' !" ' !" !* !* !! !! % $ !# ) !# ( !& !& !+' *+!" !"#$#%&'()*+,&-.&"#/-0#.$)(&'#%-1#.&& "#$% & !% !$ !% !$ &+!! #+!* % $ % $ 2-$"&+3+.&)4#5.$&2"-$+&%')*+& ) ( ) ( %+!& !! $+!# ' !" ' !" &7+$2++.&'"#$#%&).,&"+),+/& )+!% (+!$ "#$% & !* !* !! !! % $ !# ) ( !# !& !& !% !% !$ !$ Fault-tolerant quantum computing with color codes Andrew J. Landahl with Jonas T. Anderson and Patrick R. Rice. arXiv:1108.5738 89:;:88& This work was supported in part by the Laboratory Directed Research and Development program at Sandia National Laboratories. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000. <=>88?&@+*#.,&A.$+/.)1#.)(&>#.6+/+.*+&#.&<5).$54&=//#/&>#//+*1#.&
Photos placed in horizontal posi1on with even amount of white space between photos and header Fault-tolerant quantum computing with color codes Andrew J. Landahl with Jonas T. Anderson and Patrick R. Rice. arXiv:1108.5738 12/8/11 This work was supported in part by the Laboratory Directed Research and Development program at Sandia National Laboratories. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000. QEC11: Second Interna1onal Conference on Quantum Error Correc1on
Color codes The 2D topological The three semiregular 2D topological color codes subsystem color code [Bombin & Mar1n‐Delgado, PRL 97 , 180501 (2006)] [Bombin, PRA 80 , 032301 (2010)] 4.8.8 6.6.6 4.6.12 3.4.6.4 S is transversal S is transversal Fewest qubits/distance Checks: X f , Z f Two‐body checks suffice Planar color codes: 3 m corners These codes are naturally suited to 2D quantum technologies in which long‐distance quantum transport is imprac1cal. The 2D surface code has many promising features for fault‐tolerant quantum compu1ng, including a high accuracy threshold and no need for syndrome ancilla dis4lla4on. How do 2D color codes compare? QEC11: Second Interna1onal Conference on Quantum Error Correc1on 3
Control & noise models Control model: (Faulty) gate basis: { I, X, Z, H, S, S † , CNOT , M Z , M X , | 0 � , | + � , | π/ 4 �} • Standard assumpDons: Parallel opera1on, refreshable ancilla, fast classical computa1on, equal‐1me gates. • Locality assumpDons: 2D layout, local quantum processing. • BP channel: Bit‐flip channel B(p) followed by phase‐flip channel Φ(p). DP channel: Applies each two‐qubit (“double Pauli”) product with probability p/16. Noise model: Standard assumpDons: No leakage, reliable classical computa1on. • 1. Circuit‐based noise model Each prepara1on and one‐qubit gate followed by BP ( p ). • Each CNOT gate followed by DP ( p ). • Each measurement preceded by BP ( p ) and result flipped with probability p . • 2. Phenomenological noise model Same, except each syndrome‐bit extrac1on circuit modeled “phenomenologically” as a measurement that • fails with probability p ; ignores noise propaga1on between data and ancilla. Gates only appear in encoded computa1on. 3. Code‐capacity noise model Same as phenomenological model, except syndrome measurements are perfect. • QEC11: Second Interna1onal Conference on Quantum Error Correc1on 4
Decoders & thresholds OpDmal decoder: Returns recovery most likely to succeed given the syndrome. MLE decoder: Returns most likely error that occurred given the syndrome. 7.8% [A] 0.9% [B] [A: Sarvepalli & Raussendorf, arXiv:1111.0831] [B: Fowler, Whiteside, and Hollenberg, arXiv:1110.5133] [See our paper 1108.5738 for other references.] QEC11: Second Interna1onal Conference on Quantum Error Correc1on 5
Syndrome extraction XZ sequenDal schedule XZ interleaved schedule | 0 � ���� ���� ���� ���� ���� ���� ���� ���� �� � � M Z • • • • � �� � | + � • M X • • • ���� ���� ���� ���� ���� ���� ���� ���� Example of error propagaDon • X error on X ‐check bit (red circle) between 1me steps 5 and 6. • Propagates to 3 data errors; detected correctly by 3 Z ‐check bits (yellow) QEC11: Second Interna1onal Conference on Quantum Error Correc1on 6
Decoding Code‐capacity MLE decoder: (Works for all CSS codes.) OpDmizaDon problem Integer program over GF(2) Integer program over the reals min 1 T x z := s + 2 t 1 + 4 t 2 + 8 t 3 � min c T y min x v y := ( x T , t T 1 , t T 2 , t T 3 ) T v sto A y = z sto H x = s mod 2 � x v = s f ∀ f sto x ∈ B n y ∈ B n 1 T , 0 T , 0 T , 0 T � T � c := v ∈ f A := ( H | − 2 I | − 4 I | − 8 I ) x v ∈ B := { 0 , 1 } Phenomenological MLE decoder: (Works for all CSS codes.) Integer program over the reals ∆z := ∆s + 2 t 1 + 4 t 2 + 8 t 3 min c T y y := ( x T data , x T synd , t T 1 , t T 2 , t T 3 ) T sto A y = ∆ z y ∈ B n 1 T , 1 T , 0 T , 0 T , 0 T � T � c := H I H I I Measurement Data A := − 2 I − 4 I − 8 I . ... ... error error H I I QEC11: Second Interna1onal Conference on Quantum Error Correc1on 7
Code-capacity threshold Exact curves found up to d = 7. Monte‐Carlo esDmate for d = 9. � � p (est) 1 − p (est) � p | E | (1 − p ) n −| E | , p fail = = N fail p (est) fail fail fail ) (est) = ( σ 2 fail failing patterns E N N 0.45 d=3 d=5 (a) Threshold by scaling Ansatz fit, d=7 0.4 0.165 d=9 not curve crossing esDmate. 0.16 0.35 [Wang, Harrington, & Preskill, 0.155 Ann. Phys. 303 , 31 (2003)] 0.15 0.3 Theory: • 0.145 P Failure 0.14 0.25 ξ ∼ | p − p c | − ν 0 0.135 p fail = ( p − p c ) d 1 /ν 0 0.2 0.13 (b) 0.125 Fit: • 0.15 0.101 0.103 0.105 0.107 0.109 0.111 0.1422 0.142 p fail = A + B ( p − p c ) d 1 /ν 0 0.1418 0.1 0.1416 0.1414 0.05 0.1412 p th = 10 . 56(1)% 0.141 0.1054 0.1055 0.1056 0.1057 0 0 0.05 0.1 0.15 0.2 N.B. Finite size effects may ma:er. P Error QEC11: Second Interna1onal Conference on Quantum Error Correc1on 8
Phenomenological threshold 0.2 d=5 d=7 d=9 Algorithm 1 : p fail ( p ) by Monte Carlo 4 ( d + 1) 2 − 1. 1: n faces ← 1 0.15 2: for i = 1 to N do P Failure // Generate data and syndrome errors for d time slices. 3: 0.1 for t = 1 to d do 4: for j = 1 to n do 5: E [ t, j ] ← 1 with probability p . // Data errors. 6: end for 7: 0.05 for j = n + 1 to n + 1 + n faces do 8: E [ t, j ] ← 1 with probability p . // Synd. errors. 9: end for 10: 0 end for 0.02 0.025 0.03 0.035 0.04 11: P Error E min ← Decode(Syndrome( E )). // 3D error volume. 12: 0.2 E ′ ← L d=5 t E [ t ] ⊕ E min [ t ]. // 2D error plane. 13: d=7 d=9 E ′ min ← Decode(Syndrome( E ′ )). // Ideal decoding . 0.18 14: i E ′ [ i ] ⊕ E ′ if ( L min [ i ] = 1) then 15: 0.16 N fail ← N fail + 1. 16: end if 17: 0.14 P Failure 18: end for 0.12 19: return p (est) = N fail /N . fail 0.1 0.08 p th = 3 . 05(4)% 0.06 0.04 0.024 0.026 0.028 0.03 0.032 0.034 0.036 P Error QEC11: Second Interna1onal Conference on Quantum Error Correc1on 9
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