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Doctoral dissertation defense slides from Matt Purkeypile's defense, June 26, 2009.

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- 1. Cove: A Practical Quantum Computer Programming Framework Matt Purkeypile Doctorate of Computer Science Dissertation Defense June 26, 2009
- 2. Outline • This presentation will cover the following: – A brief introduction to quantum computing. – Walking through a simple factoring example. – Programming quantum computers. – Cove: A new solution for programming quantum computers. – Questions
- 3. Quantum Computing • Existing computers (classical) operate on bits, which can hold the value of 0 or 1. • Quantum computers operate on qubits, which can hold the value of 0, 1, or a combination of the two. – Utilizes probability amplitudes, which means they can reinforce or cancel out. • What known problems can quantum computers do better? – Factor numbers, which means RSA can be cracked. • A simple example will be shown. – Simulate quantum systems. – Unsorted searches.
- 4. Classical and quantum comparison • The bit is just the poles of a qubit. • The probabilistic bit is just a line through the poles of a qubit.
- 5. Mathematically • General state of an arbitrary qubit: α 0 ψ = α 0 0 + α1 1 = α 1 2 2 α 0 + α1 = 1 • α1 and α2 are complex numbers and represent probability amplitudes. – Hence the total of 1. – cos(θ ) 0 + eiϕ sin(θ ) 1 in polar form, not commonly used. 2 2 • n qubits are described by 2n complex numbers. • Operations on n qubits are described by a 2n x 2n matrix of complex numbers.
- 6. Limitations of quantum computers • There are several limitations of quantum computers. – Although qubits can hold many possible values, only one classical result can be obtained from every run. • Hence the output is probabilistic. • Repeated runs may be necessary to obtain the desired result. – The computation must be reversible. – It is impossible to copy qubits (no-cloning theorem)
- 7. Practical Example: Factoring • Shor’s algorithm for factoring (1994) is perhaps the most famous practical quantum computing example. – It is exponentially faster than the classical solution. – A quantum computer is utilized for only part of the algorithm. • This means you still have to do classical computation. • Factoring means you can break codes such as RSA. – RSA is frequently utilized. – If N=pq, it is easy to calculate N when given p and q, but very hard to determine p and q when only given N. • Also known as a one-way function.
- 8. High Level View of Factoring • Except for step 2, the algorithm is carried out classically. • A probabilistic algorithm: may have to repeat runs until the answer is achieved.
- 9. Trivial Example • Goal: Factor 15. ● Result is 3 and 5. ● This has been done on quantum computers in the lab. ● Can be worked out by hand. • Step 1, let: ● N = 15 (the number we are factoring) ● n = number of (qu)bits needed to express N, in this case 4. ● m = 8 (a randomly selected number between 1 and N)
- 10. Step 2 • Calculate: f ( x) = m x mod N ● Need to calculate with enough x’s to find the period. ● x = 0,1, 2,... ● In general, go to at least N2 values. ● It seems like guessing would be faster, but isn’t. ● For this example we’ll just do 0 – 15. • Given this we can find the period (P). ● Essentially where f ( x) repeats. ● In other words f ( x + P ) = f ( x) for every x. • Performing all these calculations where we need only one answer (P) is how we can exploit a quantum computer.
- 11. Result f (0) = 80 mod15 = 1 f (1) = 81 mod15 = 8 f (2) = 82 mod15 = 4 f (3) = 83 mod15 = 2 f (4) = 84 mod15 = 1 f (5) = 85 mod15 = 8 f (6) = 86 mod15 = 4 f (7) = 87 mod15 = 2 f (8) = 88 mod15 = 1 f (9) = 89 mod15 = 8 f (10) = 810 mod15 = 4 f (11) = 811 mod15 = 2 f (12) = 812 mod15 = 1 f (13) = 813 mod 15 = 8 f (14) = 814 mod15 = 4 f (15) = 815 mod15 = 2
- 12. Can easily see the period graphically
- 13. Using the period (P) • The period is 4 ● It repeats 1, 8, 4, 2,… ● This concludes step 2 • Step 3: is P even? ● If not we start over using a different randomly selected m, however in this case it is even. • Step 4: Utilize P: 8 + 1 = 8 + 1 = 65 4/2 2 8 − 1 = 8 − 1 = 63 4/2 2
- 14. Check the result • gcd(65, 15) = 5 and gcd(63, 15) = 3 ● Can be done efficiently on classical computers [1]. • Step 5: we have found the factors 5 and 3. ● May only obtain one of the factors. ● Simple to obtain the second factor if not found. ● Basic algebra: pq=N, we know N and either p or q. ● Start over with a different m if the gcd of the results are 1. [1] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 1 ed. Cambridge, UK: Cambridge University Press, 2000.
- 15. How does a quantum computer help? • A quantum computer speeds things up by doing step 2 (finding the period) efficiently. ● Qubits are put in a superposition to represent all possible x’s at once (in the first register). ● In the case of factoring 15 we need 12 qubits (2(4) + 4, as we need two registers) [2] • Next f ( x) is performed on the qubits in superposition. ● One calculation on a quantum computer, many more classically. ● The result is put in the second register. • Measure Register 2- collapses the superpositions. • The period is then obtained via the Quantum Fourier Transform (QFT) followed by a measurement. • The rest of the algorithm is done classically. [2] N. S. Yanofsky and M. A. Mannucci, Quantum Computing for Computer Scientists, 1 ed. New York, NY: Cambridge University Press, 2008.
- 16. What is really happening after first measurement? f(x) for m=8, N=15 after measurement of 4 5 4 3 f(x) 2 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 x
- 17. How about QFT and the second measurement?
- 18. Scaling • 15 is a trivial example, how about a 128 bit number? • We need at least 384 qubits (128 * 3) to do the quantum part of the algorithm. (scratch qubits not accounted for) ● The quantum operations that are performed are done once, just on more qubits. ● Similar to adding two integers: same technique, more bits. • If we do it classically we have to calculate f(x) many times. ● It isn’t how easy it is to calculate f(x), it is how many times. ● Need to go from 0 to N2 , this is a huge number of calculations for a 128 bit number! This could be 2(2*128) or ~1.16 x 1077 ● The results have to be stored somewhere (taking up memory) and then we still have find the period! ● Or we can just use 384 qubits and run through a set of quantum operations once per attempt, so the quantum computer scales quite well. • Likewise, Quantum Fourier Transform also finds the period in one operation.
- 19. What do you need to program quantum computer? • Fundamentally, there are only three things needed to perform quantum computation: – Initialization of a register (collection of multiple qubits) to a classical value. – Manipulation of the register via (reversible) operations. – Measurement, which “collapses” the system to a classical result. • Hence input and outputs are classical values. • Like programming classical computers, this is harder than it sounds.
- 20. Programming Quantum Computers? • Quantum computers hold immense power, but how do you program them? – The operate fundamentally different from classical computers, so classical techniques don’t work. • With the exception of one technique [3], all existing proposals are new languages. – New languages may be able to perform quantum computation, but lack power for classical computation. – Quantum computing is typically only part of the solution, as in factoring. – Often geared more towards mathematicians and physicists more than programmers. [3] S. Bettelli, "Towards an architecture for quantum programming," in Mathematics. vol. Ph.D. Trento, Italy: University of Trento, 2002, p. 115.
- 21. Grover’s algorithm in Bettelli’s:
- 22. Deutsch’s algorithm in Tafliovich’s:
- 23. A new solution: Cove • Cove is a framework for programming quantum computers. – This means classical computation is handled by the language it is built on (C#) – It designed to be extended by users. – Key concept: programming against interfaces, not implementations. • The current work includes a simulated quantum computer to execute code. – All simulations of quantum computers experience an exponential slow down.
- 24. Why is Cove a new contribution? • Provides extensibility not present in Bettelli’s solution. – Like Bettelli, classical computation is handled by the existing language. • Provides an object oriented approach for quantum computing. • Documentation is as important as the framework. – Available online, within code, intellisense, and a help file. • Attempts to avoid numerous usability flaws that are present in all existing proposals to various degrees.
- 25. Example: Entanglement • Measurement of one qubit impacts the state of another. – This doesn’t happen in a classical computer, bits are manipulated independently- no impact on other bits.
- 26. Example: Implementation of Sum (documentation of method excluded)
- 27. Reflections • Unit testing led to a much more solid design and implementation. – Forced code to be written that utilized Cove. – Takes hours to run tests with just a handful of qubits. • Implementation of the local simulation was much harder than anticipated. – Many problems with implementation aren’t documented well: • Reordering operations. • Expanding operations to match register size. – Memory and time constraints limit what can be done. • Ran into memory constraints early on. • Applying an operation to a 20 qubit register requires 220 + (220)2 =1,099,512,676,352 complex numbers! • Makes debugging difficult.
- 28. Areas for future work • Make the prototype implementation more robust and complete. – Utilize remote resources? • Investigation into the expanded QRAM model. – Essentially how classical and quantum computers interact. • Provide solutions for other algorithms such as Grover’s (unsorted search). • The number of quantum algorithms is small, so that is an area for work as well.
- 29. Conclusion • Quantum computers can carry out tasks that can never be done on classical computers, no matter how fast or powerful they become. • Existing quantum programming techniques suffer from numerous flaws. • Cove is a new method of programming quantum computers that tries to avoid flaws of existing techniques.
- 30. Questions? https://cove.purkeypile.com/ (Source code, documentation, dissertation, presentations and more) Matt Purkeypile mpurkeypile@acm.org

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