Implementations of Quantum Information I

1. Implementations of Quantum Information I Barry C. Sanders IQIS, University of Calgary, www.iqis.org CQCT, Macquarie University, Sydney, Australia, www.qcaustralia.org Montreal August 2005

2. Implementations Aim: to implement quantum information processing, i.e. realize quantum communication and quantum information processing. There are many challenges because of imperfections in systems and decoherence, but there are promising techniques such as decoherence-free subspaces and quantum error correction to overcome these problems. There are many candidates for physical quantum information processing, and we study some of these here.

3. I. Introduction • General qubit state: • General multiqubit state: • Density matrix:

4. One Universal Set of Gates Identity and Not gates: Hadamard gate: Phase gates: Controlled phase gate (equivalent to ^X=CNOT under local unitaries):

5. Goals • Encode qubits in physical system. • Process these qubits. • Universal set of gates for quantum computation. • Qubit-specific readout. • Store qubits. • Minimize decoherence. • Correct errors.

6. Problems • Qubit may be in q Hilbert space that is larger than two dimensions: truncate! • Coupling to environment ==> decoherence. • Imperfect gates so they don’t effect precisely the desired transformation • Preparation and readout

7. 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.

8. Investigate proposals • Trapped ions • Nuclear spins • Spin-based quantum dot qubit • Photons

9. 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:

10. The Hamiltonian • The Hamiltonian operator is a function of operators concerning degrees of freedom (dynamical variables) of the system. • Eg. if quantum information is encoded in positions x1and x2of two particles, then with … representing other relevant operators. • Momentum p is conjugate to position: • Real systems are highly complicated, and creating an effective model is high art!

11. Harmonic 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).

12. Phonons • Number of quanta are increased or decreased by creation or annihilation operators: • The position operator can be represented by • The conjugate momentum operator can be represented by • Quantum information can be encoded as a “qudit” in the lowest energy levels of oscillator.

13. The Environment • The Hamiltonian generates unitary evolution, which corresponds to dynamics in a closed system, but 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 puck’s environment.

14. Growing the Hamiltonian • The Hamiltonian for the entire model should 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.

15. The Reduced State • 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 environmental degrees of freedom: • The state of the system is, in general, mixed. • Decoherence-free subspaces and quantum error correction are designed to protect purity.

16. 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 Hamiltonian, 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.

17. II. Trapped ions

18. Trapped Ions • The trapped 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. • Ion-to-ion coupling is obtained via collective harmonic motion of the lattice, which is quantized (with the quanta called photons).

19. 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 notation: • For L, use S for L=0, P for L=1, D for L=2, … • Transitions between levels by photon loss or gain: chain energy and angular momentum.

20. Photon absorption from ground to excited state. Spontaneous emission from ground to excited state. Atom-Field interactions Coherent Stimulated emission from ground to excited state. Emitted photon is a copy of the “trigger” photon. Incoherent Emitted photon is random in direction and phase.

21. 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 than for y,z. Excitation of alkali ion dipole-forbidden transition

22. 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 motion with energy restricted to zero or one phonon.

23. 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 is • For evolution time t=kp/W,

24. 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.

25. Unitary Evolution • Apply the laser beam for time • Evolution is given by • This transformation doesn’t change • Effects transformation

26. Two-Qubit Gate: ^Z

27. Universal Quantum Computation • The Cirac-Zoller scheme enables universal quantum computation for ions by combining single-qubit rotations with the two-qubit ^Z. • 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.

28. XX

29. Independent of Phonon Number • Two-photon transition via intermediate states • Two-photon 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.

30. Architecture for a Large-Scale Computer • Kielpinski-Monroe-Wineland, Nature 2002. • Static ion traps may be limited to a few dozen ions so KMW 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 ion in the neighbourhood.)

31. Electrode Segments Memory region Interaction Region

32. 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 ion’s path, resulting in random fluctuations 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.

33. Summary The Cirac-Zoller proposal for quantum computation in an ion trap is one of 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 arguments for static ions by using multiple registers, quantum communication, and decoherence-free subspaces.

34. III. Various Proposals

35. Various proposals • In ion-trap quantum computation, quantum information is stored in electronic levels, and laser fields drive transitions with vibrations acting as the bus. • 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.

36. Nuclear Spin Quantum Computation • Nuclear spins - DiVincenzo Science 1995, Cory et al and Gershenfeld & Chuang both 1995. • Qubit is realized as a nucleus with gm its magnetic dipole moment, in a static magnetic field B0along z-axis and an alternating time-dependent magnetic field B1with angular frequency w along y axis; the Hamiltonian is • Couple to specific qubit by tuning w.

38. 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; unfortunately initialization is difficult as the density matrix is always close to identity. • Scalability may be resolved using solid state.

39. 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 by inter-dot electron quantum tunneling. • A quantum dot confines an electron in all 3 dimensions: artificial atom. • Qubit corresponds to spin state of quantum dot’s excess electron. • Higher energies ignored. Effects gate operation:

40. A-Gates BAC (10-3) T B (2 T) J-Gates Barrier e- e- Kane’s Proposal 1998 Si Substrate 31P+ The J-gate controls e-e collisions, hence the coupling of 31P’s. 31P+

41. 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 photons. • 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).

42. Dual-Rail Photonic Qubit • A beam splitter produces the unitary operation • The beam splitter rotates the dual-rail qubit. • Ideal “Kerr nonlinear medium” • The Kerr interaction provides the two-qubit operation sufficient for universal circuits. • Unfortunately c is miniscule or, if enhanced, accompanied by unacceptably high photon losses. Chuang & Yamamoto 1995

43. Knill-Laflamme-Milburn 2001 • A solution to weak nonlinearity 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 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.

44. One-way computation • More recently Raussendorf and Briegel introduced one-way quantum computation with cluster states, based on processing highly entangled states via successive single qubit measurements. • This approach seems promising for realizing quantum computation with photons. Raussendorf and Briegel 2001

45. 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.