1、Barry C. Sanders IQIS, University of Calgary, www.iqis.org CQCT, Macquarie University, Sydney, Australia, www.qcaustralia.org,Implementations of Quantum Information I,Montreal August 2005,Implementations,One task in physics is to implement quantum information processing, i.e. realize quantum communi
2、cation and quantum information processing.There are many challenges because of imperfections in systems and decoherence, but there are promising techniques such as quantum error correction to overcome these problems.Many candidates for physical quantum information processing, and we study some of th
3、ese here.,I. Introduction,General qubit state:General multiqubit state:Density matrix:,One Universal Set of Gates,Identity and Not gates:,Hadamard gate:,Phase gates:,Controlled phase gate (equivalent to X=CNOT under local unitaries:,Goals,Encode qubits in physical system. Process these qubits. Unive
4、rsal set of gates for quantum computation. Qubit-specific readout. Store qubits. Minimize decoherence. Correct errors.,Problems,Qubit may in Hilbert space larger than two dimensions: truncate! Coupling to environment = decoherence. Imperfect gates so they dont effect precisely the desired transforma
5、tion Preparation and readout,DiVincenzo Criteria,Scalability. Ability to initialize. Long decoherence times. Universal set of quantum gates. Qubit measurement capability.,Additional Criteria,Interconvertibility between physical qubits. Faithfully transmit flying qubits.,Investigate proposals,Trapped
6、 ions Nuclear spins Spin-based quantum dot qubit Photons,Our General Approach,Identify the physical qubits in a given system. Determine the Hamiltonian(s) governing dynamics of the qubits:The Hamiltonian generates the unitary evolution operator, which performs processing:,The Hamiltonian,The Hamilto
7、nian operator is a function of operators concerning degrees of freedom of the system. If quantum information is encoded in positions x1 and x2 of two particles, thenwith representing other relevant operators. Real systems are highly complicated, and creating an effective model is high art!,Harmonic
8、Oscillator,A simple harmonic oscillator in one dimension is described by The particle has mass m and angular frequency w, which is independent of amplitude. Number operator has spectrum 0,1,2,.Eigenstates corresponding to number of quanta are |n, e.g. photons (for light) or phonons (for vibrations).
9、,Phonons,Number of quanta are increased or decreased by creation or annihilation operators:The position operator can be represented byThe conjugate momentum operator can be represented by,The Environment,The Hamiltonian generates unitary evolution, which corresponds to dynamics in a closed system, b
10、ut the system must be open for preparation and readout. The openness is the coupling of the system to the environment; e.g. a puck sliding on ice is slowed by frictional coupling to ice and air resistance so ice and air are part of the pucks environment.,Growing the Hamiltonian,The Hamiltonian for t
11、he entire model must include system and environment. If the environment has dynamical degrees of freedom ci, these are included in the Hamiltonian; extend previous Hamiltonian:The system+environment state evolves according to unitary evolution generated by this bigger Hamiltonian.,The Reduced State,
12、The state of the system+environment is not useful to us; we just want to know the state of the system. We discard all information about the environment by tracing the density matrix for system+environment over environment degrees of freedom:The state of the system is, in general mixed. Decoherence-f
13、ree subspaces and quantum error correction are designed to protect purity.,Summary,Goals are to encode quantum information in a physical system and realize quantum gates and single qubit measurements, perhaps with subsequent dynamics dependent on these measurement results. Dynamics determined by Ham
14、iltonian, which generates the evolution operator describing the gates and circuits. Systems are necessarily coupled to the environment, and decoherence-free subspaces and quantum error correction are designed to protect against environment-induced degradation.,II. Trapped ions,Trapped Ions,The trapp
15、ed ion system is an early and promising medium for realizing quantum information processing. Ions are charged atoms, and electric fields are used to confine or move these ions in a lattice. Quantum information is encoded in the electron energy level. Coupling is obtained via collective motion, which
16、 is quantized (with the quanta called photons).,Ion,Charged atom - number of electrons is greater than or less than number of protons. Concerned with outermost electron orbiting “shielded” nucleus. Angular momentum J is a vector sum of spin s and orbital angular momentum L: J=s+L. Spectroscopic nota
17、tion: For L, use S for L=0, P for L=1, D for L=2, ,Atom-Photon interactions,Stimulated emission from ground to excited state.,Spontaneous emission from ground to excited state.,Coherent,Incoherent,Emitted photon is random in direction and phase. For S-P transitions, rate is once per nanosecond and,
18、for S-D, rate is once per second: S-D is better.,Emitted photon is a copy of the “trigger” photon.,Cirac-Zoller (1995) proposal: N ions in a linear trap, each interacting with a separate laser beam. Ions are confined by harmonic potentials in each of x, y, z directions with x frequency much less tha
19、n for y,z.,Excitation of alkali ion dipole-forbidden transition,Driving the Ion,Each laser beam acts on one ion located at the node of the laser field standing wave. There are two excited states, with transition to q=0 or q=1 determined by laser polarization. Ions share a collective centre-of-mass m
20、otion with energy restricted to zero or one phonon.,Single-Qubit Rotation,Laser field is tuned to the |g|e0 transition with the laser polarization set to q=0. The Rabi frequency W is the strength of the atom-field interaction; f is the laser phase. The Hamiltonian for the nth ion isFor evolution tim
21、e t=kp/W,Interaction Hamiltonian,Laser acts on nth ion and is detuned by centre-of-mass motion angular frequency nx. Change of electronic energy is accompanied by creation or annihilation of one phonon. For q the angle between laser beam and x-axis, and k = 2p/l for l the laser field wavelength.,Uni
22、tary Evolution,Apply the laser beam for time Evolution is given by This transformation doesnt change Effects transformation,Two-Qubit Gate: Z,Universal Quantum Computation,The Cirac-Zoller scheme enables universal quantum computation for ions by combining single-qubit rotations with the two-qubit Z.
23、 Requires atoms to be cold. Sorensen and Molmer introduced a bichromatic off-resonant driving schemes that allows ions to be warm: two-photon processes interfere to minimize sensitivity of ions to vibrational state.,XX,Independent of Phonon Number,Two-photon transition via intermediate states Two-ph
24、oton transition rate for |ggn |een:Same transition rate for |egn |gen. Two-photon transitions interfere to remove n from the effective two-photon Rabi frequency.,Architecture for a Large-Scale Computer,Kielpinski-Monroe-Wineland, Nature 2002. Static ion traps may be limited to a few dozen ions so KM
25、W suggested multiple quantum registers with quantum communication between these registers. Trapped ions are stored in quantum memory registers, with ions shuttled between registers. Ions are transported quickly and then recooled by sympathetic cooling (via cooling process on a different species of i
26、on in the neighbourhood.),Memory region,Interaction Region,Electrode Segments,Decoherence-Free Subspace,Ion transport presents another problem: qubit evolves according to |g+|e|g+eia|e. The parameter a is random and is due to varying magnetic field strength along the ions path, resulting in random f
27、luctuations in the energy separation of|g and |e. Reduce this decoherence by encoding the logical qubit as |0=|ge and |1=|eg. The logical qubits are invariant (up to unimportant phase) under stochastic magnetic fields.,Summary,The Cirac-Zoller proposal for quantum computation in an ion trap is one o
28、f the first and influential, especially in the conception of creating two-qubit unitary gates.The Sorensen-Molmer proposal relaxes constraints on the temperature of the ions by using an off-resonant bichromatic driving field.The Kielpinski-Monroe-Wineland proposal shows how to surpass the scaling ar
29、guments for static ions by using multiple registers, quantum communication, and decoherence-free subspaces.,III. Various Proposals,Various proposals,In ion-trap quantum computation, quantum information is stored in electronic levels, and laser fields drive transitions with vibrations acting as the b
30、us. Another early and promising proposal concerns encoding quantum information onto nuclei and using magnetic fields for readout. Here we discuss this nuclear magnetic resonance technique plus coupled quantum dots and also photons.,Nuclear Spin Quantum Computation,Nuclear spins - DiVincenzo Science
31、1995, Cory et al and Gershenfeld the Hamiltonian isCouple to specific qubit by tuning w.,Two-Qubit Coupling,This Hamiltonian generates a unitary evolution that is a combination of single-qubit rotation (by tuning w ) and two-qubit operations. Quantum computing has been performed in liquid state; unf
32、ortunately initialization is difficult as the density matrix is always close to identity. Scalability may be resolved using solid state.,Loss and DiVincenzo QDot QComputer,The T-gate is a magnetic gate that rotates the electron spin: single-qubit rotations. The J-gate is the coupling term effected b
33、y inter-dot electron quantum tunneling.,A quantum dot confines an electron in all 3 dimensions: artificial atom. Qubit corresponds to spin state of quantum dots excess electron. Higher energies ignored.,Effects gate operation:,31P+,31P+,Barrier,Si Substrate,BAC,B,J-Gates,e-,e-,A-Gates,(2 T),(10-3) T
34、,Kanes Proposal 1998,The J-gate controls e-e collisions, hence the coupling of 31Ps.,Photons,The electromagnetic field consists of modes identified, for example, by frequency, spatial characteristics, and polarization. Each mode is a harmonic oscillator, and the number state |n corresponds to n phot
35、ons. A qubit can be a superposition of no photon and one photon in a mode, a|0+b|1 (single-rail qubit), or exaclty one photon in a superposition of two modes, a|01+b|10 (dual-rail qubit).,Dual-Rail Photonic Qubit,The Kerr interaction provides the two-qubit operation sufficient for universal circuits
36、. Unfortunately c is miniscule or, if enhanced, accompanied by unacceptably high photon losses.,A beam splitter produces the unitary operation The beam splitter rotates the dual-rail qubit. Ideal “Kerr nonlinear medium”,Chuang & Yamamoto 1995,Knill-Laflamme-Milburn 2001,A solution to weak nonlineari
37、ty provided by GC teleportation gate, which teleports input to “processed” output by feeding entangled ancillas produced offline.,Gottesman & Chuang 1999,KLM developed a circuit to produce Kerr nonlinearity rarely, but successes are signaled by photodetection events. Successful outputs are provided
38、as ancillas for a GC type of teleporter; by only supplying proper states, the quantum computation is not hindered by the local success rate of the nonlinear gate.,One-way computation,More recently Raussendorf and Briegel introduced one-way quantum computation with cluster states, based on processing
39、 highly entangled states via successive single qubit measurements. This approach seems promising for realizing quantum computation with photons.,Raussendorf and Briegel 2001,Summary,Surveyed a few proposals for quantum computation using ions, nuclei, quantum dots, and photons.Ignored schemes such as optical cavities, superconducting qubits, and fullerene.The theory can be understood once the model Hamiltonian is determined, and a lot of work is underway to realize such Hamiltonian dynamics and understand the environment.,
