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Trugenberger’s Quantum Optimization Algorithm Overview and Application

Overview  Inspiration  Basic Idea  Mathematical and Circuit Realizations  Limitations  Future Work

Overview  Inspiration  Basic Idea  Mathematical and Circuit Realizations  Limitations  Future Work

Two Main Sources of Inspiration  Exploiting Quantum Parallelism  Analogy 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  Inspiration  Basic Idea  Mathematical and Circuit Realizations  Limitations  Future Work

Basic Idea  Use 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 algorithm  Use 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  Inspiration  Basic Idea  Mathematical and Circuit Realizations  Limitations  Future 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

Cost Functions in General  should return a cost for that data element  In this algorithm we will want to minimize cost  Data elements with lower cost are better solutions

Skeleton of the U operator  The imaginary exponential of the cost function is the main engine of the quantum optimization

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

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

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>

Control Bits also need some modification  Control bit always starts out in |0> state  Before applying Ucs, we run the control bit through a Hadamard gate  After applying Ucs, we run it through another Hadamard gate  This gives us a nice super-position of minimal and maximal cost elements

Matlab results for Graph Coloring Data element Probability amplitude i* i* i* i* i* 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

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 000i* i* i* i* i* i*0.3536

Measurement  A 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 amplitude

Life Isn’t Fair  We don’t always get a |0> for all the control qubits when we measure  Some of the qubits are bound to be measured in the |1> state  Upon 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 Tradeoff  However, 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 news  As 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:

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 k  k can be seen as a certain subset of the |S> super-position of data elements  For the graph coloring problem, k=3  More generally for other problems, k can vary from 1 to K where K > 1

Equations affected by generalization  Cnor changes:

Equations effected by generalization  U changes (this in turn changes Ucs which utilizes U):

Overview  Inspiration  Basic Idea  Mathematical and Circuit Realizations  Limitations  Future Work

U operator  Constructing the U operator may itself be exponential in the number of qubits  Perhaps 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 Heuristic  Trugenberger’s algorithm may not get the exact minimal solution  Although, keeping in mind the tradeoff, more control qubits can be added to increase the odds of a good solution

Overview  Inspiration  Basic Idea  Mathematical and Circuit Realizations  Limitations  Future Work

Future Work  Look into physical feasibility of cost function and construction of Ucs  Run more simulations on various problems and compare against classical heuristics  Compare with Grover’s algorithm

Reference  Quantum Optimization by C.A. Trugenberger, July 22, 2001 (can be found on LANL archive)