Download Qunatum extractors and the quantum entropy difference problem

Quantum expanders:
motivation and
constructions
Avraham Ben-Aroya
Oded Schwartz
Amnon Ta-Shma
Based on
arXiv:quant-ph/0702129
and
arXiv:0709.0911
Tel-Aviv University
1
Motivating
problems
2
Entropies

Entropy of a mixed state 
 von-Neumann:
 Rényi:

S() = -Tr( log ) = -i log i
H2() = -log (Tr(2)) = -log (i2)
Central notion in information theory and
computer science
Positive semi-definite
Eigenvalues: 1,…,n0
Tr() = i = 1
3
What would we like to do?
Estimate entropy
 Compare entropies
 Manipulate entropy

4
Estimating entropy

Given  specified by a quantum circuit

0
Discard
Goal: Estimate S()
 Decision
S() < t-1
version: decide whether S() > t or
5
Estimating entanglement
Entropy is a natural measure of
entanglement of bipartite pure states
 Equivalent problem: Given  on AB,
specified by a circuit, estimate the
entanglement between the two systems

0

A
B
6
Comparing entropies

0
Given 1, 2 specified by circuits
1
Discard
0
2
Discard
decide whether S(1) > S(2)+1 or S(2) > S(1)+1

Equivalently: Which of the pure states is
more entangled
7
Manipulating entropy


It will turn out understanding these questions
requires a way of manipulating entropies
Informally: A quantum transformation  that
adds a fixed amount of entropy
 For
any  with not-too-high entropy, () has more
entropy than 
 For any , the entropy () is never much larger
than the entropy of 
Lets start by looking at a classical
counterpart of such a transformation
8
Classical
expanders
9
Classical expanders


Highly connected graphs
with a low degree
Possible definitions:
 Vertex
expansion:
every set expands
 Algebraic
expansion:
adjacency matrix has large
spectral gap
0
0
…
0
0
1/D
0
…
0
1/D
0
0
…
0
0
0
1/D
…
1/D
0
0
0
…
0
0
1 = 1
|2|  
|3|  


|n | 
10
Classical expanders

Let G be a graph with a normalized adjacency
matrix 
maps a probability distribution (over the graph’s
vertices) to the distribution given by taking a
random step over the graph


G is -expanding if
 (Un)
= Un
 All other singular values are bounded by 

G is (D,) expander if it is -expanding and has
degree D
11
Classical expanders manipulate
entropies

A (2d,) expander solves the entropy
manipulation problem in the classical setting:
G
is -expanding  for every classical distribution
: H2(()) >= H2()
 Taking
a random step over a graph of degree 2d
requires d random bits   can never add more than
d bits of entropy

This is exactly what we required
12
Concluding the motivation for
quantum expanders
•
•
Fault-tolerant networks (e.g., [Pin73,Chu78,GG81])
Sorting in parallel [AKS83]
Complexity theory [Val77,Urq87]
Derandomization [AKS87,INW94,Rei05,…]
Randomness extractors [CW89,GW94,TUZ01,…]
Ramsey theory [Alo86]
Error-correcting codes [Gal63,Tan81,SS94,Spi95,LMSS01]
Distributed routing in networks [PU89,ALM96,…]
Data structures [BMRS00]
Distributed storage schemes [UW87]
Hard tautologies in proof complexity [BW99,ABRW00,…]
Other areas of Math [KR83,Lub94,Gro00,LP01]
We want to solve certain entropyrelated questions in the quantum
setting
More importantly, classical expanders
are extremely useful objects in
classical CS. It seems plausible that
their quantum counterparts may also be
useful.
13
Outline
•
•
Definition of quantum expanders
Constructions
Non-explicit bounds
 Explicit constructions

•
Applications
14
Definition
of quantum
expanders
15
Quantum expanders

An admissible superoperator
 I.e.:
:

a physicallyrealizable quantum transformation

n
2
L(C )
n
2
L(C ),
Satisfying some algebraic condition
16
Quantum expanders –
spectral gap

 is -expanding if
= Î (where Î =
is the completely
mixed state)
 All other singular values are bounded by 
 (Î)
-2n
2 I
17
What is the degree of a quantum
expander?
Without “degree” bound  can simply
always output the completely-mixed state
 In the classical setting,  corresponds to
a graph. Hence, it is clear how to define
the degree of .


There is an equivalent way to define a Dregular graph
18
Quantum expanders – degree
A classical graph G is D-regular if
(v) = D-1 iPiv where Pi is a
permutation
 A quantum superoperator is D-regular if
() = D-1 iUi  Ui* where Ui is unitary

 (Can
be generalized to an arbitrary sum of
D Kraus operators)
19
(D,) Quantum expander

An Admissible superoperator
n
n
2
2
 : L(C )  L(C )
 Degree
D
 All singular values except first are bounded
by 

[B-TaShma07]
and independently [Hastings07]
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Non-explicit
bounds
21
Ramanujan bounds

Classical expanders:

Quantum expanders:
 All
D-regular graphs [AlonBoppana91]:
 >= 2/D
 Random D-regular graphs [Friedman04]:
 < 2/D
 All
D-regular quantum expanders [Hastings07]:
 >= 2/D
 The average of D random unitaries [Hastings07]:
 < 2/D

Completely different technique
22
Explicit
constructions
23
Explicit constructions
Const.
degree
Classical counterpart
[AmbainsSmith04]
No
Cayley Z2n
[B-TaShma07]
Yes
[B-Schwartz
TaShma07]
Yes
[Harrow07]
Yes
Cayley Sn [Kassabov05]
[GrossEisert07]
Yes
[Margulis73]
Cayley PGL(2,q)
[LubotzkyPhilipsSarnak86]
Remark
s
1, 2
Zig-Zag
[ReingoldVadhanWigderson00]
3
1) Only mildly-explicit because no efficient QFT over PGL(2,q)
2) Gives an explicit construction for any group with QFT and an extra property
3) Gives an explicit construction for any group with QFT and large irreps
24
The Zig-Zag construction

A quantum version of the Zig-Zag
product [ReingoldVadhanWigderson00]
 Relatively
simpler to “quantize” than other
constructions
 Very important notion in classical CS
25
The approach

Find a good constant-size quantum
expander, 
 Using
exhaustive search
 Existence guaranteed by [Hastings]

Iteratively construct larger expanders
26
The building blocks
Operation
Qubits
Wanted goal

Degree

Same
Same
Tensor
n->n2
D->D2
Same
Squaring
Same
D->D2
->2
z
Zig-Zag 
n->nD
D4->D
->2
The composition (roughly): ()2 z 
27
The replacement product
28
The replacement product
29
The classical Zig-Zag product


Vertices: same as in replacement product
Edges: (v,u)E  there is a path of length 3
on the replacement product such that:
 The
first step is on the small graph
 The second step is on the large graph
 The third step is on the small graph
30
The classical Zig-Zag product
u
v
Example:
v and u are connected
31
The quantum Zig-Zag: setup

Large quantum expander: 1 : L(V1)  L(V1)
 dim(V1)

= N1
Small quantum expander: 2 : L(V2)  L(V2)
 dim(V2)
= N2  N1
 However, dim(V2) = deg(1)

z
The Zig-Zag product: 1
2 : L(V1V2) 
L(V1V2)
Which cloud
Position inside cloud
32
The quantum Zig-Zag: steps


Small step: I2
Large step:
 1
is D1-regular
 1()
= D1-1 iUiUi*
 TG1(ab) = (Ub a)b
Move to a different cloud,
according to the current
position within the cloud
33
The quantum Zig-Zag product

The product is composed of 3-steps
A
small step
 A large step
 Another small step
z  = (I ) (I )
1 
2
2 G
2
1
Degree: Deg(2)2
 Spectral gap?

34
Spectral gap of the Zig-Zag
product
In the classical setting we analyze some
n
2
operator over the Hilbert space C
n
2
 In the quantum setting - L(C )
 The analysis works on this space as well

 (Although
this is not guaranteed a-priori)
35
Applications
36
Applications


The complexity of comparing/approximating
entropies [B-TaShma07]
Short quantum one-time pads [AmbainisSmith04]
 Implicitly

used a quantum expander
Construction of one-dimensional Hamiltonians
with extremal properties [Hastings07]
37
Quantum Entropy Difference
(QED)
Input:
0
1
Discard
0
2
Discard
Yes: S(1) > S(2)+1
No: S(2) > S(1)+1
38
Quantum Entropy Difference
QED is QSZK-complete
 QSZK = Quantum Statistical Zero
Knowledge

 Languages
with quantum interactive proofs,
in which the verifier doesn’t “learn” anything
during the proof
39
Quantum Statistical Zero
Knowledge
Quantum analogue of SZK
 Studied by [Watrous02], [Watrous06]
 Has many properties analogous to SZK

 Closed
under complement
 Honest verifier = Dishonest verifier
 Public coins = Private coins
 A natural complete problem
40
Quantum State Distinguishability
(QSD)
Input:
1
0
Discard
0
2
Discard
Yes: |1 - 2|tr > 0.9
No: |1 - 2|tr < 0.1

[Watrous02]:
QSD is QSZK-complete
41
QED is QSZK-complete

Resembles the classical proof that ED is
SZK-complete
 QED

Won’t see
 QED
Now

is QSZK-hard
 QSZK
Based on QEA  QSZK
42
Quantum Entropy Approximation
(QEA)
Input: a number t and
0
Yes: S() > t
No: S() < t-1

Discard
To simplify even further,
we shall work with H2 entropy
43
Manipulating quantum entropies

If  is a (2d, ) quantum expander then
it solves the entropy manipulation
problem. Namely:
is -expanding  for every mixed state :
H2(()) >= H2()
  is 2d-regular   never adds more than d
bits of entropy

44
QEA  QSZK

A reduction to QSD:
Given  on n qubits and a threshold t output
(() , Î)
is an expander that adds  n-t bits of entropy
and has degree 2n-t




If H2() > t then H2(())n and is close to Î
If H2() < t-1 then H2(())  n-1 and is far
from Î
That’s it
45
Open problems

Classical expanders have many applications
Fault-tolerant networks (e.g., [Pin73,Chu78,GG81])
Sorting in parallel [AKS83]
Complexity theory [Val77,Urq87]
Derandomization [AKS87,INW94,Rei05,…]
Randomness extractors [CW89,GW94,TUZ01,…]
Ramsey theory [Alo86]
Error-correcting codes [Gal63,Tan81,SS94,Spi95,LMSS01]
Distributed routing in networks [PU89,ALM96,…]
Data structures [BMRS00]
Distributed storage schemes [UW87]
Hard tautologies in proof complexity [BW99,ABRW00,…]
Other areas of Math [KR83,Lub94,Gro00,LP01]

Find more applications for quantum expanders
46