Quantum Computing

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Quantum Computing

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Title: Quantum Computing

1
Quantum Computing

Intro. OverviewWhats Quantum Computing All
About?

2
Classical vs. Quantum Computing

For any digital computer, its set of
computational states is some set of mutually
distinguishable abstract states.
The specific computational state that is in use
at a given time represents the specific digital
data currently being processed within the
machine.
Classical computing is computing in which
All of the computational states (at all times)
are stable pointer states of the computer
hardware.
Quantum computing is computing in which
The computational state is not always a pointer
state.

3
What is Quantum Computing?

Non-pointer-state computing.
Harnesses these quantum effects on a large,
complex scale
Computational states that are not just pointer
states, but also, coherent superpositions of
pointer states.
States having non-zero amplitude in many pointer
states at the same time! Quantum parallelism.
Entanglement (quantum correlations)
Between the states of different subsystems.
Unitary (thus reversible) evolution through time
Interference (reinforcement and cancellation)
Between convergent trajectories in pointer-state
space.

4
Why Quantum Computing?

It is, apparently, exponentially more
time-efficient than any possible classical
computing scheme at solving some problems
Factoring, discrete logarithms, related problems
Simulating quantum physical systems accurately
This application was the original motivation for
quantum computing research first suggested by
famous physicist Richard Feynman in the early
80s.
However, this has never been proven yet!
If you want to win a sure-fire Nobel prize
Find a polynomial-time algorithm for accurately
simulating quantum computers on classical ones!

5
Status of Quantum Computing

Theoretical experimental progress is being
made, but slowly.
There are many areas where much progress is still
needed.
Physical implementations of very small (e.g.,
7-bit) quantum computers have been tested and
work as predicted.
However, scaling them up is difficult.
There are no known fundamental theoretical
barriers to large-scale quantum computing.
Guess It will be a real technology in 20 yrs.
or so.

6
Early History

Quantum computing was largely inspired by
reversible computation work from the 1970s
Bennett, Fredkin, and Toffoli
Early quantum computation pioneers (1980s)
Early models not using quantum parallelism to
gain performance
Benioff 80, 82 - Quantum serial TM models
Feynman 86 - Q. models of serial reversible
circuits
Margolus 86,90 - Q. models of parallel rev.
circuits
Performance gains w. quantum parallelism
Feynman 82 - Suggested faster quantum sims with
QC
Deutsch 85 - Quantum-parallel Turing machine
Deutsch 89 - Quantum logic circuits

7
More Recent History

There was a rapid ramp-up of quantum computing
research throughout the 1990s.
Some developments, 1989-present
Refining quantum logic circuit models
What is a minimal set of universal gates for QC?
Algorithms Shor factoring, Grover search, etc.
Developing quantum complexity theory
What is the ultimate power of quantum
computation?
Quantum information theory
Communications, Cryptography, etc.
Error correcting codes, fault tolerance, robust
QC
Physical implementations
Numerous few-bit implementations demonstrated

8
Quantum Logic Networks

Invented by Deutsch (1989)
Analogous to classical Boolean logic networks
Generalization of Fredkin-Toffoli reversible
logic circuits
System is divided into individual bits, or qubits
2 orthogonal states of each bit are designated as
the computational basis states, 0 and 1.
Quantum logic gates
Local unitary transforms that operate on only a
few state bits at a time.
Computation via predetermined seq. of gate
applications to selected bits.

9
Gates without Superposition

All classical input-consuming reversible gates
can be represented as unitary transformations!
E.g., input-consuming NOT gate (inverter)

in out0 11 0
in
out
in
out
10
Controlled-NOT

Remember the CNOT (input-consuming XOR) gate?

A
A
A
A
B
B A?B
B
B A?B
Example
A B
A B
11
Toffoli Gate (CCNOT)
A B C A B C0 0 0 0 0 00 0 1
0 0 10 1 0 0 1 00 1 1 0
1 11 0 0 1 0 01 0 1 1 0
11 1 0 1 1 01 1 1 1 1 1
A
AA
B
BB
A
A
B
B
C
C C?AB
C
C
(XOR)
Now, what happens if the unitary matrix elements
are not always 0 or 1?
12
The Square Root of NOT

If you put in either basis state (0 or 1) you get
a state that appears random when measured
But if you feed the output back into another N1/2
without measuring it, you get the inverse of the
original value!
How is thatpossible?

0 (50)
0 (50)
0
1
N1/2
N1/2
1 (50)
1 (50)
0 (50)
0
1
N1/2
N1/2
1 (50)
0 (50)
0
0
N1/2
N1/2
1 (50)
13
NOT1/2 Unitary implementation
Prob. ½
Prob. ½
14
The Hadamard Transform

A randomizing square root of identity gate.
Used frequently in quantum logic networks.

15
Another NOT1/2

This one negates the phase of the state if the
input state was 0?.

16
Optical Implementation of N1/2

Beam splitters (semi-silvered mirrors) form
superpositions of reflected and
transmittedphoton states.

1
1
1
1
0
0
0
laser
1
17
Deutschs Problem

Given a black-box function f0,1?0,1,
Determine whether f(0)f(1),
But you only have time to call f once!

H
H
f
(N)1/2
18
Extended Deutschs Problem

Given black-box f0,1n?0,1,
and a guarantee that f is either constant or
balanced (1 on exactly ½ of inputs)
Which is it?
Minimize number of calls to f.
Classical algorithm, worst-case
Order 2n time!
What if the first 2n-1 cases examined are all 0?
Function could still be either constant or
balanced.
Case number 2n-11 if 0, constant if 1,
balanced.
Quantum algorithm is exponentially faster!
(Deutsch Jozsa, 1992.)

19
Universal Q-Gates History

Deutsch 89
Universal 3-bit Toffoli-like gate.
diVincenzo 95
Adequate set of 2-bit gates.
Barenco 95
Universal 2-bit gate.
Deutsch et al. 95
Almost all 2-bit gates are universal.
Barenco et al. 95
CNOT set of 1-bit gates is adequate.
Later development of discrete gate sets...

20
Deutsch Gen. 3-bit Toffoli gate

The following gate is universal

(Where ? is any irrational number.)
a bb a
21
Barencos 2-bit gen. CNOT gate

where ?,?,?,? are relatively irrational
Also works, e.g., for ??, ??/2

U
22
Barenco et al. 95 results

Universality of CNOT 1-bit gates
2-bit Barenco gate already known universal
4 1-bit gates 2 CNOTs suffice to build it
Construction of generalized Toffoli gates
3-bit version via five 2-bit gates
n-bit version via O(n2) 2-bit gates
No auxilliary bits needed for the above
All operations done in place on input qubits.
n-bit version via O(n) 2-bit gates, given 1 work
bit

23
Quantum Complexity Theory

Early developments
Deutchs problem (from earlier) Slight speedup
Deutsch Jozsa Exponential speed-up
Important quantum complexity classes
EQP Exact Quantum Polynomial - like P.
Polynomial time, deterministic.
ZQP Zero-error Quantum Polynomial - like ZPP.
Probabilistic, expected polynomial-time, zero
errors.
BQP Bounded-error Quantum Poly. - like BPP.
Probabilistic, bounded probability of errors.
All results relativized, e.g.,
?O EQPO ? (NPO ? co-NPO)

Given a certain black-box, quantumcomputers can
solve a certain problemfaster than a classical
computer caneven check the answer!
24
Quantum Algorithms

Part I Unstructured Search

25
Unstructured Search Problem

Given a set S of N elements and a black-box
function fS?0,1, find an element x?S such that
f(x)1, if one exists (or if not, say so).
Any NP problem can be cast as an unstructured
search problem.
Not necessarily the optimal approach, however.
Bounds on classical run-time
?(N) expected queries in worst case (0 or 1
solns)
Have to try N/2 elements on average before
finding soln.
Have to try all N if there is no solution.
If elements are length-? bit strings,
Expected trials is ?(2?) - exponential in ?.
Bad!

26
Quantum Unstructured Search

Minimum time to solve unstructured search problem
on a quantum computer is
?(N1/2) queries (2?/2) (21/2)?
Still exponential, but with a smaller base.
The minimum of queries can be achieved using
Grovers algorithm.

27
Grovers algorithm

1. Start w. amplitude evenly distributed among
the N elements, ?(xi)1/?N
2. In each state xi, compute f(xi)
3. Apply conditional phase shift of ? if
f(xi)1(Negate sign of solution state.)
Uncompute f.

?
x1
xN
solutionxs
?
f0
f1
x1
xN
solutionxs
28
Grovers algorithm, cont.

4. Invert all amplitudes with respect to the
average amplitude

?
x1
xN
solutionxs
29
Grovers algorithm, cont.

5. Go to step 2, and repeat 0.785 N1/2 times.

1
?(xs)
of iterations
-1
30
Shors Factoring Algorithm

Solves the gt2000-year-old problem
Given a large number N, quickly find the prime
factorization of N. (At least as old as Euclid!)
No polynomial-time (as a function of nlg N)
classical algorithm for this problem is known.
The best known (as of 1993) was a number field
sieve algorithm taking time O(exp(n1/3
log(n2/3)))
However, there is also no proof that an
(undis-covered) fast classical algorithm does not
exist.
Shors quantum algorithm takes time O(n2)
No worse than multiplication of n-bit numbers!

31
Elements of Shors Algorithm

Uses a standard reduction of factoring to another
number-theory problem called the discrete
logarithm problem.
The discrete logarithm problem corresponds to
finding the period of a certain periodic function
defined over the integers.
A general way to find the period of a function is
to perform a Fourier transform on the function.
Shor showed how to generalize an earlier
algorithm by Simon, to provide a Quantum Fourier
Transform that is exponentially faster than
classical ones.

32
Powers of numbers mod N

Given natural numbers (non-negative integers)
N?1, xltN, and x, consider the sequence
x0 mod N, x1 mod N, x2 mod N, 1, x, x2 mod
N,
If x and N are relatively prime, this sequence is
guaranteed not to repeat until it gets back to 1.
Discrete logarithm of y, base x, mod N
The smallest natural number exponent k (if any)
such that xk y (mod N).
I.e., the integer logarithm of y, base x, in
modulo-N arithmetic. Example dlog7 13 (mod N)
?

33
Discrete Log Example

N15, x7, y13.
x2 49 4 (mod 15)
x3 47 28 13 (mod 15)
x4 137 91 1 (mod 15)
So, dlog7 13 3 (mod N),
Because 73 13 (mod N).

0
1
2
3
4
7
7
7
5
6
7
8
9
12
13
14
10
11
7
34
The order of x mod N

Problem Given Ngt0, and an xltN that is relatively
prime to N, what is the smallest value of kgt0
such that xk 1 (mod N)?
This is called the order of x (mod N).
From our previousexample, the orderof 7 mod N
is?

0
1
2
3
4
7
7
7
5
6
7
8
9
12
13
14
10
11
7
35
Order-finding permits Factoring

A standard reduction of factoring N to finding
orders mod N
1. Pick a random number x lt N.
2. If gcd(x,N)?1, return it (its a factor).
3. Compute the order of x (mod N).
Let r min kgt0 xk mod N 1
4. If gcd(xr/2?1, N) ? 1, return it (its a
factor).
5. Repeat as needed.
The expected number of repetitions of the loop
needed to find a factor with probability gt 0.5 is
known to be only polynomial in the length of N.

36
Factoring Example

For N15, x7
Order of x is r4.
r/2 2.
x2 5.
In this case (we are lucky), both x21 and x2?1
are factors (3 and 5).
Now, how do we compute orders efficiently?

0
1
2
3
4
7
7
7
5
6
7
8
9
12
13
14
10
11
7
37
Quantum Order-Finding

Uses 2 quantum registers (a,b)
0 ? a lt q, is the k (exponent) used in
order-finding.
0 ? b lt n, is the y (xk mod n) value
q is the smallest power of 2 greater than N2.
Algorithm
1. Initial quantum state is 0,0?, i.e., (a0,
b0).
2. Go to superposition of all possible values of
a

38
Initial State
39
After Doing Hadamard Transform on all bits of a
40
After modular exponentiationbxa (mod N)
41
State After Fourier Transform
42
Quantum Algorithms III

Wrap-up
Classical vs. Quantum Parallelism
Quantum Physics Simulations

43
Classical Unstructured Search

The classical serial algorithm takes ?(N) time.
But Suppose we search in parallel!
Have MltN processors running in parallel.
Each searches a different subset of N/M elements
of the search space.
If processors are ballistic reversible
Can cluster them in a dense mesh of diameter
?(M1/3).
Time accounting
Computation time ?(N/M)
Communication time ?(M1/3) (at lightspeed)
Total T ? N/M M1/3 is minimized when M ?
N3/4 ? N1/4 Faster than Grovers
algorithm!

M1/3
M
44
ClassicalQuantum Parallelism

Similar setup to classical parallelism
M processors, searching N/M items each.
Except, each processor uses Grovers algorithm.
Time accounting
Computation T ? (N/M)1/2
Communication T ? M1/3 (as before)
Total T ? (N/M)1/2 M1/3
Total is minimized when M?N 3/5
Minimized total is T ? N1/5.
I.e., quantum unstructured search is really only
N1/4/N1/5 N1/20 faster than classical!

45
Simulating Quantum Physics

For n particles, takes exponential time using the
best known classical methods.
Feynman suggested use quantum computing
Takes only polynomial time on a QC.
History of some fast QC physics algorithms
Schrödinger equation (Boghosian, Yepez)
Many-body Fermi systems (Abrams Lloyd)
Eigenvalue/eigenvector computations (ditto)
Quantum lattice gases, Ising models, etc.
Totally general quantum physics
simulations(Field theories, etc.) - Lloyd 1996

46
Simulating Quantum Computers

on classical ones

47
Efficient QC Simulations

Task Simulate an n-qubit quantum computer.
Maximally stupid approach
Store a 2n-element vector
Multiply it by a full 2n2n matrix for each gate
op
Some obvious optimizations
Never store whole matrix (compute dynamically)
Store only nonzero elements of state vector
Especially helpful when qubits are highly
correlated
Do only constant work per nonzero vector element
Scatter amplitude from each state to 1 or 2
successors
Drop small-probability-mass sets of states
Linearity of QM implies no chaotic growth of
errors

48
Linear-space quantum simulation

A popular myth
Simulating an n-qubit (or n-particle) quantum
system takes e?(n) space (as well as time).
The usual justification
It takes e?(n) numbers even to represent a single
?(n)-dimensional state vector, in general.
The hole in that argument
Can simulate the statistical behavior of a
quantum system w/o ever storing a state vector!
Result BQP ? PSPACE known since BV93...
But practical poly-space sims are rarely described

49
The Basic Idea

Inspiration
Feynmans path integral formulation of QED.
Gives the amplitude of a given final
configuration by accumulating amplitude over all
paths from initial to final configurations.
Each path consists of only a single
?(n)-coordinate configuration at each time, not a
full wavefunction over the configuration space.
Can enumerate all paths, while only ever
representing one path at a time.

50
Simulating Quantum Computations

Given
Any n-qubit quantum computation, expressed as a
sequence of 1-qubit gates and CNOT gates.
An initial state s0 which is just a basis state
in the classical bitwise basis, e.g. ?00000?.
Goal
Generate a final basis state stochastically with
the same probability distribution as the quantum
computer would do.

U2
U3
U4
U1
51
Matrix Representation

Consider each gate as rank-2n unitary matrix
Each CNOT application is a 0-1 (permutation)
matrix - a classical reversible bit-operation.
With appropriate row ordering, each Ui gate
application is block-diagonal, w. each 22 block
equal to Ui.
We need never represent these full matrices!
The 1 or 2 nonzero entries in a given row can be
located computed immediately given the row id
(bit string) and Ui.

52
The Linear-Space Algorithm

Generate a random coin c?0,1.
Initialize probability accumulator p?0.
For each final n-bit string y at time t,
Compute its amplitude ?(y) as follows
Generate its possible 1 or 2 predecessor strings
x1 (and maybe x2) given the gate-op preceding t.
For each predecessor, compute its amplitude at
time t?1 recursively using this same algorithm,
unless t0, in which case ?1 if ?x?s0, 0
otherwise.
Add predecessor amplitudes, weighted by entries.
Maybe output y, using roulette wheel algorithm
Accumlate Pry into total p ? p ?(y)2
Output y and halt if pgtc.

53
A Further Optimization

Dont even have to enumerate all final states!
Instead Stochasically follow a trajectory.
Basic idea
Keep track of 1 current state its amplitude
?0.
For CNOTs Deterministically transform state.
For Us
Calculate amplitude ?1 of neighbor state w.
path-integral
Calculate amplitudes ?0 and ?1 after qubit op
Choose 1 successor as new current state, using
?2 distrib.

u00
?0
?0
Current state
u10
Possiblesuccessors
u01
?1
?1
u11
Neighbor state
54
Complexity Comparison

To simulate t gate ops (c CNOTs u 1-bit unitary
ops) of an n-qubit quantum computer
Space Time
Traditional method 2n t2n
Path-integral method tn n2t
(Actually, only the u unitary ops, not all t ops
or all n qubits, contribute to any of the
exponents here.)
Upshot
Lower space usage can allow larger systems to be
simulated, for short periods.
Run time is competitive for case when t lt n

55
Quantum Information Communication

DecoherenceQuantum Error CorrectionQuantum
Cryptography

56
Decoherence

The effect that makes macroscopic quantum systems
appear to behave classically.
Theory was developed in many papers by Zurek.
Occurs due to inevitable interactions between a
given quantum system an unknown (high-entropy)
environment.
Interaction increases (von Neumann-) entropy of
the reduced density matrix of the quantum system.
Quantum state gradually collapses or decays
to a classical statistical mixture of the pointer
states (measurement eigenstates).

57
Decoherence Breaks Interference

Quantum computation w/o decoherence

Time
Trajectoryofenvironmentsstate
(Unknown,chaotic,unpredictable)
Isolation / insulation of quantum computer from
interactions
58
Quantum Information Communication cont.

Quantum Cryptography

59
The Key Distribution Problem

How can parties A and B use a physically insecure
long-distance communications channel to
nevertheless securely exchange keys that they can
use to enable future secure, encrypted
communication between them?
Present-day solution Public-Key Cryptography
see next slide

60
Public-Key Cryptography

A and B each prepare a pair of a public key and a
corresponding private key.
Infeasible to compute private key from public
one.
They openly publish their public keys.
Anyone can now use the public key to encrypt
messages to A or B.
But only the one w. the matching private key can
decrypt the message.
Security of technique depends on existence of
one-way (a.k.a. trapdoor) functions.
? functions believed (but unproven) to be such.

61
PK Crypto vs. Q-Computing

A serious weakness in most present-day PK
cryptosystems (such as RSA)
They depend for their security on the
one-way-ness of certain functions that is due
to the hardness of the factoring and discrete
logarithm problems.
But, Shors algorithm gives a fast way to solve
these problems if we just had a quantum computer!
Large QCs may be implemented within next 10-20
years.
Therefore, data encrypted today with these
cryptosystems cannot be considered secure over
multi-decade time-frames!
But, other PK systems w/o this weakness may exist.

62
Q-Cryptography to the Rescue!

Features
Provides for secure key exchange over physically
unprotected channels w. a guarantee of detection
of any eavesdropping of the key.
Doesnt protect against denial-of-service
attacks.
Physically impossible to compromise security
(except _at_ endpoints) barring overthrow of
physics!
Provably secure under known laws
Experimentally verified to work perfectly over
gt48 km distances (so far) (Hughes 99) via
fiber-optic networks.

63
The One-Time Pad

The only known provably secure cryptosystem.
Based on a key of the same length as the data to
be encrypted.
A given key can only be used once.
The key is simply a random string of bits.
The plaintext is bitwise-XORed with the key to
produce ciphertext.
Provably secure because any plaintext is equally
likely to produce the same ciphertext.
The only problem How to send the key?

64
Outline of QC Protocol (BB84)

A chooses a random bit-string
A key for later use as a one-time pad.
A sends each bit as a qubits with a
randomly-chosen basis (out of 2 different bases).
B measures each bit in a randomly-chosen basis
(out of the same 2 bases).
A B publicly determine which bits they chose
the same bases for.
They publicly spot-check a random subset of the
bits for errors, and use remaining bits.

65
Typical Implementation Method

Any flying qubit will do.
Most common method uses polarized
photons. (Bennett Brassard 84)

1
?
0
Diffraction gratingw. vertical slits
Arbitrary choiceof basis
1
? ?/4
Diffraction gratingw. diagonal slits
0
66
QC Crypto Protocol - details
BB84

1. A chooses a random bit-string
1 0 1 1 0 0 1 0 1 1 1 1 0 1 1 0 0
2. A chooses a random string of polarization
bases, out of ,
3. A encodes bits as photons polarized in the
chosen basis

67
QC Crypto Protocol - details

2. A chooses a random string of polarization
bases, out of ,
3. A encodes bits as photons polarized in the
chosen basis
4. Photons are sent to B over open channel, w.
possible noise and/or eavesdropping

68
QC Crypto Protocol - details

3. A encodes bits as photons polarized in the
chosen basis
4. Photons are sent to B over open channel, w.
possible noise and/or eavesdropping
5. B chooses random string of polarization bases
to measure with

69
QC Crypto Protocol - details

4. Photons are sent to B over open channel, w.
possible noise and/or eavesdropping
5. B chooses random string of polarization bases
to measure with
6. B measures photon state w.r.t. his bases
1 0 1 1 0 1 0 0 1 1 0 0 0 1 1 1
0

70
QC Crypto Protocol - details

5. B chooses random string of polarization bases
to measure with
6. B measures photon state w.r.t. his bases
1 0 1 1 0 1 0 0 1 1 0 0 0 1 1 1
0
7. A B publicly compare basis choices,
determined which matched

?
?
?
?
?
?
?
?
?
?
?
?
71
QC Crypto Protocol - details

6. B measures photon state w.r.t. his bases
1 0 1 1 0 1 0 0 1 1 0 0 0 1 1 1
0
7. A B publicly compare basis choices,
determined which matched
8. Now, they expect their bits w. matching bases
to match (others are discarded)
A 1 0 1 1 0 0 1 0 1 1 1 1 0 1 1 0
0 B 1 0 1 1 0 1 0 0 1 1 0 0 0 1 1
1 0

?
?
?
?
?
?
?
?
?
?
?
?
72
QC Crypto Protocol - details

7. A B publicly compare basis choices,
determined which matched
8. Now, they expect their bits w. matching bases
to match (others are discarded)
A 1 0 1 1 0 0 1 0 1 1 1 1 0 1 1 0
0 B 1 0 1 1 0 1 0 0 1 1 0 0 0 1 1
1 0
9. They compare some of these bits publicly to
determine level of errors and/or eavesdropping
Or, compare parities of random subsets (avoids
waste)

?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
73
QC Crypto Protocol - details

8. Now, they expect their bits w. matching bases
to match (others are discarded)
A 1 0 1 1 0 0 1 0 1 1 1 1 0 1 1 0
0 B 1 0 1 1 0 1 0 0 1 1 0 0 0 1 1
1 0
9. They compare some of these bits publicly to
determine level of errors and/or eavesdropping
Or, compare parities of random subsets (avoids
waste)
10. Use remaining bits in the clean part of data
as the key for 1-time pad
1 1 0

?
?
?
?
?
Or, hash these bits down to a smaller butmore
secure bit-string (privacy amplification)
74
Quantum Crypto Scalability

Optic fiber lengths ? 60-100 km not feasible due
to attenuation.
Free-space (air/vacuum) transmission being
explored.
Useful in networks of orbiting satellites?
Given quantum computers, can build quantum
repeaters that apply quantum error correction to
clean up noisy signals?
Can then maintain secure quantum cryptography
throughout large networks (quantum internet?)
Research topic currently under investigation...