Quantum expanders: motivation and constructions

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Quantum expanders motivation and constructions

• Avraham Ben-Aroya
• Oded Schwartz
• Amnon Ta-Shma

Based on arXivquant-ph/0702129 and arXiv0709.091
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Tel-Aviv University
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Motivating problems
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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
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What would we like to do?

• Estimate entropy
• Compare entropies
• Manipulate entropy

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Estimating entropy

• Given ? specified by a quantum circuit
• Goal Estimate S(?)
• Decision version decide whether S(?) gt t or S(?)
  lt t-1

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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

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Comparing entropies

• Given ?1, ?2 specified by circuits
• decide whether S(?1) gt S(?2)1 or S(?2) gt S(?1)1
• Equivalently Which of the pure states is more
  entangled

?2
?0?
Discard
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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
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Classical expanders
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Classical expanders

• Highly connected graphs with a low degree
• Possible definitions
• Vertex expansion every set expands
• Algebraic expansion adjacency matrix has large
  spectral gap

?1 1 ?2 ? ? ?3 ? ? ? ? ?n ? ?
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Classical expanders

• Let G be a graph with a normalized adjacency
  matrix ?
• ? maps a probability distribution (over the
  graphs 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

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Classical expanders manipulate entropies

• A (2d,?) expander solves the entropy manipulation
  problem in the classical setting
• G is ?-expanding ? for every classical
  distribution ? H2(?(?)) gt 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

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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.

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Outline

• Definition of quantum expanders
• Constructions
• Non-explicit bounds
• Explicit constructions
• Applications

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Definition of quantum expanders
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Quantum expanders

• An admissible superoperator
• I.e. ? L(C2n) ? L(C2n), a physically-realizable
  quantum transformation
• Satisfying some algebraic condition

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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 ?

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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

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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)

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(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

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Non-explicit bounds
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Ramanujan bounds

• Classical expanders
• All D-regular graphs AlonBoppana91
• ? gt 2/?D
• Random D-regular graphs Friedman04
• ? lt? 2/?D
• Quantum expanders
• All D-regular quantum expanders Hastings07
• ? gt 2/?D
• The average of D random unitaries Hastings07
• ? lt? 2/?D
• Completely different technique

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Explicit constructions
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Explicit constructions

• 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

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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

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The approach

• Find a good constant-size quantum expander, ?
• Using exhaustive search
• Existence guaranteed by Hastings
• Iteratively construct larger expanders

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The building blocks

• The composition (roughly) (???)2 ? ?

z
z
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The replacement product
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The replacement product
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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

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The classical Zig-Zag product
Example v and u are connected
v
u
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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
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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
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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?

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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)

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Applications
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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

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Quantum Entropy Difference(QED)

• Input
• Yes S(?1) gt S(?2)1
• No S(?2) gt S(?1)1

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Quantum Entropy Difference

• QED is QSZK-complete
• QSZK Quantum Statistical Zero Knowledge
• Languages with quantum interactive proofs, in
  which the verifier doesnt learn anything
  during the proof

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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

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Quantum State Distinguishability (QSD)

• Input
• Yes ?1 - ?2tr gt 0.9
• No ?1 - ?2tr lt 0.1
• Watrous02 QSD is QSZK-complete

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QED is QSZK-complete

• Resembles the classical proof that ED is
  SZK-complete
• QED is QSZK-hard
• Wont see
• QED ? QSZK
• Based on QEA ? QSZK

Now
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Quantum Entropy Approximation(QEA)

• Input a number t and
• Yes S(?) gt t
• No S(?) lt t-1

To simplify even further, we shall work with H2
entropy
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Manipulating quantum entropies

• If ? is a (2d, ?) quantum expander then it solves
  the entropy manipulation problem. Namely
• ? is ?-expanding ? for every mixed state ?
  H2(?(?)) gt H2(?)
• ? is 2d-regular ? ? never adds more than d bits
  of entropy

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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(?) gt t then H2(?(?))?n and is close to Î
• If H2(?) lt t-1 then H2(?(?)) ? n-1 and is far
  from Î
• Thats it

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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