Surface codes: Towards practical large-scale quantum computation

This article provides an introduction to surface code quantum computing. We first estimate the size and speed of a surface code quantum computer. We then introduce the concept of the stabilizer, using two qubits, and extend this concept to stabilizers acting on a two-dimensional array of physical qubits, on which we implement the surface code. We next describe how logical qubits are formed in the surface code array and give numerical estimates of their fault-tolerance. We outline how logical qubits are physically moved on the array, how qubit braid transformations are constructed, and how a braid between two logical qubits is equivalent to a controlled-NOT. We then describe the single-qubit Hadamard, S and T operators, completing the set of required gates for a universal quantum computer. We conclude by briefly discussing physical implementations of the surface code. We include a number of appendices in which we provide supplementary information to the main text.

[1]  Austin G. Fowler,et al.  Analytic asymptotic performance of topological codes , 2012, 1208.1334.

[2]  A. Fowler,et al.  A bridge to lower overhead quantum computation , 2012, 1209.0510.

[3]  J. Dalibard,et al.  Quantum simulations with ultracold quantum gases , 2012, Nature Physics.

[4]  Austin G. Fowler,et al.  Topological code Autotune , 2012, 1202.6111.

[5]  James R. Wootton,et al.  High threshold error correction for the surface code. , 2012, Physical review letters.

[6]  A. Fowler.  Low-overhead surface code logical H , 2012 .

[7]  Helmut G. Katzgraber,et al.  Strong resilience of topological codes to depolarization , 2012, 1202.1852.

[8]  R. Raussendorf,et al.  Efficient decoding of topological color codes , 2011, 1111.0831.

[9]  Austin G. Fowler,et al.  Towards practical classical processing for the surface code. , 2011, Physical review letters.

[10]  W. Marsden.  I and J , 2012 .

[11]  Erik Lucero,et al.  Implementing the Quantum von Neumann Architecture with Superconducting Circuits , 2011, Science.

[12]  Andrew J. Landahl,et al.  Fault-tolerant quantum computing with color codes , 2011, 1108.5738.

[13]  S. Girvin,et al.  Observation of high coherence in Josephson junction qubits measured in a three-dimensional circuit QED architecture. , 2011, Physical review letters.

[14]  Austin G. Fowler,et al.  Quantum computing with nearest neighbor interactions and error rates over 1 , 2010, 1009.3686.

[15]  Austin G. Fowler,et al.  Surface code quantum error correction incorporating accurate error propagation , 2010, Quantum Inf. Comput..

[16]  F. Nori,et al.  Natural and artificial atoms for quantum computation , 2010, 1002.1871.

[17]  Todd A. Brun,et al.  Quantum Computing , 2011, Computer Science, The Hardware, Software and Heart of It.

[18]  David Poulin,et al.  A renormalization group decoding algorithm for topological quantum codes , 2010, 2010 IEEE Information Theory Workshop.

[19]  David Poulin,et al.  Fast decoders for topological quantum codes. , 2009, Physical review letters.

[20]  Austin G. Fowler,et al.  Graphical algorithms and threshold error rates for the 2d color code , 2009, Quantum Inf. Comput..

[21]  Stephen M. Barnett,et al.  Quantum information , 2005, Acta Physica Polonica A.

[22]  David P. DiVincenzo,et al.  Fault-tolerant architectures for superconducting qubits , 2009, 0905.4839.

[23]  Vwani P. Roychowdhury,et al.  Latency in local, two-dimensional, fault-tolerant quantum computing , 2008, Quantum Inf. Comput..

[24]  A. Fowler,et al.  High-threshold universal quantum computation on the surface code , 2008, 0803.0272.

[25]  J. Clarke,et al.  Superconducting quantum bits , 2008, Nature.

[26]  R. Blatt,et al.  Entangled states of trapped atomic ions , 2008, Nature.

[27]  S. Girvin,et al.  Wiring up quantum systems , 2008, Nature.

[28]  J. Dalibard,et al.  Many-Body Physics with Ultracold Gases , 2007, 0704.3011.

[29]  N. Mermin.  Quantum Computer Science: An Introduction , 2007 .

[30]  R. Raussendorf,et al.  Topological fault-tolerance in cluster state quantum computation , 2007, quant-ph/0703143.

[31]  R. Raussendorf,et al.  Fault-tolerant quantum computation with high threshold in two dimensions. , 2006, Physical review letters.

[32]  D. DiVincenzo,et al.  Noise threshold for a fault-tolerant two-dimensional lattice architecture , 2006, Quantum Inf. Comput..

[33]  R. Raussendorf,et al.  A fault-tolerant one-way quantum computer , 2005, quant-ph/0510135.

[34]  F. Nori,et al.  Superconducting Circuits and Quantum Information , 2005, quant-ph/0601121.

[35]  Rodney Van Meter,et al.  ARCHITECTURE-DEPENDENT EXECUTION TIME OF SHOR'S ALGORITHM , 2005, quant-ph/0507023.

[36]  A. Kitaev,et al.  Universal quantum computation with ideal Clifford gates and noisy ancillas (14 pages) , 2004, quant-ph/0403025.

[37]  Thomas G. Draper,et al.  A new quantum ripple-carry addition circuit , 2004, quant-ph/0410184.

[38]  John M. Martinis,et al.  Implementing Qubits with Superconducting Integrated Circuits , 2004, Quantum Inf. Process..

[39]  Levente J. Klein,et al.  Spin-Based Quantum Dot Quantum Computing in Silicon , 2004, Quantum Inf. Process..

[40]  C. Monroe,et al.  Quantum dynamics of single trapped ions , 2003 .

[41]  H. Briegel,et al.  Measurement-based quantum computation on cluster states , 2003, quant-ph/0301052.

[42]  J. Preskill,et al.  Confinement Higgs transition in a disordered gauge theory and the accuracy threshold for quantum memory , 2002, quant-ph/0207088.

[43]  Stéphane Beauregard.  Circuit for Shor's algorithm using 2n+3 qubits , 2002, Quantum Inf. Comput..

[44]  A. Kitaev,et al.  Fault tolerant quantum computation by anyons , 1997, quant-ph/9707021.

[45]  C. Monroe,et al.  Architecture for a large-scale ion-trap quantum computer , 2002, Nature.

[46]  Mikhail N. Vyalyi,et al.  Classical and Quantum Computation , 2002, Graduate studies in mathematics.

[47]  J. Preskill,et al.  Topological quantum memory , 2001, quant-ph/0110143.

[48]  Y. Makhlin,et al.  Quantum-state engineering with Josephson-junction devices , 2000, cond-mat/0011269.

[49]  M. Freedman.  Quantum Computation and the Localization of Modular Functors , 2000, Found. Comput. Math..

[50]  I. Chuang,et al.  Quantum Computation and Quantum Information: Bibliography , 2010 .

[51]  A. Kitaev,et al.  Quantum codes on a lattice with boundary , 1998, quant-ph/9811052.

[52]  Christof Zalka.  Fast versions of Shor's quantum factoring algorithm , 1998, quant-ph/9806084.

[53]  D. Gottesman.  The Heisenberg Representation of Quantum Computers , 1998, quant-ph/9807006.

[54]  Osamu Hirota,et al.  "Quantum Communication, Computing, and Measurement" , 2012 .

[55]  A. Kitaev.  Quantum computations: algorithms and error correction , 1997 .

[56]  Lov K. Grover.  Quantum Mechanics Helps in Searching for a Needle in a Haystack , 1997, quant-ph/9706033.

[57]  Daniel Gottesman,et al.  Stabilizer Codes and Quantum Error Correction , 1997, quant-ph/9705052.

[58]  A. Steane.  Multiple-particle interference and quantum error correction , 1996, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[59]  Barenco,et al.  Quantum networks for elementary arithmetic operations. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[60]  Shor,et al.  Scheme for reducing decoherence in quantum computer memory. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[61]  J. J. Sakurai,et al.  Modern Quantum Mechanics, Revised Edition , 1995 .

[62]  Leonard M. Adleman,et al.  Algorithmic Number Theory, First International Symposium, ANTS-I, Ithaca, NY, USA, May 6-9, 1994, Proceedings , 1994, ANTS.

[63]  J. J. Sakurai,et al.  Modern Quantum Mechanics , 1986 .

[64]  Jack Edmonds,et al.  Maximum matching and a polyhedron with 0,1-vertices , 1965 .