Optimal local unitary encoding circuits for the surface code
Oscar Higgott, Matthew Wilson, James Hefford, James Dborin, Farhan Hanif, Simon Burton, Dan E. Browne
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. (2006) showed that encoding a state in the surface code using local unitary operations requires time at least linear in the lattice size <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>L</mml:mi></mml:math>, however the most efficient known method for encoding an unknown state, introduced by Dennis et al. (2002), has <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>L</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math> time complexity. Here, we present an optimal local unitary encoding circuit for the planar surface code that uses exactly <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>2</mml:mn><mml:mi>L</mml:mi></mml:math> time steps to encode an unknown state in a distance <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>L</mml:mi></mml:math> planar code. We further show how an <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>L</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> complexity local unitary encoder for the toric code can be found by enforcing locality in the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>log</mml:mi><mml:mo></mml:mo><mml:mi>L</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>-depth non-local renormalisation encoder. We relate these techniques by providing an <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>L</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> 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.