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An Introduction to Quantum Computing
For Computer Science Undergraduates
Aria Schultz
Abstract
Quantum computing is still in its infancy. The world is still waiting for the first quantum
computer to be built. It would be naive to try and speculate as to when hardware for quantum
computing will become a reality; however, it would also be naive to forgo exposing oneself to
the unique benefits and problems of quantum computing under the assumption that quantum
computers are a long ways away.
This paper will strive to provide a cursory overview of the key points of quantum computing and
is aimed at the computer science undergraduate. Because of this, many of the concepts will be
basic overviews namely the underlying mathematical basis for quantum computing.
Introduction
In order to begin exploring the world of quantum computing it is useful to first compare and
contrast it with classical computing. Within the context of this paper, “classical computing” will
refer to the standard computing model that has memory composed of a series of bits. These bits
(which shall be referred to as Cbits henceforth) have a value of either 0 or 1. The state of a
classical computer can be seen as the state of all of the Cbits in the computer's memory. When
working with Cbits in a classical computer it is a trivial operation to read the state of the Cbit and
such an operation has no effect on the value the Cbit holds.
In contrast, the idea of a state has a very different meaning for a quantum bit (which shall be
denoted as Qbit henceforth). Before delving into that, it seems important to dispel a popular
misconception that whereas a Cbit holds the state of 0 or 1, a Qbit can hold a state of any number
between 0 and 1. This idea is a fundamental misunderstanding of how a Qbit works.
To begin understanding how a Qbit works it is important to start with the key concept that it is
impossible to read the state of a Qbit. In order to get any information from a Qbit one must
invoke a process known as measuring. The act of measuring a Qbit does not yield the state of
the Qbit (as was just pointed out, this is impossible) instead the process of measuring a Qbit
causes it to collapse into either the state of 0 or 1. Therefore, it is of utmost importance to
understand that although a Qbit is capable of holding very complex information, once one tries to
retrieve this information the Qbit will collapse down to a 0 or 1.
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Quantum Superpositon
So what information does a Qbit have in its state? The answer to that is both simple, and
complex. The state of a Qbit can be 0 or 1. It can also be 0 AND 1, or 0 AND 1 AND 0, etc. In
other words a Qbit can hold multiple states through quantum superpositon. Superpositon is a
basic concept of quantum mechanics. The logic of superpositon flies in the face of everything
we experience in the macro world. Our brains are sure of the idea that an object can only be in
one definite state at any given time. A light switch is either on or off, it cannot be on and off at
the same time. This simple rule that we often take for granted fails to characterize how things
behave at a quantum level. At that level it is possible, say, for an atom to be rotating clockwise
and counter-clockwise at the same time. This is known as quantum superpositon, the concept
that at the quantum level an object may not have one simple state, but instead multiple states that
all exist concurrently. It is this fundamental idea that gives quantum computing its power.
Manipulating Qbits
In classical computing Cbits are manipulated through the application of logic gates (AND, XOR,
etc.). The majority of these logic gates cannot be reversed once they have been applied with the
exception of something like NOT which can simply be applied a second time to reverse the
operation. When working with Qbits all operations can be reversed with the exception of
measuring the Qbits. Since measurement can be put off until calculations are completed this is
not a big issue.
The exact mechanic of the Qbits logic gates all being reversible has its explanation in linear
algebra. Given that linear algebra is not a required course for computer science undergraduates,
and trying to write a summary with enough brevity to allow room for the real meat of this article
(the portion on that chiefly pertains to computer science) the underlying mathematical formulas
and their notation will have to be replaced with a bit of hand waving1.
The basic point that should be understood is that a Qbit's state is represented mathematically as
any two-dimensional unit vector spanned by the vectors (0) and (1) over the complex numbers .
Thus the state of a Qbit can be generalize as
S = α0 (0) + α1(1)
where α0 and α1 complex number coefficients. Therefore, the state S associated with a Qbit is
the superpositon of the states 0 and 1 with amplitudes α0 and α1. The state of a single Cbit can
only hold one bit of information (a 0 or 2), but a Qbit is represented by two complex numbers
1
I will try to indicate points in this paper where I oversimplify or understate what is going on “behind the scenes.”
In other words I will try to make it apparent where further explanation of linear algebra as it applies to quantum
computing has been left out in order to make this paper more comprehensible for a computer science
undergraduate.
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and in order to represent it with an arbitrary amount of precision one requires an arbitrary
number of bits to specify the two complex numbers α0 and α1 [1].
In the case of Qbits, the logic gates that are applied are unitary matrices. These matrices are
basically rotations that act on a Qbit, and as such they can all be undone easily by applying the
opposite rotation. The benefits of this will be expanded on later, when I discuss the
computational process of a quantum computer.
Given that Qbits have far more complex states it would seem as though a quantum computer has
vastly superior computational power when compared to a classical computer, but there is a major
catch. As previously stated a Qbit may contain a plethora of information there is no way to “read”
that information from the Qbit as one would with a Cbit in classical computing. You cannot get
the amplitudes α0 and α1 instead your perform something called “measuring” to obtain a value
from a Qbit. This measurement will collapse the Qbit to either a 0 or 1.
Applying a measurement to a Qbit does not determine the result held in a Qbit. Instead, the
superpositon of states that a Qbit is in determines the probability that it will collapse into a
particular state. The probability of a particular result is the square of the magnitude of the
amplitude. In the example above of a single Qbit with amplitudes α0 and α1 the probability that
measuring this Qbit will yield a 0 is | α0|
One should avoid thinking of measuring a Qbit as the same thing as measuring someone's weight.
This is not the case. It would be more correct to compare it with measuring a person’s IQ.
Having someone take an IQ test will not yield any preexisting numerical property of a person,
just what happens when you give someone an IQ test [1].
Given these conditions it may be hard to imagine how anything of significance can be computed
using a Qbit. The trick is to use the unitary transformations to most amplitudes at or near zero,
and have useful information carried in any amplitude that has a decent chance of yielding the
correct result. As such, quantum computers are most useful at finding solutions that can then be
easily checked using some other method to verify their accuracy. Additionally, note that the
chance of getting an incorrect result often has a low probability.
The Computational Process in a Quantum Computer
Having described how a Qbit stores information and how information is manipulated and
retrieved from a Qbit it is time to talk about how one computes with a Qbit. Considering what
has been said so far about the Qbit, one may be wondering how a quantum computer could offer
any advantage over a classical one. In truth, if this was all a quantum computer had to offer it
would not be worth the effort of actually trying to build one, but the true beauty of a quantum
computer comes from the interesting computations that can be performed with a Qbit that are not
possible on classical computers. By taking advantage of quantum superpositon a programmer
can sacrifice gaining one piece of information for another. This trade-off is typical of quantum
computing, and leads to its most powerful features.
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The basis of this kind of trade-off is to set the input of a function to be the superpositon of all
possible inputs the function can take, and then execute the function. Thus, the output of the
function will be all the calculations of the function. With a 100 bit input register, this would
result in 1030 evaluations for the price of one. Of course, as has been reiterated throughout this
paper, there is no way to extract that information from the Qbits. All that one can do is measure
the output of the function, which will cause the Qbits complex state to collapse down do a simple
0 or 1. Thus, one can only learn a random value of the function this way, but there are some
interesting implications to this sort of parallelism if one is willing to give up the actual results of
a computation.
Deutsch's Problem
The first example of this quantum trade-off, and the example that got the whole field of quantum
computing started, is the Deutsch's Problem. The problem goes like this. Say there is a function
f that takes a single Qbit of input and returns a single Qbit as output. The function f will perform
some action on the input in order to produce an output. On a classical computer if one wanted to
know the result of f(0) they would have to run the function and check the output (which would be
a 0 or 1), and similarly one would call f(1) to obtain that result.
Things work the same on a quantum computer as well. Calling f(0) will return the computed
value of f(0). Now suppose all one wanted to know was whether f(0) was equal to f(1). On a
classical computer there would be no choice but to call f twice, passing 0 and 1, then compare the
results. A quantum computer, on the other hand, can make the determination with only one call
to f!
Of course, there is the quantum trade-off to keep in mind. In this case, what is lost is the actual
values of f(0) and f(1), but that information is irrelevant to answering the question of whether or
not the values are equal. In this way a quantum computer can trade one piece of information for
another.
The way one accomplishes this feat is by using superpositon to set the input register to be 0 and
1. Now if f is executed on the input that is a superpositon of both the inputs we wish to check an
interesting thing happens. The output register will just have a copy of the input register from
before f was called, and the input register will have either a 0 or a 1 depending on whether f(0)
== f(1) [4].
Simon's Problem
The next problem that we are going to examine is called Simon's problem. It is an extension of
the trick used in Deutsch's problem with the added benefit of being able to clearly see the speed
up that a quantum computer can offer over a classical machine.
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The Problem
In Simon's problem we have a function f(x) that takes an integer x as its input. The function then
computes the bitwise modulo-2 sum2 of x and sum unknown integer a. The goal is to find the
value of a or in other words find the period of f.
In order to solve this problem on a classical computer all that can really be done is start putting
in values for x and keep going until you find a value for f(xi) that is equal to some previous value
f(xj) obtained from the function. Then you would know that a equals the bitwise modulo-2
addition of xi and xj. The number of times you must call this method in order to have an
appreciable chance of finding the value of a grows exponentially with the number of bits in a. If
a had 100 bits a classical computer would have to call f about 1015 times. At ten million calls per
second, it would take about three years [2].
The Solution
A quantum computer can determine the value of a with high probability (less than one in a
million chance of failing) by calling f about 120 times given an a with 100 bits. This can be
accomplished by the trick we used in Deutsch's problem and set all the bits of the input x to be a
superpositon of all possible inputs3. Now when the function f is run and we measure the input
register we will learn (with equal probability) a random number y for which the modulo-2 sum of
a and y is 0. This value y gives us a subset of the bits in a hose modulo-2 sum is 0. Therefore, if
n is the number of bits in a we can lean the value of a with high probability by invoking the
function n + 20 times. This is a clear case where a quantum computer calculates the correct
result with a high probability and the result can be confirmed by a classical computer in far less
time than a classical computer could ever calculate the result.
Implications of Quantum Computing Techniques
These two problems may seem a bit artificial; however, they are intentionally simple in order to
give you an idea of the types of things one can do on a quantum computer, and how this can lead
to massive performance increases. The trick of using quantum superpositon to perform
calculations in parallel is at the heart of what gives quantum computing its allure. Always keep
in mind though, that the results of a parallel computation cannot be obtained, but we can set
things up so that by giving up learning the results of any of the calculations we can learn other
information about a function.
The concept of quantum parallelism can be used in some truly amazing ways from Groover's
algorithm, a search algorithm that runs in (π/4) √N time, to an algorithm that utterly defeats the
2
If you are unsure what is meant by bitwise modulo-2 addition have no fear it is equivalent to the XOR bitwise
logical operator.
3
Like in Deutsch's problem, there is a bit of additional overhead besides setting the input register as the
superpositon of all possible inputs. Again this amount to nothing more than applying some unitary logic gates to
the input register.
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one of the most widely used encryption scheme today RSA4. This is only a quick overview of
the usefulness of quantum computation and how one computes at the quantum level. From here
we can begin discussing some of the recent progress that has been made in the hardware side of
the field.
Quantum Hardware
During the past several years there have been a number of very important break-throughs in the
quest to build the first true quantum computer. These discoveries give a decent picture of what
the current issues with building a quantum computer are. They highlight the progress that has
been made, and the obstacles that need to be overcome. The also demonstrate the major issues
that must be solved in order to build a full scale quantum computer.
Problems
The main issue that one must contend with when they try to build a quantum computer is how to
make a Qbit. The majority of problems dealing with quantum hardware involve creating and
maintaining a Qbit. There are many different ways a Qbit can be created, and the only real
restriction is that you need something that can be in a superpositon of states [3]. Current
physical implementations are based on a variety of approaches including superconductors, ions,
the magnetic resonance of molecules in a solution, and the list goes on. The basic problem that
all these methods run into though is not how to create a Qbit, but how to get that Qbit to maintain
its state for sufficiently long enough time to perform calculations.
In order for a Qbit to properly hold its state, it must be completely isolate from the outside world.
Any interference will cause the state of the Qbit to decay, and become unusable. IBM has been
working on an implementation of Qbits. The IBM group announced earlier this year that they
have made a number of break-throughs which have allowed them to keep a Qbit free from any
outside interference for a longer period of time allowing the group to run some simple
calculations on them.
Another group of researchers at Yale created the first solid-state quantum processor back in 2009.
The chip had a 2-Qbit register and the Qbits could only maintain their states for a microsecond
which is considerably longer than what previous Qbits were capable of in the past.
I highlight these two groups because I think they illustrate the problem with building a full scale
quantum computer quite well. Even though both groups have very different approaches they
have the same basic problem: the Qbits are being interfered with by the outside environment
which causes them to decay and become useless. As you add more and more Qbits to a machine,
the likelihood that any one of those Qbits will decay before your calculation completes increases
dramatically.
4
Unfortunately we again have reached the boarders of this paper. Quantum computer science is an amazingly vast
field and many of the topics that have been covered here deserve a paper all to themselves.
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Controversies
Currently there is a lot of controversy surrounding a company named D-Wave. D-Wave claims
that they have built a full scale quantum computer; however, the company has not allowed for
any peer review of this so called quantum computer. As a result there is a great deal of debate in
the scientific community about whether this machine is truly a quantum computer or not.
Though the machine certainly has an interesting story, there is no consensus on whether D-Wave
Systems has actually produced a quantum computer or not.
Recently a lead scientist for the aerospace company Lockheed Martin gave the group at D-Wave
code from their F-16 fighter jet. The group at Lockheed knew there was a bug in the code, and
they wanted to see if D-Wave’s quantum computer could analyze the code and find the bug.
After taking the code from Lockheed D-Wave came back and announced they had found the bug.
Lockheed was so impressed by this that they paid $10 million for a quantum computer that is
now set up in a Lockheed research lab.
Looking Ahead
So that is a glimpse into current attempts to build a quantum computer. As was mentioned at the
beginning of this paper, there is no clear idea of when the first true quantum computer will be
built. There are many research groups working hard on the problems that face quantum
computer hardware.
Implications
While quantum computers may still be a purely theoretical concept many groups have shown a
great deal of interest in acquiring a quantum computer. The biggest of these groups would be
national governments who are interested in acquiring a quantum computer in order to decrypt
data from other nations.
Decryption
Because of the advantages that a quantum computer has over a classical computer in the speed
at which it can perform calculations, a quantum computer is very capable of factoring large
numbers. This means that quantum computers can break encryption schemes making them very
valuable to intelligence agencies across the world.
Simulation
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Another potential use for quantum computers is in simulating quantum chemistry. Classical
computers simply do not have the necessary memory or speed to simulate how atoms and
molecules behave. A quantum computer, on the other hand, has at the core of its functionality,
quantum mechanics. As a result a quantum computer is uniquely able to run simulations for
quantum chemists [3].
Conclusion
The first quantum computers will not be around any time soon, but it is not unreasonable to think
they will be a reality within 50 years (or less, by some estimates). This paper has tried to cover
(with much brevity) the way in which a quantum computer performs calculations and how those
calculations can be performed with amazing speed when compared to a classical computer. As a
result, many governments are interested in quantum computing as a way to break encryption.
Others are interested in quantum computing's potent ability to simulate quantum mechanics, a
task not feasible on classical computers. Given all this interest it is not hard to imagine how the
demand for computer scientists capable of developing programs for a quantum computer will
increase dramatically once these machines become available. Having a basic understanding of
how a quantum computer works and how it matches up to a classical computer can be a great
benefit in the future.
References
[1] Mermin, N. D. (2007). Quantum Computer Science. Cambridge: Cambridge University Press.
[2] Nielsen, M. A., & Chuang, I. L. (2000). Quantum Computation and Quantum Information.
Cambridge: Press Syndicate of the University of Cambridge.
[3] Le, B. M. (2006). A short introduction to quantum information and quantum computation.
Cambridge: Cambridge University Press.
[4] Yanofsky, N. S., & Mannucci, M. A. (2008). Quantum Computing for Computer Scientist.
Cambridge: Cambridge University Press.
Acknowledgments
Quantum Computing. (n.d.). Retrieved from http://en.wikipedia.org/wiki/Quantum_computer
Poeter, D. (2012, February 28). IBM says it's 'on the cusp' of building a quantum computer.
Retrieved from http://www.pcmag.com/article2/0,2817,2400930,00.asp
Muzzin, S. (2009, June 28). Scientists create first electronic quantum processor. Retrieved from
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http://news.yale.edu/2009/06/28/scientists-create-first-electronic-quantum-processor
Smalley, E. (2012, February 22). D-wave defies world of critics with ‘first quantum cloud’.
Retrieved from http://www.wired.com/wiredenterprise/2012/02/dwave-quantum-cloud/