Title: Liquid state NMR quantum computation
1Liquid 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
2Conditions 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.
3A 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.
4The basis of NMR
5The physical apparatus of NMR
6Cartoon 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.
7NMR signal free induction decay
Fourier transform of free induction decay
8A 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.
9A 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.
10Chemical 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
11Isolating 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.
12Effect 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.
13More 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.
14Spin-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
15J 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.
17From 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.
18The 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
19The 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
20The 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
22Spin-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
25Decoherence
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..
26Problems 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.
27How 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.
28Pseudo-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.
29PHYSICAL REVIEW A 57, 1998, p.3348
30A 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.
31Creating 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.
32Pseudo-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
33PHYSICAL REVIEW A 57, 1998, p.3348
34Pseudo-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!
35PHYSICAL 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
36The 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.
37Grovers 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.
383 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.
39Computation
- 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.
40Quantum 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?
41PHYSICAL REVIEW LETTERS 80, 1998, p. 3408, Isaac
L. Chuang, Neil Gershenfeld, and Mark Kubinec
42Conclusions
- 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.