# Optimization of Lattice Surgery is NP-Hard

## TL;DR

This paper proves that optimizing quantum circuits in the lattice surgery model, an alternative to braid-based surface code methods, is an NP-hard problem, highlighting the computational difficulty of resource optimization.

## Contribution

It introduces an optimality condition for lattice surgery and proves that circuit optimization in this model is NP-hard, a novel complexity result.

## Key findings

- Optimization of lattice surgery circuits is NP-hard.
- Lattice surgery achieves universal quantum computation without defects.
- Classical complexity of braid-based optimization remains open.

## 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.