Download CSE 506/606 NSC Nonstandard Computation Winter Quarter 2004

Quantum Computing
Part 2: Computation
Based on the paper:
E. Rieffel & W. Polak,
An introduction to Quantum Computing for Non-Physicists.
ACM Computing Surveys, 32(3) 300-335, 2000.
And ppt by Melanie Mitchell, Portland State University
Bra/Ket notation
Ket:
x
• Column vector, used to describe quantum states
• E.g.,
 0,1 
can be written as
 (1, 0)
• In general,
x  a 0  b 1  (a, b)T
Bra: x
• Conjugate transpose of
x  (a , b )
x
T
, (0,1)T

Quantum bits (“qubits”)
A qubit is a unit vector in a two-dimensional complex vector
space.
Assume a particular basis, denoted by
0,1 
These basis states represent the classical bit values 0 and 1.
However, unlike classical bits, qubits can be in a
superposition of these two states, e.g.,
a 0 b1
such that
a  b 1
2
2
• Suppose a qubit (a, b)T is measured with respect to the basis
 0,1 
Probability that 0 will be measured is |a|2, and
Probability that 1 will be measured is |b|2.
• When a qubit is measured, its state becomes the basis state that
was measured.
• All information about the original superposition is lost.
Several qubits
• The system of n qubits "contain" 2n classical bits (basis
states)
• Thus the potential of a quantum computer grows
exponentially
• We can measure individual qubits in the multi-qubit system
– For example, in a two-qubit system we can measure the state of
first or second qubit, or both
• The results of measurement are probabilistic
• After the measurement, the system collapses in the
corresponding state
Example of using qubits: Quantum key distribution
(Bennett & Brassard, 1987)
•
Solves problem of secure key distribution for private key
cryptography
•
Public key cryptography using RSA is great, but relies on
factoring being a hard problem.
•
Private key cryptography is in principle more secure, but
relies on private key being securely transmitted.
•
Quantum method gives perfect security in private key
transmission
Alice wants to send a key to Bob by encoding each bit as
the quantum state of a photon.
She secretly encodes each bit by randomly choosing one of
two basis:
0
1
or
0
1
E.g.,
1
1
0
1
1 1 0 0 0 0
Bob measures the state of each photon he receives by
randomly picking either basis, up-down (ud), or diagonal
(diag).
What Alice sent
1
1
0
1
1 1 0 0 0 0
What Bob measured
Basis:
Measurement:
ud
1
diag ud diag ud ud diag ud diag diag
0
0
1
1 1
1 0 1 0
Now Alice and Bob communicate what bases they used for
each measurement. They save only the bits for which they
used the same bases (approx. 50%). These bits form the
private key.
Bob measures the state of each photon he receives by
randomly picking either basis, up-down (ud), or diagonal
(diag).
What Alice sent
1
1
0
1
1 1 0 0 0 0
What Bob measured
Basis:
Measurement:
ud
1
diag ud diag ud ud diag ud diag diag
0
0
1
1 1
1 0 1 0
Now Alice and Bob communicate what bases they used for
each measurement. They save only the bits for which they
used the same bases (approx. 50%). These bits form the
private key.
Now, suppose a third person, Eve, intercepts the stream of
photons from Alice, measures their states, and sends new
photons with the measured states to Bob.
She will choose the wrong basis 50% of the time.
So on the photons for which Bob chooses the same basis as
Alice, 25% of them will be incorrect.
Bob and Alice can detect this error by sending sample
“correct” (parity) bits to each other over open communication
line. Thus they can detect an eavesdropper.
Multiple qubits
• Suppose you have n particles whose states are each vectors
in a 2-D vector space (e.g., x, y position)
• How many dimensions do you need to describe complete
state of the system?
– Particles have coordinates (x1, y1), (x2, y2), ..., (xn, yn)
– Complete state is simply a list of all the coordinates:
(x1, y1, x2, y2, ..., xn, yn)
– i.e., 2n dimensions
• Quantum case is different due to “entanglement”.
• Again, suppose you have n particles whose states are each
vectors in a 2-D vector space.
– Particles have coordinates (x1, y1), (x2, y2), ..., (xn, yn)
– Complete state is more complicated, due to
entanglement of states:
(x1x2x3...xn-1xn, x1x2x3...xn-1yn, x1x2x3...yn-1xn, ...)
– i.e., 2n dimensions
y
(1, 2)
Example: n = 2
(2, 1)
– Classical basis vectors: (1, 0), (0, 1)
– Classical state of system: ( (1, 2), (2, 1) )
x
– Quantum basis vectors (Tensor Product):
(1, 0)  (1,0) = (1, 0, 0, 0)
(1, 0)  (0,1) = (0, 0, 1, 0)
(0, 1)  (1,0) = (0, 1, 0, 0)
(0, 1)  (0, 1)= (0, 0, 0, 1)
– Quantum state of system: Linear combination of those
basis vectors
y
(1, 2)
(2, 1)
x
Interpretation: In quantum system you now need four
numbers to give the state of the two-particle system.
Each dimension describes a particular entanglement of
the original dimensions of the individual particles.
In quantum computation, the “particles” are the qubits.
Each qubit has a state defined by two complex numbers:
  a 0  b 1 , where
 1
 0
0    and 1   
 0
 1
A two-qubit state is defined by four complex numbers:
q  a 00  b 01  c 10  d 11 , where
 1
 0
 0
 0
 
 
 
 
 0
 0
 1
 0
00    , 01    , 10    , 11   
0
1
0
0
 
 
 
 
 0
 0
 0
 1
 
 
 
 
• Recall the notion of superposition:
– A single qubit is in a superposition of two basis states:
  a 0 b1
– A 3-qubit system is in a superposition of eight basis
states:
  a1 000  a2 001  a3 010  a4 011  a5 100  a6 101  a7 110  a8 111
where a1, a2, ..., an are complex numbers such that
a1  a2  a3  a4  a5  a6  a7  a8  1
2
2
2
2
2
2
2
2
• In general, an n-qubit system is in a superposition of 2n
basis states.
• This exponential increase in size of state space as n
increases is why classical computers cannot efficiently
simulate quantum systems.
• This was Feynman’s impetus for proposing quantum
computers.
Measurement in entangled systems
• Example 1: The state
1
( 00  11 )
2
is entangled:
– If no bits have yet been measured, the probability of
measuring the first (or second) bit as 0 is 0.5.
– However, if the second (or first) bit has been measured,
the probability is 0 or 1, depending on what the second
(or first) bit was measured to be.
1
( 00  01 ) is not entangled:
• Example 2: The state
2
Can factor out first bit: it will always be measured as 0 .
Measuring it does not affect the second bit.
EPR paradox
(Einstein, Podolsky, Rosen, “Can quantum-mechanical description of
physical reality be considered complete?” (1935))
• Imagine a source that generates two maximally entangled
1
particles:
( 00  11 )
2
• One particle is sent to Alice, and the other to Bob.
• Suppose Alice measures her particle and observes state 0 .
• Then the combined state will now be 00 . This means
Bob will also measure 0 .
• The change in state occurs instantaneously, no matter how
far away Alice and Bob are from each other.
• Does this allow communication faster than the speed of
light?
• EPR’s explanation:
– Each particle has a “hidden” internal state that determines what the
result of any given measurement will be.
– Called “local hidden variable theory”.
– Einstein: “God does not play dice.”
• Problem with this explanation:
– Doesn’t explain measurements with respect to a different basis.
– Bell’s theorem: predicts an inequality that must be satisfied if
hidden variable theory is correct.
– Experiments show it to be incorrect.
• Cause and effect explanation:
– Somehow Alice’s measurement causes the result found by Bob,
• Problems with this explanation:
– Violates relativity theory: Can set up experiment where one
observer sees Alice do measurement first, while other observer
sees Bob do measurement first. This does not change the results
of the measurements.
– This also shows that Alice and Bob can’t use their particles to
communicate faster than the speed of light.
– “All that can be said is that Alice and Bob will observe the same
random behavior.”
• From Black et al., “Quantum computing and
communications” (2002):
“The ugly truth is that general relativity and quantum
mechanics are not consistent. That is, our current
formulations of general relativity and quantum mechanics
give different predictions for extreme cases. We assume
there is a ‘Theory of Everything’ that reconciles the two,
but it is still very much an area of thought and research.
Since relatively is not needed in quantum computing, we
ignore this problem.”
Schor’s Algorithm
• Breaking RSA reduces to finding the prime factors of a
large integer N
• Problem solved by period-finding
• E.g. sequence of integers : the powers of two.
2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, …
• Look at the powers of 2 “mod 15″: in other words, the
remainder when 15 divides each power of 2.
2, 4, 8, 1, 2, 4, 8, 1, 2, 4, …
• Taking the powers of 2 mod 15 gives a periodic sequence,
whose period (i.e., how far you have to go before it starts
repeating) is 4.
Euler in the 1760’s
• Let N be a product of two prime numbers, p and q, and consider the
sequence
x mod N, x2 mod N, x3 mod N, x4 mod N, …
• Provided x is not divisible by p or q, the above sequence will repeat
with some period that evenly divides (p-1)(q-1).
– e.g., if N=15, then the prime factors of N are p=3 and q=5, so (p1)(q-1)=8. The period of the sequence was 4, which divides 8.
– If N=21, then p=3 and q=7, so (p-1)(q-1)=12. And indeed, the
period was 6, which divides 12.
• If we could learn several random divisors of (p-1)(q-1) (e.g., by trying
different random values of x),
– then with high probability we could put those divisors together to
learn (p-1)(q-1) itself.
– once we knew (p-1)(q-1), we could then use some more little tricks
to recover p and q, the prime factors we wanted.
Problem
Even though the sequence
x mod N, x2 mod N, x3 mod N, x4 mod N, …
will eventually start repeating itself,
the number of steps before it repeats could be almost as large
as N itself — and N might have hundreds or thousands of
digits!
This is why finding the period doesn’t seem to lead to a fast
classical factoring algorithm.
Quantum Computer Solution
Suppose we could create an enormous quantum superposition
over all the numbers in our sequence:
x mod N, x2 mod N, x3 mod N, x4 mod N, …
There may be some quantum operation we could perform on
that superposition that would reveal the period
To get this period-finding idea to work, we’ll have to answer two
questions :
1.
Using a quantum computer, can we quickly create a
superposition over
x mod N, x2 mod N, x3 mod N, x4 mod N, …?
2.
Supposing we did create such a superposition, how would
we figure out the period?
1.
Create a superposition over all integers r, from 1 up to N or
so by using repeated squaring.
e.g. Suppose N=17, x=3, and r=14. Then the first step is to
represent r as a sum of powers of 2:
r = 23 + 22 + 2 1 .
Then
3 + 2 2 + 21
3 22 2 1
r
14
2
2
x =3 =3
=3 3 3
Also, notice that we can do all the multiplications mod N, thereby
preventing the numbers from growing out of hand at
intermediate steps. This yields the result
314 mod 17 = 2.
We can create a quantum superposition over all pairs of integers
of the form (r, xr mod N), where r ranges from 1 up to N
2.
Given a superposition over all the elements of a periodic
sequence, how do we extract the period of the sequence?
To get the period out, Shor uses something called the quantum
Fourier transform, or QFT.
The QFT is a linear transformation (indeed a unitary
transformation) that maps one vector of complex numbers to
another vector of complex numbers.
It is a linear transformation that maps a quantum state encoding a
periodic sequence, to a quantum state encoding the period of
that sequence.
Another way to think about this is in terms of interference
Key point about quantum mechanics — the thing that makes it different from
classical probability theory —
probabilities are always nonnegative
amplitudes in quantum mechanics can be positive, negative, or even complex.
amplitudes corresponding to different ways of getting a particular answer can
“interfere destructively” and cancel each other out.
For all periods other than the “true” one, these contributions point in different
directions and therefore cancel each other out.
Only for the “true” period do the contributions from different solutions all point in
the same direction.
And that’s why, when we measure at the end, we’ll find the true period with high
probability.
• The QFT closely related to the classical fast Fourier transform (FFT),
but can be computed exponentially faster.
• The FFT takes as input a string of K complex numbers, xk, and produces
as output another string of K numbers, yk, with
For an input string of K numbers, which repeat themselves with period r,
the FFT produces an output string with period K/r, as illustrated in the
following examples for the case of K = 8).
1
1
1
1
0
0
0
1
0
0
1
1
0
0
0
1
0
1
1
1
0
0
0
1
0
0
1
1
0
0
0
1
→
→
→
→
1
1
1
1
1
0
0
0
1
1
0
0
1
0
0
0
1
1
1
0
1
0
0
0
1
1
0
0
1
0
0
0
In case r divides K with a remainder, the inversion of the period is only
approximate.
The FFT turns shifts in the input string into phase factors in the output
string:
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
→
→
→
→
1
1
1
1
0
0
0
0
1
–i
–1
i
0
0
0
0
1
–1
1
–1
0
0
0
0
1
i
–1
–i
0
0
0
0
• The QFT performs exactly the same transformation as the FFT, but the
complex numbers are stored in the amplitude and phase of the terms in
a superposition.
Shor’s Quantum Factoring
• Quantum operation takes polynomial time O(N3)
• Best known traditional algorithm takes O(en1/3)
– ƒWe have an exponential speed up!
• The fastest supercomputer in the world can’t compete with
a quantum computer for certain problems!
– ƒMany encryption algorithms rely on the difficulty of factoring
large numbers.
– ƒMotivation to build one!
5-qubit example
• A 5-qubit example shows how quantum parallelism and use of the
QFT make it possible to efficiently find the period of a function. For
clarity, the states are written in decimal instead of binary notation, for
example, |010〉 will be denoted |2〉.
• Consider two registers (groups of qubits) in which the first register
contains three qubits, each initialized to an equal superposition of |0〉
and |1〉, and with the second register consisting of two qubits set to |0〉.
Suppressing normalization constants for clarity, the state of the system
is thus written as
(|0〉 + |1〉 + |2〉 + |3〉 + |4〉 + |5〉 + |6〉 + |7〉)|0〉
Practical Requirements for Experimental
Implementation
1. Need a system of qubits.
2. The qubits must be individually addressable and must
interact with each other to provide for a universal set of
logic gates.
3. It must be possible to initialize them to a known state
because the result of a computation generally depends on
its input state.
4. Must be able to extract a computation result from the
qubits by some measurement.
5. A long coherence time compared with an average logic
gate's duration, such that many logic gates can be
implemented within the coherence time.
Nuclear magnetic resonance quantum
computing (NMR QC)
• Many atoms such as 1H, 13C, and 19F have a spin-1/2 nucleus.
• A nuclear spin-1/2 can be thought of as a tiny bar magnet spinning
about its own axis, with two well-defined states; it can be aligned or
anti-aligned with respect to an external magnetic field (spin up or spin
down) and thus can represent logical zero and one.
• Since a nuclear spin is extremely small, it's a quantum mechanical
object and can exist in a superposition of up and down.
• An atomic nucleus' spin can thus serve as a quantum bit.
• An NMR quantum computer consists of several individual atoms with
a spin-1/2 nucleus.
Nuclear Magnetic Resonance
• RF pulse can effect the nuclear spin
state
• Quantum NOT operation
– ƒA properly timed and calibrated RF
pulse flips the spin state from spin
up to down
• Halving the RF pulse can achieve a
superposition dual 0/1 state
• Different qubits are ‘set’ to a value via
a unique RF pulse at a specific
frequency
• J-coupling - indirect dipole dipole coupling
– ƒThe coupling between two nuclear
spins due to the influence of bonding
electrons on the magnetic field running
between the two nuclei.
• Enables Quantum Conditional NOT
• Conditional on J-coupling between other
nearby qubit state (0 or 1)
– ƒA nearby qubit state cancels RF pulse
– ƒJ-coupling enables conditional not
(CNOT)
– ƒThis operation enables a universal set
J-coupling
J-coupling
Alternate CNOT
1. Solid line |0> dashed line |1>
2. RF pulse rotates spin from +z to –y
3. Freely evolve for ½ J seconds (coupling)
• Rotated to either +x or –x based on other spin state
4. A 90o pulse on spin rotates back to +z or –z
A 5 Qubit Molecule
• A molecule with 5
Fluorine ½ spin molecules
• Each has a unique
resonance frequency (f1,
f2, f3, f4, f5)
• Allows each qubit to be
individually addressed
• Frequency difference
much larger than J
coupling
Table of the relative chemical shifts of the 19F spins at 11.7
T [Hz], and the J couplings [Hz]
Initialize State of Qubits
• The thermal equilibrium of nuclear spin at room
temperature is highly random
• Initialize all qubits by creating a effectively “pure”
state
• Main conceptual breakthrough that made NMR
quantum computing possible
Qubit Reading of State
• A specific read out pulse rotates spin to the +x –x axis
• The resulting counter spins create a time-varying magnetic
field.
• This magnetic field is measured, and integrated to obtain a
value.
– Positive was in |0> spin
– Negative was in |1> spin
• Learn the state of each spin or qubit in system
Schematic of NMR spectrometer
•
•
•
•
Sample tube contains 1012 dissolved NMR molecules
Placed in superconducting solenoid, high current, very strong Magnetic field
RF coil for pulsing the molecules.
Computer controls an amplifier for controlling RF pulse, and reading state
Quantum Experiment
Results
Experimentally measured spectra of spin 1, acquired after executing the orderfinding algorithm, for four different cases. The respective permutations are shown
in inset, with the starting element in bold. The markers above the spectra indicate
the position of the lines in the full multiplet.