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.
Nature 479, 345–353 (2011)
Read out
The state of the donor nuclear spin can also be measured as a result of its hyperfine coupling to the electron. For example, a charge-trap mechanism can be used; this results in a reduction in photocurrent only if the electron is spin down. By using the hyperfine interaction to selectively rotate the electron based on the spin of the nucleus, the current reflects the nuclear state.
Nature 479, 345–353 (2011)
16 Kane Quantum Computer • Semiconductor substrate with embedded electron donors (31P) • Electron wave functions manipulated by changing gate voltages • Most easily scalable
Cross-section of Kane Quantum Computer Potential wells in Kane Quantum Computer www.lanl.gov/physics/quantum/ MRS, February 2005, Kane
17 Kane Quantum Computer
The nuclear spin (I=1/2) of 31P is a natural two- level system embedded in a spin-free substrate of 28Si (I=0).
The nuclear spins of 31P donors are separated by approximately 20 nm and there is a hyperfine interaction between donor electron spin and nuclear spin (qubit).
Interaction between qubits is mediated through the donor-electron exchange interaction.
Exchange coupling: basic physical interaction between the spins of electrons whose wave functions overlap, arising from the Pauli exclusion principle.
The spins are maintained at millikelvin (mK) temperatures in an external magnetic field of several Tesla, perpendicular to the plane of the substrate.
Nanoscale surface A and J gates control the hyperfine and exchange interactions at qubit sites.
Logic Gate Operations
Nuclear spins alone will not interact significantly with other nuclear spins 20 nm away.
Nuclear spin is useful to perform single-qubit operations, but to make a quantum computer, two-qubit operations are also required.
This is the role of electron spin in this design.
Under A-gate control, the spin is transferred from the nucleus to the donor electron. Then, a potential is applied to the J gate, drawing adjacent donor electrons into a common region, greatly enhancing the interaction between the neighbouring spins. By controlling the J gate voltage, two-qubit operations are possible.
18 Kane Quantum Computer: qubits P nucleus – Spin mediated by electron spin through hyperfine interaction – Controlled and measured by varying voltages in top gates – Long decoherence times ~1018 s
Cross-sections of Kane Quantum Computer www.lanl.gov/physics/quantum/
19 Kane Quantum Computer Future • Further research into semiconductor materials • Smaller technology while approaching limit by Moore’s law
http://qso.lanl.gov/qc
Quantum dot:
A confining structure for electrons, which can be designed to stably hold a small number of electrons.
20 What is a Quantum Dot?
• A quantum dot is a structure in which the particles are confined in all spatial
dimensions Lx,Ly,Lz ¼ (DeBroglie wave length). • The dimensions of the quantum dots range from 10nm to 1m (103 to 109 atoms). • The energy levels in a quantum dot are quantized – they are called sometimes macroscopic atoms.
QDs in heterostructures
Depending on the fabrication technique there are different kinds of quantum A semiconductor dots: heterostructure is a stack of •Self-assembled layers made of different materials. Due to the •Vertical (etched) mismatch of the Fermi levels, •Lateral the electrons are confined at the interface of the two different semiconductors.
21 Vertically Etched QDs
The vertical quantum dots can be fabricated by wet etching (chemically washing away the semiconductor layers) or dry etching (reactive ion etching).
Self-assembled QDs on the top of semiconductor layer
22 Lateral Quantum Dots
The 2DEG is confined by modifying the potential of the gates on the top of the heterostructure.
Qubits in QD
• Spin qubit – one dot
• Charge qubit – two dots
• Quantum Ground State qubit – 2 (N + 1) dots for N-step evolution
23 DiVincenzo Criteria
• Scalable system with • The qubit is one spin QD, well-characterized or two charge QDs, or qubits 2(N+1) QDs • Initialization • Spin/charge injection •Long decoherence time • Decoherence time w.r.t. the gate-operation ~100μsec; gate-operation time time ~1nsec • Universal set of gates • Only single-qubit gates are demostrated so far • Read-out of the result • Depends on the system
Loss-DiVincenzo Proposal
The use of the spins of the electrons confined in quantum dots as qubits was motivated by the fact that spins are natural two-level systems and the possibility of building scalable arrays of quantum dots.
The interaction between neighboring electron spins is time dependent Heisenberg coupling. By controlling the strength of this interaction, we can implement the quantum computing operations.
24 Quantum-dot array proposal
Charge density maps in solid state quantum computer
25 Loss- DiVincenzo proposal
Advantages • The spins have long decoherence times (up to 1ms) compared to the gate operation times (of the order of 1ns). • The quantum dot arrays can be fabricated with the current experimental techniques. • Specific qubits can be addressed in order to perform quantum gate operations. • Charge control and readout has been achieved in double dots, as well as single shot spin readout
Loss - DiVincenzo proposal
Disadvantages
• Spin injection in semiconductors is not efficient with the current experimental techniques. • Decoherence due to the hyperfine coupling with the atoms nuclei in semiconductor is unavoidable • The control over the Heisenberg interaction is not yet fully understood
26 Special strengths
1. Semiconductor systems (GaAs, Si, SiGe,) offer inherent scalability. Established and new semiconductor patterning processes allow for the construction of submicron 2-D arrays of qubits.
2. Semiconductor systems have compatibility with existing microelectronics industry and have high potential for development of integrated on-chip devices.
3 Spin qubits in semiconductors are well-defined “native” qubits (two-level systems).
4. Spin qubits in semiconductors (single donor or QDs) can be decoupled from charge fluctuations, leading to long decoherence times (from s to ms) compared with practical gate operation times (from ps to ns).
5. Charge qubits in semiconductors (e.g., electron position in double quantum wells) offer potential for extremely fast qubit (ps) operations.
Unknowns, weaknesses
1. Background impurity levels and disorder in semiconductor systems may lead to difficulties in device reproducibility. These issues are common with sub-100-nm devices in conventional microprocessors.
2. For spin qubits—single-spin readout is challenging.
3. For spin qubits in Si—the exchange interaction is predicted to oscillate as a function of donor separation, which may place stringent requirements on nanofabrication accuracy.
4. For charge qubits—decoherence likely to be dominated by voltage fluctuations on control gates and may be fast. Experiments on GaAs dots indicate dephasing times on the order of ns.
5. Most semiconductor-based schemes are based on linear qubit arrays. The extension to 2-D arrays will require via-gate techniques on the sub-100-nm scale, which is challenging.
6. A number of solid-state schemes are still at the conceptual phase. Detailed fabrication strategies still to be developed.
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