Genetic braid optimization: A heuristic approach to compute quasiparticle braids

In topologically-protected quantum computation, quantum gates can be carried out by adiabatically braiding two-dimensional quasiparticles, reminiscent of entangled world lines. Bonesteel et al. [Phys. Rev. Lett. 95, 140503 (2005)], as well as Leijnse and Flensberg [Phys. Rev. B 86, 104511 (2012)] recently provided schemes for computing quantum gates from quasiparticle braids. Mathematically, the problem of executing a gate becomes that of finding a product of the generators (matrices) in that set that approximates the gate best, up to an error. To date, efficient methods to compute these gates only strive to optimize for accuracy. We explore the possibility of using a generic approach applicable to a variety of braiding problems based on evolutionary (genetic) algorithms. The method efficiently finds optimal braids while allowing the user to optimize for the relative utilities of accuracy and/or length. Furthermore, when optimizing for error only, the method can quickly produce efficient braids.

[1]  L. Balents,et al.  Fractionalization in an easy-axis Kagome antiferromagnet , 2002 .

[2]  S. Simon,et al.  Topological quantum computing with Read-Rezayi states. , 2008, Physical review letters.

[3]  R. Moessner,et al.  Three-dimensional resonating-valence-bond liquids and their excitations , 2003 .

[4]  Michael A. Nielsen,et al.  The Solovay-Kitaev algorithm , 2006, Quantum Inf. Comput..

[5]  H. Rieger,et al.  New Optimization Algorithms in Physics , 2004 .

[6]  Giuseppe Mussardo,et al.  Topological quantum hashing with the icosahedral group. , 2010, Physical review letters.

[7]  Thierry Paul,et al.  Quantum computation and quantum information , 2007, Mathematical Structures in Computer Science.

[8]  Leon Balents,et al.  Numerical evidences of fractionalization in an easy-axis two-spin heisenberg antiferromagnet. , 2005, Physical review letters.

[9]  P Zoller,et al.  Atomic quantum simulator for lattice gauge theories and ring exchange models. , 2005, Physical review letters.

[10]  Matthias Troyer,et al.  Engineering exotic phases for topologically protected quantum computation by emulating quantum dimer models , 2007, 0708.0191.

[11]  Alexei Kitaev,et al.  Anyons in an exactly solved model and beyond , 2005, cond-mat/0506438.

[12]  Michel Ferrero,et al.  Zero-temperature properties of the quantum dimer model on the triangular lattice , 2005, cond-mat/0502294.

[13]  S. Simon,et al.  Braid topologies for quantum computation. , 2005, Physical review letters.

[14]  H. Bombin,et al.  Topological quantum distillation. , 2006, Physical review letters.

[15]  Xin Wan,et al.  Constructing functional braids for low-leakage topological quantum computing , 2008, 0802.4213.

[16]  E. Belmont-Moreno.  THE ROLE OF MUTATION AND POPULATION SIZE IN GENETIC ALGORITHMS APPLIED TO PHYSICS PROBLEMS , 2001 .

[17]  Steven H. Simon,et al.  BRAIDING AND ENTANGLEMENT IN NONABELIAN QUANTUM HALL STATES , 2009 .

[18]  Alex Bocharov,et al.  Resource-optimal single-qubit quantum circuits. , 2012, Physical review letters.

[19]  Parsa Bonderson,et al.  Detecting non-Abelian statistics in the nu = 5/2 fractional quantum hall state. , 2006, Physical review letters.

[20]  D. Rokhsar,et al.  Superconductivity and the quantum hard-core dimer gas. , 1988, Physical review letters.

[21]  L. Landau.  Fault-tolerant quantum computation by anyons , 2003 .

[22]  Frank Wilczek,et al.  Fractional Statistics and the Quantum Hall Effect , 1984 .

[23]  F. E. Camino,et al.  Transport in the Laughlin quasiparticle interferometer: Evidence for topological protection in an anyonic qubit , 2006 .

[24]  Gregory W. Moore,et al.  Nonabelions in the fractional quantum Hall effect , 1991 .

[25]  Martin Leijnse,et al.  Hybrid topological-spin qubit systems for two-qubit-spin gates , 2012, 1206.2455.

[26]  S V Isakov,et al.  Spin-liquid phase in a spin-1/2 quantum magnet on the kagome lattice. , 2006, Physical review letters.

[27]  Wei Zhou,et al.  Aharonov-Bohm superperiod in a Laughlin quasiparticle interferometer. , 2005, Physical review letters.

[28]  P. Zoller,et al.  A toolbox for lattice-spin models with polar molecules , 2006 .

[29]  S. Simon,et al.  Topological quantum compiling , 2006, quant-ph/0610111.

[30]  R. Moessner,et al.  Resonating valence bond phase in the triangular lattice quantum dimer model. , 2001, Physical review letters.

[31]  P. Zoller,et al.  Condensed Matter Physics with Cold Polar Molecules , 2008, 0805.1896.

[32]  G Misguich,et al.  Quantum dimer model on the kagome lattice: solvable dimer-liquid and ising gauge theory. , 2002, Physical review letters.

[33]  Haitan Xu,et al.  Unified approach to topological quantum computation with anyons: From qubit encoding to Toffoli gate , 2010, 1001.4085.

[34]  L. B. Ioffe,et al.  Topologically protected quantum bits from Josephson junction arrays , 2022 .

[35]  Chetan Nayak,et al.  Extended hubbard model with ring exchange: a route to a non-Abelian topological phase. , 2005, Physical review letters.