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Quantum information and computation: Why, what, and how

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Quantum processes can provide extraordinary benefits for information processing, communication and security, offering striking novel features beyond the possibilities of standard (classical) paradigms. ...

https://dblp.org/db/journals/qic/qic7

Published in: Science
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Quantum information and computation: Why, what, and how

  1. 1. Quantum information and computation: Why, what, and how I. Introduction II. Qubitology and quantum circuits III. Entanglement and teleportation IV. Quantum algorithms V. Quantum error correction VI. Physical implementations Carlton M. Caves University of New Mexico http://info.phys.unm.edu SFI Complex Systems Summer School 2006 June Quantum circuits in this presentation were set using the LaTeX package Qcircuit, developed by Bryan Eastin and Steve Flammia. The package is available at http://info.phys.unm.edu/Qcircuit/ .
  2. 2. I. Introduction In the Sawtooth range Central New Mexico
  3. 3. A new way of thinking Quantum information science Computer science Computational complexity depends on physical law. Old physics Quantum mechanics as nag. The uncertainty principle restricts what can be done. New physics Quantum mechanics as liberator. What can be accomplished with quantum systems that can’t be done in a classical world? Explore what can be done with quantum systems, instead of being satisfied with what Nature hands us. Quantum engineering
  4. 4. Quantum information science A way of doing Experiment A way of thinking Theory Atomic, molecular, optical physics Super- conductivity physics Metrology Nuclear magnetic resonance (NMR) Condensed- matter physics
  5. 5. Classical information Stored as string of bits Few electrons on a capacitor Pit on a compact disk 0 or 1 on the printed pageSmoke signal on a distant mesa Quantum information Stored as quantum state of string of qubits Spin-1/2 particle Two-level atom Photon polarization Physical system with two distinguishable states Pure quantum state Qubits whole story much more
  6. 6. Classical information Stored as string of bits Quantum information Stored as quantum state of string of qubits Transmission of (qu)bits (communication, dynamics) Readout of (qu)bits (measurement) Distinguishability of bit states Quantum states are not distinguishable, except for orthogonal states Bit states can be copied. Qubit states cannot be copied, except for orthogonal states Manipulation of (qu)bits (computation, dynamics) Unitary operations U (reversible) Bit transformations (function computation) All functions can be computed reversibly.
  7. 7. Classical information Stored as string of bits Quantum information Stored as quantum state of string of qubits Manipulation of (qu)bits (computation, dynamics) Transmission of (qu)bits (communication, dynamics) Readout of (qu)bits (measurement) Quantum mechanics as liberator. Classical information processing is quantum information processing restricted to distinguishable (orthogonal) states. Superpositions are the additional freedom in quantum information processing.
  8. 8. Classical information Stored as string of bits Quantum information Stored as quantum state of string of qubits Manipulation of (qu)bits (computation, dynamics) Transmission of (qu)bits (communication, dynamics) Readout of (qu)bits (measurement) Error correction (copying and redundancy OR nonlocal storage of information) Quantum error correction (entanglement OR nonlocal storage of quantum information) Correlation of bit states Quantum correlation of qubit states (entanglement) anticorrelation Bell inequalities Analogue vs. digital
  9. 9. II. Qubitology and quantum circuits Albuquerque International Balloon Fiesta
  10. 10. Qubitology. States Bloch sphere Spin-1/2 particle Direction of spin Pauli representatio n
  11. 11. Qubitology. States Abstract “direction” Polarization of a photon Poincare sphere
  12. 12. Qubitology. States Abstract “direction” Bloch sphere Two-level atom
  13. 13. Qubitology Single-qubit states are points on the Bloch sphere. Single-qubit operations (unitary operators) are rotations of the Bloch sphere. Single-qubit measurements are rotations followed by a measurement in the computational basis (measurement of z spin component). Platform-independent description: Hallmark of an information theory
  14. 14. Single-qubit gates Qubitology. Gates and quantum circuits Phase flip Hadamard
  15. 15. More single-qubit gates Qubitology. Gates and quantum circuits Phase-bit flip Bit flip
  16. 16. Control-target two-qubit gate Control Target Qubitology. Gates and quantum circuits Control Target
  17. 17. Qubitology. Gates and quantum circuits Universal set of quantum gates ● T (45-degree rotation about z) ● H (Hadamard) ● C-NOT
  18. 18. Another two-qubit gate Control Target Control Target Qubitology. Gates and quantum circuits
  19. 19. Qubitology. Gates and quantum circuits C-NOT as measurement gate C-NOT as parity check Circuit identity
  20. 20. Qubitology. Gates and quantum circuits Making Bell states using C-NOT Bell states parity bit phase bit
  21. 21. Qubitology. Gates and quantum circuits Making cat states using C-NOT GHZ (cat) state
  22. 22. III. Entanglement and teleportation Oljeto Wash Southern Utah
  23. 23. Entanglement and teleportation Alice Bob 2 bits
  24. 24. Classical teleportation Demonstration Teleportation of probabilities
  25. 25. Entanglement and teleportation Bob Alice Alice Bob 2 bits
  26. 26. Entanglement and teleportation Bob Alice Bob Alice
  27. 27. Entanglement and teleportation Standard teleportation circuit Error correction Bob Alice Coherent teleportation circuit Bob Alice
  28. 28. IV. Quantum algorithms Truchas from East Pecos Baldy Sangre de Cristo Range Northern New Mexico
  29. 29. Quantum algorithms. Deutsch-Jozsa algorithm Boolean function Promise: f is constant or balanced. Problem: Determine which. Classical: Roughly 2N-1 function calls are required to be certain. Quantum: Only 1 function call is needed. work qubit
  30. 30. Quantum algorithms. Deutsch-Jozsa algorithm work qubit Example: Constant function
  31. 31. Quantum algorithms. Deutsch-Jozsa algorithm work qubit Example: Constant function
  32. 32. Quantum algorithms. Deutsch-Jozsa algorithm work qubit Example: Balanced function
  33. 33. Quantum algorithms. Deutsch-Jozsa algorithm Problem: Determine whether f is constant or balanced. quantum interference phase “kickback” quantum parallelism work qubit N = 3
  34. 34. Quantum interference in the Deutsch-Jozsa algorithm N = 2 Hadamards Constant function evaluation Hadamards
  35. 35. Quantum interference in the Deutsch-Jozsa algorithm N = 2 Hadamards Constant function evaluation Hadamards
  36. 36. Quantum interference in the Deutsch-Jozsa algorithm N = 2 Hadamards Balanced function evaluation Hadamards
  37. 37. Quantum interference in the Deutsch-Jozsa algorithm Quantum interference allows one to distinguish the situation where half the amplitudes are +1 and half -1 from the situation where all the amplitudes are +1 or -1 (this is the information one wants) without having to determine all amplitudes (this information remains inaccessible).
  38. 38. Entanglement in the Deutsch-Jozsa algorithm This state is globally entangled for some balanced functions. Implementations Example N = 3
  39. 39. V. Quantum error correction Aspens Sangre de Cristo Range Northern New Mexico
  40. 40. Classical error correction Correcting single bit flips code words N o error Bit flip on second bit Bit flip on first bitBitflip on third bit Copying Redundancy: majority voting reveals which bit has flipped, and it can be flipped back.
  41. 41. N o error Bit flip on second qubit Bit flip on first qubitBitflip on third qubit Quantum error correction Correcting single bit flips Four errors map the code subspace unitarily to four orthogonal subspaces. code states Parity of pairs 12 and 23 Error syndrome Even Even Odd Even Odd Odd Even Odd No need for copying. Redundancy replaced by nonlocal storage of information.
  42. 42. Quantum error correction ancilla qubits Single bit flip correction circuit Error correction Syndrome measuremen t ancilla qubits Coherent version
  43. 43. Quantum error correction Other quantum errors? Entangleme nt phase error Z code states Shor’s 9-qubit code Corrects all single-qubit errors
  44. 44. Quantum error correction Correcting single qubit errors using Shor’s 9-qubit code 27 errors plus no error map the code subspace unitarily to 22 orthogonal subspaces. What about errors other than bit flips, phase flips, and phase-bit flips? code subspac e N o error Phase flip Bit flip on first qubit Phase-bitflip on sixth qubit on ninth qubit 13 other errors 11 other errors Shor-code circuits
  45. 45. VI. Physical implementations Echidna Gorge Bungle Bungle Range Western Australia
  46. 46. 1. Scalability: A scalable physical system made up of well characterized parts, usually qubits. 2. Initialization: The ability to initialize the system in a simple fiducial state. 3. Control: The ability to control the state of the computer using sequences of elementary universal gates. 4. Stability: Decoherence times much longer than gate times, together with the ability to suppress decoherence through error correction and fault-tolerant computation. 5. Measurement: The ability to read out the state of the computer in a convenient product basis. Implementations: DiVincenzo criteria Strong coupling between qubits and of qubits to external controls and measuring devices Weak coupling to everything else Many qubits, entangled, protected from error, with initialization and readout for all.
  47. 47. Original Kane proposal Implementations Qubits: nuclear spins of P ions in Si; fundamental fabrication problem. Single-qubit gates: NMR with addressable hyperfine splitting. Two-qubit gates: electron-mediated nuclear exchange interaction. Decoherence: nuclear spins highly coherent, but decoherence during interactions unknown. Readout: spin-dependent charge transfer plus single-electron detection. Scalability: if a few qubits can be made to work, scaling to many qubits might be easy.
  48. 48. Ion traps Implementations Qubits: electronic states of trapped ions (ground-state hyperfine levels or ground and excited states). State preparation: laser cooling and optical pumping. Single-qubit gates: laser-driven coherent transitions. Two-qubit gates: phonon-mediated conditional transitions. Decoherence: ions well isolated from environment. Readout: fluorescent shelving. Scalability: possibly scalable architectures, involving many traps and shuttling of ions between traps, are being explored.
  49. 49. Implementations Qubits Trapped ions Electronic states AMO systems Trapped neutral Electronic atoms states Linear optics Photon polarization or spatial mode Superconducting Cooper pairs or circuits quantized flux Condensed Doped Nuclear spins systems semiconductors Semiconductor Quantum dots heterostructures NMR Nuclear spins (not scalable; high temperature prohibits preparation of initial pure state) Controllabilty Coherence Readout Scalability
  50. 50. Implementations ARDA Quantum Computing Roadmap, v. 2 (spring 2004) By the year 2007, to ● encode a single qubit into the state of a logical qubit formed from several physical qubits, ● perform repetitive error correction of the logical qubit, ● transfer the state of the logical qubit into the state of another set of physical qubits with high fidelity, and by the year 2012, to ● implement a concatenated quantum error correcting code. It was the unanimous opinion of the Technical Experts Panel that it is too soon to attempt to identify a smaller number of potential “winners;” the ultimate technology may not have even been invented yet.
  51. 51. Bungle Bungle Range Western Australia That’s all, folks.
  52. 52. Entanglement, local realism, and Bell inequalities Entangled state (quantum correlations) Bell entangled state A B
  53. 53. Bell entangled state Entanglement, local realism, and Bell inequalities
  54. 54. Local hidden variables (LHV) and Bell inequalities Back Bell entangled state Entanglement, local realism, and Bell inequalities QM: The quantum correlations cannot be explained in terms of local, realistic properties. LHV:
  55. 55. C-NOT as measurement gate: circuit identity Back
  56. 56. Shor code encoding circuit Quantum error correction
  57. 57. Shor code correction circuit (coherent version) ancilla qubits Quantum error correction Bit-flip syndrome detection Phase-flip syndrome detection Bit-flip correction Phase-flip correction Back

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