1. Quantum Network Coding Harumichi Nishimura 西村治道 Graduate School of Science, Osaka Prefecture University 大阪府立大学理学系研究科 March 30, 2011, Institute of Network Coding

2. Why Quantum Network Coding? Power of quantum information processing –Fast algorithms Factoring large integers Database search –Secure cryptosystems Unconditionally secure key distribution Public key cryptosystems –Communication-efficient multi-party protocols Communication complexity Leader election  But quantum channel is "expensive" Q. Can we reduce the amount of quantum communication using the idea of network coding?

3. Outline of This Talk Basic of quantum information –States –Measurements –Transformations Quantum network coding –Negative results –Positive results

4. Basic of Quantum Information

5. Mathematical Representation of Quantum Mechanics Quantum mechanics Representations quantum statevector vector space projection unitary state space measurement evolution

6. Qubit Bit ：＝ Basic unit of information and computation Quantum bit (qubit) ：＝ Basic unit of "quantum" information and computation –allows a superposed state of the 2 basis states which corresponds "0'' and "1" |0>+|1> Implemented by various micro systems: Nuclear spin of an atom ("up" or "down") Polarization of the light （ "vertical" or "horizontal" ）

7. Qubit Represented by a unit vector of the two dim. complex-valued vector space ket (probability) amplitude basis state orthogonal a b If a, b are real, a state is represented in 2D plane If a, b are complex, it is identified as a vector on the unit ball, called "Bloch sphere" andrepresent the same state

8. To Get Classical Information Measurement After the measurement 0 1 prob. The state becomes a b Measurement a b Measurement is represented by a set of projections The state becomes projection on

9. Multiple Qubits A m-qubit state is in the 2 m dim. vector space that is the tensor product of m 2-dim. spaces. 2-qubit state Measurement After the measurement prob. The state becomes normalization

10. Entanglement Entanglement is one of important key words in the quantum world, which is a quantum correlation between two (or more) qubit states. Mathematically, a two-qubit state is called entangled if it cannot be "represented" by any tensor product of two single qubits Ex. A quantum state on two quantum registers (=quantum systems) R and Q are called entangled if RQ

11. Partial Measurement 2-qubit state State after getting 0: ＝ Measure 2nd qubit Result 0 (prob. 2/3) Result 1 (prob. 1/3)

12. To Transform Quantum States A measurement collapses a superposition Measurement On the contrary, the quantum mechanics allows us to transform a superposed state into another state, which is mathematically described as a unitary transformation. Quantum gate Important quantum gates Hadamard transformation Controlled NOT (Controlled) Phase-shift

13. Hadamard Transformation Hadamard gate matrix representation Hadamard gate is essential to prepare the uniform superposed state of all n classical bits Apply H for each qubit n qubits

14. Controlled NOT Controlled NOT gate matrix representation control part target part Notice: CN is essentially "classical" transformation since it flips the target part if the control part is 1. But the fact that CN can apply to any superposed state is very important in quantum information processing. Toffoli gate (Controlled-Controlled NOT) Basic Fact Any function f computed efficiently by a classical computer can be computed efficiently using only Toffoli gates, which means that the transformation can be efficiently implemented by quantum computing

15. Phase-Shift Phase-shift gate It does not change classical states but change superposed states! 1/√2 Apply Z at 2nd qubit matrix representation

16. How to Process Quantum Information  Prepare quantum bits as much as you need  Choose a few qubits and apply  unitary transformation  measurement  You can introduce a "fresh" qubit if you want  You can use previous measurement results for choosing your operations  The final result may be classical or quantum (depending on your task)

17. Ex.1: Controlled Phase-Shift Controlled phase-shift gate matrix representation control part target part Similarly, is efficiently implementable CZ can be implemented using Toffoli and Hadamard. H on the 3rd qubit Toffoli input output control parttarget part

18. Ex.2: Quantum Teleportation B pre-shared state (prior "entanglement") AB output A input only local operation & classical communication Quantum teleportation is sending an unknown state from A to B under only local operations and classical communication with the assistance of pre-shared state between A and B

19. Quantum Teleportation: A's Local Transformation control part target part A applies CN A applies Hadamard on 1st qubit AB pre-shared input AB output local operation & classical communication

20. Quantum Teleportation: A's Measurement A measures the two qubits AB pre-shared input AB output local operation & classical communication

21. Quantum Teleportation: Classical Communication + B's Local Transformation A sends 2bits c and d B applies CN target partcontrol part target part B applies Controlled phase-shift output AB pre-shared input AB output local operation & classical communication

22. Quantum Network Coding

23. Network Coding Problems in This Talk  We consider the "solvability."  We consider the multiple unicast problem. Instance: –a directed acyclic graph G=(V,E), where each edge has a unit capacity –k source-target pairs (s 1,t 1 ),...,(s k,t k ) where each s j has a message x j An instance is solvable if there is a network coding protocol which sends x j from s j to t j for every j.  For simplicity, we assume that the alphabet is binary

24. Revisiting Butterfly x y y = x ⊕ x ⊕ y x x y y x ⊕ y x = y ⊕ x ⊕ y POINT 1 Information can be copied POINT 2 Information can be encoded [Ahlswede-Li-Cai-Yeung 2000]

25. Quantum Butterfly |Ψ1>|Ψ1> |Ψ2>|Ψ2> POINT 1 Quantum information cannot be copied Information to be sent is quantum states Quantum operation is possible at each node Every channel is quantum Source nodes may have an entangled state Q. If an instance is classically solvable, is quantum also? POINT 2 How quantum information is encoded?

26. Quantum Information cannot be Copied (No-cloning theorem: Wootters-Zurek) An "unknown" quantum state cannot be cloned. X

27. Multiple Unicast is a Natural Target Multicast Multiple Unicast |φ 1 >, |φ 2 >, |φ 3 > |φ1>|φ1>|φ2>|φ2> |φ2>|φ2> |φ1>|φ1> c.f. Shi-Soljanin 2006, Kobayashi et al. 2010

28. How Quantum Information is Encoded? |Ψ1>|Ψ1> Sending |Ψ 1 > and b simultaneously seems to be impossible b |Ψ1>|Ψ1> b Apply U if b=0, V if b=1 We cannot whether b=0 or b=1 when U|Ψ 1 > ＝ V |Ψ 1 >

29. Negative Results |φ1>|φ1>|φ2>|φ2> |φ2>|φ2> |φ1>|φ1> １ qubit Unsolvable under a single use of the network (one shot) [Hayashi-Iwama-N-Raymond-Yamashita07] |φ1>|φ1>|φ2>|φ2> |φ2>|φ2> |φ1>|φ1> m qubits Unsolvable even asymptotically, i.e., under the condition m/n goes to 1 asymptotically [Hayashi07, Leung-Oppenheim-Winter10] Use of network n timesone shot

30. Under Additional Resources Entanglement –among sources [Hayashi07] –among neighboring nodes [Leung-Oppenheim-Winter10] Classical channel [LOW10, Kobayashi-Le Gall-N-Roetteler09 & 11] –Much cheaper than quantum: LOCC (Local Operation & Classical Communication) is easier than quantum communication. Q. If an instance is classically solvable, then is quantum also under free classical communication?

31. Our Question sources sinks Classical classical channel message is classical Quantum sources sinks quantum channel message is quantum free classical communication solvable! ? Q. If an instance is classically solvable, then is quantum also under free classical communication?

32. Our Result If there is a classical coding protocol for an instance, then there is also a quantum coding protocol for the corresponding quantum instance. [Kobayashi-Le Gall-N-Roetteler 2011] Our previous results was: If there is a classical linear coding protocol for an instance, then there is also a quantum coding protocol for the corresponding quantum instance. [Kobayashi-Le Gall-N-Roetteler 2009]

33. Idea of Our Protocol Our protocol consists of three stages 1.Node-by-node simulation – 1 qubit for each edge 2.Removal of internal registers – 1 bit backward for each edge 3.Removal of initial registers – 1bit forward for each edge

34. Stage1: Node-by-node Simulation u1 u2 um R v Classical Quantum P1 P2 Pm R v 1-1. Receive registers P1,...,Pm 1-2. Introduce fresh register R and apply the unitary transformation on registers P1, P2,..., Pm and R. 1-3. Send R to w. 1 qubit for each edge w u1 u2 um w

35. Stage1: Node-by-node Simulation P1 R2R4 R1 R5 R3 R6 R7 Q2Q1 P2 R2R4 R1 R5 R3R6R7

36. Stage2: Removal of Internal Registers P1, P2,..., Pm R v 2-1. Apply the Hadamard transformation on R, and measure it. 2-2. Send the measurement value backward 2-3. Erase ''phase error" using phase-shift transformation Ignore R since no correlation with other registers!! 2. Do the following for internal registers in the inverse topological order 1 bit backward for each edge w After 2-3 After 2-1

37. Stage2: Removal of Internal Registers P1 R2R4 R1 R5 R3R6R7 Q2Q1 P2 2-1. Apply the Hadamard transformation on R7, and measure it 2-2. Send the measurement value backward. 2-3. Erase phase error by ignore this After 2-3 After 2-1

38. Stage2: Removal of Internal Registers P1 R2R4 R1 R5 R3R6R7 Q2 Q1 P2 2-1. Apply the Hadamard transformation on R6, and measure it 2-2. Send the measurement value backward. 2-3. Erase phase error by ignore this After 2-1 After 2-3

39. Stage2: Removal of Internal Registers P1 R2R4 R1 R5 R3R6R7 Q2 Q1 P2 2-1. Apply the Hadamard transformation on R6, and measure it 2-2. Send the measurement value backward. 2-3. Erase phase error by By continuing these, we have After 2-3 After 2-1

40. Stage3: Removal of Initial Registers 1 bit forward for each edge sources sinks P1P2Pk Q1Q2 Qk After Stage 2 3-1. Apply the Hadamard transformations on the initial registers P1,....,Pk, and measure them. After Stage 3.1 ignore these 3-2. Send the measurement values using the classical network coding protocol. 3-3. Erase ''phase error" using phase-shift transformation.

41. Stage3: Removal of Initial Registers P1 Q2 Q1 P2 After Stage 2 3-1. Apply the Hadamard transformations on P1,P2, and measure them. ignore 3-2. Send the measurement values using the classical network coding protocol. 3-3. Erase ''phase error" by Finally, we obtain

42. Comments for Free Classical Communication KLNR11 reduces the amount of classical communication compared to KLNR09 –k*m*#(node) where m:=max fan-in of all nodes [KLNR09] –1 bit forward +1 bit backward for each edge = total 2*#(edge) [KLNR11] Sending classical bits backward is necessary –Quantum butterfly is not solvable even for the case where free classical communication is allowed in the direction of edges. [Leung-Oppenheim-Winter 2010]

43. Summary No additional assistance –Butterfly is not solvable [HINRY07, LOW10, H07] –Routing is optimal for a few cases [LOW10] 2 source-sink pairs shallow networks (including butterfly) Under free classical communication –If an instance is classically solvable, then the corresponding instance is also quantumly solvable [KLNR11] –Additional classical communication is efficient –Outer/inner bound for a few cases [LOW10]

44. Future Work [No additional resources] Q. If an instance is quantumly solvable with network coding, then is it solvable with routing? [LOW10] A. Unknown. Yes under only a few special cases Generally; Advantage from classical, say, for security or complexity Lossy quantum channels Application (such as wireless communication in classical case), etc. An instance is classically solvable The corresponding quantum instance is quantumly solvable ? [Under free classical communication] [KLNR11]

45. Future Work An instance is classically solvable The corresponding quantum instance is quantumly solvable X [Under free classical communication] [KLNR11] In case where underlying graphs are directed classically not solvable quantumly solvable since sending two bits backward enables us to reverse the direction of the edge by quantum teleportation!! An instance is classically solvable The corresponding quantum instance is quantumly solvable ? [Under free classical communication] [KLNR11] In case where underlying graphs are undirected