1. Quantum expanders: motivation and constructions
Avraham Ben-Aroya
Oded Schwartz
Amnon Ta-Shma
Talk a bit about the recent work (new topic, several papers, I’ll describe the recent progress…)
Based on
arXiv:quant-ph/
and
arXiv:
Tel-Aviv University

2. Motivating
problems

3. Positive semi-definite
Entropies
Entropy of a mixed state 
von-Neumann: S() = -Tr( log ) = -i log i
Rényi: H2() = -log (Tr(2)) = -log (i2)
Central notion in information theory and computer science
Positive semi-definite
Eigenvalues: 1,…,n0
Tr() = i = 1

4. What would we like to do? Estimate entropy Compare entropies
Manipulate entropy

5. Estimating entropy Given  specified by a quantum circuit 
Goal: Estimate S()
Decision version: decide whether S() > t or S() < t-1 
Discard
0
Say noticeable gap

6. 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
Emphasize canonical B
0 A


7. Comparing entropies 1 2 Given 1, 2 specified by circuits
decide whether S(1) > S(2)+1 or S(2) > S(1)+1
Equivalently: Which of the pure states is more entangled 1
Discard
0 2
0
Discard

8. 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 
Read the slide
Say: forget about previous problems (we’ll come back to them later)
Lets start by looking at a classical
counterpart of such a transformation

9. Classical
expanders

10. Classical expanders Highly connected graphs with a low degree
Many neighbors
Highly connected graphs with a low degree
Possible definitions:
Vertex expansion: every set expands
Algebraic expansion: adjacency matrix has large spectral gap …
1/D
1 = 1
|2|  
|3|   
|n|  

11. 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

12. 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
Say low entropy

13. 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 entropy-related 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.

14. Outline Definition of quantum expanders Constructions Applications
Non-explicit bounds
Explicit constructions
Applications

15. Definition of quantum expanders

16. Quantum expanders An admissible superoperator
I.e.:  : L(C2n)  L(C2n), a physically-realizable quantum transformation
Satisfying some algebraic condition
Talk a bit about L and density m.

17. Quantum expanders – spectral gap
 is -expanding if
(Î) = Î (where Î = 2-2n I is the completely mixed state)
All other singular values are bounded by 

18. 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 D-regular graph

19. 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)

20. (D,) Quantum expander An Admissible superoperator  : L(C2n)  L(C2n)
Degree D
All singular values except first are bounded by 
[B-TaShma07] and independently [Hastings07]

21. Non-explicit bounds

22. Ramanujan bounds Classical expanders: Quantum expanders:
All D-regular graphs [AlonBoppana91]:
 >= 2/D
Random D-regular graphs [Friedman04]:
 < 2/D
Quantum expanders:
All D-regular quantum expanders [Hastings07]:
The average of D random unitaries [Hastings07]:
Completely different technique

23. Explicit constructions

24. Explicit constructions
Const. degree
Classical counterpart
Remarks
[AmbainsSmith04] No
Cayley Z2n
[B-TaShma07]
Yes
Cayley PGL(2,q) [LubotzkyPhilipsSarnak86]
1, 2
[B-Schwartz TaShma07]
Zig-Zag [ReingoldVadhanWigderson00]
[Harrow07]
Cayley Sn [Kassabov05] 3
[GrossEisert07]
[Margulis73]
Say about combinatroail/algebraic
Say that aram will speak about algebraic
Say “in some aspects is the best known construction”
Only mildly-explicit because no efficient QFT over PGL(2,q)
Gives an explicit construction for any group with QFT and an extra property
Gives an explicit construction for any group with QFT and large irreps

25. 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

26. The approach Find a good constant-size quantum expander, 
Using exhaustive search
Existence guaranteed by [Hastings]
Iteratively construct larger expanders
Say constant size domain issue

27. The building blocks The composition (roughly): ()2   Operation
Qubits
Degree 
Wanted goal
Same
Tensor
n->n2
D->D2
Squaring
->2
Zig-Zag 
n->nD
D4->D
->2
The composition (roughly): ()2   z z

28. The replacement product

29. The replacement product

30. 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

31. The classical Zig-Zag product
Example:
v and u are connected u v

32. 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)
The Zig-Zag product: 12 : L(V1V2)  L(V1V2) z
Which cloud
Position inside cloud

33. 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

34. The quantum Zig-Zag product
The product is composed of 3-steps
A small step
A large step
Another small step
Degree: Deg(2)2
Spectral gap?
1  2 = (I2)G1(I2) z

35. Spectral gap of the Zig-Zag product
In the classical setting we analyze some operator over the Hilbert space C2n
In the quantum setting - L(C2n)
The analysis works on this space as well
(Although this is not guaranteed a-priori)

36. Applications

37. 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]

38. Quantum Entropy Difference (QED)
Input:
Yes: S(1) > S(2)+1
No: S(2) > S(1)+1 1
Discard
0 2
Discard
0

39. 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

40. 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

41. Quantum State Distinguishability (QSD)
Input:
Yes: |1 - 2|tr > 0.9
No: |1 - 2|tr < 0.1
[Watrous02]: QSD is QSZK-complete 1
Discard
0 2
Discard
0

42. QED is QSZK-complete
Resembles the classical proof that ED is SZK-complete
QED is QSZK-hard
Won’t see
QED  QSZK
Based on QEA  QSZK
Now

43. Quantum Entropy Approximation (QEA)
Input: a number t and
Yes: S() > t
No: S() < t-1 
Discard
0
To simplify even further,
we shall work with H2 entropy

44. 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

45. 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

46. Open problems Classical expanders have many applications
Find more applications 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]