# Consequences and Limits of Nonlocal Strategies

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```					       Introduction to
Quantum Information Processing
QIC 710 / CS 667 / PH 767 / CO 681 / AM 871

Lectures 17-19 (2011)
Richard Cleve
DC 2117
cleve@cs.uwaterloo.ca
1
Complexity class NP

2
Complexity classes
Recall:
• P (polynomial time): problems solved by O(nc)-size
classical circuits (decision problems and uniform circuit
families)
• BPP (bounded error probabilistic polynomial time):
problems solved by O(nc)-size probabilistic circuits that
err with probability £ ¼
• BQP (bounded error quantum polynomial time):
problems solved by O(nc)-size quantum circuits that err
with probability £ ¼
• PSPACE (polynomial space): problems solved by
algorithms that use O(nc) memory.                           3
Summary of previous containments
P Í BPP Í BQP Í PSPACE Í EXP
EXP
We now consider further
structure between P and                      PSPACE
PSPACE

Technically, we will restrict                  BQP
our attention to languages
(i.e. {0,1}-valued problems)                   BPP

Many problems of interest can                  P
be cast in terms of languages
For example, we could define
FACTORING = {(x,y) : \$ 2 £ z £ y, such that z divides x}   4
NP
Define NP (non-deterministic polynomial time) as
the class of languages whose positive instances have
“witnesses” that can be verified in polynomial time
Example: Let 3-CNF-SAT be the language consisting of all
3-CNF formulas that are satisfiable
3-CNF formula:

is satisfiable iff there exists
such that

No sub-exponential-time algorithm is known for 3-CNF-SAT
But poly-time verifiable witnesses exist (namely, b1, ..., bn)   5
Other “logic” problems in NP
• k-DNF-SAT:

* But, unlike with k-CNF-SAT, this one is known to be in P

• CIRCUIT-SAT:
1    Λ    Ø     Λ     Ø          Ø         output
bit
0    Ø     Λ          Ø    Λ     Λ
Ø          Ø               * All known
1    Λ     Λ          Λ          Λ
algorithms
1    Ø    Ø     Λ          Λ               exponential-
0          Λ          Λ    Ø     Λ         time
6
“Graph theory” problems in NP

• k-COLOR: does G have a k-coloring ?
• k-CLIQUE: does G have a clique of size k ?
• HAM-PATH: does G have a Hamiltonian path?
• EUL-PATH: does G have an Eulerian path?

7
“Arithmetic” problems in NP
• FACTORING = {(x, y) : \$ 2 £ z £ y, such that z divides x}

• SUBSET-SUM: given integers x1, x2 , ..., xn, y, do there exist
i1, i2 , ..., ik Î{1, 2,... , n} such that xi1+ xi2 + ... + xik = y?

• INTEGER-LINEAR-PROGRAMMING: linear programming
where one seeks an integer-valued solution (its existence)

8
P vs. NP
All of the aforementioned problems have the property that
they reduce to 3-CNF-SAT, in the sense that a polynomial-
time algorithm for 3-CNF-SAT can be converted into a poly-
time algorithm for the problem

Example:         algorithm for 3-COLOR

algorithm for 3
-CNF-SAT

If a polynomial-time algorithm is discovered for 3-CNF-SAT
then a polynomial-time algorithm for 3-COLOR follows
And this holds for any problem X Î NP
9
P vs. NP
For any problem X Î NP
algorithm for X

algorithm for 3
-CNF-SAT

A problem that has this property is said to be NP-hard

Polynomial-time algorithm any NP-Hard problem implies P=NP

NP-hard problems: 3-CNF-SAT, CIRCUIT-SAT, 3-COLOR ,
k-CLIQUE, HAM-PATH, SUBSET-SUM, INT-LIN-PROG, …
Some problems in P:
k-DNF-SAT, 2-COLOR, 3-CLIQUE, EUL-PATH, ...       10
FACTORING vs. NP
Is FACTORING NP-hard too?
PSPACE
If so, then every problem in
NP is solvable by a poly-time
quantum algorithm!          3-CNF-SAT
NP            co-NP

But FACTORING has
not been shown to be
NP-hard                   FACTORING
P
Moreover, there is “evidence”
that it is not NP-hard:
FACTORING Î NPÇco-NP

If FACTORING is NP-hard then NP = co-NP                       11
FACTORING vs. co-NP
FACTORING = {(x, y) : \$ 2 £ z £ y, s.t. z divides x}

co-NP: languages whose negative
PSPACE
instances have “witnesses” that can
be verified in poly-time                     NP        co-NP

Question: what is a
good witness for the
negative instances?              FACTORING
P
p1, p2 , ..., pm of x will work

Can verify primality and compare
p1, p2 , ..., pm with y, all in poly-time
12
Grover’s quantum
search algorithm

13
Quantum search problem
Given: a black box computing      f : {0,1}n à {0,1}
Goal: determine if   f is satisfiable (if \$x Î {0,1}n s.t. f(x) = 1)
In positive instances, it makes sense to also find such a satisfying
assignment x

n
Classically, using probabilistic procedures, order 2 queries are
necessary to succeed—even with probability ¾ (say)

n
Grover’s quantum algorithm that makes only O(Ö2             ) queries
Query:    |x1ñ                |x1ñ

|xnñ
Uf        |xnñ
[Grover ’96]             |yñ                  |y Å f(x1,...,xn)ñ        14
Applications of quantum search
The function f could be realized as a 3-CNF formula:

PSPACE
Alternatively, the search could
be for a certificate for any
problem in NP                                         NP            co-NP
3-CNF-SAT

The resulting quantum
algorithms appear to be
than the best classical                                     P
algorithms known*

Subtlety in that the search space might
have to be redefined to achieve this                                        15
Prelude to Grover’s algorithm:
two reflections = a rotation
Consider two lines with intersection angle q:

reflection 2
q2
q2
q1     reflection 1
q                 q1

Net effect: rotation by angle 2q, regardless of starting vector
16
Grover’s algorithm: description I
Basic operations used:

|x1ñ            |x1ñ
Uf |xñ|-ñ = (-1) f(x) |xñ|-ñ
|xnñ
Uf      |xnñ
|yñ              |y Å f(x1,...,xn)ñ

Implementation?            X          X
|x1ñ           |x1ñ                                              X          X
X          X
|xnñ
U0      |xnñ
|yñ             |y Å [x = 0...0]ñ      U0 |xñ|-ñ = (-1) [x = 0...0]|xñ|-ñ
H
H
H
17
Grover’s algorithm: description II
iteration 1                   iteration 2    ...
|0ñ

|0ñ
H       Uf       H      U0         H     Uf    H         U0     H
|-ñ

•   construct state H |0...0ñ|-ñ
•   repeat k times:
apply -H U0 H Uf to state
n
3. measure state, to get xÎ{0,1} , and check if    f (x) =1
(The setting of k will be determined later)
18
Grover’s algorithm: analysis I
Let A = {x Î{0,1}n : f (x) = 1} and B = {x Î{0,1}n : f (x) = 0}
and N = 2 and a = |A| and b = |B|
n

Let                            and

Consider the space spanned by |Añ and |Bñ

|Añ ß goal is to get close to this state

H|0...0ñ
|Bñ
Interesting case: a << N   19
Grover’s algorithm: analysis II
|Añ

Algorithm:   (-H U0 H Uf )k H |0...0ñ

H|0...0ñ
|Bñ

Observation:
Uf   is a reflection about |Bñ:   Uf |Añ = -|Añ   and   Uf |Bñ = |Bñ
Question: what is -H U0 H ?              U0     is a reflection about   H|0...0ñ
Partial proof:
-H U0 H H|0...0ñ = -H U0 |0...0ñ = -H (- |0...0ñ) = H |0...0ñ
20
Grover’s algorithm: analysis III
|Añ

Algorithm:   (-H U0 H Uf )k H |0...0ñ
2q
2q
2q
2q       H|0...0ñ
q   |Bñ

Since -H U0 H Uf is a composition of two reflections, it is a rotation
by 2q, where sin(q)=Öa/N                 » Öa/N
When a = 1, we want             (2k+1)(1/ÖN) » p/2 , so k » (p/4)ÖN
More generally, it suffices to set k          » (p/4)ÖN/a
Question: what if a is not known in advance?                                     21
Unknown number of solutions
1 solution                         2 solutions   3 solutions
1
probability
success

0 number of iterations
p√N/2

4 solutions                       6 solutions   100 solutions

success probability
very close to zero!

Choose a random k in the range to get success probability > 0.43
22
Optimality of
Grover’s algorithm

23
Optimality of Grover’s algorithm I
Theorem: any quantum search algorithm for           f : {0,1}n à {0,1} must make
( n) queries to f
W Ö2                   (if f is used as a black-box)

Proof (of a slightly simplified version):

f(x)
Assume queries are of the form               |x ñ        f        (-1)          |x ñ

and that a k-query algorithm is of the form

|0...0ñ     U0     f      U1       f         U2       f       U3     f      Uk
where U0, U1, U2, ..., Uk, are arbitrary unitary operations
24
Optimality of Grover’s algorithm II
n
Define fr : {0,1} à {0,1} as fr (x) = 1 iff x = r

Consider

|0ñ    U0     fr       U1     fr     U2       fr      U3       fr     Uk      |ψr,kñ

versus

|0ñ    U0      I       U1      I     U2       I       U3        I     Uk      |ψr,0ñ

We’ll show that, averaging over all r Î {0,1} , || |ψr,kñ   - |ψr,0ñ || £ 2k /
n

Ö2n
25
Optimality of Grover’s algorithm III
Consider

|0ñ    U0          I        U1          I        U2         fr        U3           fr   Uk   |ψr,iñ

k-i                                          i
Note that

|ψr,kñ - |ψr,0ñ = (|ψr,kñ - |ψr,k-1ñ) + (|ψr,k-1ñ - |ψr,k-2ñ) + ... + (|ψr,1ñ - |ψr,0ñ)

which implies
|| |ψr,kñ - |ψr,0ñ || £ || |ψr,kñ - |ψr,k-1ñ || + ... + || |ψr,1ñ - |ψr,0ñ ||
26
Optimality of Grover’s algorithm IV
query   i        query i+1

|0ñ   U0       I        U1       I        U2     fr        U3   fr   Uk    |ψr,iñ

query   i         query i+1

|0ñ   U0       I        U1       I        U2      I        U3   fr   Uk |ψr,i-1ñ

|| |ψr,iñ - |ψr,i-1ñ || = |2ai,r|, since query only negates |rñ
Therefore, || |ψr,kñ   - |ψr,0ñ || £
27
Optimality of Grover’s algorithm V
n
Now, averaging over all r Î {0,1} ,

(By Cauchy-Schwarz)

n
( n)
Therefore, for some r Î {0,1} , the number of queries k must be W Ö2 ,
in order to distinguish fr from the all-zero function
This completes the proof                                            28

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