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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,…,n0 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 AB, 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] 20 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(V1V2) L(V1V2) Which cloud Position inside cloud 32 The quantum Zig-Zag: steps Small step: I2 Large step: 1 is D1-regular 1() = D1-1 iUiUi* TG1(ab) = (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