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/0702129 and arXiv:0709.0911 Tel-Aviv University

Motivating problems

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

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

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

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 

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

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

Classical expanders

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|  

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

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

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.

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

Definition of quantum expanders

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.

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 

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

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)

(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]

Non-explicit bounds

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

Explicit constructions

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

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

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

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

The replacement product

The replacement product

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

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

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

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

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

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)

Applications

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]

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

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

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

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

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

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

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

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

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]