What is a quantum computer, and how do we build one?

The DiVincenzo criteria for implementing a quantum compute r have been seminal in focussing both experimental and theoretical research in quantum information processing. These criteria were formulated specifically for the circuit model of quantum computing. However, several new models for quantum computing (paradig ms) have been proposed that do not seem to fit the criteria well. The question is therefore what are the general criteria for implementing quantum computers. To this end, a formal opera tion l definition of a quantum computer is introduced. It is then shown that according to th is definition a device is a quantum computer if it obeys the following criteria: Any quantum com puter must consist of a quantum memory, with additional structure that (1) facilitates a co ntr lled quantum evolution of the quantum memory; (2) includes a method for information theor etic cooling of the memory; and (3) provides a readout mechanism for subsets of the quant um memory. The criteria are met when the device is scalable and operates fault-tolerant y. We discuss various existing quantum computing paradigms, and how they fit within this fra mework. Finally, we present a decision tree for selecting an avenue towards building a qua ntum computer. This is intended to help experimentalists determine the most natural paradi gm given a particular physical implementation. PACS numbers: 03.67.Lx 03.67.Ac 03.67.Pp 03.67.Hk Submitted to:New Journal of Physics ‡ c.perez@sheffield.ac.uk § p.kok@sheffield.ac.uk What is a quantum computer, and how do we build one? 2

[1]  Robert Raussendorf,et al.  Fault-tolerant quantum computation with high threshold in two dimensions. , 2007, Physical review letters.

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

[3]  Seth Lloyd,et al.  Universal Quantum Simulators , 1996, Science.

[4]  DiVincenzo.  Two-bit gates are universal for quantum computation. , 1994, Physical review. A, Atomic, molecular, and optical physics.

[5]  Y. Lim,et al.  Repeat-until-success quantum computing using stationary and flying qubits (14 pages) , 2005, quant-ph/0508218.

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

[7]  K. Mølmer,et al.  QUANTUM COMPUTATION WITH IONS IN THERMAL MOTION , 1998, quant-ph/9810039.

[8]  Simon C. Benjamin,et al.  Quantum error correction in globally controlled arrays , 2003 .

[9]  Barenco,et al.  Conditional Quantum Dynamics and Logic Gates. , 1995, Physical review letters.

[10]  Peter W. Shor,et al.  Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer , 1995, SIAM Rev..

[11]  Hans-J. Briegel,et al.  Computational model underlying the one-way quantum computer , 2002, Quantum Inf. Comput..

[12]  N. Cerf,et al.  Quantum search by local adiabatic evolution , 2001, quant-ph/0107015.

[13]  S. C. Benjamin,et al.  Quantum Computing in Arrays Coupled by 'Always On' Interactions , 2004, quant-ph/0401071.

[14]  T. Ralph,et al.  Universal quantum computation with continuous-variable cluster states. , 2006, Physical review letters.

[15]  Virgílio A. F. Almeida,et al.  Performance by Design - Computer Capacity Planning By Example , 2004 .

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

[17]  Simon C Benjamin,et al.  Quantum computing with an always-on Heisenberg interaction. , 2002, Physical review letters.

[18]  Simon C Benjamin,et al.  Measurement-based entanglement under conditions of extreme photon loss. , 2007, Physical review letters.

[19]  C. P. Sun,et al.  Quantum computation based on d-level cluster state (11 pages) , 2003, quant-ph/0304054.

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

[21]  Shor,et al.  Simple proof of security of the BB84 quantum key distribution protocol , 2000, Physical review letters.

[22]  Michael A. Nielsen,et al.  Fault-tolerant quantum computation with cluster states , 2005 .

[23]  Paola Cappellaro,et al.  Single spin measurement using cellular automata techniques. , 2006, Physical review letters.

[24]  R. Feynman.  Simulating physics with computers , 1999 .

[25]  G. Vidal.  Efficient classical simulation of slightly entangled quantum computations. , 2003, Physical review letters.

[26]  R. Jozsa,et al.  On the role of entanglement in quantum-computational speed-up , 2002, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[27]  Amr F. Fahmy,et al.  Ensemble Quantum Computing by Nuclear Magnetic Resonance Spectroscopy , 1997 .

[28]  Dorit Aharonov,et al.  Fault-tolerant quantum computation with constant error , 1997, STOC '97.

[29]  D. Gottesman.  Theory of fault-tolerant quantum computation , 1997, quant-ph/9702029.

[30]  Robert Raussendorf,et al.  Topological fault-tolerance in cluster state quantum computation , 2007 .

[31]  Orlando,et al.  Josephson Persistent-Current Qubit , 2022 .

[32]  F. Yamaguchi,et al.  Crystal lattice quantum computer , 1999 .

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

[34]  Lov K. Grover.  A fast quantum mechanical algorithm for database search , 1996, STOC '96.

[35]  Donny Cheung,et al.  Local unitary quantum cellular automata , 2007, 0709.0006.

[36]  D. DiVincenzo.  TOPICS IN QUANTUM COMPUTERS , 1996, cond-mat/9612126.

[37]  N. Gershenfeld,et al.  Bulk Spin-Resonance Quantum Computation , 1997, Science.

[38]  David P. DiVincenzo,et al.  Dogma and heresy in quantum computing , 2001, Quantum Inf. Comput..

[39]  R. Jozsa,et al.  SEPARABILITY OF VERY NOISY MIXED STATES AND IMPLICATIONS FOR NMR QUANTUM COMPUTING , 1998, quant-ph/9811018.

[40]  U. Toronto,et al.  Analysis of generalized Grover quantum search algorithms using recursion equations , 2000, quant-ph/0010077.

[41]  D. DiVincenzo,et al.  Quantum computation with quantum dots , 1997, cond-mat/9701055.

[42]  D. Deutsch.  Quantum computational networks , 1989, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[43]  Pieter Kok,et al.  Efficient high-fidelity quantum computation using matter qubits and linear optics , 2005 .

[44]  Isaac L. Chuang,et al.  Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations , 1999, Nature.

[45]  A. Steane.  Overhead and noise threshold of fault-tolerant quantum error correction , 2002, quant-ph/0207119.

[46]  John Watrous,et al.  On one-dimensional quantum cellular automata , 1995, Proceedings of IEEE 36th Annual Foundations of Computer Science.

[47]  H. Briegel,et al.  Persistent entanglement in arrays of interacting particles. , 2000, Physical review letters.

[48]  Mark S. Sherwin,et al.  Quantum Computation with Quantum Dots and Terahertz Cavity Quantum Electrodynamics , 1999 .

[49]  H. S. Allen.  The Quantum Theory , 1928, Nature.

[50]  G. Milburn,et al.  Linear optical quantum computing with photonic qubits , 2005, quant-ph/0512071.

[51]  R. Cleve,et al.  Quantum algorithms revisited , 1997, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[52]  P. Shor,et al.  Error Correcting Codes For Adiabatic Quantum Computation , 2005, quant-ph/0512170.

[53]  Alastair Kay.  Error Correcting the Control Unit in Global Control Schemes , 2005 .

[54]  Alastair Kay.  Deriving a Fault-Tolerant Threshold for a Global Control Scheme , 2007 .

[55]  M. Nielsen.  Optical quantum computation using cluster States. , 2004, Physical review letters.

[56]  A. Ekert,et al.  Universality in quantum computation , 1995, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[57]  G. Schoen,et al.  Quantum Manipulations of Small Josephson Junctions , 1997, cond-mat/9706016.

[58]  Markus Grassl,et al.  Grover's quantum search algorithm for an arbitrary initial amplitude distribution , 1999 .

[59]  Lloyd,et al.  Almost any quantum logic gate is universal. , 1995, Physical review letters.

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

[61]  J. Preskill.  Reliable quantum computers , 1997, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[62]  Dan E. Browne,et al.  Brokered graph-state quantum computation , 2005, quant-ph/0509209.

[63]  Barenco,et al.  Elementary gates for quantum computation. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[64]  Daniel A. Lidar,et al.  Towards fault tolerant adiabatic quantum computation. , 2007, Physical review letters.

[65]  D. V. Averin,et al.  Adiabatic quantum computation with Cooper pairs , 1998 .

[66]  E. Knill,et al.  A scheme for efficient quantum computation with linear optics , 2001, Nature.

[67]  Michele Mosca,et al.  On quantum algorithms , 1998 .

[68]  Seth Lloyd,et al.  Adiabatic quantum computation is equivalent to standard quantum computation , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[69]  S Lloyd,et al.  A Potentially Realizable Quantum Computer , 1993, Science.

[70]  Poul S. Jessen,et al.  Quantum Computing with Neutral Atoms in An Optical Lattice , 2000, quant-ph/0003022.

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

[72]  Frédéric Magniez,et al.  Self-testing of Quantum Circuits , 2006, ICALP.

[73]  Unruh.  Maintaining coherence in quantum computers. , 1994, Physical review. A, Atomic, molecular, and optical physics.

[74]  David P. DiVincenzo,et al.  A pr 2 00 0 The Physical Implementation of Quantum Computation , 2000 .

[75]  Peter W. Shor,et al.  Fault-tolerant quantum computation , 1996, Proceedings of 37th Conference on Foundations of Computer Science.

[76]  E. Gauger,et al.  Robust adiabatic approach to optical spin entangling in coupled quantum dots , 2008, 0804.2139.

[77]  Lov K. Grover.  Quantum Computers Can Search Rapidly by Using Almost Any Transformation , 1998 .

[78]  V. Fock,et al.  Beweis des Adiabatensatzes , 1928 .

[79]  Simon C. Benjamin.  Multi-qubit gates in arrays coupled by 'always-on' interactions , 2004 .

[80]  Andrew Steane,et al.  Active Stabilization, Quantum Computation, and Quantum State Synthesis , 1997 .

[81]  Steane,et al.  Error Correcting Codes in Quantum Theory. , 1996, Physical review letters.

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

[83]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[84]  T. Rudolph,et al.  Resource-efficient linear optical quantum computation. , 2004, Physical review letters.

[85]  Gardiner,et al.  Decoherence, continuous observation, and quantum computing: A cavity QED model. , 1995, Physical review letters.

[86]  S. Benjamin,et al.  Optical quantum computation with perpetually coupled spins , 2004, quant-ph/0407063.

[87]  Jason Twamley,et al.  Globally controlled quantum wires for perfect qubit transport, mirroring, and computing. , 2006, Physical review letters.

[88]  Mark I. Dykman,et al.  Quantum computing with electrons floating on liquid helium , 1999, Science.

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

[90]  S. Braunstein,et al.  Quantum computation over continuous variables , 1998 .

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

[92]  E. Knill.  Quantum computing with realistically noisy devices , 2005, Nature.

[93]  J. Cirac,et al.  Quantum Computations with Cold Trapped Ions. , 1995, Physical review letters.

[94]  J. I. Cirac,et al.  Reversible universal quantum computation within translation-invariant systems , 2006 .

[95]  Péter Gács,et al.  Reliable computation with cellular automata , 1983, J. Comput. Syst. Sci..

[96]  B. E. Kane.  A silicon-based nuclear spin quantum computer , 1998, Nature.

[97]  J. Eisert,et al.  Multiparty entanglement in graph states , 2003, quant-ph/0307130.

[98]  Markus Grassl,et al.  Exact Solution of Grover's Quantum Search Algorithm for Arbitrary Initial Amplitude Distribution , 1998 .