Matt Purkeypile's Doctoral Dissertation Defense Slides
Cove: A Practical Quantum
Doctorate of Computer Science
June 26, 2009
• 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
• 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
– Utilizes probability amplitudes, which means they can
reinforce or cancel out.
• What known problems can quantum computers
– 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.
• General state of an arbitrary qubit:
ψ = α 0 0 + α1 1 =
α 0 + α1 = 1
• α1 and α2 are complex numbers and represent
– Hence the total of 1.
– cos(θ ) 0 + eiϕ sin(θ ) 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.
Limitations of quantum computers
• There are several limitations of quantum
– 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
– The computation must be reversible.
– It is impossible to copy qubits (no-cloning
Practical Example: Factoring
• Shor’s algorithm for factoring (1994) is perhaps
the most famous practical quantum computing
– It is exponentially faster than the classical solution.
– A quantum computer is utilized for only part of the
• This means you still have to do classical computation.
• Factoring means you can break codes such as
– 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
• 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.
• 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
m = 8 (a randomly selected number between 1 and N)
• 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
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
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
8 − 1 = 8 − 1 = 63
Check the result
• gcd(65, 15) = 5 and gcd(63, 15) = 3
Can be done efficiently on classical computers .
• 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
 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
• 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) 
• 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.
 N. S. Yanofsky and M. A. Mannucci, Quantum Computing for Computer
Scientists, 1 ed. New York, NY: Cambridge University Press, 2008.
What is really happening after first
f(x) for m=8, N=15 after measurement of 4
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
• 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
● 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
• Likewise, Quantum Fourier Transform also finds the
period in one operation.
What do you need to program
• 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)
– Measurement, which “collapses” the system to a
• Hence input and outputs are classical values.
• Like programming classical computers, this is
harder than it sounds.
• 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 , 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
– Often geared more towards mathematicians and physicists more
 S. Bettelli, "Towards an architecture for quantum programming," in Mathematics. vol.
Ph.D. Trento, Italy: University of Trento, 2002, p. 115.
A new solution: Cove
• Cove is a framework for programming quantum
– 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
• 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
– Like Bettelli, classical computation is handled by the
• Provides an object oriented approach for
• Documentation is as important as the
– Available online, within code, intellisense, and a help
• Attempts to avoid numerous usability flaws that
are present in all existing proposals to various
• Measurement of one qubit impacts the state of
– 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)
• Unit testing led to a much more solid design and
– 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
– 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
– Utilize remote resources?
• Investigation into the expanded QRAM model.
– Essentially how classical and quantum computers
• 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.
• Quantum computers can carry out tasks
that can never be done on classical
computers, no matter how fast or powerful
• 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.
(Source code, documentation, dissertation,
presentations and more)