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Quantum Dots NMR Silicon quantum computing Quantum dots Practical realization of Quantum Computation QC implementation proposals Bulk spin Optical Atoms Solid state Resonance (NMR) Linear optics Cavity QED Trapped ions Optical lattices Electrons on He Semiconductors Superconductors Nuclear spin Electron spin Orbital state Flux Charge qubits qubits qubits qubits qubits http://courses.washington.edu/bbbteach/576/ 1 What is NMR (Nuclear magnetic resonance) ? NMR: study of the transitions between the Zeeman levels of an atomic nucleus in a magnetic field. • If put a system of nuclei with non-zero spin in E static magnetic field, there 2 E will be different energy RF 0 levels. 0 E2 E1 • If radio frequency wave with is applied, RF 0 E1 transitions will occur. Nuclear Energy Levels 2 Nuclear magnetic resonance NMR is one of the most important spectroscopic techniques available in molecular sciences since the frequencies of the NMR signal allow study of molecular structure. Basic idea: use nuclear spins of molecules as qubits Liquid NMR: rapid molecular motion in fluids greatly simplifies the NMR Hamiltonian as anisotropic interactions can be replaced by their isotropic averages, often zero. The signals are often narrow (high-resolution NMR). It is impossible to pick individual molecule. The combined signal from all molecules is detected (“bulk” NMR). Liquid Solution NMR Billions of molecules are used, each one a separate quantum computer. Most advanced experimental demonstrations to date, but poor scalability as molecule design gets difficult. Qubits are stored in nuclear spin of molecules and controlled by different frequencies of magnetic pulses. Vandersypen, 2000 Used to factor 15 experimentally. 3 4 Bulk NMR Major problem: it is very difficult to place an NMR quantum computer in a well defined state. Advantage: Extensive technical development of NMR spectrometers (very complex but easily controlled devices). Quantum gates One qubit gates: rotation of a single spin within its own Hilbert space is done using RF (radio frequency) fields. Two-qubit gates: use scalar coupling (J-coupling) between two nuclei H = JIS IZ SZ Bulk NMR: readout Typically, readout in quantum computing is reading values of one or more qubits which finish the calculation in terms of eqigenstates |0> or |1>. It is not practical in NMR. Instead, 90° excitation pulse is applied to create superposition of |0> and |1> which then oscillates in a magnetic field. The relative population can then be determined by observing size and phase of the oscillatory signal. It is possible to include reference signal to include relative phases. It is possible in certain cases to see result directly from NMR signal. 5 Quantum computation with chloroform NMR 1H Cl13C Cl Cl Deutsch algorithm demonstrated. 5 qubit 215 Hz Q. Processor ( Vandersypen, Steffen, Breyta, Yannoni, Cleve, Chuang, July 2000 Physical Rev. Lett. ) • 5-spin molecule synthesized • First demonstration of a fast 5-qubit algorithm • Pathway to more qubits 6 Solid State Systems All-Silicon Quantum Computer 10529Si atomic chains in 28Si matrix work like molecules in solution NMR QC. Many techniques used for solution NMR QC are available. 2 1 4 3 6 Copies 5 8 7 No impurity dopants or A large field gradient separates electrical contacts are Larmor frequencies of the nuclei 9 needed. within each chain. T.D.Ladd et al., Phys. Rev. Lett. 89, 017901 (2002) 7 Fabrication: 29Si Atomic Chain Regular step arrays on slightly miscut 28Si(111)7x7 surface (~1º from normal) Steps are straight, with about 1 kink in 2000 sites. 29Si 28Si STM image J.-L.Lin et al., J. Appl. Phys 84, 255 (1998) Natural Silicon: 28Si – 92% Silicon quantum computing 29Si – 4.7% I=1/2 30Si – 3.1% Silicon 28Si spin 0 – does not couple to qubit’s spin. Silicon’s nuclear-spin-bearing isotope has a low natural abundance (only 5% of natural silicon is 29Si, which can be removed by isotopic enrichment). Then, long spin coherence times. The advanced state of silicon electronics offers a common platform for spin qubits alongside sophisticated classical integrated circuits. 8 Qubits: donor electron spin in Si:P or quantum dots Donor electron spin in Si:P Structure Si atom (group-IV) Diamond crystal structure Natural Silicon: 5.43Å 28Si – 92% 29Si – 4.7% I=1/2 30Si – 3.1% 31P electron spin (T=4.2K) T1~ min T2~ msecs P atom (group-V) b ≈ 15 Å + = a ≈ 25 Å Natural Phosphorus: 31P – 100% I=1/2 1 ( x2 y2 ) / a2 z2 / b2 In the effective mass approximation electron wave function is s-like: F(r) e ab 18 9 Semion Saikin’s talk Donor electron spin in Si:P Application for QC Bohr Radius: A -gate J -gate Si: a ≈ 25 Å b ≈ 15 Å SixGe1- Ge: a ≈ 64 Å Si x Si1- b ≈ 24 Å xGex B.E.Kane, Nature 393 133 (1998) R.Vrijen, E.Yablonovitch, K.Wang, H.W.Jiang, A.Balandin, V.Roychowdhury, T.Mor, D.DiVincenzo, 31 P donor Phys. Rev. A 62, 012306 (2000) Qubit – nuclear spin Qubit-qubit inteaction – electron spin 31P donor Qubit – electron spin HEx Qubit-qubit inteaction – electron spin S1 J -gate S2 HHf A -gate H S1 Ex S2 I1 I2 19 Qubit 1 Qubit 2 Qubit 1 Qubit 2 Electrons are localized using phosphorus P donors. The electron spin eigenstates (red), which are energetically separated by a Zeeman interaction with an external magnetic field, can be used to encode the |0〉 and |1〉 states of a quantum bit. Donor electron spins have an additional hyperfine interaction with the 31P nuclear spin (blue), which can be exploited for both storing information in the nuclear spin and for read-out. Nature 479, 345–353 (2011) 10 Donor electron spin in Si:P Spin Hamiltonian 28 Si e nucl HSpin H Z H Z (i) H Hf (i) H Dip (i, j) i i i j H e- 31P Effect of Electron- Nuclei- external field nuclei 29Si nuclei interaction interaction e Electron spin Zeeman term: H Z gHS Effective Bohr radius ~ 20-25 Å Lattice constant = 5.43 Å nulc Nuclear spin Zeeman term: H Z (i) iHI In a natural Si crystal the donor electron interacts with ~ 80 nuclei of 29Si i System of 29Si nuclear spins can be Hyperfine electron-nuclear spin interaction: H Hf (i) SAiI considered as a spin bath Dipole-dipole nuclear spin interaction: i j H Dip (i, j) I DijI 21 Semion Saikin’s talk Donor electron spin in Si:P Energy level structure (high magnetic field) H e H Z gH zSz P P H Z PH z I z P P P - 31P electron spin H Hf AzzSz I z Si1 - 31P nuclear spin H Z H Si1 - 29Si nuclear spin Hf … 22 11 Quantum computing architectures usually incorporate buried arrays of donors or quantum dots with control gates on the surface of the silicon. Nature 479, 345–353 (2011) Scanning probe techniques, such as scanning tunnelling microscopy (STM), can be used to fabricate arrays of donors with atomic precision (top, inset), as well as the electrical leads and gates with which to control and address them (top, main image). Nature 479, 345–353 (2011) 12 Decoherence times The corruption of the state of the electron spin is characterized by two timescales: T1 describes spin relaxation from the spin-up to the spin-down state (or the loss of classical information stored in the spin), whereas T2 is the coherence time, which characterizes spin decoherence (or the loss of the phase information that is necessary to store quantum information) Nature 479, 345–353 (2011) Quantum computation Decoherence. Interaction with macroscopic environment. Markov process Non-exponential decay T1 T2 concept t T1 11 22 ~ e ~ et T2 12 0 t t Semion Saikin’s talk 13 Quantum computation Measure of Decoherence • Basis independent. • Additive for a few qubits. • Applicable for any timescale and complicated system dynamics. S ideal real ideal max1 , 2 ... S real ~ Sreal Sideal 11 12 A. Fedorov, L. Fedichkin, 21 22 V. Privman, cond-mat/0401248 27 Semion Saikin’s talk Decoherence times Nature 479, 345–353 (2011) 14 Manipulating spins Spins (whose eigenstates are separated in energy by ΔE) can be coherently manipulated using resonant microwave pulses of frequency, f = ΔE/h. A pulse of a given magnetic field strength, B, and duration, , will drive a spin from the |0ۧ to the |1ۧ state, whereas a pulse of half this duration will leave the spin in an equal superposition of the two states. The evolution of the spin magnetization, M, under an applied pulse can be understood as a rotation around the Bloch sphere (green trajectory) perpendicular to the direction of B. This can be measured as coherent oscillations in the z component of the magnetization as the duration of the pulse is increased. Nature 479, 345–353 (2011) Read out An entire class of quantum information processing (using cluster states) can be implemented by global manipulation, if individual donors can be selectively read out. Read-out seems a simple problem: many thousands of spin-resonance experiments are undertaken every day using read-out by magnetic induction, and single spins can be measured in a similar way. The difficulty arises in that the magnetic signal from a single spin is extremely small, requiring a long time for a single measurement. To overcome this challenge, alternative techniques have been developed. These are based on converting the information from the spin to another property that is more readily measurable, such as current (charge) or light. Nature 479, 345–353 (2011) 15 Read out An electron may be induced to tunnel from a donor onto the island of a nearby single electron transistor modifying its current.
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