Hardness of braided quantum circuit optimization in the surface code

TL;DR

This paper demonstrates that optimizing a subset of braided quantum circuits in surface codes is computationally hard (NP-hard), highlighting the complexity challenges in fault-tolerant quantum circuit compilation.

Contribution

It proves that the optimization problem for a specific class of braided quantum circuits is NP-hard through a polynomial-time reduction to Planar Rectilinear 3SAT.

Findings

Optimization of braided quantum circuits is NP-hard.

Highlights the computational difficulty in fault-tolerant quantum circuit design.

Provides a formal complexity classification for quantum circuit optimization.

Abstract

Large-scale quantum information processing requires the use of quantum error correcting codes to mitigate the effects of noise in quantum devices. Topological error-correcting codes, such as surface codes, are promising candidates as they can be implemented using only local interactions in a two-dimensional array of physical qubits. Procedures such as defect braiding and lattice surgery can then be used to realize a fault-tolerant universal set of gates on the logical space of such topological codes. However, error correction also introduces a significant overhead in computation time, the number of physical qubits, and the number of physical gates. While optimizing fault-tolerant circuits to minimize this overhead is critical, the computational complexity of such optimization problems remains unknown. This ambiguity leaves room for doubt surrounding the most effective methods for compiling fault-tolerant circuits for a large-scale quantum computer. In this paper, we show that the optimization of a special subset of braided quantum circuits is NP-hard by a polynomial-time reduction of the optimization problem into a specific problem called Planar Rectilinear 3SAT.