# 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…

## 21 Citations

Entangling logical qubits with lattice surgery

- PhysicsNature
- 2021

Entanglement between two logical qubits is demonstrated and logical state teleportation between them and the demonstration of these operations—fundamental building blocks for quantum computation—through lattice surgery represents a step towards the efficient realization of fault-tolerant quantum computation.

Formal Methods in Quantum Circuit Design

- Computer Science
- 2019

The development of a formal model of quantum circuits with ancillary bits based on the Feynman path integral, along with a concrete verification algorithm, which doubles as a natural specification language for quantum computations.

A Game of Surface Codes: Large-Scale Quantum Computing with Lattice Surgery

- Computer Science, PhysicsQuantum
- 2019

No knowledge of quantum error correction is necessary to understand the schemes in this paper, but only the concepts of qubits and measurements, which are based on surface-code patches.

Optimal Space-Depth Trade-Off of CNOT Circuits in Quantum Logic Synthesis

- Computer ScienceSODA
- 2020

An asymptotically optimal space-depth trade-off for the design of CNOT circuits is established, and it is proved that for any $m\geq0$-qubit CNOT circuit can be parallelized to O(m) depth, with ancillae, showing that this bound is tight by a counting argument.

Quantum circuit optimizations for NISQ architectures

- Computer Science, PhysicsQuantum Science and Technology
- 2020

This work constructs a circuit synthesis scheme that takes as input the qubit connectivity graph and a quantum circuit over the gate set generated by { CNOT, R Z } and outputs a circuit that respects the connectivity of the device.

Optimized Surface Code Communication in Superconducting Quantum Computers

- Physics, Computer Science2017 50th Annual IEEE/ACM International Symposium on Microarchitecture (MICRO)
- 2017

This paper evaluates two established quantum error correction codes-planar and double-defect surface codes-using a set of compilation, scheduling and network simulation tools, and finds that the simpler planar codes are sometimes more favorable for implementation on superconducting quantum computers, especially under conditions of high communication congestion.

NISQ circuit compilation is the travelling salesman problem on a torus

- Computer Science
- 2020

This work bridges a theoretical and practical gap between classical circuit design automation and the emerging field of quantum circuit optimisation with a novel approach to quantum circuit compilation (QCC).

On the controlled-NOT complexity of controlled-NOT–phase circuits

- Computer ScienceQuantum Science and Technology
- 2018

It is proved that CNOT minimization over CNOT and phase gates is at least as hard as synthesizing a CNOT -optimal circuit computing a set of parities of its inputs, and presented an efficient heuristic algorithm for synthesizing circuits over C NOT and Z-basis rotations with small CNOT cost.

Blueprint for a Scalable Photonic Fault-Tolerant Quantum Computer

- PhysicsQuantum
- 2021

The proposed architecture enables exploiting state-of-the-art procedures for the non-deterministic generation of bosonic qubits combined with the strengths of continuous-variable quantum computation, namely the implementation of Clifford gates using easy-to-generate squeezed states.

On the CNOT-complexity of CNOT-PHASE circuits

- Computer Science
- 2017

It is proved that CNOT minimization over CNOT and phase gates is at least as hard as synthesizing a CNOT-optimal circuit computing a set of parities of its inputs as well as NP-complete for two restricted cases where all CNOT gates are required to have the same target.

## References

SHOWING 1-10 OF 38 REFERENCES

Surface code quantum computing by lattice surgery

- Computer Science
- 2012

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

- Computer Science, Physics
- 2013

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

- Computer Science
- 2016

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

- Physics
- 2002

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

- Physics, Computer Science
- 2012

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

- Physics, Computer Science
- 2014

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

- Computer Science
- 2016

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

- Physics, Computer ScienceNature
- 2014

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

- Computer Science, Physics
- 2009

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

- PhysicsScience Advances
- 2015

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.