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Noisy Interactive Quantum Communication

Full version: arxiv.org/abs/1309.2643 Dave Touchette touchetd@iro.umontreal.ca

Laboratoire d’informatique th´ eorique et quantique D´ epartement d’informatique et de recherche op´ erationnelle Universit´ e de Montr´ eal Joint work with Gilles Brassard, Ashwin Nayak, Alain Tapp and Falk Unger

6 November 2013, INTRIQ meeting

Dave Touchette (LITQ) Noisy Interactive Quantum Communication 6 November 2013, INTRIQ meeting 1 / 12

Description of the Problem

Goal: simulate interactive quantum protocols over noisy channels

◮ with positive communication rate? ◮ while tolerating positive adversarial error rate? |Ψ A U1 C B C C U2

Alice Bob

U3

...

A B Dave Touchette (LITQ) Noisy Interactive Quantum Communication 6 November 2013, INTRIQ meeting 2 / 12

Noiseless Interactive Quantum Protocols

Well-studied research area: Quantum communication complexity

◮ 2 Models for computing classical f (xA, xB)

|Ψ TA M1 TB C C M2

Alice Bob

M3

...

TA TB

Cleve-Buhrman

Pre-shared Entanglement Classical Communication

|0 A U1 C B C C U2

Alice Bob

U3

...

A B

Yao

No Pre-shared Entanglement Quantum Communication

|0 |0 x x y y x x

Exponential separations in communication complexity

◮ Quantum communication as a resource: classical vs. quantum ◮ Interaction as a resource: N-rounds vs. N+1-rounds Dave Touchette (LITQ) Noisy Interactive Quantum Communication 6 November 2013, INTRIQ meeting 3 / 12

Noisy Quantum Communication

Well-studied for unidirectional data transmission Quantum information theory: Random noise, ` a la Shannon Quantum coding theory: Adversarial noise, ` a la Hamming Transmission rate R = k/n Error rate δ = t/n

... |Ψ E k qubits

Alice Bob

N

Eve

... N N N D ... ... n qubits n qubits k qubits |Ψ ?

Dave Touchette (LITQ) Noisy Interactive Quantum Communication 6 November 2013, INTRIQ meeting 4 / 12

Naive Strategy

Encode each transmission into a QECC Worst case interaction: 1 qubit communication

◮ Random noise: communication rate → 0 ◮ Adversarial noise: tolerable error rate → 0

Classical protocols: positive communication and error rates

|Ψ A U1 C B C C U2

Alice Bob

U3

...

A B E D N

Eve

E D N E D N

Dave Touchette (LITQ) Noisy Interactive Quantum Communication 6 November 2013, INTRIQ meeting 5 / 12

Classical Simulation Protocols

Tree representation for communication protocols

1 1 1 1 1 1 Partial Transcript: 100 ... Final Transcript: 100... 01

Tree codes

◮ Online codes ◮ Self-healing property

Classical strategy: Simulate evolution in protocol tree

◮ If error, go back to last agreement point Dave Touchette (LITQ) Noisy Interactive Quantum Communication 6 November 2013, INTRIQ meeting 6 / 12

Problems for Quantum Simulation

For quantum protocols, no protocol tree to synchronize on

◮ Entanglement between local and communication registers

Cannot restart with a copy of previous state (no-cloning)

◮ Need to rewind unitaries, leading to more errors

|Ψ A U1 C B C C U2

Alice Bob

U3

...

A B

Dave Touchette (LITQ) Noisy Interactive Quantum Communication 6 November 2013, INTRIQ meeting 7 / 12

Problems for Quantum Simulation

Classical information can be copied

◮ Can be resent if destroyed by noise

By no-cloning theorem, quantum information cannot In Cleve-Buhrman model, communication is classical

◮ However, quantum measurements are irreversible

Can we do better than naive strategy?

|Ψ TA M1 TB C C M2

Alice Bob

M3

...

TA TB

Cleve-Buhrman

Pre-shared Entanglement Classical Communication

|0 A U1 C B C C U2

Alice Bob

U3

...

A B

Yao

No Pre-shared Entanglement Quantum Communication

|0 |0 x x y y x x

Dave Touchette (LITQ) Noisy Interactive Quantum Communication 6 November 2013, INTRIQ meeting 8 / 12

Noiseless Teleportation Protocol

Input: |Ψ Bell Measurement

Alice Bob

EPR pair: |Ψ Z X Output: |Ψ

State after Bell measurement: X xZ z |ψ x, z known to Alice, unknown to Bob Alice communicate classical information x, z to Bob

Dave Touchette (LITQ) Noisy Interactive Quantum Communication 6 November 2013, INTRIQ meeting 9 / 12

Solutions to Quantum Simulation Problems

Make everything coherent: measurement → pseudo-measurement Use teleportation to avoid losing quantum information Everything on joint register is a sequence of reversible operations Evolve sequence of noiseless unitaries

|Ψ TA M1 TB C C M2

Alice Bob

M3

...

TA TB

Cleve-Buhrman

Pre-shared Entanglement Classical Communication

|0 A U1 C B C C U2

Alice Bob

U3

...

A B

Yao

No Pre-shared Entanglement Quantum Communication

|0 |0 x x y y x x

Dave Touchette (LITQ) Noisy Interactive Quantum Communication 6 November 2013, INTRIQ meeting 10 / 12

Quantum Simulation Protocol

To distribute EPR pairs, use tools from quantum coding theory For interaction, use tools from classical interactive coding. Classical transcript not sufficient

◮ Contains mostly random teleportation outcomes ◮ Must carefully design classical strategy

1 1 1 1 1 1 ...

Dave Touchette (LITQ) Noisy Interactive Quantum Communication 6 November 2013, INTRIQ meeting 11 / 12

Results and Further Research Directions

3 Noisy communication models

◮ Noisy quantum communication, no shared entanglement ◮ Noisy classical communication, perfect shared entanglement ◮ Noisy classical communication, noisy shared entanglement

Simulations in all 3 models

◮ Positive communication rate ◮ Tolerate positive adversarial error rate ◮ Interactive analog of good quantum code

Tolerate maximal error in perfect shared entanglement model. Upcoming: Adaptation of classical results to quantum realm

◮ Computationally efficient protocols against adversarial noise ◮ Tight characterization of best communication rates for random noise

Open: Develop a fully quantum approach Integration into larger fault-tolerant framework

Dave Touchette (LITQ) Noisy Interactive Quantum Communication 6 November 2013, INTRIQ meeting 12 / 12