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

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

Jan 18, 2024 221 likes β€’668 views

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:

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SLIDE 1

Augmented Index and Quantum Streaming Algorithms for DYCK(2)

ASHWIN NAYAK, AND DAVE TOUCHETTE

CCC 2017, Riga, Latvia 8 July 2017 University of Waterloo, Perimeter Institute

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SLIDE 2

Communication Complexity

  • Communication Complexity setting:
  • 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?

𝑆𝐡𝑆𝐢 π‘Œ, 𝑍 𝐷1 = 𝑔

1(π‘Œ, 𝑆𝐡)

𝐷2 = 𝑔

2(𝑍, 𝑆𝐢, 𝐷1)

𝐷𝑁 = 𝑔

𝑁(𝑍, 𝑆𝐢, 𝐷<𝑁)

… Output: f(x,y)

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SLIDE 3

Communication Complexity

  • Communication Complexity setting:
  • 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?

𝑆𝐡𝑆𝐢 π‘Œ, 𝑍 𝐷1 = 𝑔

1(π‘Œ, 𝐡, 𝑆𝐡)

𝐷2 = 𝑔

2(𝑍, 𝑆𝐢, 𝐷1)

𝐷𝑁 = 𝑔

𝑁(𝑍, 𝑆𝐢, 𝐷<𝑁)

… Output: f(x,y) πœšπ‘— 𝐡𝑗 πœšπ‘— 𝐢𝑗

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SLIDE 4

Communication Complexity

  • Communication Complexity setting:
  • 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?

𝑆𝐡𝑆𝐢 π‘Œ, 𝑍 𝜚1 𝐷1 … Output: f(x,y) 𝜚2 𝐷2 πœšπ‘ 𝐷𝑁 πœšπ‘— 𝐡𝑗 πœšπ‘— 𝐢𝑗

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SLIDE 5

Communication Complexity

  • Communication Complexity setting:
  • 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?

|πœ”βŒͺ π‘Œ, 𝑍 𝜚1 𝐷1 … Output: f(x,y) 𝜚2 𝐷2 πœšπ‘ 𝐷𝑁 πœšπ‘— 𝐡𝑗 πœšπ‘— 𝐢𝑗

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SLIDE 6

Quantum Communication Complexity

𝑉1 𝑉2 𝑉3 𝑉

𝑔

𝑉𝑁 𝜈 Protocol Ξ : π‘Œ 𝑍 𝐡0 | πœ”βŒͺ 𝐢0 π‘Œπ΅1 𝐷1 𝐷2 𝐷3 𝑍𝐢2 π‘Œπ΅2 π‘Œπ΅3 π·π‘βˆ’1 𝐷𝑁 π‘Œπ΅π‘ π‘πΆπ‘βˆ’1 π‘Œ 𝐡𝑔 𝐢𝑔 𝑍 Output: f(X,Y)

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SLIDE 7

Th.1: Streaming Algorithms for DYCK(2)

  • Streaming algorithms: Attractive model for early Quantum Computers
  • Some exponential advantages possible for specially crafted problems [LeG06, GKKRdW07]
  • DYCK 2 = πœ— + 𝐸𝑍𝐷𝐿 2

+ 𝐸𝑍𝐷𝐿 2 + 𝐸𝑍𝐷𝐿 2 β‹… 𝐸𝑍𝐷𝐿 2

  • Classical bound: 𝑑 𝑂 ∈ Ξ©

𝑂 π‘ˆ

[MMN10, JN10, CCKM10]

  • Two-way classical algorithm: 𝑑 𝑂 ∈ O(π‘žπ‘π‘šπ‘§π‘šπ‘π‘•(𝑂))

|0𝑑(𝑂)βŒͺ 𝑃𝑦1 𝑦1 𝑃𝑦2 𝑦2 … 𝑃𝑦𝑂 𝑦𝑂 Repeat T times Pre

Post

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SLIDE 8

Th.1: Streaming Algorithms for DYCK(2)

  • Th. 1: Any T-pass 1-way qu. streaming algo. for DYCK(2) needs space 𝑑 𝑂 ∈ Ξ©(

𝑂 π‘ˆ3) 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 β€œeasy distribution”
  • Subtlety for non-unitary streaming operations 𝑃
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SLIDE 9

Th.2: Augmented Index

  • Index 𝑦1 … 𝑦𝑗 … π‘¦π‘œ, 𝑗 = 𝑦𝑗
  • Augmented Index: π΅π½π‘œ 𝑦1 … π‘¦π‘œ, 𝑗, 𝑦1 … 𝑦<𝑗, 𝑐

= 𝑦𝑗 βŠ• 𝑐 𝑦1 𝑦2 …

π‘¦π‘—βˆ’1

𝑦𝑗 …

π‘¦π‘œβˆ’1

π‘¦π‘œ 𝑦1 𝑦2 …

π‘¦π‘—βˆ’1

𝑦 𝑧 𝑗 ∈ [π‘œ] b∈ {0, 1} 𝑦𝑗 βŠ• 𝑐

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SLIDE 10

Th.2: Augmented Index

  • Th. 2.2: For any r-round protocol Ξ  for π΅π½π‘œ, either
  • 𝑅𝐽𝐷

𝐡→𝐢 Ξ , 𝜈0 ∈ Ξ© π‘œ 𝑠2

  • r
  • 𝑅𝐽𝐷𝐢→𝐡 Ξ , 𝜈0 ∈ Ξ©

1 𝑠2

with

  • 𝜈0 the uniform distribution on zeros of π΅π½π‘œ (β€œeasy distribution”)
  • Classical bounds:
  • 𝐽𝐷

𝐡→𝐢 Ξ , 𝜈0 ∈ Ξ© π‘œ 2𝑛 or

  • 𝐽𝐷𝐢→𝐡 Ξ , 𝜈0 ∈ Ξ© 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)
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SLIDE 11

Warm-up: Disjointness

πΈπ‘—π‘‘π‘˜π‘œ(𝑦, 𝑧) = Β¬Ϊ€π‘—βˆˆ[π‘œ](𝑦𝑗 ∧ 𝑧𝑗) 𝐷𝐷 πΈπ‘—π‘‘π‘˜π‘œ ∈ Ξ©(π‘œ)

𝑦1 𝑦2 …

π‘¦π‘—βˆ’1

𝑦𝑗 …

π‘¦π‘œβˆ’1

π‘¦π‘œ 𝑦 𝑧1 𝑧2 …

π‘§π‘—βˆ’1

𝑧𝑗 …

π‘§π‘œβˆ’1

π‘§π‘œ 𝑧 AND

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SLIDE 12

Warm-up: Disjointness

CC πΈπ‘—π‘‘π‘˜π‘œ β‰₯ 𝐽𝐷0 πΈπ‘—π‘‘π‘˜π‘œ = π‘œ 𝐽𝐷0 𝐡𝑂𝐸 [BJKS02] 𝐽𝐷 ≀ 𝐷𝐷, 𝐽𝐷 satisfies direct sum property (needs private and public randomness) 𝐽𝐷0(𝐡𝑂𝐸) =

2 3 𝐽 π‘Œ: 𝑁 𝑍 = 0 + 2 3 𝐽(𝑍: 𝑁|π‘Œ = 0)

Comparing message transcript 𝑁 on 01, 00, 10 inputs: ?

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SLIDE 13

Tool: Average Encoding Theorem

  • Average encoding theorem [KNTZ01]: Eπ‘Œ β„Ž2 π‘π‘Œ, π‘βˆ—

≀ 𝐽(π‘Œ: 𝑁)

  • π‘βˆ— = Eπ‘Œ[π‘π‘Œ], average message
  • h2 𝑁1 𝑁2 , Heilinger distance
  • Follows from Pinsker’s inequality
  • Low information messages are close to average message
  • For AND, 𝑍 = 0: 1

2 β„Ž2 𝑁00, π‘βˆ—0 + 1 2 β„Ž2 𝑁10, π‘βˆ—0 ≀ 𝐽(π‘Œ: 𝑁|𝑍 = 0)

  • Using Jensen and triangle inequality: 1

4 β„Ž2 𝑁00, 𝑁10 ≀ 𝐽 π‘Œ: 𝑁 𝑍 = 0

  • Similarly, for X=0: 1

4 β„Ž2 𝑁00, 𝑁01 ≀ 𝐽 𝑍: 𝑁 π‘Œ = 0

  • Comparing 01, 10 inputs:

1 8 β„Ž2 𝑁10, 𝑁01 ≀ 𝐽 π‘Œ: 𝑁 𝑍 = 0 + 𝐽 𝑍: 𝑁 π‘Œ = 0 = 3 2 𝐽𝐷0(𝐡𝑂𝐸)

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SLIDE 14

Warm-up: Disjointness

CC πΈπ‘—π‘‘π‘˜π‘œ β‰₯ 𝐽𝐷0 πΈπ‘—π‘‘π‘˜π‘œ = π‘œ 𝐽𝐷0 𝐡𝑂𝐸 [BJKS02] 𝐽𝐷 ≀ 𝐷𝐷, 𝐽𝐷 satisfies direct sum property, needs private and public randomness 𝐽𝐷0(𝐡𝑂𝐸) =

2 3 𝐽 π‘Œ: 𝑁 𝑍 = 0 + 2 3 𝐽(𝑍: 𝑁|π‘Œ = 0)

Comparing message transcript 𝑁 on 01, 00, 10 inputs:

1 12 β„Ž2 𝑁10, 𝑁01 ≀ 𝐽𝐷0(𝐡𝑂𝐸)

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SLIDE 15

Warm-up: Disjointness

CC πΈπ‘—π‘‘π‘˜π‘œ β‰₯ 𝐽𝐷0 πΈπ‘—π‘‘π‘˜π‘œ = π‘œ 𝐽𝐷0 𝐡𝑂𝐸 [BJKS02] 𝐽𝐷 ≀ 𝐷𝐷, 𝐽𝐷 satisfies direct sum property, needs private and public randomness 𝐽𝐷0(𝐡𝑂𝐸) =

2 3 𝐽 π‘Œ: 𝑁 𝑍 = 0 + 2 3 𝐽(𝑍: 𝑁|π‘Œ = 0)

Comparing message transcript 𝑁 on 01, 00, 10 inputs:

1 12 β„Ž2 𝑁10, 𝑁01 ≀ 𝐽𝐷0(𝐡𝑂𝐸)

Comparing 𝑁 on 00, 11 inputs: ?

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SLIDE 16

Tool: Cut-and-Paste Lemma

  • Consider input subset {𝑦1, 𝑦2} Γ— {𝑧1, 𝑧2}

𝑦1 𝑦2 𝑧1 𝑧2 𝑦1 𝑦2 𝑧1 𝑧2 + 𝑦1 𝑦2 𝑧1 𝑧2 β‡’ 𝑦1 𝑦2 𝑧1 𝑧2

  • Triangle inequality implies for 𝑁 on 𝑦1, 𝑧2 and (𝑦2, 𝑧1):
  • β„Ž 𝑁𝑦1𝑧2, 𝑁𝑦2𝑧1 ≀ β„Ž 𝑁𝑦1𝑧1, 𝑁𝑦1𝑧2 + β„Ž(𝑁𝑦1𝑧1, 𝑁𝑦2𝑧1)

𝑦1 𝑦2 𝑧1 𝑧2 β‡’ 𝑦1 𝑦2 𝑧1 𝑧2

  • What about 𝑁 on 𝑦1, 𝑧1 and (𝑦2, 𝑧2): ?
  • Cut-and-paste Lemma [BJKS02]: β„Ž 𝑁𝑦1𝑧1, 𝑁𝑦2𝑧2 = β„Ž 𝑁𝑦1𝑧2, 𝑁𝑦2𝑧1
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SLIDE 17

Warm-up: Disjointness

CC πΈπ‘—π‘‘π‘˜π‘œ β‰₯ 𝐽𝐷0 πΈπ‘—π‘‘π‘˜π‘œ = π‘œ 𝐽𝐷0 𝐡𝑂𝐸 [BJKS02] 𝐽𝐷 ≀ 𝐷𝐷, 𝐽𝐷 satisfies direct sum property, needs private and public randomness 𝐽𝐷0(𝐡𝑂𝐸) =

2 3 𝐽 π‘Œ: 𝑁 𝑍 = 0 + 2 3 𝐽(𝑍: 𝑁|π‘Œ = 0)

Comparing message transcript 𝑁 on 01, 00, 10 inputs:

1 12 β„Ž2 𝑁10, 𝑁01 ≀ 𝐽𝐷0(𝐡𝑂𝐸)

Comparing 𝑁 on 00, 11 inputs:

1 12 β„Ž2 𝑁00, 𝑁11 = 1 12 β„Ž2 𝑁10, 𝑁01 ≀ 𝐽𝐷0(𝐡𝑂𝐸)

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SLIDE 18

Warm-up: Disjointness

CC πΈπ‘—π‘‘π‘˜π‘œ β‰₯ 𝐽𝐷0 πΈπ‘—π‘‘π‘˜π‘œ = π‘œ 𝐽𝐷0 𝐡𝑂𝐸 [BJKS02] 𝐽𝐷 ≀ 𝐷𝐷, 𝐽𝐷 satisfies direct sum property, needs private and public randomness 𝐽𝐷0(𝐡𝑂𝐸) =

2 3 𝐽 π‘Œ: 𝑁 𝑍 = 0 + 2 3 𝐽(𝑍: 𝑁|π‘Œ = 0)

Comparing message transcript 𝑁 on 01, 00, 10 inputs:

1 12 β„Ž2 𝑁10, 𝑁01 ≀ 𝐽𝐷0(𝐡𝑂𝐸)

Comparing 𝑁 on 00, 11 inputs:

1 12 β„Ž2 𝑁00, 𝑁11 = 1 12 β„Ž2 𝑁10, 𝑁01 ≀ 𝐽𝐷0(𝐡𝑂𝐸)

Statistical interpretation: β„Ž 𝑁1, 𝑁2 β‰₯

1 4 | 𝑁1 βˆ’ 𝑁2 |π‘ˆπ‘Š

||𝑁00 βˆ’ 𝑁11||π‘ˆπ‘Š ∈ Ξ© 1 since for AND, 𝑁00 β†’ 0, 𝑁11 β†’ 1 𝐷𝐷 πΈπ‘—π‘‘π‘˜π‘œ ∈ Ξ©(π‘œ) Quantum?

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SLIDE 19

Warm-up: Disjointness

𝑅𝐷𝐷 πΈπ‘—π‘‘π‘˜π‘œ ∈ Θ( π‘œ) 𝑅𝐷𝐷𝑠 πΈπ‘—π‘‘π‘˜π‘œ ∈ O(

π‘œ 𝑠), r round protocols

QCCr πΈπ‘—π‘‘π‘˜π‘œ β‰₯

1 𝑠 ΰ·ͺ

𝑅𝐽𝐷0

𝑠 πΈπ‘—π‘‘π‘˜π‘œ = 1 𝑠 π‘œ ΰ·ͺ

𝑅𝐽𝐷0

𝑠 𝐡𝑂𝐸 [JRS03]

ΰ·ͺ 𝑅𝐽𝐷 ≀ 𝑠 𝑅𝐷𝐷, ΰ·ͺ 𝑅𝐽𝐷 satisfies direct sum property, requires private and public β€œrandomness” Comparing 𝑁 on 01, 00, 10 inputs: ?

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SLIDE 20

Tool: Quantum Average Encoding Theorem

  • Average encoding theorem [KNTZ01, JRS03]: Eπ‘Œ β„Ž2 𝜍𝐢

π‘Œ, 𝜍𝐢

≀ 𝐽 π‘Œ: 𝐢 𝜍

  • πœπ‘ŒπΆ = σ𝑦 π‘žπ‘Œ 𝑦 𝑦 βŒ©π‘¦|π‘Œ βŠ— 𝜍𝐢

𝑦

  • 𝜍𝐢 = Eπ‘Œ[𝜍𝐢

π‘Œ], average state

  • h2 𝜏, πœ„ = 1 βˆ’ 𝐺(𝜏, πœ„), Bures distance, with 𝐺 𝜏, πœ„ = ||

𝜏 πœ„||1

  • Follows from Pinsker’s inequality
  • For AND: 1

4 β„Ž2 πœπΆπ‘—π·π‘— 00 , πœπΆπ‘—π·π‘— 10

≀ 𝐽 π‘Œ: 𝐢𝑗𝐷𝑗 𝑍 = 0 πœπ‘— = ΰ·ͺ 𝑅𝐽𝐷𝑗, odd i

  • Similarly, for X=0: 1

4 β„Ž2 πœπ΅π‘—π·π‘— 00 , πœπ΅π‘—π·π‘— 01

≀ 𝐽 𝑍: 𝐡𝑗𝐷𝑗 π‘Œ = 0 πœπ‘— = ΰ·ͺ 𝑅𝐽𝐷𝑗 , even i

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SLIDE 21

Warm-up: Disjointness

𝑅𝐷𝐷 πΈπ‘—π‘‘π‘˜π‘œ ∈ Θ( π‘œ) 𝑅𝐷𝐷𝑠 πΈπ‘—π‘‘π‘˜π‘œ ∈ O(

π‘œ 𝑠), r round protocols

QCCr πΈπ‘—π‘‘π‘˜π‘œ β‰₯

1 𝑠 ΰ·ͺ

𝑅𝐽𝐷0

𝑠 πΈπ‘—π‘‘π‘˜π‘œ = 1 𝑠 π‘œ ΰ·ͺ

𝑅𝐽𝐷0

𝑠 𝐡𝑂𝐸 [JRS03]

ΰ·ͺ 𝑅𝐽𝐷 ≀ 𝑠 𝑅𝐷𝐷, ΰ·ͺ 𝑅𝐽𝐷 satisfies direct sum property, requires private and public β€œrandomness” Comparing 𝑁 on 01, 00, 10 inputs:

  • 1

4 β„Ž2 πœπΆπ‘—π·π‘— 00 , πœπΆπ‘—π·π‘— 10

≀ ΰ·ͺ 𝑅𝐽𝐷𝑗 , odd i

  • 1

4 β„Ž2 πœπ΅π‘—π·π‘— 00 , πœπ΅π‘—π·π‘— 01

≀ ΰ·ͺ 𝑅𝐽𝐷𝑗 , even i

  • ΰ·ͺ

𝑅𝐽𝐷0

𝑠 𝐡𝑂𝐸 = σ𝑗 ΰ·ͺ

𝑅𝐽𝐷𝑗

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SLIDE 22

Warm-up: Disjointness

𝑅𝐷𝐷 πΈπ‘—π‘‘π‘˜π‘œ ∈ Θ( π‘œ) 𝑅𝐷𝐷𝑠 πΈπ‘—π‘‘π‘˜π‘œ ∈ O(

π‘œ 𝑠), r round protocols

QCCr πΈπ‘—π‘‘π‘˜π‘œ β‰₯

1 𝑠 ΰ·ͺ

𝑅𝐽𝐷0

𝑠 πΈπ‘—π‘‘π‘˜π‘œ = 1 𝑠 π‘œ ΰ·ͺ

𝑅𝐽𝐷0

𝑠 𝐡𝑂𝐸 [JRS03]

ΰ·ͺ 𝑅𝐽𝐷 ≀ 𝑠 𝑅𝐷𝐷, ΰ·ͺ 𝑅𝐽𝐷 satisfies direct sum property, requires private and public β€œrandomness” Comparing 𝑁 on 01, 00, 10 inputs:

  • 1

4 β„Ž2 πœπΆπ‘—π·π‘— 00 , πœπΆπ‘—π·π‘— 10

≀ ΰ·ͺ 𝑅𝐽𝐷𝑗 , odd i

  • 1

4 β„Ž2 πœπ΅π‘—π·π‘— 00 , πœπ΅π‘—π·π‘— 01

≀ ΰ·ͺ 𝑅𝐽𝐷𝑗 , even i

  • ΰ·ͺ

𝑅𝐽𝐷0

𝑠 𝐡𝑂𝐸 = σ𝑗 ΰ·ͺ

𝑅𝐽𝐷𝑗

Comparing 𝑁 on 00, 11 inputs: ?

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SLIDE 23

Tool: Quantum Cut-and-Paste Lemma

  • Variant of a tool developed in [JRS03, JN10]
  • Consider input subset {𝑦1, 𝑦2} Γ— {𝑧1, 𝑧2}
  • Lemma: If

for odd i and for even i, then β„Ž π‘Š

𝐡𝑒 𝑦1→𝑦2π‘Š 𝐢𝑒 𝑧1→𝑧2(πœπ΅π‘’πΆπ‘’π·π‘’ 𝑒, 𝑦1𝑧1), πœπ΅π‘’πΆπ‘’π·π‘’ 𝑒, 𝑦2𝑧2

≀ 2 Οƒπ‘˜β‰€π‘’ πœ€

π‘˜

𝑦1, 𝑦2 𝑧1 β„Ž πœπΆπ‘—π·π‘—,

𝑗, 𝑦1𝑧1πœπΆπ‘—π·π‘— 𝑗, 𝑦2𝑧1

= πœ€π‘— 𝑦1 𝑧1, 𝑧2 β„Ž πœπ΅π‘—π·π‘—,

𝑗, 𝑦1𝑧1πœπ΅π‘—π·π‘— 𝑗, 𝑦1𝑧2

= πœ€π‘—

slide-24
SLIDE 24

Tool: Quantum Cut-and-Paste Lemma

  • Consider input subset {𝑦1, 𝑦2} Γ— {𝑧1, 𝑧2}

𝑦1 𝑦2 𝑧1 𝑧2 𝑦1 𝑦2 𝑧1 𝑧2 + 𝑦1 𝑦2 𝑧1 𝑧2 β‡’ 𝑦1 𝑦2 𝑧1 𝑧2

  • Hybrid argument (up to correction unitaries π‘Š

𝐡𝑒 𝑦1→𝑦2, π‘Š 𝐢𝑒 𝑧1→𝑧2, with previous round dependence)

𝑦1 𝑦2 𝑧1 𝑧2 𝑦1 𝑦2 𝑧1 𝑧2 𝑦1 𝑦2 𝑧1 𝑧2 , , , .

slide-25
SLIDE 25

Warm-up: Disjointness

𝑅𝐷𝐷 πΈπ‘—π‘‘π‘˜π‘œ ∈ Θ( π‘œ) 𝑅𝐷𝐷𝑠 πΈπ‘—π‘‘π‘˜π‘œ ∈ O(

π‘œ 𝑠), r round protocols

QCCr πΈπ‘—π‘‘π‘˜π‘œ β‰₯

1 𝑠 ΰ·ͺ

𝑅𝐽𝐷0

𝑠 πΈπ‘—π‘‘π‘˜π‘œ = 1 𝑠 π‘œ ΰ·ͺ

𝑅𝐽𝐷0

𝑠 𝐡𝑂𝐸 [JRS03]

ΰ·ͺ 𝑅𝐽𝐷 ≀ 𝑠 𝑅𝐷𝐷, ΰ·ͺ 𝑅𝐽𝐷 satisfies direct sum property, requires private and public β€œrandomness” Comparing 𝑁 on 01, 00, 10 inputs:

  • 1

4 β„Ž2 πœπΆπ‘—π·π‘— 00 , πœπΆπ‘—π·π‘— 10

≀ ΰ·ͺ 𝑅𝐽𝐷𝑗 , odd i

  • 1

4 β„Ž2 πœπ΅π‘—π·π‘— 00 , πœπ΅π‘—π·π‘— 01

≀ ΰ·ͺ 𝑅𝐽𝐷𝑗 , even i

  • ΰ·ͺ

𝑅𝐽𝐷0

𝑠 𝐡𝑂𝐸 = σ𝑗 ΰ·ͺ

𝑅𝐽𝐷𝑗

Comparing 𝑁 on 00, 11 inputs:

1 4 β„Ž2 πœπ·π‘— 00, πœπ·π‘— 11 ≀ 4 𝑠 σ𝑗 ΰ·ͺ

𝑅𝐽𝐷𝑗= 4 r ΰ·ͺ 𝑅𝐽𝐷0 𝐡𝑂𝐸

slide-26
SLIDE 26

Warm-up: Disjointness

𝑅𝐷𝐷 πΈπ‘—π‘‘π‘˜π‘œ ∈ Θ( π‘œ) 𝑅𝐷𝐷𝑠 πΈπ‘—π‘‘π‘˜π‘œ ∈ O(

π‘œ 𝑠), r round protocols

QCCr πΈπ‘—π‘‘π‘˜π‘œ β‰₯

1 𝑠 ΰ·ͺ

𝑅𝐽𝐷0

𝑠 πΈπ‘—π‘‘π‘˜π‘œ = 1 𝑠 π‘œ ΰ·ͺ

𝑅𝐽𝐷0

𝑠 𝐡𝑂𝐸 [JRS03]

ΰ·ͺ 𝑅𝐽𝐷 ≀ 𝑠 𝑅𝐷𝐷, ΰ·ͺ 𝑅𝐽𝐷 satisfies direct sum property, requires private and public β€œrandomness” Comparing 𝑁 on 01, 00, 10 inputs:

  • 1

4 β„Ž2 πœπΆπ‘—π·π‘— 00 , πœπΆπ‘—π·π‘— 10

≀ ΰ·ͺ 𝑅𝐽𝐷𝑗 , odd i

  • 1

4 β„Ž2 πœπ΅π‘—π·π‘— 00 , πœπ΅π‘—π·π‘— 01

≀ ΰ·ͺ 𝑅𝐽𝐷𝑗 , even i

  • ΰ·ͺ

𝑅𝐽𝐷0

𝑠 𝐡𝑂𝐸 = σ𝑗 ΰ·ͺ

𝑅𝐽𝐷𝑗

Comparing 𝑁 on 00, 11 inputs:

1 4 β„Ž2 πœπ·π‘— 00, πœπ·π‘— 11 ≀ 4 𝑠 σ𝑗 ΰ·ͺ

𝑅𝐽𝐷𝑗= 4 r ΰ·ͺ 𝑅𝐽𝐷0 𝐡𝑂𝐸 Gives ΰ·ͺ 𝑅𝐽𝐷0

𝑠 𝐡𝑂𝐸 ∈ Ξ© 1 𝑠 , 𝑅𝐷𝐷𝑠 πΈπ‘—π‘‘π‘˜π‘œ ∈ Ξ©( π‘œ 𝑠2 + 𝑠) [JRS03]

slide-27
SLIDE 27

Augmented Index

𝑦1 𝑦2 …

π‘¦π‘—βˆ’1

𝑦𝑗 …

π‘¦π‘œβˆ’1

π‘¦π‘œ 𝑦1 𝑦2 …

π‘¦π‘—βˆ’1

𝑦 𝑧 𝑗 ∈ [π‘œ] b∈ {0, 1} 𝑦𝑗 βŠ• 𝑐

π΅π½π‘œ 𝑦1 … π‘¦π‘œ, 𝑗, 𝑦1 … π‘¦π‘—βˆ’1, 𝑐 = 𝑦𝑗 βŠ• 𝑐 𝜈0: uniform distribution on 𝑦𝑗 βŠ• 𝑐 = 0 Either 𝐽𝐷𝐢→𝐡 Ξ , 𝜈0 ∈ Ξ©(1)

slide-28
SLIDE 28

Augmented Index

π΅π½π‘œ 𝑦1 … π‘¦π‘œ, 𝑗, 𝑦1 … π‘¦π‘—βˆ’1, 𝑐 = 𝑦𝑗 βŠ• 𝑐 𝜈0: uniform distribution on 𝑦𝑗 βŠ• 𝑐 = 0 Either 𝐽𝐷𝐢→𝐡 Ξ , 𝜈0 ∈ Ξ©(1) or 𝐽𝐷

𝐡→𝐢 Ξ , 𝜈0 ∈ Ξ©(π‘œ)

𝑦1 𝑦2 …

π‘¦π‘—βˆ’1

𝑦𝑗 …

π‘¦π‘œβˆ’1

π‘¦π‘œ 𝑦1 𝑦2 …

π‘¦π‘—βˆ’1

𝑦 𝑧 𝑗 ∈ [π‘œ] b∈ {0, 1} 𝑦𝑗 βŠ• 𝑐

slide-29
SLIDE 29

Augmented Index

Direct sum approach [JN10] Avoid switching Bob’s prefix: π‘¦π‘š for π‘š ∈ [

π‘œ 2 + 1, π‘œ]

𝑦1 … …

π‘¦π‘œ/2

π‘¦π‘œ/2+1

π‘¦π‘š

…

π‘¦π‘œ 𝑦1 𝑦2 …

π‘¦π‘—βˆ’1

𝑦 𝑧 𝑗 ∈ {𝑙, π‘š} b∈ {0, 1} 𝑦𝑗 βŠ• 𝑐

{ {

π‘š ∈ [π‘œ 2 + 1, π‘œ] 𝑙 ∈ [1, π‘œ 2] ΰ·€ π‘¦π‘š

slide-30
SLIDE 30

Augmented Index

Direct sum approach [JN10] Avoid switching Bob’s prefix: π‘¦π‘š for π‘š ∈ [

π‘œ 2 + 1, π‘œ]

𝑦1 … …

π‘¦π‘œ/2

π‘¦π‘œ/2+1

π‘¦π‘š

…

π‘¦π‘œ 𝑦1 𝑦2 …

π‘¦π‘—βˆ’1

𝑦 𝑧 𝑗 ∈ {𝑙, π‘š} b∈ {0, 1} 𝑦𝑗 βŠ• 𝑐

{ {

π‘š ∈ [π‘œ 2 + 1, π‘œ] 𝑙 ∈ [1, π‘œ 2]

Bob’s one bit input: 𝑑

  • 𝑑 = 0 ↔ 𝑗 = 𝑙, 𝑐 = 𝑦𝑙
  • 𝑑 = 1 ↔ 𝑗 = π‘š, 𝑐 = ΰ·€

π‘¦π‘š

Alice’s one bit input: 𝑏

  • π‘¦π‘š = ΰ·€

π‘¦π‘š βŠ• 𝑏

Output 𝑦𝑗 βŠ• 𝑐 = 1 ↔ 𝑏, 𝑑 = (1, 1)

ΰ·€ π‘¦π‘š 𝑦𝑙

slide-31
SLIDE 31

Augmented Index

Classical: Uses Average Encoding Theorem and Cut-and-Paste Lemma Quantum?

𝑦1 … …

π‘¦π‘œ/2

π‘¦π‘œ/2+1

π‘¦π‘š

…

π‘¦π‘œ 𝑦1 𝑦2 …

π‘¦π‘—βˆ’1

𝑦 𝑧 𝑗 ∈ {𝑙, π‘š} b∈ {0, 1} 𝑦𝑗 βŠ• 𝑐

{ {

π‘š ∈ [π‘œ 2 + 1, π‘œ] 𝑙 ∈ [1, π‘œ 2]

Bob’s one bit input: 𝑑

  • 𝑑 = 0 ↔ 𝑗 = 𝑙, 𝑐 = 𝑦𝑙
  • 𝑑 = 1 ↔ 𝑗 = π‘š, 𝑐 = ΰ·€

π‘¦π‘š

Alice’s one bit input: 𝑏

  • π‘¦π‘š = ΰ·€

π‘¦π‘š βŠ• 𝑏

Output 𝑦𝑗 βŠ• 𝑐 = 1 ↔ 𝑏, 𝑑 = (1, 1)

ΰ·€ π‘¦π‘š 𝑦𝑙

slide-32
SLIDE 32

Augmented Index

  • ΰ·ͺ

𝑅𝐽𝐷𝐡→𝐢 and ΰ·ͺ 𝑅𝐽𝐷𝐢→𝐡 allows for reduction between DYCK(2) and Augmented Index [JN10]

  • But Quantum Average Encoding Theorem + Cut-and-Paste approach breaks down
  • ΰ·ͺ

𝑅𝐽𝐷𝐡→𝐢 and ΰ·’ 𝑅𝐽𝐷𝐢→𝐡 allows for Quantum Average Encoding Theorem + Cut-and-Paste [JN10]

  • But reduction between DYCK(2) and Augmented Index breaks down
  • ΰ·’

𝑅𝐽𝐷𝐢→𝐡 does not satisfy direct sum, ΰ·’ 𝑅𝐽𝐷𝐢→𝐡 ≀ 𝑅𝐷𝐷

  • Looking for alternative QIC that trades-off between these
  • Subtle issues about dealing with distributions vs. dealing with superpositions, private randomness
slide-33
SLIDE 33

Superposition vs. Distribution

  • Input distribution: πœπ‘Œπ‘

𝜈 = σ𝑦𝑧 𝜈 𝑦, 𝑧

𝑦𝑧 βŒ©π‘¦π‘§|π‘Œπ‘

  • Superposition: 𝜍𝜈 π‘Œπ‘π‘†π‘Œπ‘†π‘ = σ𝑦𝑧

𝜈 𝑦, 𝑧 𝑦𝑧𝑦𝑧 π‘Œπ‘π‘†π‘Œπ‘†π‘

  • π‘ˆπ‘ π‘†π‘Œπ‘†π‘ πœπ‘Œπ‘π‘†π‘Œπ‘†π‘

𝜈

= πœπ‘Œπ‘

𝜈

  • Superposition can leak information

π‘†π‘Œπ‘†π‘

slide-34
SLIDE 34

Superposition vs. Distribution

  • Extension of Input distribution: ΰ·€

πœπ‘Œπ‘πΉ

𝜈

= σ𝑦𝑧𝑓 ΰ·€ 𝜈 𝑦, 𝑧, 𝑓 𝑦𝑧𝑓 βŒ©π‘¦π‘§π‘“|π‘Œπ‘πΉ s.t. π‘ˆπ‘ πΉ ΰ·€ πœπ‘Œπ‘πΉ

𝜈

= πœπ‘Œπ‘

𝜈

  • Alternative Superposition: ΰ·€

𝜍𝜈 π‘Œπ‘π‘†π‘Œπ‘†π‘π‘†πΉ1𝑆𝐹2 = σ𝑦𝑧𝑓 ΰ·€ 𝜈 𝑦, 𝑧, 𝑓 𝑦𝑧𝑦𝑧𝑓𝑓 π‘Œπ‘π‘†π‘Œπ‘†π‘π‘†πΉ1𝑆𝐹2

  • π‘ˆπ‘ π‘†π‘Œπ‘†π‘π‘†πΉ2 ΰ·€

πœπ‘Œπ‘π‘†π‘Œπ‘†π‘π‘†πΉ1𝑆𝐹2

𝜈

= 𝐽𝐹→𝑆𝐹1[ ΰ·€ πœπ‘Œπ‘πΉ

𝜈

]

  • Isometry π‘Š

π‘†π‘Œπ‘†π‘β†’π‘†π‘Œπ‘†π‘π‘†πΉ1𝑆𝐹2 maps 𝜍𝜈 to ΰ·€

𝜍𝜈 π‘†π‘Œπ‘†π‘π‘†πΉ1𝑆𝐹2

slide-35
SLIDE 35

Superposition for Augmented Index

  • Augmented Index: 𝑦 = 𝑦1 … π‘¦π‘œ, 𝑧 = (𝑗, 𝑦1 … π‘¦π‘—βˆ’1, 𝑐), 𝑐 = 𝑦𝑗 under 𝜈0
  • Extension: 𝐿 βˆˆπ‘† 1, π‘œ

2 , 𝑀 βˆˆπ‘† π‘œ 2 + 1, π‘œ , 𝐷 βˆˆπ‘† 0,1 , π‘Ž βˆˆπ‘† 0,1 𝑀, 𝑋 βˆˆπ‘† 0,1 π‘œβˆ’π‘€

  • π‘Œ = π‘Žπ‘‹, 𝐽 ∈ 𝐿, 𝑀 according to 𝐷
  • Z = Zβ€² Zβ€²β€² with |π‘Žβ€²| = 𝐿
  • 𝐹 = πΏπ‘€π‘Žπ‘‹π·
  • ΰ·€

𝜍𝜈0 𝑆𝐹1𝑆𝐹2π‘†π‘Œπ‘†π‘π‘Œπ‘ = Οƒπ‘™π‘šπ‘¨π‘₯ … |π‘™π‘™π‘šπ‘šπ‘¨π‘¨π‘₯π‘₯βŒͺ (|00βŒͺ 𝑨π‘₯ π‘˜π‘¨β€² 𝑨π‘₯ π‘Œ π‘˜π‘¨β€² 𝑍 + 11 𝑨π‘₯ π‘šπ‘¨ 𝑨π‘₯ π‘Œ π‘šπ‘¨ 𝑍)

  • How to deal with such superposition?
slide-36
SLIDE 36

Quantum Information Complexity (QIC)

  • QIC(Ξ , 𝜈) = σ𝑗 𝑝𝑒𝑒 𝐽( π‘†π‘Œπ‘†π‘: 𝐷𝑗 𝑍𝐢𝑗 + σ𝑗 π‘“π‘€π‘“π‘œ 𝐽( π‘†π‘Œπ‘†π‘: 𝐷𝑗 π‘Œπ΅π‘—
  • Motivated by quantum state redistribution, with π‘†π‘Œπ‘†π‘ purifying the π‘Œπ‘ input registers: 𝜍𝜈 π‘†π‘Œπ‘Œπ‘†π‘π‘ = σ𝑦,𝑧

𝜈 𝑦, 𝑧 𝑦𝑦𝑧𝑧 π‘†π‘Œπ‘Œπ‘†π‘π‘

  • Invariant under choice of purifying register.. Works also with ΰ·€

𝜍𝜈 π‘Œπ‘π‘†π‘Œπ‘†π‘π‘†πΉ1𝑆𝐹2!

πœ” 𝐡𝐢𝐷𝑆 𝑆 Referee 𝐡 𝐷 𝐢 𝐷 π‘†π‘Œπ‘†π‘

slide-37
SLIDE 37

Quantum Information Complexity (QIC)

  • QIC(Ξ , 𝜈) = σ𝑗 𝑝𝑒𝑒 𝐽( π‘†π‘Œπ‘†π‘: 𝐷𝑗 𝑍𝐢𝑗 + σ𝑗 π‘“π‘€π‘“π‘œ 𝐽( π‘†π‘Œπ‘†π‘: 𝐷𝑗 π‘Œπ΅π‘—
  • Motivated by quantum state redistribution, with π‘†π‘Œπ‘†π‘ purifying the π‘Œπ‘ input registers: 𝜍𝜈 π‘†π‘Œπ‘Œπ‘†π‘π‘ = σ𝑦,𝑧

𝜈 𝑦, 𝑧 𝑦𝑦𝑧𝑧 π‘†π‘Œπ‘Œπ‘†π‘π‘

  • Invariant under choice of purifying register.. Works also with ΰ·€

𝜍𝜈 π‘Œπ‘π‘†π‘Œπ‘†π‘π‘†πΉ1𝑆𝐹2!

  • Properties [T15]:
  • Additivity
  • QIC ≀ QCC
  • QIC allows for reduction between DYCK(2) and Augmented Index
  • What about Quantum Average Encoding Theorem + Cut-and-Paste approach?
  • Not directly.. Superposition-Average Encoding Theorem!
slide-38
SLIDE 38

Recoverability

  • Recoverability [FR15]
  • There exists a recovery map π‘ˆπΆβ†’πΆπ· such that
  • β„Ž2 (πœπ‘†πΆπ·, π‘ˆπΆβ†’πΆπ·(πœπ‘†πΆ)) ≀ 𝐽 𝑆: 𝐷 𝐢 𝜍

πœπ‘†πΆπ· πœπ‘†πΆπ· π‘ˆ 𝑆 𝐷 𝐢 𝑆 𝐷 𝐢 𝐷 𝐢 β‰ˆ πœ” 𝐡𝐢𝐷𝑆 𝑆 Referee 𝐡 𝐷 𝐢 𝐷

slide-39
SLIDE 39

Tool: Superposition-Average Encoding Th.

  • Average encoding theorem [KNTZ01]: Eπ‘Œ β„Ž2 𝜍𝐢

π‘Œ, 𝜍𝐢

≀ 𝐽 π‘Œ: 𝐢 𝜍

  • What about superposition over (part of) X?
  • Recall [FR15]’s recoverability theorem
  • There exists a recovery map π‘ˆπΆβ†’πΆπ· such that β„Ž2(πœπ‘†πΆπ·, π‘ˆπΆβ†’πΆπ·(πœπ‘†πΆ)) ≀ 𝐽 𝑆: 𝐷 𝐢 𝜍
  • Theorem: If for odd i

then β„Ž2 πœπ‘†π‘Œπ‘†π‘π‘πΆπ‘’

𝑔

, πœπ‘†π‘Œπ‘†π‘π‘πΆπ‘’

𝑔

≀ 𝑁 σ𝑗≀𝑒 πœπ‘— 𝐽 π‘†π‘Œπ‘†π‘: 𝐷𝑗 𝑍 𝐢𝑗 = πœπ‘— πœπ‘†π‘Œπ‘†π‘π‘πΆπ‘”

𝑔

πœπ‘†π‘Œπ‘†π‘π‘πΆπ‘”

𝑔

𝑆𝑗: F-R maps

slide-40
SLIDE 40

Quantum Information Complexity (QIC)

  • QIC(Ξ , 𝜈) = σ𝑗 𝑝𝑒𝑒 𝐽( π‘†π‘Œπ‘†π‘: 𝐷𝑗 𝑍𝐢𝑗 + σ𝑗 π‘“π‘€π‘“π‘œ 𝐽( π‘†π‘Œπ‘†π‘: 𝐷𝑗 π‘Œπ΅π‘—
  • Motivated by quantum state redistribution, with π‘†π‘Œπ‘†π‘ purifying the π‘Œπ‘ input registers: 𝜍𝜈 π‘†π‘Œπ‘Œπ‘†π‘π‘ = σ𝑦,𝑧

𝜈 𝑦, 𝑧 𝑦𝑦𝑧𝑧 π‘†π‘Œπ‘Œπ‘†π‘π‘

  • Invariant under choice of purifying register.. Works also with ΰ·€

𝜍𝜈 π‘Œπ‘π‘†π‘Œπ‘†π‘π‘†πΉ1𝑆𝐹2!

  • Properties [T15]:
  • Additivity
  • QIC ≀ QCC
  • QIC allows for reduction between DYCK(2) and Augmented Index
  • What about Quantum Average Encoding Theorem + Cut-and-Paste approach?
  • Superposition average encoding theorem allows to deal with such superpositions
  • How else?
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SLIDE 41

Tool: Information Flow Lemma

  • Lemma [LT17]: 𝐽 𝑆𝐹: π‘Œπ΅π‘”|𝑆𝐺 βˆ’ 𝐽 𝑆𝐹: π‘Œ|𝑆𝐺 = σ𝑗 π‘“π‘€π‘“π‘œ 𝐽 𝑆𝐹: 𝐷𝑗 π‘Œπ΅π‘—π‘†πΊ

βˆ’ σ𝑗 𝑝𝑒𝑒 𝐽 𝑆𝐹: 𝐷𝑗 π‘Œπ΅π‘—π‘†πΊ

  • 𝑆𝐹, 𝑆𝐺 arbitrary quantum extension to inputs π‘Œπ‘
  • ΰ·«

𝑅𝐽𝐷𝑗

β€² = 𝐽(𝑆𝐽𝑆𝐿1𝑆𝐷1: 𝑆𝑋

1𝑆𝑋 2𝑋𝐡𝑗𝐷𝑗 𝑆𝑀1π‘Ž ΰ·₯

πœπ‘— = 𝐽(𝑆𝐽𝑆𝐿1𝑆𝐷1: π‘‹π‘Žπ΅π‘—π·π‘— 𝑆𝑋

1𝑆𝑋 2𝑆𝑀1π‘†π‘Ž1 ΰ·₯

πœπ‘— ≀ 𝑅𝐽𝐷

  • Handles superposition such that it does not leak information about preparation
  • Allows for Average Encoding Theorem

𝑆𝐹 | 𝑆𝐺

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SLIDE 42

Outlook

  • Information-Theoretic Tools for Interactive Quantum Protocols
  • Superposition-average encoding theorem
  • Quantum Cut-and-Paste Lemma
  • Information Flow Lemma to handle superpositions
  • Applications
  • Space lower bound on quantum streaming algorithms for DYCK(2)
  • Streaming with non-unitary operations
  • Quantum information trade-off for Augmented Index
  • Superposition vs. Distributions
  • Open Questions
  • Remove dependence on number of round in Augmented Index trade-off
  • Obtain better quantum round elimination
  • Obtain stronger trade-off, 𝑛 vs.

π‘œ 2𝑛

  • Obtain trade-off for ΰ·«

𝑅𝐽𝐷 𝐢→𝐡 (i.e. only limit how much information about index 𝑗 flows to Alice)

  • Find Further applications of tools…
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SLIDE 43

References

[MMN10] - Magniez, Mathieu and Nayak. STOC'10. [JN10] - Jain and Nayak. ECCC'10, IEEE TIT'14. [CCKM10] - Chakrabarti, Cormode, Kondapally and McGregor. FOCS'10. [LeG06] - Le Gall. SPAA'06. [GKKRdW06] - Gavinsky, Kempe, Kerenidis, Raz and de Wolf. STOC'07. [CK11] - Chakrabarti and Kondapally. APPROX'11/RANDOM'11. [BJKS02] - Bar-Yossef, Jayram, Kumar, and Sivakumar. FOCS'02. [KNTZ01] - Klauck, Nayak, Ta-Shma and Zuckerman. STOC'01. [JRS03] - Jain, Radhakrishnan and Sen. FOCS'03. [T15] - T. STOC'15. [FR15] - Fawzi, Renner. CMP'15. [ML17] - Lauriere, T. ITCS'17.

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SLIDE 44

ASCENSION

[MMN10]