An Introduction
to
Quantum Computation
and
Quantum Communication
 Rob Pike
Bell Labs
Lucent Technologies
rob@plan9.belllabs.com
June 23, 2000
Introduction
An analogy:
Newtonian physics is an approximation to Einsteinian physics (general relativity).
Classical physics is an approximation to quantum mechanics.
Classical information is an approximation to quantum information.
In each case, the approximation excludes important details but serves
well for many purposes.
In each case, removing the approximation requires deeper understanding
and harder math, but results in a truer picture of Nature and may enable
new technologies.
Yes, Nature: we're beginning to understand that
information is a physical concept.
What approximation do we remove?
Relativity: we remove (among others) the approximation that we are traveling
much slower than light.
Quantum mechanics: we remove (among others) the approximation that
we are manipulating things much larger than atoms.
Quantum computation: we remove (among others) the approximation
that the elements of information are independently manipulable.
Why would we care?
That approximation means that we can look at one bit in a register
without affecting the other bits.
Why remove that approximation?
Because it limits the power of the computer.
(Keep in mind the analogies.)
Also, getting ahead of ourselves, that approximation turns out to
be troublesome in representing information quantum mechanically.
Why would we do that?
We're running out of particles.
The insulators in CMOS transistors can't get much smaller
or the insulating layers will stop insulating (at around 6 atoms thick).
(Maybe before 2010.)
In optical fiber, we use ten thousand or so photons to represent a bit.
There's a Moore's law for fiber, too, and we'll soon run out of photons.
(Maybe before 2010.)
Quantum mechanical effects will become important in just a few years!
Currently, we work in the classical information regime. That won't last.
We'd better come to understand quantum information.
Of course, this version of the story isn't how quantum computation came to be.
(Keep in mind the analogies.)
So let's back up and tell a more historical story, to introduce the ideas.
Feynman's Question
In a couple of papers in the 1980s,
Feynman asked and began to answer
the following question:


Is it feasible for a computer to simulate a physical system perfectly?
The answer appears to be, "No".
A classical computer seems to need time exponential in
n
to predict precisely the behavior of a general quantum mechanical system
of
n
particles.
(Yet nature manages to do it in real time.)
Briefly, a quantum mechanical system of
n
particles
is represented by a wave function in a Hilbert space of dimension exponential
in
n.
We really do need that much dimensionality to represent
all possible behaviors of the system.
Less briefly...
The Nature of Quantum Reality
The twoslit experiment.
1. Single photon still produces interference pattern!
2. Ask which slit photon passes  pattern disappears!
Interpretation
The photon can go through either slit, or both; its state
embodies both possibilities.
If we ask which slit it went through, there must be an answer,
and the system must decide:


Asking the question changes the
state of the system from both possibilities to exactly one.
The Quantum Measurement Postulate


When you make a measurement, the system makes a random
selection among the possible answers and chooses one.
After the measurement, the system is in the state
that always gives that answer; the possibility of other answers is gone.
Do the measurement again (sufficiently quickly) and you'll
get the same answer.
Quantum mechanics in two slides (I)
The state of a QM system is described by its
wave
function,
an oscillating complexvalued function defined over all of space.
can interfere with itself.
Quantum mechanics is linear. We can create
by linear combination, e.g.:


For welldefined states, e.g. up,
we use the notation
as in


The
are complex coefficients that must normalize;
if the
>
states are orthogonal, the
must satisfy:


These are called
probability
amplitudes
and
(note the square) is the probability that
if we make a measurement of the system, we will find it in state
Quantum mechanics in two slides (II)
The quantum measurement postulate in math:
If we make a measurement on a system
with wave function


and find it's in state
i,
the wave function is now


Math aside: the
i
are the eigenvalues corresponding to eigenvectors
of the operator (e.g. energy) defining the measurement.
See for yourself
Light can be
linearly
polarized:
its vibrations can lie in a plane, say horizontally or vertically.
Represent these two possibilities as
and
We call these orthogonal states a
basis
of the system.
Plain light is a mixture of these polarizations and in fact a single
photon can be a mixture. For example,
light polarized at 45° is
Light can also be circularly polarized: circular polarization
can be created from linear as follows:




This is another orthogonal basis of the polarization.
We can demonstrate that light obeys the quantum
measurement postulate using three linear polarizing
filters and an overhead projector....
A few points about wave functions
1. Quantum mechanics is a linear theory: we can create
linear superpositions of wave functions, provided we
keep the probability amplitudes normalized.
2. The quantum measurement postulate can be described
as the wave function `collapsing' to the basis state
corresponding to the outcome of the experiment.
3. We cannot discover the full quantum state of
a system, only the
squared
probability amplitudes
The
are the projections
of the system onto the basis states and are complexvalued.
4: We cannot clone an unknown quantum state.
There are no quantum wires. (Proof a little later.)
Bits and Qubits
A bit is in one of two states, 0 and 1,
represented by e.g. the state of a switch or a voltage.
To map this to quantum mechanics, choose two orthogonal
states (e.g. horizontal and vertical polarization) and label these
and
The state maps to a Boolean 0 or 1.
A
qubit
is a parcel of information represented by such a system.
Because quantum mechanics is linear,
unlike Boolean algebra, a
qubit
can be not just the value
and
but any complex
linear superposition that satisfies the normalization condition.
For example, a qubit might be
a horizontally polarized
photon; or it might be
a vertically polarized one,
or it might be
a right circularly
polarized one, or any other linear combination with appropriate
normalization.
The interpretation of Qubits
A bit represents one of two points, but
a qubit represents any point on the unit circle in the complex plane.
To ask the state of the qubit is to ask whether it is
or
and by QMP it must decide.
Therefore, when we measure a qubit, we can only ever get
or
corresponding to Boolean 0 or 1.
But
until we ask, it can be an arbitrary mixture of
and
Multiple bits and qubits
N
bits can represent
integer values.
N
qubits can represent any complex vector of unit length
in
dimensions, one
dimension
for each possible classical state.
A
spectacularly
larger set of values!
3 bits can represent any one of 000, 001, 010, ..., 111.
3 qubits can represent any value of the form


as long as


For example, a 3qubit register might have the value
There is no classical analogue of this sort of state.
The register represents two (or up to 2^N) different values
simultaneously!
The register could be in the `pure' state
or
but the overwhelming majority of possible states
are not pure.
Entanglement
Two classical bits can be 00 or 01 or 10 or 11.
We can ask the value of the first bit without affecting the second bit.
Two qubits could be in the state


The first qubit is neither
nor
It's not even a superposition of
and
because the state is not separable: the value of the first qubit
is
entangled
with the value of the second.
We can't discover value of first qubit
affecting the second.
Say we measure it and get 0; by QMP that means the state
of the system is now
and therefore the second qubit
is now
But it wasn't
before; it was entangled with the first qubit.
This is another very different feature of quantum information.
Two points about entanglement
1.
An entanglement by definition involves multiple qubits;
this is not an entangled state:


2. A superposition is not necessarily entangled.
Consider


We can measure the first qubit without affecting the second.
Compare the two above with this truly entangled state:


Proof of the no cloning theorem
Proof by contradiction. Assume we have a box that
will take an arbitrary qubit and create a copy.
Given
the result will be
given
the result will be
Given the arbitrary state


we want as output two separable qubits like this:


But quantum mechanics is linear, so applying the box to our state
will produce
Unless one of
or
is zero, this is not the desired state; it is entangled.
Therefore the cloning box cannot exist.
Similarly, an unknown quantum state cannot be deleted
without affecting the rest of the system.
Conservation of information.
This theorem means: no wires, no oscilloscope probes,
no debugging print statements.
Note: this theorem doesn't apply once we measure the qubits,
since the result is a pure state.
Computation
Classically, we put
n
bits into a calculation and get
m
bits out:
A quantum computer
can't create or destroy qubits during the calculation,
so
m
must equal
n:
The quantum computer is an operator
that maps
n
input qubits to
n
output qubits.
Recall that
n
qubits represent a unit vector pointing to the
surface of a sphere in complex space of
dimensions.
Therefore the QC is a kind of rotation;
it can be represented by a rotation matrix in complex
space; such matrices are called
unitary.
Quantum systems evolve by unitary operations, and all
steps in a quantum calculation must be unitary.
The final measurement step does not need to be unitary,
since we can throw data away at the end.
A quantum gate
A quantum gate is a unitary operator, so number of bits in equals
number of bits out.
No AND or OR, but instead e.g., a controlledNOT,
which inverts
B
if
A
is 1:
It's a rotation, so reversible: given the output, we can recover the input.
Other quantum gates include controlledcontrolledNOT,
square root of NOT, and other exotica.
Reversibility has the side effect that, in principle, it means a
quantum gate can use zero energy (but might take arbitrarily long).
Reversibility has the undesired side effect that we are forbidden from
using latches, feedback, or rewritable memory.
The big picture
A quantum computer looks like this, taking
n
input qubits,
the register
V,
and producing
n
output qubits, the register
W:
The input register can be prepared as a superposition of states, e.g.
an equal superposition of
all
integers from 0 to


The computer then calculates in parallel the function
applied to all
integers simultaneously.
From QMP,
when we measure
W,
it will choose a Boolean for each
bit of the output register according to the resulting
entangled
wave function
of the output qubits.
Design
F
so that it maximizes the probability
that the output we measure is the answer we want.
The big picture continued
Measuring the output collapses
the wave function: get
Boolean values for all the qubits in
W.
The result is one of the possible outputs.
Imagine that
F
is (integer) square root
Prepare
V
as the superposition
of all integers from 0 to
run the computer, then measure
W.
Result will square root of
some
number between 0 and
The square root of
any
such number, with equal probability.
F
calculates the square roots of all the integers in parallel,
but QMP says we can only find out about one.
For real problems,
arrange
F
so the probability amplitudes of the output
state strongly favor the desired output from
F.
Recall the doubleslit experiment. Quantum computers are like
huge multidimensional arrays of slits that generate interference
patterns in the wave functions.
Design the array right, and the pattern solves your problem.
A quantum computer is
probabilistic:
we may need to run it multiple times before we get the answer we want.
Shor's algorithm, simplified (I)
Peter Shor showed how to design a quantum computer
to calculate the factors of an integer in polynomial time,
theoretically breaking RSA.
We want to factor
N,
that is, find
A
and
B
such that
AB
=
N.
Trick: find distinct
x
and
y
such that


Then


so one must contain
a factor, which we can find by e.g. gcd(xy, N).
Next, take
y
to be 1, so if
and
r
is even then


Then
r
is the period of
the function
in
a.
Shor's algorithm, simplified (II)
Looking for pair
x,
r
such that
Greatly simplifying,
algorithm builds a superposition of all integers
x
<
N,
then calculates
for all
a
in parallel.
Discover the periods using a (quantum) FFT on the resulting entanglement.
The final state is (sort of) an entanglement of all valid
x,
r
pairs.
Finally, measure the output register.
QMP says it must choose one
x,
r
pair,
and we can factor.
With small probability the output may be zero; if so, we run it again.
Not much like a regular computer program!
What else can we do?
Shor's algorithm


factors in polynomial time.
(We expect to get a nonzero result in a small number of runs.)
It's dramatically faster than any known classical algorithm;
Entanglement
gives us exponential parallelism.
A few other quantum algorithms go faster than classical.
Most are obscure but one is important:
Lov Grover's algorithm searches an unordered database of
N
elements
to finds an element satisfying a given condition in square root of
N
time.
In other
words,


it searches a linear list in square root time.
Not as dramatic as the exponential speedup in Shor's algorithm,
but remarkable and possibly even practical.
Decoherence
Factoring a 200digit number using Shor
requires 3500 perfectly wellbehaved qubits.
(Current state of the art is four entangled qubits.)
But that's not the hard part.
The challenge is
decoherence:
the `leakage' of quantum state into the environment.
Actually, it is entanglement with the environment.
(Believed to be
the explanation for why the macroscopic world behaves classically).
QC must be run in a sealed box
without any interaction with the outside world.
Otherwise the qubits will be contaminated.
(This is another reason debugging could be hard.)
The required isolation is extreme;
today's entangled atomic states in the lab last for about 10ns,
and decoherence proceeds exponentially fast in the number
of particles.
Error correction
Decoherence would be the death knell for QC, except that
Shor and others discovered
quantum error correction.
Like classical error correction,
but QEC can correct an
arbitrary
error in a qubit, even if we don't know its state!
(Much more astonishing than repairing a bit flip in a classical message.)
Many such codes exist, e.g.
7 qubits can fully repair damage to any one qubit in the message.
QEC could compensate for decoherence and other losses
if they're at a low enough rate.
(Current theory ranges from .000001 to .01.)
Error correcting a
tstep
computation involves
overhead polynomial in log
t.
Using Shor to factor 200 digits requires 3500 perfect qubits,
100,000 if error correction is involved.
Quantum communication
Computation may be extremely hard because it involves many qubits.
But
communication
can work one qubit at a time.
Errorcorrecting such states might be practical.
Experiments have reliably transmitted kiloqubits per second
over many kilometers of fiber, and in one case a mile of open air!
Is this a solution to the running out of photons problem? An open question.
Several ways to communicate:


C:
Send classical bits.


Q:
Send qubits.


Send qubits but also use twoway classical
communication to assist.


Send qubits but first prepare them by prior
entanglement between sender and receiver.
Channel capacities:


Entanglement is again the source of power.
EPR pairs
Einstein, Podolsky and Rosen proposed a thought experiment to show
that quantum mechanics was crazy.
Today we can do the EPR experiment and Einstein
would have hated the result: QM
is
crazy.
Based on EPR, we can do stuff like teleportation, unbreakable key exchange,
and highefficiency communication.
Electronpositron annihilation produces two photons:
The two must have entangled states: the polarization of one must correlate
with the polarization of the other.
What if Alice measures using plane polarization? Then if Bob measures
using plane polarization, he must get same answer. Ditto for
circular. But.... what if Alice doesn't tell until after Bob measures?
A classical channel can be used to report how the measurement was done,
and Alice and Bob can compare notes.
Using EPR pairs
Quantum key exchange:
Exchange a bunch of EPR pairs.
Alice reports the basis she used for her measurements;
Bob checks against his, and the photons measured in the
same basis get the same answer.
Cannot be tapped because tapping will
destroy the entanglement. Alice can add extra `check' bits; Bob can check them to see if
key has been tampered with.
After sharing an EPR pair, two classical bits can send
an
arbitrary
quantum state from Alice to Bob.
Alice combines her half of the pair with the state (say an atom),
does a measurement, and sends the result to Bob.
Bob uses the bits to entangle his half of the pair
and the destination atom.
Result is to transfer
the unknown state to the atom: Teleportation!
More about EPR pairs
In a similar experiment,
after prior sharing an EPR pair, Alice can send Bob two classical
bits in one qubit.
This is called superdense coding.
Time is important: Alice and Bob can share and separate
months before Alice decides which bits to send.
EPR pairs are a new kind of data communication.
There's nothing like them in classical information theory.
Quantum computation can reduce the time complexity of some calculations.
Quantum communication can reduce the communication
complexity of some calculations.
Physical Realizations
Far from an exhaustive list.
Photons:
23 km. of fiber under lake Geneva, reflecting with polarization change
at the end back to sender. (Gisin, U. Geneva).
Electrons:
Floating on liquid helium with electrodes underneath; too early to tell (Platzman & Dykman, Bell Labs).
Atoms:
Ion traps, controlling quantum state by external laser pulses (Cirac & Zoller, Austria; Kimble, Caltech).
Passing qubit from atom to photon is work in progress.
Molecules:
NMR on an ensemble of molecules (e.g. chloroform, trichlorethylene) (Gershenfeld, MIT).
All these have limitations. Current status: a few qubits.
Solid state: In the future.
Quantum dots, ultrasmall Josephson junctions, semiconductor microcavities, ...
Conclusions
As computational elements get smaller and smaller,
quantum mechanical effects will become important.
Somewhat to our surprise, this may turn out to be a good thing.
Information is a physical variable, and we can use the
properties of its physical manifestation to our advantage.
Quantum mechanical information has deeper structure
and greater power than classical information.
Studying information as a physical notion helps us understand.
For example, to understand what can and cannot travel
faster than light, say this: information cannot travel faster than light.
A final thought:
Sixty years ago classical computers seemed as remote as quantum computers do today.
Summary
Table adapted from Bennett & DiVincenzo:
Quantum speedups:
Factoring: exponential; Search: quadratic; Iteration, parity: no speedup; Simulation of quantum systems: up to exponential.
References
Richard. P. Feynman,
"Simulating Physics with Computers",
International Journal of Theoretical Physics,
1982,
Vol. 21,
No. 6/7,
pp. 467488.
R. P. Feynman,
"Quantum Mechanical Computers",
Foundations of Physics,
1986,
Vol. 16,
No. 6,
pp. 507531.
C. H. Bennett and G. Brassard and A. K. Ekert,
"Quantum Cryptography",
Scientific American,
October 1992,
pp. 5057.
P. W. Shor,
"Quantum Computing",
Documenta Mathematica,
Extra Volume ICM 1998 I,
pp. 467480.
Three good overviews:
Charles H. Bennett & David P. DiVincenzo,
``Quantum information and computation'',
Nature,
Vol. 404, 16 Mar 2000, pp. 247255.
A. Steane,
"Quantum Computing",
Reports on Progress in Physics,
1998,
Vol. 61,
pp. 117173,
https://xxx.lanl.gov/abs/quantph/9708022.
E. G. Rieffel and W. Polak,
"An Introduction to Quantum Computing for NonPhysicists",
1998,
https://xxx.lanl.gov/abs/quantph/9809016.
Copyright © 2000 Lucent Technologies Inc. All rights reserved.