Quantum Error Correction for Long-Distance Quantum Communication - PowerPoint PPT Presentation

Quantum Error Correction for Long-Distance Quantum Communication Institute of Physics, University of Mainz Peter van Loock

Quantum Error Correction for Long-Distance Quantum Communication Institute of Physics, University of Mainz Peter van Loock, Fabian Ewert, Marcel Bergmann

Overview  Old versus New Quantum Repeaters: QED vs. QEC  Photon Loss Codes  Ultrafast Long-Distance Quantum Communication

Overview  Old versus New Quantum Repeaters: QED vs. QEC  Photon Loss Codes  Ultrafast Long-Distance Quantum Communication with Linear Optics

Classification of Quantum Repeaters 1.) Original Quantum Repeaters (Briegel et al., DLCZ,…): use entanglement distribution, swapping, purification (loss, local errors) 2.) Quantum repeaters with purification (loss) and QECC (local errors) 3.) Quantum repeaters with QECC only (loss and local errors) S. Muralidharan, J. Kim, N. Lütkenhaus, M.D. Lukin, and L. Jiang , PRL 112 , 250501 (2014)

Original Quantum Repeaters: Quantum Error Detection for Long-Distance Quantum Communication

Direct Transmission of Flying Qubits      0 1 in L   in out     2            H . c . 1 0 0 1 0 out      / exp L L att

Direct Transmission of Flying Qubits      10 01 in L   in out           1 00 00 in in out           exp / L L F in in att out

Direct Transmission of Flying Qubits      10 01 in L   PS QED in out      PS in in out           PS / exp Tr L L P att succ out

QED on Flying Qubits      10 01 in  ? QED in

QED on Flying Qubits      10 01 in  ? QED in ….need to detect the qubit non-destructively

QED on Flying Qubits Bell measurement detects syndrome and „ recovers “ in one step: no loss = 2-photon detection, photon lost =1-photon detection L L L  0 0 0 in BM       classical channel

QED on Flying Qubits L L L  0 0 0 in BM       classical channel Complications:  on-demand generation of local Bell states  Bell measurement with unit success probability  never beats direct transmission

QED on Flying Qubits L L L  0 0 0 in BM       classical channel       / L L     0 / / exp exp P P L L L L succ 0 att att BM L ( for any ) 0

Original Quantum Repeater Essence of subexponential scaling: some form of quantum error detection and quantum memories

Original Quantum Repeater  distribute known, entangled states  distribute different copies in each segment  QED/entanglement purification  quantum memories  two-way classical communication

With Memories: Quantum Repeater P P distr distr L L 0 0 P swap L log ( / ) L L   2 0 2   log ( 2 / 3 ) P   2 swap Rate ~ ~ / P P L L distr swap 0   3

Without Memories: Quantum Relay P P distr distr L L 0 0 P swap L  /  L L / 1 L L Rate ~ 0 0 P P distr swap

Original Quantum Repeater  Entanglement Distribution  Entanglement Purification (Quantum Error Detection)  Entanglement Swapping  Quantum Memories

DLCZ Quantum Repeater L.M. D uan, M.D. L ukin, J.I. C irac, P. Z oller, Nature 414 , 413 (2001)         $ 0 0 r $ 0 0 r 1 1 1 1    0 1 1 0     ( ) 2 2 ~ ; 1 P r F O r distr

DLCZ Quantum Repeater no-loss space        1100 0011 ( 2 ) 0000 r O r         $ 0 0 r $ 0 0 r 1 1 1 1    0 1 1 0     ( ) 2 2 ~ ; 1 P r F O r distr

DLCZ Quantum Repeater loss space: only QED, not QEC!        2 1100 0011 ( ) 0000 r O r         $ 0 0 r $ 0 0 r 1 1 1 1    0 1 1 0     ( ) 2 2 ~ ; 1 P r F O r distr

Original Quantum Repeater  distribute known, entangled states  distribute different copies in each segment  QED/entanglement purification  quantum memories  two-way classical communication Problems: very slow, limited by CC rates, good memories required ~ 1 / O(poly( )) e.g. 100Hz/1000km L

New Quantum Repeaters: Quantum Error Correction for Long-Distance Quantum Communication

Encoded Quantum Repeaters: Local Errors   0 111 000 1 , etc. 2. Encoded Connection 3. Pauli Frame ~ 1 / O(poly(log ( ) )) L L. Jiang et al., Phys. Rev. A 79 , 032325 (2009); W.J. Munro et al., Nat. Photon. 4, 792 (2010) implementation-independent secret key rates in QKD S. Bratzik, H. Kampermann, and D. Bruß, PRA 89 , 032335 (2014) HQR with encoding N.K. Bernardes and P.v.L ., PRA 86, 052301 (2012)

Encoded Quantum Repeaters: Loss Errors topological surface codes A.G. Fowler et al., Phys. Rev. Lett. 104 , 180503 (2010) parity loss codes W.J. Munro et al., Nature Photon. 6 , 777 (2012) K. Azuma, K. Tamaki, and H.-K. Lo, arXiv: 1309.7207 cluster states and feedforward

Photon Loss Codes  04 40   , 1 22 0 exact Leung‘s bosonic code: 2 Leung‘s AD code: [ 4 , 1 ]  1111 0000 approximate  , 0 2  1100 0011  1 2

Photon Loss Codes Quantum Parity Code (QPC):  n    m m   0 1       ( n , m ) m m m   m   , with , 0 10 1 01 n 2           ( , ) ( , ) ( , ) ( , ) n m n m n m n m     ( n , m ) ( n , m )       , 1 0 2 2 QPC( n,n ) corrects ( n – 1 ) photon losses T.C. Ralph, A.J.F. Hayes, and A. Gilchrist ., PRL 95, 100501 (2005)

Photon Loss Codes QPC(1,1):   ,  ( ) ( ) 1 , 1 1 , 1   01 10 , 01 10 1 ( ) 0 1 , 1   2  10101010 01010101 ( 2 , 2 ) QPC(2,2):  , 0 2  10100101 01011010 ( 2 , 2 )  1 2   C C C (is exact!)  QPC ( 2 , 2 ) [ 4 , 1 ] Dual Rail

Photon Loss Codes Quantum Parity Code (QPC):    1 ... , 1 ...( 1 ) Z Z i n j m  , 1 ij i j m    stabilizers for 1 ...( 1 ) X X i n  ij i 1 , j physical Pauli operators  1 j      ( 1 ) 1 1 n m n nm independent stabilizers QPC(2,2):  , , , , Z Z Z Z X X X X ZZII IIZZ XXXX 11 12 21 22 11 21 12 22 [ 4 , 1 ] like code S. Muralidharan, J. Kim, N. Lütkenhaus, M.D. Lukin, and L. Jiang , PRL 112 , 250501 (2014)

Ultrafast Quantum Communication …replace DR - qubit/Bell states/BM‘s by QPC - encoded qubit/Bell states/BM‘s, use stabilizer formalism and exploit transversality of QPC code as a CSS code L L  L ( n , m ) 0 0 0 in BM  ( n , m ) ( , )  ( n , m )  n m    classical channel S. Muralidharan, J. Kim, N. Lütkenhaus, M.D. Lukin, and L. Jiang , PRL 112 , 250501 (2014)

Ultrafast Quantum Communication  ( n , m ) L L L 0 0 0 in BM  ( n , m ) ( , )  ( n , m )  n m    classical channel … many physical BM‘s for one logical BM via many physical CNOTs and many physical Hadamards: need nonlinear operations, matter-light interactions ,…

Ultrafast Long-Distance Quantum Communication with Linear Optics

Linear-Optics Quantum Communication …replace matter -qubit-based QPC-Bell states by optical QPC-Bell states and nonlinear light-matter interactions by static linear optics L L  L ( n , m ) 0 0 0 in BM  ( n , m ) ( , )  ( n , m )  n m    classical channel

Linear-Optics Quantum Communication …replace matter -qubit-based QPC-Bell states by optical QPC-Bell states and nonlinear light-matter interactions by static linear optics L L  L ( n , m ) 0 0 0 in BM  ( n , m ) ( , )  ( n , m )  n m    classical channel / L L 0     nm nm         nm l l   ( 1 ) P P     succ BM , l       l  0 / exp  L L 0 l att

Linear-Optics Quantum Communication …replace matter -qubit-based QPC-Bell states by optical QPC-Bell states and nonlinear light-matter interactions by static linear optics L L  L ( n , m ) 0 0 0 in BM  ( n , m ) ( , )  ( n , m )  n m    classical channel P What is ? BM , l Can we again exploit „ transversality “?

Linear-Optics Quantum Communication BM of QPC(2,2) encoded Bell states:

Linear-Optics Quantum Communication BM of QPC(2,2) encoded Bell states: