Optimization of lattice surgery is NP-hard

@article{Herr2017OptimizationOL,
  title={Optimization of lattice surgery is NP-hard},
  author={Daniel Herr and Franco Nori and Simon J. Devitt},
  journal={npj Quantum Information},
  year={2017},
  volume={3},
  pages={1-5}
}
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… 
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References

SHOWING 1-10 OF 38 REFERENCES
Surface code quantum computing by lattice surgery
TLDR
This paper introduces a new technique enabling the coupling of two planar codes without transversal operations, maintaining the 2DNN of the encoded computer, and shows how lattice surgery allows us to distribute encoded GHZ states in a more direct manner, and how a demonstration of an encoded CNOT between two distance-3 logical states is possible with 53 physical qubits.
Quantum circuit optimization by topological compaction in the surface code
TLDR
This work examines the problem of minimizing computation time on a two-dimensional qubit lattice of arbitrary, but fixed dimension, and proposes two algorithms for doing so.
Poking holes and cutting corners to achieve Clifford gates with the surface code
TLDR
It is shown how all of the Clifford gates can be implemented with the planar code without loss of distance using code deformations, thus offering an attractive alternative to ancilla-mediated schemes to complete the Clifford group with lattice surgery.
Topological quantum memory
We analyze surface codes, the topological quantum error-correcting codes introduced by Kitaev. In these codes, qubits are arranged in a two-dimensional array on a surface of nontrivial topology, and
Surface codes: Towards practical large-scale quantum computation
TLDR
The concept of the stabilizer, using two qubits, is introduced, and the single-qubit Hadamard, S and T operators are described, completing the set of required gates for a universal quantum computer.
Logic gates at the surface code threshold: Superconducting qubits poised for fault-tolerant quantum computing
TLDR
The results demonstrate that Josephson quantum computing is a high-fidelity technology, with a clear path to scaling up to large-scale, fault-tolerant quantum circuits.
Lattice surgery translation for quantum computation
TLDR
A method for a compiler to translate any non fault tolerant quantum circuit to the geometric representation of the lattice surgery error-correcting code using inherent merge and split operations is outlined.
Superconducting quantum circuits at the surface code threshold for fault tolerance
TLDR
The results demonstrate that Josephson quantum computing is a high-fidelity technology, with a clear path to scaling up to large-scale, fault-tolerant quantum circuits.
MEASUREMENT-BASED QUANTUM COMPUTATION WITH CLUSTER STATES
TLDR
The one-way quantum computer is described, a scheme of universal quantum computation that consists entirely of one-qubit measurements on a highly entangled multiparticle state, i.e. the cluster state, which proves the universality of the , and establishes the link to the network model — the common model of quantum computation.
A surface code quantum computer in silicon
TLDR
A scalable shared-control architecture for silicon-based quantum computing using topological quantum error correction and a new pathway for large-scale quantum information processing in silicon and potentially in other qubit systems where uniformity can be exploited.
...
...