Cove: A Practical Quantum Computer Programming

Framework

Matt PurkeypileDoctorate of Computer Science

Dissertation DefenseJune 26, 2009

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

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.

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.

Mathematically

• General state of an arbitrary qubit:

• α1 and α2 are complex numbers and represent probability amplitudes. – Hence the total of 1.– in polar form, not commonly used.

• n qubits are described by 2n complex numbers.• Operations on n qubits are described by a 2n x

2n matrix of complex numbers.

1

010 10

12

1

2

0

cos( ) 0 sin( ) 12 2

ie

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)

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.

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.

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)

Step 2

• Calculate:– Need to calculate with enough x’s to find the period. – – 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 repeats.– In other words for every x.

• Performing all these calculations where we need only one answer (P) is how we can exploit a quantum computer.

( ) modxf x m N

0,1,2,...x

( )f x( ) ( )f x P f x

Result0

1

2

3

4

5

6

7

8

9

10

11

12

13

(0) 8 mod15 1

(1) 8 mod15 8

(2) 8 mod15 4

(3) 8 mod15 2

(4) 8 mod15 1

(5) 8 mod15 8

(6) 8 mod15 4

(7) 8 mod15 2

(8) 8 mod15 1

(9) 8 mod15 8

(10) 8 mod15 4

(11) 8 mod15 2

(12) 8 mod15 1

(13) 8 mod

f

f

f

f

f

f

f

f

f

f

f

f

f

f

14

15

15 8

(14) 8 mod15 4

(15) 8 mod15 2

f

f

Can easily see the period graphically

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: 4/2 28 1 8 1 65 4/2 28 1 8 1 63

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.

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 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.

( )f x

What is really happening after first measurement?

f(x) for m=8, N=15 after measurement of 4

0

1

2

3

4

5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

x

f(x

)

How about QFT and the second measurement?

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.

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.

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.

Grover’s algorithm in Bettelli’s:

Deutsch’s algorithm in Tafliovich’s:

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.

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.

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.

Example: Implementation of Sum

(documentation of method excluded)

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.

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.

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.

Questions?

https://cove.purkeypile.com/

(Source code, documentation, dissertation, presentations and more)

Matt Purkeypilempurkeypile@acm.org