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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)