Decodiy for surface. Quantum Error Correction. path integral. theory versus. Decoding for GKP code. experiment. Correcting

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1 Quantum Error Correction 1 3X 1100 ( 230 ntro : theory versus experiment Quantum Error i Bosonic Codes # } \ Correcting Conditions stalilizer Codes ( including conthuoqsoyueagjask \ # Decoding Decoding for GKP code : path integral minimum action ( Decodiy for surface GKP code )

2 Short QEC timeline Shor 9 qubit Code 1996 Shor fault tolerant correcting quantum t error computing 1996 Quantum ewor Correction conditions 5 C Knill 1 Laflamme Beane # Divincenzo lsmolin / Wootten ) qusit code correcting Terror 1997 Gottesman lots of theory : Stsilizer codes Kitaeo : toric code 2001 Topological Quantum memory ( Dennis ( Kitaeu Landau 1 Preshill ) { lots of theory

3 Code Experimental Status What has been done ( no bosonic codes on this list ) 3bit repetition code bit repetition code qubit error detection code [[422]] which can detect a single error 5qubit code which corrects a single error [[513]] 7qubit Steane code [[713]] which corrects a single error smallest color code 9qubit surface code (plus 8 ancilla qubits) Early liquidnmr & iontrap qubit experiments Lifetime preservation of NVcenter in diamond qubits (Delft 2015) & superconducting qubits (Google/UCSB 2014) Evidence that 5 qubit code performs better than 3qubit by Google/UCSB BM Quantum Experience (BM & Vuillot 2017) on trap qubits (Maryland/onQ 2017) Not yet Faulttolerant circuits for code are larger ontrap qubits (nnsbruck 2014): logic on encoded block but no evidence for lifetime improvement Planned for superconducting qubits by eg BM Google & Delft (2018+)

4 Theory Versus Experiment : where do your errors from? come 3 p noise Depolarizing pts model : toy to E p ) + Px p Xp X + py + Z pz Z p [ tpt Pi ii X 4 Z general toy model : positigecpyap 1 More 4 p4 S (e) tip Eit E Eitti i S : trace preserving completely ±

5 models Conversion between toy via Pauli twirling 4 P 's ' e) Ppg J pl?$(pippi)pij( : depolarising channel Exercise Hint : use Prove this ps ( as ) ( ) Ps < S ḅ ± Use other Bell States kto )

6 5 Zoo of error strengths : Depot channel : easy probability of error Entanglement fidelity ( # ol 05) ( 140 )( Eol KO ) Diamond Purity norm 11 SK ) Eipeit Write Ei xi t Pi Sho m x Eeise (es)( LEKED HAHtrTrfAAF U 14 ) lose Ei 14) lise $ \* system environment U )E di 14) li )e + / error (4) > A lerror (4) ) 11 : total error amplitude may

7 quit 6 Advantage able to extract : Z X etc spay : Example single channel error amplitude ( biased noise leakage extract Z error ) amplitude Ez Write Ei ; dijpj for i o 3 ( why at most a : 4 4 matrix collect terms which act like 4 or Z on qusit 4 Kraus operators Ei : exercise ) Ez myax 11 ( dizli )E Y + dis lise Z ) t )s 11 Take ts ) leto > so that 4 KO ) 1 Z loto ) kotloo)+h D at M xnt + staff fpj + fpj Pyipz?

8 7 Twirl Channel depolarizing channel ( px Pypz ) of X matrix E P ; ; p PK Ekizl X matrix if E ' twirled channel is diagonal and Pytpztztpypj py Pz i 1431 ' coherent versus errors pytpz stochastic errors

9 Errors can be Correlated in space & time H 0 11 ' Z AH on local depends electric or Time Dynamical magnetic field ( of ) ) o ( Atl alt ' ) ) f ( t t ) Founiertrafo ( ft ' 1) F ( ) w noise spectral density tw ' Sloulg varying Yf noise decoupling techniques more useful than QEC via redundancy spin echo : X R+H)XRzl ) Rtl ) Rz ( )

10 talk 9 Spay D Usual Model 2 qusit fate is assumed to have correlated 2 quit errors e g But interactions happen due to coupling between qlits in real space and resonances in frequency space : cross all over

11 Superconducting Qusits Lectures Divincenzo see 10 ) the in in harmonic Surface ) qwbitt 10 ) Ghsitz harmonic 17 tu Delft Versluis et al

12 Hou well do concepts of fault Work in practice for small codes? µ Example : SWAP Fault tolerant 2 avoiding qusi 't error no cross talk allowed between qclitt 2 & ' But if dummy 3 *3 9ns 't Let a single swap fate fail ie * D say swap l gate PP ' : Paulis the first gate ) only quit t XDor 2 will have Pauli error

13 tolerant But if 11 there is cross Talk then this eh concept may not be useful direct SWAP between quit T and 2 takes T SWAP gate Two gait error rate P fault SWAP takes 3 SWAP fates and ancilla qusit it T SWAP gate has cross talk error rate pl } for error on qusit & spectator quit then two qusit error rate is also p