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Liquid state NMR quantum computation

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Title: Liquid state NMR quantum computation


1
Liquid state NMR quantum computation
A good resource Introduction to NMR Quantum
Information Processing http//www.c3.lanl.gov/kn
ill/qip/nmrprhtml/nmrprhtml.html
Karen Sauer Rm. 309, ST 1 (703)993-1281 ksauer_at_phy
sics.gmu.edu
2
Conditions for quantum computation
  • Represents quantum information set of accessible
    states should be finite. Spin-state of a
    spin-1/2 particle is ideal.
  • Single-spin operations and controlled-NOT gates
    are sufficient to perform any quantum
    computation.
  • Preparation of desired initial state need
    reproducibility and low entropy.
  • Ideally, 00000gt
  • Measurement of output result.

3
A controlled- NOT gate
A gate defined as follows

where ? is the gate operator, ??c is the control
qubit and ??t is the target qubit. In simple
terms, if the control qubit is 0?c then the
target qubit is left alone, if the control qubit
is 1?c then the target qubit is flipped.
4
The basis of NMR
5
The physical apparatus of NMR
6
Cartoon Version of NMR
1. Magnetic Field aligns the magnetic moments
2. Radio-frequency pulses at just the Larmor
frequency, ??B tip the magnetic moments out of
alignment.
3. The transverse magnetization now rotates
at the Larmor frequency and is detected by a coil
using Faradays law.
4. This coil is a part of a tuned circuit which
is only sensitive to a limited band of
frequencies.
7
NMR signal free induction decay
Fourier transform of free induction decay
8
A 2-qubit NMR computer
1H, spin a
Cl
B
13C, spin b
13C-labeled chloroform
00gt ? ? gt 01gt ? ? gt 10gt ? ? gt 11gt
? ? gt
In an 11.7 Tesla Magnet, Precession frequency of
hydrogen is 500 MHz Precession frequency of 13C
is 125 MHz. (Remember??B). Other spin-1/2 nuclei
19F, 15N, and 31P.
9
A 3-qubit NMR computer
B
Trichloroethylene using the one hydrogen and
two 13C atoms. There is a frequency difference
(called the chemical shift'') of 600-900 Hz
between the two carbon atom at 11.7 Tesla.
10
Chemical shifts
Chemical shifts arise because of the simultaneous
interaction of a nucleus with an electron and
that of the electron with an applied field.
A look into an alanine crystal carbon (green),
hydrogen (red), oxygen (purple), and nitrogen
(blue). The arrows show how the electrons flow
when a magnetic field is applied perpendicular to
the cutting plane.
http//www.nersc.gov/research/annrep97/louie.html
11
Isolating the computers
Molecules are dissolved in a solvent such that
inter-molecular interactions become
negligible. Solvent typically is deuterated (like
acetone-d6) so that no NMR signal arises from the
solvent.
12
Effect of resonant RF-rotation of the
magnetization
Single bit rotation around the x-axis in the
rotating frame. A weak field (typically less than
.001 of the external field) is applied along the
x axis of the rotating frame.
13
More quantum mechanically
While the RF field is applied, the qubit's state
evolves as Rxe(-i?xXt/2) . The strength of the
pulse is characterized by ?x, the nutation
frequency. For example, with the initial spin
state being purely 0gt,
the final state can be any superposition of 0gt
and 1gt, depending on the angle of rotation ?
?xt. Similarly Ry e(-i?yYt/2), represents
rotation around the y-axis.
These are the single-spin operators we need.
14
Spin-spin couplings
  • Direct dipolar coupling
  • the effect of the magnetic field of one nucleus
    on an other
  • largely averaged away in a liquid
  • Indirect through-bond coupling (J coupling)
  • magnetic field seen by one nucleus is perturbed
    by the state of its electronic cloud which
    interacts with another nucleus through Fermi
    contact interaction
  • key to doing a controlled-not gate

15
J couplings
When the difference of the precession frequencies
between the coupled nuclear spins is large
compared to the strength of the coupling, it is a
good approximation to write the coupling
Hamiltonian as
So that
where J is the coupling constant in Hz.
16
J-coupling
With a positive coupling constant, the coupling
between two spins can be interpreted as an
increase in precession frequency of the spin 2
when the spin 1 is up and a decrease when spin
1 is down.
17
From Quantum computation and quantum
information, M.A. Nielsen and I.L. Chuang
The coupling constants in trichloroethylene at
11.7 T are close to 100 Hz between the two
carbons, 200 Hz between the proton and the
adjacent carbon, and 9 Hz between the proton and
the far carbon.
18
The basic NMR-controlled not gate
Spin 1
Ry
Spin 2
Rx
RF pulses
System evolving under J-coupling Z-Z 90
10?s
5ms for J 100 Hz
10?s
19
The controlled-NOT gate
1gt 0gt
1. Initial state 0gt0gt
2. After Ry
3. System allowed to evolve freely for t 1/2J
4. After last Rx Final state 0gt0gt
1gt 1gt
20
The controlled-NOT gate
1gt 1gt
1. Initial state 0gt1gt
2. After Ry
3. System allowed to evolve freely for t 1/2J
4. After last Rx Final state 0gt1gt
1gt 0gt
21
Field inhomogeneities causes loss of signal.
initial pulse
22
Spin-echoes stopping the evolution
Runners begin.
At time t ?? the runners suddenly turn around
and start running in the opposite direction.
If they do not change their speed, they will meet
each other at t 2?.
23
p pulse reverses the order of the spins
t t-
t t
24
?-pulse also reverses J-coupling
25
Decoherence
T1 - time it takes for the longitudinal
magnetization to completely reappear (reach
thermal equilibrium). Energy exchange is
required with the lattice.
T2 - time it takes for the transverse
magnetization to disappear, neglecting effects
due to magnetic field inhomogeneities.
In a liquid, T2 T1 order of 1- 100 seconds..
26
Problems with NMR Quantum computing
  • Quantum computing best done on a single molecule,
    but NMR needs 108 molecules to see a signal.
  • Solution Consider the large collection of
    molecules as an ensemble of identical systems,
    read out collective answer.
  • NMR too weak to determine the outcome and cause
    the state's collapse into 0gt or 1gt for each
    molecule.
  • Solution Often good enough to observe a NMR
    signal that represents the average over all the
    molecules of the probability that 1gt would be
    the outcome of a projective measurement.
  • The equilibrium states of the molecules' nuclear
    spins are nearly random, with only a small
    fraction pointing in the right direction.
  • Solution Temporal, spatial, or logic
    labelling, methods for singling out the small
    fraction of the observable signal that represents
    the desired initial state.

27
How to begin?
Quantum logic gates transform qubits from one
state to another, but this is only useful if the
qubits start off in some well defined initial
state (or a pure state) like 0gt 00gt. However,
in liquid state NMR at room temperature, because
the energy difference between the nuclear spins
up and down states is so small compared to room
temperature, the equilibrium distribution of
states is nearly random.
The nth diagonal element of the density matrix
tells us the probability of finding the system in
the ngt pure state. Note h?/kT 810-5 for
protons in an 11.7 Tesla magnet.
28
Pseudo-pure states
Do not despair! We can transform the system to a
pseudo-pure state
For n spins all but one of the 2n populations are
equal. The identity density matrix is not
observable in NMR since only population
differences are observed, and furthermore does
not transform under unitary evolutions (UIU
I). Thus the visible signal is produced solely
by the one distinct population.
29
PHYSICAL REVIEW A 57, 1998, p.3348
30
A pseudo-pure state of a 2-qubit system
This pseudo-pure state behaves identically to the
pure state 00gt ??gt even though it is still
highly mixed.
31
Creating a pseudo-pure state
  • temporal averaging schemes.
  • spatially averaging uses pulse sequences and
    field gradients.
  • logical labeling uses a subset of the energy
    levels in a more complex spin system (ancilla
    spins label the correct subspace).
  • combination of the above methods.

32

Pseudo-pure states by temporal averaging
-Perform many different experiments, each with a
different initial state. -The different initial
states are obtained using controlled-NOT
gates. -The final spectra from the experiments
are then added and the result is equivalent to a
single experiment starting from a pseudo-pure
state
33
PHYSICAL REVIEW A 57, 1998, p.3348
34

Pseudo-pure states by logical labeling
density matrix at thermal equilibrium
after manipulation (unitary operations)
2
6
Relevant submatrix can be written as
A pseudo-pure state!
35
PHYSICAL REVIEW LETTERS 83, 1999,
p.3085 Realization of Logically Labeled Effective
Pure States for Bulk Quantum Computation Lieven
M. K. Vandersypen, Costantino S. Yannoni, Mark H.
Sherwood, and Isaac L. Chuang
36
The problem with pseudo-pure states
  • With the present methods, the amount of
    pseudo-pure state which can be extracted from an
    NMR system falls off exponentially with number of
    spins in the system. Or in other words all spins
    have intensity decreasing exponentially with
    number of qubits, so that liquid state NMR is
    limited to about ten qubits.
  • Higher polarization (gt1/3) could solve both
    objections.

37
Grovers Search Algorithm
Finding the needle in a haystack of N
elements Classical computing requires on O(N)
queries Quantum computing can find with only
O(N1/2) attempts. Simple example 4 elements,
one needle f(xneedle) 1 for only one of 4
possible elements f(xhay) 0 for the other
three. Classically evaluate f(x) until you can
find the needle, takes an average of 9/4 2.25
evaluations. Grovers algorithm 1) use 4 states
(00gt, 01gt, 10gt, and 11gt) to represent the 4
elements 2) simultaneously processing a
superposition of states to find the needlein
one evaluation.
38
3 steps of bulk computation
  • Preparation transform the input to a pseudo-pure
    state, 00gt.
  • Computation create a superposition of all four
    states, then pick out the state corresponding to
    xneedle.
  • Readout postprocessing of the algorithm's output
    which terminates in the measurement of the
    observable Iza and Izb, the spins along the
    z-axis of the first and second qubit.

39
Computation
  • Create a uniform superposition of the four
    possible input (00gt01gt10gt11gt)/2.
  • Identify xneedle with a negative phase using a
    propagator to implement f(x) xgt ? (-1)f(x)xgt
    . If xneedle 11gt , output is
    (00gt01gt10gt-11gt)/2.
  • Convert the phase difference to an amplitude
    difference by inversion around the average.
    Inversion around the average takes in a
    superposition and reflects the amplitude of each
    component around the average amplitude of all the
    components.
  • Above example (xneedle 11gt) average amplitude
    is 1/4 reflecting 1/2 around 1/4 gives 0, while
    reflecting -1/2 around 1/4 gives 1.
  • This operation acts to concentrate all the
    amplitude on one member of the superposition,
    giving a final state of just 11gt.

40
Quantum circuit for Grovers algorithm (xneedle
11gt)
Rx exp(i?X/4) Ry exp(-i?Y/4)
(00gt01gt10gt11gt)/2
(00gt01gt10gt-11gt)/2
spin 1
0gt
1gt
Phase shifter
Oracle
spin 2
0gt
1gt
H Rx2 Ry Hadamard gates transform 0gt to
(0gt1gt)/21/2 for each spin.
Oracle Ry1 Rx1 Ry1Ry2Rx2Ry2 ? where ? denotes a
time t 1/2J of evolution without RF during
which the system undergoes J-coupling such that
11gt is replaced by -11gt .
Inverts the states around their mean with Phase
shifter Ry1 Rx1 Ry1 Ry2 Rx2 Ry2?
41
PHYSICAL REVIEW LETTERS 80, 1998, p. 3408, Isaac
L. Chuang, Neil Gershenfeld, and Mark Kubinec
42
Conclusions
  • With liquid state NMR it is possible to perform
    computations in fewer steps than is possible
    using any classical machine.
  • However it is unlikely that liquid state NMR
    could ever be use to solve problems faster than
    any classical machine (need to speed up gate
    times and increase the number of gates possible
    within the coherence time).
  • Furthermore NMR approaches are likely to be
    limited to computers containing 10-20 qubits
    this is significantly smaller that estimates of
    the size required to perform useful computations
    (50-300 qubits).
  • Hopefully other quantum computer implementations
    will benefit from the ideas, concepts and
    solutions which arise from liquid state NMR
    experiments.
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