Thesis defense slides

Introduction NMR 2-D Ion Arrays Electronic Networking Conclusions
An investigation of precision and scaling issues in
trapped ion and nuclear spin quantum simulators
Robert J. Clark
Thesis Defense
Department of Physics
Massachusetts Institute of Technology
11 March 2009
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Introduction NMR 2-D Ion Arrays Electronic Networking Conclusions
Simulation: a very useful tool
Idea:
Simulate one system with another that is easier to control.
Target system
↓
Model system
A great approach!
Much control over
parameters.
Space, time resources scale
linearly with system size.
But with limitations...
Noise
Control precision
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Introduction NMR 2-D Ion Arrays Electronic Networking Conclusions
Digital simulation
Analog simulation is largely replaced by digital: more resilience to
noise with an affordable increase in resources!
http://dcoward.best.vwh.net/
Digital v. analog
→
Analog: total error limited by accumulation of intrinsic errors.
Digital: precision may be increased efficiently.
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Introduction NMR 2-D Ion Arrays Electronic Networking Conclusions
Quantum simulation
Quantum systems hard to simulate!
Simulation of n two-level systems (qubits) on a classical computer
requires ∝ 2 n bits just to write the state down!
Classical computers currently limited to ∼ 36 qubits...
DeRaedt et al., Comput. Phys. Commun. 176, 121 (2007).
A possible way out: quantum simulation
Quantum simulation maps a target quantum system to a more
controllable model system.
Feynman, Int. J. Theor. Phys. 21, 467 (1982); Lloyd, Science 273, 1073 (1996).
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Introduction NMR 2-D Ion Arrays Electronic Networking Conclusions
Why is quantum simulation so interesting?
Resources required scale efficiently with system size!
Digital
Analog
Digital quantum simulation
Discrete pulses.
Error correction possible.
Classical Quantum
Space: 2n Time: T 22n Space: n
Time: T n2 Space: 2n Space: n
Time: T Time: T
Here n is the number of qubits simulated
Analog quantum simulation
Continuous controls.
Sensitive to noise.
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Introduction NMR 2-D Ion Arrays Electronic Networking Conclusions
Quantum simulation: prior art
Analog quantum simulation
Much success with neutral atoms.
Greiner et al., Nature 415, 39 (2002).
Digital quantum simulation
Shin et al., Nature 451, 689 (2008).
NMR with small qubit numbers (examples to follow).
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Introduction NMR 2-D Ion Arrays Electronic Networking Conclusions
Problems for quantum simulation
1. Precision
Quantum simulation generally inefficient w.r.t. precision.
Why? Every time you measure, state collapses.
Digital
Analog
Resources required:
Classical Quantum
Space: 2n O(log 1/ǫ)
Time: T22n Space: 2n Time: T
Precision: O(1/ǫfix)
Space: n
Time: Tn 2 O(1/ǫ)
Space: n
Time: T O(1/ǫ)
Here n is qubit number, T is simulation time ǫ is the error, and ǫfix is the fixed error
of an analog device.
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Introduction NMR 2-D Ion Arrays Electronic Networking Conclusions
Problems for quantum simulation
2. Scalability
It’s hard to build a reliable large system from faulty components.
Task grows harder, the more control is required.
For instance:
Optical lattice: many particles, global control.
Trapped ions: ≤ 8 particles (so far), exquisite individual ion
control.
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Introduction NMR 2-D Ion Arrays Electronic Networking Conclusions
Thesis outline
Precision: NMR simulation of BCS model
How much precision can be obtained experimentally using digital
quantum simulation?
Scaling: 2-D ion arrays
How can we build an ion trap for analog quantum simulations of
frustrated spin systems?
Scaling: Electronic networking of ions
Can electronic networking be used to scale up ion trap quantum
simulation?
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Introduction NMR 2-D Ion Arrays Electronic Networking Conclusions
Thesis outline
Precision: NMR simulation of BCS model
How much precision can be obtained experimentally using digital
quantum simulation?
Scaling: 2-D ion arrays
How can we build an ion trap for analog quantum simulations of
frustrated spin systems?
Scaling: Electronic networking of ions
Can electronic networking be used to scale up ion trap quantum
simulation?
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Introduction NMR 2-D Ion Arrays Electronic Networking Conclusions
Motivation: find the spectrum of Hamiltonian HT
1. Initialize
{|ΨT 〉} → {|ΨM〉}
Prepare |Ψ〉 = |E〉 + |G〉
Goal: calculate ∆ = 〈E|HT |E〉 − 〈G|HT |G〉
3. Measure some M for each t.
∆ = 2
2. Approximate HT
Do |Ψ(t)〉 = U |Ψ0〉.
(U = exp(−itHT/))
4. Fourier transform
∆ = 2
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Introduction NMR 2-D Ion Arrays Electronic Networking Conclusions
Digital quantum simulation: prior art
Most digital quantum simulations have used nuclear spins with a
small number of qubits.
Truncated harmonic oscillator (2
qubit)
Observed oscillations at
simulated frequency.
Somaroo et al., PRL 82, 5381 (1999).
Fano-Anderson model. (2 qubit)
Eigenvalues determined from this
spectrum.
Negrevene et al, PRA 71, 032344 (2005).
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Introduction NMR 2-D Ion Arrays Electronic Networking Conclusions
Problem: what limits the precision?
Previously known: Fourier sampling rate
Uncertainty ǫFT ∼ 1/N.
N limited by coherence time.
Consequence of wave function
collapse.
Unknown: Effect of control errors
How do control errors affect ǫ?
Where do errors come from?
Can they be efficiently
compensated?
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Introduction NMR 2-D Ion Arrays Electronic Networking Conclusions
The NMR experiment
Our experiment:
Experimental system:
Strong static field B0 ˆz.
Rotations about ˆx and ˆy with rf
radiation.
Inductive readout of sample
magnetization.
Calculate ∆ = E(|E1〉) − E(|G〉) for BCS
Hamiltonian:
HBCS = n νm
m=1 2 (−Zm) + Vml
m

Introduction NMR 2-D Ion Arrays Electronic Networking Conclusions
Experiment: results
One Hamiltonian simulated:
1 Requires large number of
single-qubit “refocusing”
pulses.
2 Method W2 compensates
for evolution during pulses
(W1 does not).
3 Error within ǫFT only with
error compensation!
The source of these control errors:
They arise from “always-on” scalar coupling during
single-qubit pulses.
For another Hamiltonian with fewer control pulses, method
W1 resulted in error ǫFT.
Brown, Clark, and Chuang, Phys. Rev. Lett. 97, 050504 (2006).
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Introduction NMR 2-D Ion Arrays Electronic Networking Conclusions
Discussion
Can’t we use quantum error correction?
Yes. It will cost us time ∼ O(1/ǫ r ),r ≥ 2. Why?
Trotter approximation needed to approximate HA, HB for
[HA,HB] = 0.
With this approximation, ǫ ∼ 1/N, with N number of gates.
Fault-tolerant gates require roughly equal time.
Can we efficiently compensate systematic errors?
Yes, we need poly(n) time for practical purposes (but not to
arbitrary order). Brown, Harrow, and Chuang, PRA 70 052318 (2004).
Even so, compensating gates → fewer operations due to
coherence time.
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Introduction NMR 2-D Ion Arrays Electronic Networking Conclusions
Summary: the precision of digital quantum simulation
Digital
Analog
Resources required:
Classical Quantum
Space: 2nO(log 1/ǫ)
Time: T22n Space: 2n Time: T
Precision: O(1/ǫfix)
Space: n
Time: Tn 2 O(1/ǫ r )
Space: n
Time: T O(1/ǫ)
Here n is qubit number, T is simulation time ǫ is the error, and ǫfix is the fixed error
of an analog device.
Main results
Quantum simulation is inefficient w.r.t. precision.
Errors lead generally to Time ∼ O(1/ǫ r ), r ≥ 2.
This constrains the utility of digital quantum simulation.
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Introduction NMR 2-D Ion Arrays Electronic Networking Conclusions
Thesis outline
Precision: NMR simulation of BCS model
How much precision can be obtained experimentally using digital
quantum simulation?
Error correction requires time ∼ 1/ǫ r , r ≥ 2. Control errors can dominate.
Scaling: 2-D ion arrays
How can we build an ion trap for analog quantum simulations of
frustrated spin systems?
Scaling: Electronic networking of ions
Can electronic networking be used to scale up ion trap quantum
simulation?
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Introduction NMR 2-D Ion Arrays Electronic Networking Conclusions
Thesis outline
Precision: NMR simulation of BCS model
How much precision can be obtained experimentally using digital
quantum simulation?
Error correction requires time ∼ 1/ǫ r , r ≥ 2. Control errors can dominate.
Scaling: 2-D ion arrays
How can we build an ion trap for analog quantum simulations of
frustrated spin systems?
Scaling: Electronic networking of ions
Can electronic networking be used to scale up ion trap quantum
simulation?
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Introduction NMR 2-D Ion Arrays Electronic Networking Conclusions
Motivation: a scalable, controllable system
Why ions?
1 Potentially long (> seconds) coherence times.
2 Exquisite control:
> 99% two-qubit operation fidelity.
> 99% measurement fidelity.
3 Many ideas for scalability:
Moving ions.
Photonic networking.
Electronic networking.
Our goal: analog simulation with ions
Quantum spin models, Bose-Hubbard physics, quantum fields, ...
May require scaling to “only” tens of ions!
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Introduction NMR 2-D Ion Arrays Electronic Networking Conclusions
Analog simulations with trapped ions: prior art
Several interesting proposals:
Effective spin systems Porras & Cirac PRL 92 207901 (2004).
Bose-Hubbard model Porras & Cirac PRL 92 263602 (2004).
Quantum fields in an expanding universe Alsing et al., PRL 94
220401 (2005).
But so far, few experiments:
Nonlinear interferometer
Leibfried et al., PRL 89, 247901 (2002).
Two-spin Ising model
Friedenauer et al., Nature Physics 4, 757
(2008).
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Introduction NMR 2-D Ion Arrays Electronic Networking Conclusions
Specific goal: quantum simulation of frustrated spins
Analog simulation of spin models
Spin states mapped to internal states of ions
State-dependent forces from lasers + Coulomb =
Effective spin system!
Spin frustration
observable in
2-D.
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Introduction NMR 2-D Ion Arrays Electronic Networking Conclusions
2-D Ion Arrays
Linear, mm-scale ion traps are well-developed. But analog
quantum simulation of spin frustration requires two dimensions!
Penning trap
Itano et al., Science
279 5351 (1998).
Some previous work:
Single Paul trap
Block et al., J. Phys.
B 33 L375 (2000).
Proposed Paul trap array
Chiaverini & Lybarger, PRA 77 022324
(2008).
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Introduction NMR 2-D Ion Arrays Electronic Networking Conclusions
Outline of 2-D ion array work
Our goal
Design a trap for 2-D ion arrays that is suitable for simulating spin
frustration.
How to evaluate suitability
We focus on transverse Ising model: H =
i,j Ji,jZiZj +
i BiXi.
We require the ability to tune J; J ∝ 1/(ω4d 3 ).
J related to ion motional coupling rate ωex ∝ 1/(ωd3 ).
We look at two paradigms:
Array of individual Paul traps.
Ion crystal in a single trapping region.
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Introduction NMR 2-D Ion Arrays Electronic Networking Conclusions
2-D Lattice ion trap
The “classic” ring
Paul trap.
Lattice trap advantages:
→ Remove top endcap to “infinity”; iterate to
create a lattice.
Lattice geometry determined by trap electrodes and choice of
sites loaded.
Amenable to proposal for using rf and magnetic fields
(eschewing most lasers).
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Introduction NMR 2-D Ion Arrays Electronic Networking Conclusions
Lattice ion trap experimental setup
A test trap was mounted in a UHV system (O(10 −9 torr)) for
trapping 88 Sr + .
Steel mesh (d = 1.64 mm) mounted above CPGA with UHV epoxy and glass spacers.
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Introduction NMR 2-D Ion Arrays Electronic Networking Conclusions
Lattice ion trap measurements
Trapping observed; motional frequencies match simulations.
Ion images: cloud and crystal
Clark, Lin, Brown, and Chuang, J. Appl. Phys. 105 013114 (2009).
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Introduction NMR 2-D Ion Arrays Electronic Networking Conclusions
Lattice ion trap measurements
Trapping observed; motional frequencies match simulations.
Secular frequency measurement
Clark, Lin, Brown, and Chuang, J. Appl. Phys. 105 013114 (2009). (28/50)

Introduction NMR 2-D Ion Arrays Electronic Networking Conclusions
Coupling rates I: motional coupling rate
Next step: evaluate the design for quantum simulation.
We need to calculate the motional coupling rate ωex vs. d.
Comparison of ωex for lattice and linear traps.
Stiffness of lattice trap leads to lowered coupling.
Clark, Lin, Brown, and Chuang, J. Appl. Phys. 105 013114 (2009).
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Introduction NMR 2-D Ion Arrays Electronic Networking Conclusions
Coupling rates II: simulated J-coupling
How does this coupling translate into a simulated coupling rate?
J vs. d for lattice and linear trap.
Ion confined too weakly for Jlattice → Jlinear.
Clark, Lin, Brown, and Chuang, J. Appl. Phys. 105 013114 (2009).
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Introduction NMR 2-D Ion Arrays Electronic Networking Conclusions
Surface-electrode elliptical traps
Problem:
Lattice trap coupling rates scale poorly with trap size.
Solution idea:
Form 2-D crystal in the same trap region.
One example: surface-electrode (SE)
elliptical trap
Three nondegenerate motional
frequencies.
SE amenable to microfabrication.
−→ Higher coupling, lower micromotion
Inclusion of B-field gradients?
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Introduction NMR 2-D Ion Arrays Electronic Networking Conclusions
Experimental apparatus: 4 K closed-cycle cryostat
Reasons to use a cryostat:
Heating rates greatly reduced at 4 K.
Cryopumping enables rapid pumpdown to UHV.
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Introduction NMR 2-D Ion Arrays Electronic Networking Conclusions
Experimental measurements: ion crystals
Small ion crystals have been confined in the elliptical trap. Two
examples:
Measured separations
Two ions Four ions
Two ions: dy = 16.5 µm (Camera magnification 4.5)
Four ions: dy = 28±3 µm, dx = 17±3 µm.
Four ions (theory): dy = 29.6 µm, dx = 17.1 µm.
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Introduction NMR 2-D Ion Arrays Electronic Networking Conclusions
Micromotion and quantum simulation
Micromotion is
a small-amplitude, driven oscillation of the ion at the rf drive
frequency.
problematic, because it modulates the laser phase at the ion
position, increases Doppler shifts, and in an ion cloud, it
causes heating.
How we model it:
J coupling depends on d: J ∼ 1/(ω 4 d 3 ).
Numerically integrate equations of motion (with two ions).
Simulate H = J(t) (Z1Z2) + B (X1 + X2).
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Introduction NMR 2-D Ion Arrays Electronic Networking Conclusions
Micromotion effects for two qubit Ising model
Note: in the axial (ˆz) direction, all micromotion can be
compensated! We focus on the radial (ˆx,ˆy) directions.
Plot of a single trajectory of
〈Z1 + Z2〉, J = −B = 1 kHz.
Same plot with
Javg = 1.035 kHz, B = -1 kHz.
Micromotion leads to control errors that may be calculated and corrected
if effective controls are available for each pair of ions.
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Introduction NMR 2-D Ion Arrays Electronic Networking Conclusions
Summary of 2-D ion array work
Two new paradigms for trapping 2-D arrays of ions
1 Lattice trap
2 Surface-electrode elliptical trap
Lattice trap:
Well-defined lattice structure.
Some good fabrication ideas −→
Coupling rates poor compared to
linear/ elliptical trap.
Ziliang Lin, Yufei Ge
Elliptical trap:
Stronger coupling rates than lattice.
Micromotion → systematic errors. (36/50)

Introduction NMR 2-D Ion Arrays Electronic Networking Conclusions
Thesis outline
Precision: NMR simulation of BCS model
How much precision can be obtained experimentally using digital
quantum simulation?
Error correction requires time ∼ 1/ǫ r , r ≥ 2. Control errors can dominate.
Scaling: 2-D ion arrays
How can we build an ion trap for analog quantum simulations of
frustrated spin systems?
Ions in arrays of individual traps have poor interaction rates, rates are high with ions
in the same trap (e.g. elliptical trap), as long as micromotion control errors can be
compensated.
Scaling: Electronic networking of ions
Can electronic networking be used to scale up ion trap quantum
simulation?
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Introduction NMR 2-D Ion Arrays Electronic Networking Conclusions
Thesis outline
Precision: NMR simulation of BCS model
How much precision can be obtained experimentally using digital
quantum simulation?
Error correction requires time ∼ 1/ǫ r , r ≥ 2. Control errors can dominate.
Scaling: 2-D ion arrays
How can we build an ion trap for analog quantum simulations of
frustrated spin systems?
Ions in arrays of individual traps have poor interaction rates, rates are high with ions
in the same trap (e.g. elliptical trap), as long as micromotion control errors can be
compensated.
Scaling: Electronic networking of ions
Can electronic networking be used to scale up ion trap quantum
simulation?
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Introduction NMR 2-D Ion Arrays Electronic Networking Conclusions
Motivation
2-D static arrays only semi-scalable
Electronic networking
Concept: use image charges induced in a wire to connect ions in
different traps.
Proposed Penning trap array, with focus on
interactions between single electrons
(Stahl et al., EJPD 32 139 (2005)).
Proposed interface between ion and
superconducting qubits
(Tian et al., EJPD 32 201 (2005)).
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Introduction NMR 2-D Ion Arrays Electronic Networking Conclusions
Problem Statement
Research questions
Two trapped ions, one wire.
1 What are the coupling rates (ωex,J) and decoherence rates in
theory?
2 How does the wire affect the electric potential seen by a single
ion?
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Introduction NMR 2-D Ion Arrays Electronic Networking Conclusions
The circuit model
They derived:
Heinzen and Wineland treated this problem:
PRA 42, 2977 (1990).
For L1 = L2 = L, C1 = C2, ω = 1/ √ LC1,
Motional coupling rate ωex given by
ωex =
1 1/2
2ωLC
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Introduction NMR 2-D Ion Arrays Electronic Networking Conclusions
Estimating the coupling rate
System model in detail:
Derivation of constants in circuit model
L ∝ mH2
ec ≈ 6 × 104 H.
C1,2 = 1/(ω 2 L) ≈ 4 × 10 −19 F.
Then C ≈ 10 −15 F gives ωex ≈ 1 kHz, J ≈ 200 Hz.
Daniilidis, Lee, Clark, Narayanan, Häffner, submitted to J. Phys. B
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Introduction NMR 2-D Ion Arrays Electronic Networking Conclusions
Decoherence
Consider two main sources:
1 Dissipation of the current.
Current is ∼ 1 fA, → 2 × 10 5 s needed to dissipate one
quantum with R = 0.6 Ω.
2 Motional heating.
1 Johnson noise: Estimated at ∼ 0.1 quantum/s at 300 K and
ion-wire distance d ≡ H − h = 50 µm.
2 Anomalous heating: Much higher heating rates:
O(10 6 quanta/s) at 300 K, in larger traps.
Again, cryogenic cooling probably needed.
Daniilidis, Lee, Clark, Narayanan, Häffner, submitted to J. Phys. B
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Introduction NMR 2-D Ion Arrays Electronic Networking Conclusions
Experimental apparatus
1 Microfabricated surface-electrode gold trap.
2 25 µm diameter gold wire.
3 Stack of four piezoelectric nanopositioners.
In collaboration with the University of Innsbruck, Austria.
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Introduction NMR 2-D Ion Arrays Electronic Networking Conclusions
Experimental measurements
We probe the dc and ac effects of the wire on the trap potential by
measuring:
Vertical compensation voltage vs. d.
Indicates unknown stray charge on the
wire.
Two secular frequencies vs. d. Capacitive
coupling between wire and rf drive has a
significant effect.
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Introduction NMR 2-D Ion Arrays Electronic Networking Conclusions
Summary: Electronic networking
Theoretical results
1 Coupling rates of ωex ∼ 1 kHz, J ∼ 200 Hz calculated.
2 Decoherence (in theory) sufficiently low at 4 K.
Experimental results
1 Wire strongly influences potential:
Frequencies ωx,y ∝ d −2 .
Strong dc effect too.
2 But this is also an opportunity!
Measure charge, capacitance of the wire using the ion?
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Introduction NMR 2-D Ion Arrays Electronic Networking Conclusions
Conclusions
How much precision can be obtained experimentally using digital
quantum simulation?
Quantum simulation generally inefficient w.r.t. precision.
Random and systematic errors lead to time ∼ 1/ǫ r , r ≥ 2.
How can we build an ion trap for analog quantum simulations of
frustrated spin systems?
Ions in arrays of individual traps have poor J-coupling rates.
J is high with ions in the same trap (e.g. elliptical trap), as long as
micromotion control errors can be compensated.
Can electronic networking be used to scale up ion trap quantum
simulation?
Coupling and decoherence rates promising in theory.
Use a single ion as a probe of a macroscopic conductor!
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Introduction NMR 2-D Ion Arrays Electronic Networking Conclusions
Outlook
Digital quantum simulation
Inefficiency → massive resource requirements.
Solving hard problems basically requires large-scale quantum
computer.
Analog quantum simulation with ions
A way to solve hard problems without the resources required
for digital quantum simulation.
Ions need to occupy the same trap → limited system size.
A promising solution: interconnect ions in different traps
Transmit information using photons or (in this thesis)
electrons.
Unlimited system size, no fundamental obstacle to scaling.
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Introduction NMR 2-D Ion Arrays Electronic Networking Conclusions
Publications
In this presentation
1 Brown, Clark, and Chuang, Phys. Rev. Lett. 97 050504 (2006).
2 Clark, Lin, Brown, and Chuang, J. Appl. Phys. 105 013114 (2009).
3 Daniilidis, Lee, Clark, Narayanan, and Häffner, submitted to J.
Phys. B.
4 Clark, Diab, Lin, and Chuang, in preparation.
5 Clark, Daniilidis, Narayanan, and Häffner, in preparation.
Others
1 Brown, Clark, Labaziewicz, Richerme, Leibrandt, and Chuang, Phys.
Rev. A. 75 015401 (2007).
2 Leibrandt, Clark, Labaziewicz, Antohi, Bakr, Brown, and Chuang,
Phys. Rev. A 76 055403 (2007).
3 Leibrandt, Labaziewicz, Clark, Chuang, Pei, Low, Frahm, and
Slusher, in preparation.
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Introduction NMR 2-D Ion Arrays Electronic Networking Conclusions
Acknowledgements
My thesis committee
Isaac Chuang, Wolfgang Ketterle, Leonid Levitov
Many collaborators
Ken Brown, Jarek Labaziewicz, Dave Leibrandt, Tongyan Lin,
Ziliang Lin, Kenan Diab, Tony Lee, Nikos Daniilidis, Sankar
Narayanan, Hartmut Häffner, Rainer Blatt, Paul Antohi, Yufei Ge,
Shannon Wang, ...
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