolerant quantum computing.Two-qubit gates may be realised in these codes using only nearest-neighbour interactions, either by lattice surgery [27] or by braiding defects around each other [43].The effect of lattice surgery operations may be simply described [3] using the ZX-calculus: a graphical language that has proven effective for program design and optimisation (see e.g.[23]).In this work, we formalise a similar description via the ZX-calculus of defect braiding, as it is conventionally described.We define a graphical calculus KNOT, denoting the logical effects (in the absence of byproduct operations) of defect braiding in surface codes: we show how these effects may be described via a fragment of ZX-calculus which we call the (0, π)-fragment.We then use a 'doubling' construction to define a subtheory of KNOT, more specialised to standard encoding techniques in the defect braiding literature.Within this subtheory, we encompass standard braiding techniques by families of 'ribbon-like' and 'tangle-like' diagrams, each with semantics distinct from KNOT, in terms of the (0, π)-fragment of ZX diagrams (again in the absence of byproducts).These subtheories may be used interoperably, and are each sound and complete for the (0, π)-fragment of ZX diagrams.This provides a starting point to use the formal diagrammatics to analyse the operational effects of defect braiding procedures.* Corresponding author; author order is reverse-alphabetical, with the purpose of sowing confusion.

arXiv:2508.14672•Electronic Proceedings in Theoretical Computer Science•Aug 19, 2025•1 citation
We present a systematic method for translating fault-tolerant quantum circuits between their braiding and lattice surgery (LS) representations within the surface code. Our approach employs the ZX calculus to establish an equivalence between these two paradigms, enabling verified, bidirectional conversion of arbitrary surface-code-level circuits. We show that both braiding and LS operations can be uniformly expressed as compositions of multibody measurements and demonstrate that the Raussendorf compression rule encompasses all known braid and bridge optimizations. We also introduce a novel CNOT circuit with LS. Our framework provides a foundation for the automated verification, compilation, and benchmarking of large-scale surface code computations, advancing toward a unified formal language for topological quantum computation.
The traditional method for computation in either the surface code or in the Raussendorf model is the creation of holes or “defects” within the encoded lattice of qubits that are manipulated via topological braiding to enact logic gates. However, this is not the only way to achieve universal, fault-tolerant computation. In this work, we focus on the lattice surgery representation, which realizes transversal logic operations without destroying the intrinsic 2D nearest-neighbor properties of the braid-based surface code and achieves universality without defects and braid-based logic. For both techniques there are open questions regarding the compilation and resource optimization of quantum circuits. Optimization in braid-based logic is proving to be difficult and the classical complexity associated with this problem has yet to be determined. In the context of lattice-surgery-based logic, we can introduce an optimality condition, which corresponds to a circuit with the lowest resource requirements in terms of physical qubits and computational time, and prove that the complexity of optimizing a quantum circuit in the lattice surgery model is NP-hard.
Mapping a quantum algorithm to any practical large-scale quantum computer will require a sequence of compilations and optimizations.At the level of fault-tolerant encoding, one likely requirement of this process is the translation into a topological circuit, for which braided circuits represent one candidate model.Given the large overhead associated with encoded circuits, it is paramount to reduce their size in terms of computation time and qubit number through circuit compression.While these optimizations have typically been performed in the language of three-dimensional diagrams, such a representation does not allow an efficient, general, and scalable approach to reduction or verification.We propose the use of the ZX-calculus as an intermediate language for braided circuit compression, demonstrating advantage by comparing results using this approach with those previously obtained for the compression of jAi and jYi state distillation circuits.We then provide a benchmark of our method against a small set of Clifford þ T circuits, yielding compression percentages of ∼77%.Our results suggest that the overheads of braided, defect-based circuits are comparable to those of their lattice-surgery counterparts, restoring the potential of this model for surface-code quantum computation.
Abstract Lattice surgery is a method to perform quantum computation fault-tolerantly by using operations on boundary qubits between different patches of the planar code. This technique allows for universal planar code computation without eliminating the intrinsic two-dimensional nearest-neighbor properties of the surface code that eases physical hardware implementations. Lattice surgery approaches to algorithmic compilation and optimization have been demonstrated to be more resource efficient for resource-intensive components of a fault-tolerant algorithm, and consequently may be preferable over braid-based logic. Lattice surgery can be extended to the Raussendorf lattice, providing a measurement-based approach to the surface code. In this paper we describe how lattice surgery can be performed on the Raussendorf lattice and therefore give a viable alternative to computation using braiding in measurement-based implementations of topological codes.
Abstract The traditional method for computation in either the surface code or in the Raussendorf model is the creation of holes or “defects” within the encoded lattice of qubits that are manipulated via topological braiding to enact logic gates. However, this is not the only way to achieve universal, fault-tolerant computation. In this work, we focus on the lattice surgery representation, which realizes transversal logic operations without destroying the intrinsic 2D nearest-neighbor properties of the braid-based surface code and achieves universality without defects and braid-based logic. For both techniques there are open questions regarding the compilation and resource optimization of quantum circuits. Optimization in braid-based logic is proving to be difficult and the classical complexity associated with this problem has yet to be determined. In the context of lattice-surgery-based logic, we can introduce an optimality condition, which corresponds to a circuit with the lowest resource requirements in terms of physical qubits and computational time, and prove that the complexity of optimizing a quantum circuit in the lattice surgery model is NP-hard.
Lattice surgery is a leading approach for implementing fault-tolerant logical operations in surface code quantum computing, but compiling efficient lattice surgery layouts remains challenging. Existing compilers are largely circuit-centric and operate directly on gate sequences, limiting their ability to exploit the topological flexibility of merge-split operations and minimize space--time volume. We present TopoLS, a topology-centric compiler that uses ZX diagrams as an intermediate representation for lattice surgery compilation. TopoLS combines semantic-preserving ZX-level program transformations, including spider fusion and topology-aware slicing, with a Monte Carlo Tree Search (MCTS)-based synthesis procedure that constructs pipe-diagram embeddings by jointly optimizing placement and routing in 3D space--time. To scale to large circuits, TopoLS further introduces topology-aware partitioning that decomposes the compilation task into bounded subproblems and limits the routing frontier during embedding. Across evaluated benchmarks, TopoLS achieves an average 46%46\%46% reduction in space--time volume over prior circuit-centric compilers, with improvements ranging from 25%25\%25% to 90%90\%90%, and exhibits strong empirical scalability on large benchmark families. Compared with SAT-based formulations that become intractable on larger instances, TopoLS offers a practical end-to-end solution for optimized lattice surgery compilation. TopoLS has been integrated into the TQEC ecosystem, enabling downstream circuit-level simulation and resource estimation workflows.
Surface codes are a popular choice for implementing fault-tolerant quantum computing.Two-qubit gates may be realised in these codes using only nearest-neighbour interactions, either by lattice surgery [27] or by braiding defects around each other [43].The effect of lattice surgery operations may be simply described [3] using the ZX-calculus: a graphical language that has proven effective for program design and optimisation (see e.g.[23]).In this work, we formalise a similar description via the ZX-calculus of defect braiding, as it is conventionally described.We define a graphical calculus KNOT, denoting the logical effects (in the absence of byproduct operations) of defect braiding in surface codes: we show how these effects may be described via a fragment of ZX-calculus which we call the (0, π)-fragment.We then use a 'doubling' construction to define a subtheory of KNOT, more specialised to standard encoding techniques in the defect braiding literature.Within this subtheory, we encompass standard braiding techniques by families of 'ribbon-like' and 'tangle-like' diagrams, each with semantics distinct from KNOT, in terms of the (0, π)-fragment of ZX diagrams (again in the absence of byproducts).These subtheories may be used interoperably, and are each sound and complete for the (0, π)-fragment of ZX diagrams.This provides a starting point to use the formal diagrammatics to analyse the operational effects of defect braiding procedures.* Corresponding author; author order is reverse-alphabetical, with the purpose of sowing confusion.