Noisy Interactive Quantum Communication

Transcription

1 Noisy Interactive Quantum ommunication arxiv: Gilles rassard, shwin Nayak, lain Tapp, Dave Touchette and Falk Unger arcelona, QIP 2014 Noisy Interactive Quantum ommunication arcelona, QIP / 20

2 Problem Simulate highly interactive quantum protocols over noisy channels Positive communication rate Positive adversarial error rate U1 U3 lice Ψ ob U2 Noisy Interactive Quantum ommunication arcelona, QIP / 20

3 Problem Simulate highly interactive quantum protocols over noisy channels Positive communication rate Positive adversarial error rate U1 U3 Ψ lice Eve N N N ob U2 touchetd@iro.umontreal.ca Noisy Interactive Quantum ommunication arcelona, QIP / 20

4 Noiseless Interactive Quantum Protocols Well-studied: Quantum communication compleity 2 Models for computing classical f : X Y Z lice ob U1 Yao U2 y No Entanglement Quantum ommunication U3 Ψ leve-uhrman lice ob M1 Entanglement Eponential separations in communication compleity lassical vs. quantum N-rounds vs. N+1-rounds 0 M2 M3 y lassical ommunication touchetd@iro.umontreal.ca Noisy Interactive Quantum ommunication arcelona, QIP / 20

5 Noisy Quantum ommunication Well-studied for unidirectional data transmission lice Eve ob N Ψ E N N D Ψ? N k qudits n qudits n qudits k qudits n channels t errors Quantum information theory: Random noise, à la Shannon ommunication rate R = k/n Quantum coding theory: dversarial noise, à la Hamming Error rate δ = t/n touchetd@iro.umontreal.ca Noisy Interactive Quantum ommunication arcelona, QIP / 20

6 Noisy Interactive Quantum ommunication ommunication rate R = k/n Error rate δ = t/n U1 U3 E1 E3 Ψ lice ob U2 k qudits Ψ lice Eve ob E2 N N N n qudits t errors n channels Noiseless protocol Simulation protocol touchetd@iro.umontreal.ca Noisy Interactive Quantum ommunication arcelona, QIP / 20

7 Naive Strategy Encode each transmission into a QE U1 U3 lice E E E Ψ Eve N N N ob D D D U2 Worst case interaction: 1 qubit communication Random noise: communication rate 0 dversarial noise: tolerable error rate 0 lassical protocols: same problems but [Schulman 96] Simulation protocols with positive communication and error rates touchetd@iro.umontreal.ca Noisy Interactive Quantum ommunication arcelona, QIP / 20

8 Problems for Quantum Simulation lassical information can be copied and resent if destroyed by noise Yao model problem: no-cloning theorem leve-uhrman model: communication is classical Problem: quantum measurements are irreversible an we do better than naive (block coding) strategy? lice ob U1 Yao U2 y No Entanglement Quantum ommunication U3 Ψ leve-uhrman 0 lice ob M1 Entanglement M2 M3 y lassical ommunication touchetd@iro.umontreal.ca Noisy Interactive Quantum ommunication arcelona, QIP / 20

9 Noisy ommunication Models onsider 3 distinct noisy communication models Noisy quantum communication, no shared entanglement Noisy analogue to the Yao model Noisy classical communication, perfect shared entanglement Noisy analogue to the leve-uhrman model Noisy classical communication, noisy shared entanglement Noisy EPR pairs (Werner states) touchetd@iro.umontreal.ca Noisy Interactive Quantum ommunication arcelona, QIP / 20

10 Results Simulations in all 3 models Positive communication rates Yao model: O( 1 Q ) overhead over depolarizing channel leve-uhrman: O( 1 ) overhead over binary symmetric channel Tolerate positive adversarial error rates Yao model: 1 6 ɛ leve-uhrman model: 1 2 ɛ, optimal First interactive analogue of good quantum code touchetd@iro.umontreal.ca Noisy Interactive Quantum ommunication arcelona, QIP / 20

11 Results Noisy entanglement: Simulation for any non-separable Werner state leve-uhrman model: O( 1 ) overhead is optimal Yao model: O( 1 Q ) overhead is not Simulation for some Q = 0 depolarizing channel! touchetd@iro.umontreal.ca Noisy Interactive Quantum ommunication arcelona, QIP / 20

12 Main Ingredient : Teleportation Protocol Input: Ψ ell Measurement z EPR pair: Ψ lice ob Z X Output: Ψ State after ell measurement: X Z z ψ ob decodes with Z z X Obtains ±X + Z z+z ψ Noisy classical communication Pauli error on ψ touchetd@iro.umontreal.ca Noisy Interactive Quantum ommunication arcelona, QIP / 20

13 Solutions to Quantum Simulation Problems leve-uhrman model: Make everything coherent Measurements pseudo-measurements Yao model: Use teleportation to avoid losing quantum information Evolve sequence of noiseless unitaries Everything on joint register is a sequence of reversible operations lice ob U1 Yao U2 y No Entanglement Quantum ommunication U3 Ψ leve-uhrman 0 lice ob M1 Entanglement M2 M3 y lassical ommunication touchetd@iro.umontreal.ca Noisy Interactive Quantum ommunication arcelona, QIP / 20

14 Quantum Simulation Protocol Yao: To distribute EPR pairs, use tools from quantum coding theory For interaction, use tools from classical interactive coding an we use classical simulation protocols: No! lassical goal: lice and ob agree on transcript Here: ontains mostly random teleportation outcomes lice's output ob's output Noisy Interactive Quantum ommunication arcelona, QIP / 20

15 Tools for lassical Simulation Protocols Tree representation for communication protocols Partial Transcript: 100 Tree codes Online codes Self-healing property lueberry codes Randomized error detection codes lassical strategy: Simulate evolution in protocol tree Error go back to last agreement point Noisy Interactive Quantum ommunication arcelona, QIP / 20

16 Further Problems for Quantum Simulation For quantum protocols, no protocol tree to synchronize on an still synchronize on sequential structure of quantum protocol annot restart with a copy of previous state (no-cloning) Need to rewind unitaries, leading to more errors U1 U3 lice Ψ ob U2 touchetd@iro.umontreal.ca Noisy Interactive Quantum ommunication arcelona, QIP / 20

17 lassical Information Sent over Noisy hannel Teleportation measurement outcome : M, z M {0, 1} Teleportation decoding operation : D, z D {0, 1} Direction for evolution of noiseless protocol : M { 1, 0, +1} Inde of noiseless protocol unitary : j [n + 1] Implicit: jl = i<l 2M i + M l (+1 for ob) M=+1, j=15 D z D =11 U 15 U 15-1 lice ZX X ob XZ Z M z M =11 U 16 I touchetd@iro.umontreal.ca Noisy Interactive Quantum ommunication arcelona, QIP / 20

18 Eample Run of Simulation Protocol U 15 I I lice ZX X XZ X ob XZ Z X Z U 16 U 16-1 U 16 touchetd@iro.umontreal.ca Noisy Interactive Quantum ommunication arcelona, QIP / 20

19 onclusion : Summary ommunication compleity robust under noisy communication Tolerate maimal error in perfect shared entanglement model Requires new bound on tree codes Positive communication rates for some Q = 0 depolarizing channel Separation between standard and interactive quantum capacity touchetd@iro.umontreal.ca Noisy Interactive Quantum ommunication arcelona, QIP / 20

20 Further Research Directions daptation of classical results to quantum realm omputationally efficient protocols against adversarial noise High communication rates for low random noise Upper bound on interactive quantum capacity Improve tolerable error rate in quantum model Possibly by developing a fully quantum approach onstruction of quantum tree codes? Integration into larger fault-tolerant framework touchetd@iro.umontreal.ca Noisy Interactive Quantum ommunication arcelona, QIP / 20