Quantum Streaming Algorithms for DYCK(2) ASHWIN NAYAK, AND DAVE - PowerPoint PPT Presentation

Augmented Index and Quantum Streaming Algorithms for DYCK(2) ASHWIN NAYAK, AND DAVE TOUCHETTE University of Waterloo, Perimeter Institute CCC 2017, Riga, Latvia 8 July 2017

Communication Complexity ◦ Communication Complexity setting: 𝑆 𝐵 𝑆 𝐶 𝑌, 𝑍 𝐷 1 = 𝑔 1 (𝑌, 𝑆 𝐵 ) 𝐷 2 = 𝑔 2 (𝑍, 𝑆 𝐶 , 𝐷 1 ) … 𝐷 𝑁 = 𝑔 𝑁 (𝑍, 𝑆 𝐶 , 𝐷 <𝑁 ) Output: f(x,y) ◦ How much communication to compute f on x, y ◦ Take information-theoretic view: Information Complexity ◦ How much information to compute f on x, y ∼ 𝜈 ◦ Information content of interactive protocols? ◦ Classical vs. Quantum?

Communication Complexity ◦ Communication Complexity setting: 𝑆 𝐵 𝑆 𝐶 𝑌, 𝑍 𝐷 1 = 𝑔 1 (𝑌, 𝐵, 𝑆 𝐵 ) 𝐷 2 = 𝑔 2 (𝑍, 𝑆 𝐶 , 𝐷 1 ) 𝜚 𝑗 𝐵 𝑗 𝜚 𝑗 𝐶 𝑗 … 𝐷 𝑁 = 𝑔 𝑁 (𝑍, 𝑆 𝐶 , 𝐷 <𝑁 ) Output: f(x,y) ◦ How much communication to compute f on x, y ◦ Take information-theoretic view: Information Complexity ◦ How much information to compute f on x, y ∼ 𝜈 ◦ Information content of interactive protocols? ◦ Classical vs. Quantum?

Communication Complexity ◦ Communication Complexity setting: 𝑆 𝐵 𝑆 𝐶 𝑌, 𝑍 𝜚 1 𝐷 1 𝜚 2 𝐷 2 𝜚 𝑗 𝐵 𝑗 𝜚 𝑗 𝐶 𝑗 … 𝜚 𝑁 𝐷 𝑁 Output: f(x,y) ◦ How much communication to compute f on x, y ◦ Take information-theoretic view: Information Complexity ◦ How much information to compute f on x, y ∼ 𝜈 ◦ Information content of interactive protocols? ◦ Classical vs. Quantum?

Communication Complexity ◦ Communication Complexity setting: |𝜔〉 𝑌, 𝑍 𝜚 1 𝐷 1 𝜚 2 𝐷 2 𝜚 𝑗 𝐵 𝑗 𝜚 𝑗 𝐶 𝑗 … 𝜚 𝑁 𝐷 𝑁 Output: f(x,y) ◦ How much communication to compute f on x, y ◦ Take information-theoretic view: Information Complexity ◦ How much information to compute f on x, y ∼ 𝜈 ◦ Information content of interactive protocols? ◦ Classical vs. Quantum?

Quantum Communication Complexity 𝑌 𝑌𝐵 1 𝑌𝐵 2 𝑌𝐵 3 𝑌𝐵 𝑁 𝑌 Protocol Π: 𝐵 𝑔 𝑉 1 𝑉 3 𝑉 𝑔 𝐵 0 Output: f(X,Y) 𝐷 3 𝐷 1 𝐷 𝑁−1 𝜈 | 𝜔〉 𝐷 𝑁 𝐷 2 𝐶 0 𝐶 𝑔 𝑉 2 𝑉 𝑁 𝑍𝐶 𝑁−1 𝑍𝐶 2 𝑍 𝑍

Th.1: Streaming Algorithms for DYCK(2) 𝑦 𝑂 𝑦 1 𝑦 2 |0 𝑡(𝑂) 〉 … 𝑃 𝑦 𝑂 𝑃 𝑦 1 𝑃 𝑦 2 Pre Post Repeat T times ◦ Streaming algorithms: Attractive model for early Quantum Computers ◦ Some exponential advantages possible for specially crafted problems [LeG06, GKKRdW07] ◦ DYCK 2 = 𝜗 + 𝐸𝑍𝐷𝐿 2 + 𝐸𝑍𝐷𝐿 2 + 𝐸𝑍𝐷𝐿 2 ⋅ 𝐸𝑍𝐷𝐿 2 𝑂 ◦ Classical bound: 𝑡 𝑂 ∈ Ω [MMN10, JN10, CCKM10] 𝑈 ◦ Two-way classical algorithm: 𝑡 𝑂 ∈ O(𝑞𝑝𝑚𝑧𝑚𝑝𝑕(𝑂))

Th.1: Streaming Algorithms for DYCK(2) 𝑂 ◦ Th. 1: Any T-pass 1-way qu. streaming algo. for DYCK(2) needs space 𝑡 𝑂 ∈ Ω( 𝑈 3 ) on length N inputs ◦ Even holds for non-unitary streaming operations 𝑃 ◦ Reduction from multi-party QCC to streaming algorithm to DYCK(2) [MMN10] ◦ Multi-party problem consists of OR of multiple instances of two-party problem ◦ Space s(N) in algorithm corresponds to communication between parties ◦ Consider T-pass, one-way quantum streaming algorithms ◦ Direct sum argument allows to reduce from a two-party problem, Augmented Index ◦ Multi-party QCC lower bounds requires two- party QIC lower bound on “easy distribution” ◦ Subtlety for non-unitary streaming operations 𝑃

Th.2: Augmented Index ◦ Index 𝑦 1 … 𝑦 𝑗 … 𝑦 𝑜 , 𝑗 = 𝑦 𝑗 ◦ Augmented Index: 𝐵𝐽 𝑜 𝑦 1 … 𝑦 𝑜 , 𝑗, 𝑦 1 … 𝑦 <𝑗 , 𝑐 = 𝑦 𝑗 ⊕ 𝑐 𝑦 𝑧 𝑦 1 𝑦 1 𝑦 2 𝑦 2 … … 𝑦 𝑗−1 𝑦 𝑗−1 𝑦 𝑗 𝑗 ∈ [𝑜] … 𝑦 𝑜−1 b ∈ {0, 1} 𝑦 𝑗 ⊕ 𝑐 𝑦 𝑜

Th.2: Augmented Index ◦ Th. 2.2: For any r-round protocol Π for 𝐵𝐽 𝑜 , either 𝑜 ◦ 𝑅𝐽𝐷 𝐵→𝐶 Π, 𝜈 0 ∈ Ω or 𝑠 2 1 ◦ 𝑅𝐽𝐷 𝐶→𝐵 Π, 𝜈 0 ∈ Ω with 𝑠 2 ◦ 𝜈 0 the uniform distribution on zeros of 𝐵𝐽 𝑜 (“easy distribution”) ◦ Classical bounds: 𝑜 ◦ 𝐽𝐷 𝐵→𝐶 Π, 𝜈 0 ∈ Ω 2 𝑛 or ◦ 𝐽𝐷 𝐶→𝐵 Π, 𝜈 0 ∈ Ω 𝑛 ◦ [MMN10, JN10, CCKM10, CK11] ◦ We Build on direct sum approach of [JN10] ◦ General approach uses two main Tools (Sup.-Average Encoding Th., Qu. Cut-and-Paste) ◦ More specialized approach uses one more Tool (Information Flow Lemma)

Warm-up: Disjointness 𝐸𝑗𝑡𝑘 𝑜 (𝑦, 𝑧) = ¬ڀ 𝑗∈[𝑜] (𝑦 𝑗 ∧ 𝑧 𝑗 ) 𝐷𝐷 𝐸𝑗𝑡𝑘 𝑜 ∈ Ω(𝑜) 𝑦 𝑧 𝑦 1 𝑧 1 𝑦 2 𝑧 2 … … 𝑦 𝑗−1 𝑧 𝑗−1 𝑦 𝑗 AND 𝑧 𝑗 … … 𝑦 𝑜−1 𝑧 𝑜−1 𝑦 𝑜 𝑧 𝑜

Warm-up: Disjointness CC 𝐸𝑗𝑡𝑘 𝑜 ≥ 𝐽𝐷 0 𝐸𝑗𝑡𝑘 𝑜 = 𝑜 𝐽𝐷 0 𝐵𝑂𝐸 [BJKS02] 𝐽𝐷 ≤ 𝐷𝐷, 𝐽𝐷 satisfies direct sum property (needs private and public randomness) 2 2 𝐽𝐷 0 (𝐵𝑂𝐸) = 3 𝐽 𝑌: 𝑁 𝑍 = 0 + 3 𝐽(𝑍: 𝑁|𝑌 = 0) Comparing message transcript 𝑁 on 01, 00, 10 inputs: ?

Tool: Average Encoding Theorem ◦ Average encoding theorem [KNTZ01]: E 𝑌 ℎ 2 𝑁 𝑌 , 𝑁 ∗ ≤ 𝐽(𝑌: 𝑁) ◦ 𝑁 ∗ = E 𝑌 [𝑁 𝑌 ] , average message ◦ h 2 𝑁 1 𝑁 2 , Heilinger distance ◦ Follows from Pinsker’s inequality ◦ Low information messages are close to average message ◦ For AND, 𝑍 = 0 : 1 2 ℎ 2 𝑁 00 , 𝑁 ∗0 + 1 2 ℎ 2 𝑁 10 , 𝑁 ∗0 ≤ 𝐽(𝑌: 𝑁|𝑍 = 0) ◦ Using Jensen and triangle inequality: 1 4 ℎ 2 𝑁 00 , 𝑁 10 ≤ 𝐽 𝑌: 𝑁 𝑍 = 0 ◦ Similarly, for X=0: 1 4 ℎ 2 𝑁 00 , 𝑁 01 ≤ 𝐽 𝑍: 𝑁 𝑌 = 0 1 8 ℎ 2 𝑁 10 , 𝑁 01 ≤ 𝐽 𝑌: 𝑁 𝑍 = 0 + 𝐽 𝑍: 𝑁 𝑌 = 0 = 3 ◦ Comparing 01, 10 inputs: 2 𝐽𝐷 0 (𝐵𝑂𝐸)

Warm-up: Disjointness CC 𝐸𝑗𝑡𝑘 𝑜 ≥ 𝐽𝐷 0 𝐸𝑗𝑡𝑘 𝑜 = 𝑜 𝐽𝐷 0 𝐵𝑂𝐸 [BJKS02] 𝐽𝐷 ≤ 𝐷𝐷, 𝐽𝐷 satisfies direct sum property, needs private and public randomness 2 2 𝐽𝐷 0 (𝐵𝑂𝐸) = 3 𝐽 𝑌: 𝑁 𝑍 = 0 + 3 𝐽(𝑍: 𝑁|𝑌 = 0) 12 ℎ 2 𝑁 10 , 𝑁 01 ≤ 𝐽𝐷 0 (𝐵𝑂𝐸) 1 Comparing message transcript 𝑁 on 01, 00, 10 inputs:

Warm-up: Disjointness CC 𝐸𝑗𝑡𝑘 𝑜 ≥ 𝐽𝐷 0 𝐸𝑗𝑡𝑘 𝑜 = 𝑜 𝐽𝐷 0 𝐵𝑂𝐸 [BJKS02] 𝐽𝐷 ≤ 𝐷𝐷, 𝐽𝐷 satisfies direct sum property, needs private and public randomness 2 2 𝐽𝐷 0 (𝐵𝑂𝐸) = 3 𝐽 𝑌: 𝑁 𝑍 = 0 + 3 𝐽(𝑍: 𝑁|𝑌 = 0) 12 ℎ 2 𝑁 10 , 𝑁 01 ≤ 𝐽𝐷 0 (𝐵𝑂𝐸) 1 Comparing message transcript 𝑁 on 01, 00, 10 inputs: Comparing 𝑁 on 00, 11 inputs: ?

Tool: Cut-and-Paste Lemma ◦ Consider input subset {𝑦 1 , 𝑦 2 } × {𝑧 1 , 𝑧 2 } 𝑦 1 𝑧 1 𝑦 2 𝑧 2 ◦ Triangle inequality implies for 𝑁 on 𝑦 1 , 𝑧 2 and (𝑦 2 , 𝑧 1 ) : ◦ ℎ 𝑁 𝑦 1 𝑧 2 , 𝑁 𝑦 2 𝑧 1 ≤ ℎ 𝑁 𝑦 1 𝑧 1 , 𝑁 𝑦 1 𝑧 2 + ℎ(𝑁 𝑦 1 𝑧 1 , 𝑁 𝑦 2 𝑧 1 ) 𝑦 1 𝑧 1 𝑦 1 𝑧 1 𝑦 1 𝑧 1 + ⇒ 𝑦 2 𝑧 2 𝑦 2 𝑧 2 𝑦 2 𝑧 2 ◦ What about 𝑁 on 𝑦 1 , 𝑧 1 and (𝑦 2 , 𝑧 2 ) : ? ◦ Cut-and-paste Lemma [BJKS02]: ℎ 𝑁 𝑦 1 𝑧 1 , 𝑁 𝑦 2 𝑧 2 = ℎ 𝑁 𝑦 1 𝑧 2 , 𝑁 𝑦 2 𝑧 1 𝑦 1 𝑧 1 𝑦 1 𝑧 1 ⇒ 𝑦 2 𝑧 2 𝑦 2 𝑧 2

Warm-up: Disjointness CC 𝐸𝑗𝑡𝑘 𝑜 ≥ 𝐽𝐷 0 𝐸𝑗𝑡𝑘 𝑜 = 𝑜 𝐽𝐷 0 𝐵𝑂𝐸 [BJKS02] 𝐽𝐷 ≤ 𝐷𝐷, 𝐽𝐷 satisfies direct sum property, needs private and public randomness 2 2 𝐽𝐷 0 (𝐵𝑂𝐸) = 3 𝐽 𝑌: 𝑁 𝑍 = 0 + 3 𝐽(𝑍: 𝑁|𝑌 = 0) 12 ℎ 2 𝑁 10 , 𝑁 01 ≤ 𝐽𝐷 0 (𝐵𝑂𝐸) 1 Comparing message transcript 𝑁 on 01, 00, 10 inputs: 12 ℎ 2 𝑁 00 , 𝑁 11 = 1 12 ℎ 2 𝑁 10 , 𝑁 01 ≤ 𝐽𝐷 0 (𝐵𝑂𝐸) 1 Comparing 𝑁 on 00, 11 inputs:

Warm-up: Disjointness CC 𝐸𝑗𝑡𝑘 𝑜 ≥ 𝐽𝐷 0 𝐸𝑗𝑡𝑘 𝑜 = 𝑜 𝐽𝐷 0 𝐵𝑂𝐸 [BJKS02] 𝐽𝐷 ≤ 𝐷𝐷, 𝐽𝐷 satisfies direct sum property, needs private and public randomness 2 2 𝐽𝐷 0 (𝐵𝑂𝐸) = 3 𝐽 𝑌: 𝑁 𝑍 = 0 + 3 𝐽(𝑍: 𝑁|𝑌 = 0) 12 ℎ 2 𝑁 10 , 𝑁 01 ≤ 𝐽𝐷 0 (𝐵𝑂𝐸) 1 Comparing message transcript 𝑁 on 01, 00, 10 inputs: 12 ℎ 2 𝑁 00 , 𝑁 11 = 1 12 ℎ 2 𝑁 10 , 𝑁 01 ≤ 𝐽𝐷 0 (𝐵𝑂𝐸) 1 Comparing 𝑁 on 00, 11 inputs: 1 Statistical interpretation: ℎ 𝑁 1 , 𝑁 2 ≥ 4 | 𝑁 1 − 𝑁 2 | 𝑈𝑊 ||𝑁 00 − 𝑁 11 || 𝑈𝑊 ∈ Ω 1 since for AND, 𝑁 00 → 0, 𝑁 11 → 1 𝐷𝐷 𝐸𝑗𝑡𝑘 𝑜 ∈ Ω(𝑜) Quantum?

Warm-up: Disjointness 𝑅𝐷𝐷 𝐸𝑗𝑡𝑘 𝑜 ∈ Θ( 𝑜) 𝑅𝐷𝐷 𝑠 𝐸𝑗𝑡𝑘 𝑜 ∈ O( 𝑜 𝑠 ) , r round protocols 1 𝑠 𝐸𝑗𝑡𝑘 𝑜 = 1 𝑠 𝐵𝑂𝐸 [JRS03] QCC r 𝐸𝑗𝑡𝑘 𝑜 ≥ 𝑠 ෪ 𝑠 𝑜 ෪ 𝑅𝐽𝐷 0 𝑅𝐽𝐷 0 ෪ ෪ 𝑅𝐽𝐷 ≤ 𝑠 𝑅𝐷𝐷, 𝑅𝐽𝐷 satisfies direct sum property, requires private and public “randomness” Comparing 𝑁 on 01, 00, 10 inputs: ?