KEG PARTY!!!!!
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Transcript of KEG PARTY!!!!!
KEG PARTY!!!!! Keg Party tomorrow night Prof. Markov will give out extra credit to
anyone who attends*
*Note: This statement is a lie
Trugenberger’s Quantum Optimization Algorithm
Overview and Application
Overview InspirationBasic IdeaMathematical and Circuit RealizationsLimitationsFuture Work
Overview InspirationBasic IdeaMathematical and Circuit RealizationsLimitationsFuture Work
Two Main Sources of Inspiration
Exploiting Quantum ParallelismAnalogy of Simulated Annealing
What is quantum parallelism?What is quantum parallelism?
We can represent super-positions of specific instances of data in a single quantum state
We can then apply a single operator to this quantum state and thereby change all instances of data in a single step
What is Simulated Annealing?Comes from physical annealing
Iteratively heat and cool a material until there’s a high probability of obtaining a crystalline structure
Can be represented as a computational algorithm Iteratively make changes to your data until
there is a high probability of ending up with the data you want
Overview InspirationBasic IdeaMathematical and Circuit RealizationsLimitationsFuture Work
Basic IdeaUse this inspiration to come up with a
more generalized quantum searching algorithm
Trugenberger’s algorithm does a heuristic search on the entire data set by applying a cost function to each element in the data set
Goal is to find a minimal cost solution
The high-level algorithmUse quantum parallelism to apply the
cost function to all elements of the data set simultaneously in one step
Iteratively apply this cost function to the data set
Number of iterations is analogous to an instance of simulated annealing
Overview InspirationBasic IdeaMathematical and Circuit RealizationsLimitationsFuture Work
Representing the Problem: Graph Coloring
Super-position of the data elements N instances Use n qubits to represent the N instances Each instance encoded as a binary
number I^k whose value is between 0 and 2^n
N
k
kIN
S1|
1|
Cost Functions in General should return a cost for that data
element In this algorithm we will want to
minimize costData elements with lower cost are better
solutions
)( kIC
Skeleton of the U operator
The imaginary exponential of the cost function is the main engine of the quantum optimization
))(2
exp( knor IallforCiU
What is Cnor?We know in general that exp(i*theta) =
cos(theta) + i*sin(theta)Since U will need the imaginary
exponential of the cost function, we want to normalize the cost function
By normalizing, we ensure that the cost function result is between 0 and pi/2
What is Cnor?
C(I^k) can at most be Cmax and is at least Cmin
Cnor is always between 1 and 0
minmax
min)()(
CC
CICIC
kk
nor
And Cmin and Cmax?Simple to determine for graph coloring
Cmin = 0 (no pair connected vertices shares the same color)
Cmax = # of edges (every pair of connected vertices shares the same color)
More general method for determining Cmin and Cmax will be introduced later
Fleshing out U for Graph Coloring
),...,()1,...,1(
2)0,...,0(
2 11 nnnor CnoriCieediagU
Still don’t quite have our magic operator
As written, U by itself will not lower the probability amplitude of bad states and increase the amplitude of good states
If we apply U now, the probability amplitudes of both the best and worst data elements will be the same and differ only in phase
Take Advantage of Phase Differences
We can accomplish the proper amplitude modifications by using a controlled form of the U gate
Can’t be an ordinary controlled gate though
Ucs: The Answer to our Problems
Ucs is a controlled gate that applies U to the data elements when the control bit is |0> and applies the inverse of U when the control bit is |1>
1|11||00| UUU cccccs
Control Bits also need some modification
Control bit always starts out in |0> stateBefore applying Ucs, we run the control
bit through a Hadamard gateAfter applying Ucs, we run it through
another Hadamard gateThis gives us a nice super-position of
minimal and maximal cost elements
Matlab results for Graph Coloring
Data element Probability amplitude000 0001 0.25010 0.3536011 0.25100 0.25101 0.3536111 0-----------------------------------------------------------------000 i*0.3536001 i*0.25010 0011 i*0.25100 i*0.25101 0110 i*0.25111 i*0.3536
Measurement If we were to measure the control bit now and
get a |0>, we’d know that the data will get the “first half” of the super-position:
Data element Probability amplitude
000 0
001 0.25
010 0.3536
011 0.25
100 0.25
101 0.3536
111 0
Measurement However if we got a |1> instead, we’d know
that the data will get the “second half” of the super-position:
Data element Probability amplitude
000 i*0.3536
001 i*0.25
010 0
011 i*0.25
100 i*0.25
101 0
110 i*0.25
111 i*0.3536
MeasurementA control qubit measurement of |0>
means we have a better chance of getting a lower cost state (a good solution)
A control qubit measurement of |1> means we have a better chance of getting a higher cost state (a bad solution)
Measurement Assume the world is perfect and we always
get a |0> when we measure the control qubit We can effectively increase our probability of
getting good solutions and decrease the probability of getting bad solutions by iterating the H,Ucs,H operations
We iterate by duplicating the circuit and adding more control qubits
Matlab Results after 26 “Ideal” Iterations
Data element Probability amplitude000 0001 0010 0.3536011 0100 0101 0.3536111 0-----------------------------------------------------------------000 0001 0010 0011 0100 0101 0110 0111 0
Life Isn’t FairWe don’t always get a |0> for all the
control qubits when we measureSome of the qubits are bound to be
measured in the |1> stateUpon measuring the control qubits we
can at least know the quality of our computation
The Tradeoff If we increase the number of control
qubits (b), then we have a chance of bumping up the probability amplitudes of the lower cost solutions and canceling out the probability amplitudes of the higher cost solutions
The TradeoffHowever, if we increase the number of
control qubits (b), we ALSO lower our chances of getting more control qubits in the |0> state
Some good newsAs mentioned earlier, the measurement
of the control qubits will tell us how good our bad a particular run was
Trugenberger gives an equation for the expected number of runs needed for a good result:
0
2
1
20
))(2(cos
1)(
))(2(cos
1
b
knor
bkb
knor
N
k
bb
NPZ
ICZ
IP
ICN
P
Analogy to Simulated Annealing
Can view b, the number of control qubits, as a sort of temperature parameter
Trugenberger gives some energy distributions based on the “effective temperature” being equal to 1/b
Simply an analogy to the number of iterations needed for a probabilistically good solution
A Whole New Meaning for kk can be seen as a certain subset of the
|S> super-position of data elementsFor the graph coloring problem, k=3More generally for other problems, k
can vary from 1 to K where K > 1
Equations affected by generalization
Cnor changes:
minmax
min)'()'(
CC
CIofsubsetthkCIofsubsetthkC
kkkkk
nor
Equations effected by generalization
U changes (this in turn changes Ucs which utilizes U):
m
k ii
kii
k
kGU
1 ......
1
1
),...,()1,...,1(
2)0,...,0(
2 11 kknork
knor CiCik eediagG
Overview InspirationBasic IdeaMathematical and Circuit RealizationsLimitationsFuture Work
U operatorConstructing the U operator may itself
be exponential in the number of qubitsPerhaps some physical process to get
around this
Cost Function Oracle?Trugenberger glosses over the
implementation of the cost function (in fact no implementation is suggested)
Some problems may still be intractable if cost function is too complicated
Only a HeuristicTrugenberger’s algorithm may not get
the exact minimal solutionAlthough, keeping in mind the tradeoff,
more control qubits can be added to increase the odds of a good solution
Overview InspirationBasic IdeaMathematical and Circuit RealizationsLimitationsFuture Work
Future WorkLook into physical feasibility of cost
function and construction of UcsRun more simulations on various
problems and compare against classical heuristics
Compare with Grover’s algorithm
ReferenceQuantum Optimization by C.A.
Trugenberger, July 22, 2001 (can be found on LANL archive)