Ankit GargPrinceton Univ.Jo int work with

Mark Braverman Young Kun Ko Pr inceton Univ. Pr inceton Univ. J ieming Mao Dave Touchette Pr inceton Univ. Univ. of Waterloo and Perimeter Inst itute

Near Optimal Bounds on Bounded-Rounded Quantum Communication

Complexity of Disjointness

Quantum Communication Complexity

Classical inputs and .Goal: Compute classical function .Communicate using quantum resources.

𝑥 𝑦

Alice Bob

Quantum Protocols

𝑥 𝑦

Alice Bob

Send C1

Quantum Protocols

𝑥 𝑦

Alice Bob

Send C2

Quantum protocols

State after round:

Total rounds.

After round: do binary measurements on partial states and get .

Total communication cost:

Quantum communication complexity [Yao ‘93]

: min communication cost of a quantum protocol that outputs for .

: min communication cost of a r-round quantum protocol that outputs for .

Disjointness

. Alice has X and Bob has Y.

0 else

[Gro `96, BCW `98, AA `03]

0100110110 001100 0001000 0101101000100101001000 0010101110100110100 00101100100 0

Disjointness

Is optimal?

Communication protocol may not look like a query protocol.

[Razborov `02, Sherstov `07]

Stronger evidence of optimality of Grover search.

Disjointness

[BCW,AA] protocols for involve rounds of interaction.

What if only rounds of interaction are allowed?

[Folklore].

0100110110 001100 0001000 0101101000100101001000 0010101110100110100 00101100100 0

𝑟2RoundsCommunication:

Disjointness

[JRS `03].

Theorem: .

Query complexity analogue: [Zalka ‘99]

Quantum Information Theory 101

• Systems with joint state .

•

•

•

• [Lieb-Ruskai ‘73]

Classical Information Cost [CSWY `01, BJKS-04, BBCR-10]

.

A B

𝑋 𝑌Protocol πProtocol transcript

𝐼 (𝜋 ,𝜇)=𝐼 (𝜋 ;𝑌∨𝑋 )+𝐼 (𝜋 ; 𝑋∨𝑌 )What Alice learns about + What Bob learns about

Quantum Information Cost

.

A B

𝑋 𝑌Protocol πQuantum Protocol

Issues:1. No concept of transcript.

2. Reversible computing.

Quantum Information Cost[Touchette `14]

Account for forgetting of information.

Quantum Information Cost[Touchette `14]

State after round:

Information sent by Alice

Information forgotten by Alice

𝐼 (𝑋 ;𝐶𝑖∨𝑌 ,𝐵𝑖)𝜓 𝑖+𝐼 (𝑌 ;𝐶𝑖∨𝑋 ,𝐴𝑖)𝜓𝑖

Quantum Information Complexity

: inf over protocols which compute w.p. at least w.r.t .

[Touchette `14]:

Quantum Information equals Amortized Communication. Quantum analogue of [Braverman-Rao `10].

Understand of .

Understand of .

• has no mass on but correct for all inputs.

What are the hard instances of

0100110111001100010100101011110 011001011010100 01010111010111110 0001011101000

0100110110 001100 0001000 0101101000100101001000 0010101110100110100 00101100100 0

QIC of AND

[Theorem]: Any -round protocol computing AND has

Matching upper bound – related to Elitzur-Vaidman bomb testing problem.

[JRS `03]: Proof can be interpreted to get .

Why is it hard?

Hard to manipulate quantum information. not well understood.Several natural conjectures only recently

resolved [FR `14].Don’t have optimal round elimination

arguments yet.

Proof idea

Go back to !

Protocol for copies of with and success prob

(impossible by direct product theorems [KSW `04, She `12])

Complete reduction

Protocol for copies of with and success prob

What is the need for quantum information theory?

Continuity of QIC

Where does the number of rounds show up?Continuity of QIC.

-round. and differ by at most .Classically pay only !

Conclusion and Open Problems

Direct sum: Does computing require times as much resources as computing

Compression: Can we compress down not-very-informative conversations?

Quantum protocol with and . Simulate with .Classically: [BBCR ‘10] [B ‘12]

Conclusion and Open Problems

[Touchette `14]:

Direct sum Compression.

Open problem 1: .

Open problem 2: .We prove this if the goal is to compute boolean functions.

Thank You