Noisy Interactive Quantum Communication Full version: - PowerPoint PPT Presentation

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

Description of the Problem Goal: simulate interactive quantum protocols over noisy channels ◮ with positive communication rate? ◮ while tolerating positive adversarial error rate? A A U1 U3 C ... C Alice C |Ψ  Bob U2 B B 6 November 2013, INTRIQ meeting 2 / Dave Touchette (LITQ) Noisy Interactive Quantum Communication 12

Noiseless Interactive Quantum Protocols Well-studied research area: Quantum communication complexity ◮ 2 Models for computing classical f ( x A , x B ) Yao Cleve-Buhrman x x x x A A TA |0  TA U1 U3 C M1 M3 |0  0 ... Alice C ... C Alice C C |Ψ  Bob Bob B U2 B TB M2 |0  TB y y No Pre-shared Quantum Pre-shared Classical Entanglement Entanglement Communication Communication Exponential separations in communication complexity ◮ Quantum communication as a resource: classical vs. quantum ◮ Interaction as a resource: N-rounds vs. N+1-rounds 6 November 2013, INTRIQ meeting 3 / Dave Touchette (LITQ) Noisy Interactive Quantum Communication 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 Alice Bob Eve N N |Ψ  E D |Ψ  ? ... ... ... ... N N k qubits n qubits n qubits k qubits 6 November 2013, INTRIQ meeting 4 / Dave Touchette (LITQ) Noisy Interactive Quantum Communication 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 A A U1 U3 C C C Alice E E E ... Eve |Ψ  N N N D D D Bob U2 B B 6 November 2013, INTRIQ meeting 5 / Dave Touchette (LITQ) Noisy Interactive Quantum Communication 12

Classical Simulation Protocols Tree representation for communication protocols 0 1 0 1 Partial Transcript: 100 0 1 0 1 ... 0 1 0 1 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 6 November 2013, INTRIQ meeting 6 / Dave Touchette (LITQ) Noisy Interactive Quantum Communication 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 A A U1 U3 C ... Alice C C |Ψ  Bob B U2 B 6 November 2013, INTRIQ meeting 7 / Dave Touchette (LITQ) Noisy Interactive Quantum Communication 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? Yao Cleve-Buhrman x x x x A A TA |0  TA U1 U3 C M1 M3 |0  0 ... Alice C ... C Alice C C |Ψ  Bob Bob B U2 B TB M2 |0  TB y y No Pre-shared Quantum Pre-shared Classical Entanglement Entanglement Communication Communication 6 November 2013, INTRIQ meeting 8 / Dave Touchette (LITQ) Noisy Interactive Quantum Communication 12

Noiseless Teleportation Protocol Bell Input: |Ψ  Measurement Alice EPR pair: |Ψ  Bob Z X Output: |Ψ  State after Bell measurement: X x Z z | ψ � x , z known to Alice, unknown to Bob Alice communicate classical information x , z to Bob 6 November 2013, INTRIQ meeting 9 / Dave Touchette (LITQ) Noisy Interactive Quantum Communication 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 Yao Cleve-Buhrman x x x x A A TA |0  TA U1 U3 C M1 M3 |0  0 ... Alice ... C C Alice C C Bob |Ψ  Bob B U2 B TB M2 |0  TB y No Pre-shared y Quantum Pre-shared Classical Entanglement Communication Entanglement Communication 6 November 2013, INTRIQ meeting 10 Dave Touchette (LITQ) Noisy Interactive Quantum Communication / 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 0 1 0 1 0 1 0 1 ... 0 1 0 1 6 November 2013, INTRIQ meeting 11 Dave Touchette (LITQ) Noisy Interactive Quantum Communication / 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 6 November 2013, INTRIQ meeting 12 Dave Touchette (LITQ) Noisy Interactive Quantum Communication / 12