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
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?
Outline of This Talk • Basic of quantum information • States • Measurements • Transformations • Quantum network coding • Negative results • Positive results
Mathematical Representation of Quantum Mechanics Representations Quantum mechanics quantum state vector state space vector space measurement projection evolution unitary
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" Implemented by various micro systems: • Nuclear spin of an atom ("up" or "down") • Polarization of the light （"vertical" or "horizontal"） |0>+|1>
Qubit Represented by a unit vector of the two dim. complex-valued vector space (probability) amplitude orthogonal basis state ket and represent the same state If a, b are complex, it is identified as a vector on the unit ball, called "Bloch sphere" b a If a, b are real, a state is represented in 2D plane
To Get Classical Information Measurement is represented by a set of projections After the measurement Measurement 0 prob. The state becomes 1 prob. projection on The state becomes Measurement b b a a
Multiple Qubits A m-qubit state is in the 2m dim. vector space that is the tensor product of m 2-dim. spaces. 2-qubit state After the measurement Measurement prob. The state becomes normalization
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 R Q
Result 0 (prob. 2/3) Result 1 (prob. 1/3) Partial Measurement State after getting 0: ＝ 2-qubit state Measure 2nd qubit
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. • Important quantum gates • Hadamard transformation • Controlled NOT • (Controlled) Phase-shift Quantum gate
Hadamard Transformation matrix representation Hadamard gate Hadamard gate is essential to prepare the uniform superposed state of all n classical bits Apply H for each qubit n qubits
Controlled NOT Controlled NOT gate matrix representation target part control 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) Any function f computed efficiently by a classical computer can be computed efficiently using only Toffoli gates, which means that the transformation Basic Fact can be efficiently implemented by quantum computing
Phase-Shift Phase-shift gate matrix representation 1/√2 It does not change classical states but change superposed states! 1/√2 Apply Z at 2nd qubit
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)
control part target part Ex.1: Controlled Phase-Shift Controlled phase-shift gate matrix representation target part control part CZ can be implemented using Toffoli and Hadamard. input H on the 3rd qubit Toffoli output Similarly, is efficiently implementable
A input Ex.2: Quantum Teleportation 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 pre-shared state (prior "entanglement") B A B output only local operation & classical communication
pre-shared A B A B local operation & classical communication input output Quantum Teleportation: A's Local Transformation target part A applies CN control part A applies Hadamard on 1st qubit
pre-shared A B A B local operation & classical communication input output Quantum Teleportation: A's Measurement A measures the two qubits
pre-shared A B A B local operation & classical communication input output Quantum Teleportation: Classical Communication + B's Local Transformation A sends 2bits c and d output target part control part B applies CN B applies Controlled phase-shift control part target part
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 (s1,t1),...,(sk,tk) where each sj has a message xj An instance is solvable if there is a network coding protocol which sends xj from sj to tj for every j. • For simplicity, we assume that the alphabet is binary
Revisiting Butterfly [Ahlswede-Li-Cai-Yeung 2000] y x y x POINT 1 Information can be copied y x x ⊕y POINT 2 Information can be encoded x ⊕y x ⊕y y = x ⊕ x ⊕ y x = y ⊕ x ⊕ y
Quantum Butterfly |Ψ2> |Ψ1> • 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 POINT 1 Quantum information cannot be copied POINT 2 How quantum information is encoded? Q.If an instance is classically solvable, is quantum also?
Quantum Information cannot be Copied (No-cloning theorem: Wootters-Zurek) An "unknown" quantum state cannot be cloned. X
Multiple Unicast is a Natural Target Multicast Multiple Unicast |φ1>, |φ2>, |φ3> |φ1> |φ2> |φ1>, |φ2>, |φ3> |φ1>, |φ2>, |φ3> |φ1>, |φ2>, |φ3> |φ1> |φ2> c.f. Shi-Soljanin 2006, Kobayashi et al. 2010
How Quantum Information is Encoded? We cannot whether b=0 or b=1 when U|Ψ1>＝V |Ψ1> |Ψ1> b |Ψ1> b Apply U if b=0, V if b=1 Sending |Ψ1> and b simultaneously seems to be impossible
Negative Results １qubit １qubit m qubits m qubits |φ1> |φ2> |φ1> |φ2> one shot Use of network n times |φ1> |φ1> |φ2> |φ2> Unsolvable under a single use of the network (one shot) [Hayashi-Iwama-N-Raymond-Yamashita07] Unsolvable even asymptotically, i.e., under the condition m/n goes to 1 asymptotically [Hayashi07, Leung-Oppenheim-Winter10]
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?
Our Question Q.If an instance is classically solvable, then is quantum also under free classical communication? Quantum Classical sources sources quantum channel message is quantum free classical communication solvable! solvable! classical channel message is classical ? sinks sinks
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]
Idea of Our Protocol • Our protocol consists of three stages • Node-by-node simulation • 1 qubit for each edge • Removal of internal registers • 1 bit backward for each edge • Removal of initial registers • 1bit forward for each edge
Stage1: Node-by-node Simulation 1 qubit for each edge Classical Quantum um u2 um u2 u1 Pm P2 u1 P1 v v R R w w 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.
Stage1: Node-by-node Simulation P1 P2 R2 R4 R1 R3 R2 R4 R5 R5 R6 R7 R1 R6 R7 R3 Q2 Q1
Stage2: Removal of Internal Registers 1 bit backward for each edge 2. Do the following for internal registers in the inverse topological order 2-1. Apply the Hadamard transformation on R, and measure it. v P1, P2,..., Pm R w 2-2. Send the measurement value backward 2-3. Erase ''phase error" using phase-shift transformation After 2-1 After 2-3 Ignore R since no correlation with other registers!!
Stage2: Removal of Internal Registers 2-1. Apply the Hadamard transformation on R7, and measure it P1 P2 R2 R4 2-2. Send the measurement value backward. 2-3. Erase phase error by R5 R1 R6 R7 R3 Q2 Q1 After 2-1 After 2-3 ignore this
Stage2: Removal of Internal Registers 2-1. Apply the Hadamard transformation on R6, and measure it P1 P2 R2 R4 2-2. Send the measurement value backward. 2-3. Erase phase error by R5 R1 R6 R7 R3 Q2 Q1 After 2-1 After 2-3 ignore this
Stage2: Removal of Internal Registers 2-1. Apply the Hadamard transformation on R6, and measure it P1 P2 R2 R4 2-2. Send the measurement value backward. 2-3. Erase phase error by R5 R1 R6 R7 R3 Q2 Q1 After 2-1 After 2-3 By continuing these, we have
Stage3: Removal of Initial Registers 1 bit forward for each edge After Stage 2 P1 P2 Pk sources 3-1. Apply the Hadamard transformations on the initial registers P1,....,Pk, and measure them. sinks After Stage 3.1 Qk Q1 Q2 ignore these 3-2. Send the measurement values using the classical network coding protocol. 3-3. Erase ''phase error" using phase-shift transformation.
Stage3: Removal of Initial Registers After Stage 2 P1 P2 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. Q2 Q1 Finally, we obtain 3-3. Erase ''phase error" by
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]
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]
[No additional resources] Q. If an instance is quantumly solvable with network coding, then is it solvable with routing? A. Unknown. Yes under only a few special cases [LOW10] [Under free classical communication] [KLNR11] The corresponding quantum instance is quantumly solvable An instance is classically solvable ? Future Work • Generally; • Advantage from classical, say, for security or complexity • Lossy quantum channels • Application (such as wireless communication in classical case), etc.
quantumly solvable since sending two bits backward enables us to reverse the direction of the edge by quantum teleportation!! classically not solvable [Under free classical communication] [Under free classical communication] [KLNR11] [KLNR11] The corresponding quantum instance is quantumly solvable The corresponding quantum instance is quantumly solvable An instance is classically solvable An instance is classically solvable X ? Future Work In case where underlying graphs are directed In case where underlying graphs are undirected