1. Debbie Leung (IQC, UW) w/ Panos Aliferis, Yingkai Ouyang, Man-Hong Yung (IBM) (IQC, UW) (UIUC) $$: NSERC, CFI, ORF, CIFAR, MITACS, QWorks, ARO, Mike Lazaridis Fellowship(s) Two measurement problems in Fault-tolerant Quantum-Computation Dec 17, 2007. QEC07, USC

2. Measurement problem 1: * In measurement based quantum computation will 1 error in a measurement outcome cascade to many, overwhelming the ability to perform QEC? Measurement problem 2: * NonMarkovian noise is modeled as a system-bath Hamiltonian evolution. Its sup-norm should be sub-threshold for FT method to work. Will a sharp measurement, with an obligatorily large amplitude for the system-bath Hamiltonian, spoil the result? * * * * * * Plan: interpret the problem away, thus, the problem and result are inevitably trivial Disclaimer -- this is not on Foundations of QM

3. Measurement problem 1: * In measurement based quantum computation will 1 error in a measurement outcome cascade to many, overwhelming the ability to perform QECC? * * * * * * Will focus on graph state quantum computation

4. Graph state Quantum Computation GSQC: RBB 0301052 Ã 1WQC: RB01 Linear optics app: N 0402005 Circuit interpretation CLN04, AL04 w/ gate-teleportation (GC99, ZLC00) (1) Produce a graph state (2) Apply single measurements that can be controlled by prior measurement outcomes Simulate a circuit element-by-elment -- can identify "regions" responsible for each

5. Graph state simulation of a circuit element O 0/1 x1x1 x2x2 xnxn |+ i... e in P ein ( in ) e out P eout (O( in )) |+ i = |0 i + |1 i = controlled- z P i = Pauli rotation indexed by i S O (e in ­ P ein ( in )) = (e out ­ P eout O( in )) (will see this is composable)

6. Composable simulation for a universal gate set £ £ |+ i MxMx abab b©dab©da XaZb|iXaZb|i X b © d Z a H| i d S(H): |+ i MxMx abab ab©cab©c XaZb|iXaZb|i X a Z b © c Z  | i c S(Z  ): Z (-1) a  H a2b2a2b2 X a 1 Z b 1 ­ X a 2 Z b 2 | i S( CP ): a1b1a1b1 a 1 b 1 © a 2 a 2 b 2 © a 1 X a 1 Z b 1 © a 2 ­ X a 2 Z b 2 © a 1 (CP| i )

7. Goal: perform Z rotation e iZ

8. H c |0 i |i|i Zc|iZc|i Goal: perform Z rotation e iZ Z-Telep (ZT)

9. H c |0 i |i|i Zc|iZc|i Goal: perform Z rotation e iZ c Z c e i(-1) a Z X a Z b | i XaZb|iXaZb|i H |0 i e i(-1) a Z Z-Telep (ZT) Input state = e i(-1) a Z X a Z b | i

10. H c |0 i |i|i Zc|iZc|i Goal: perform Z rotation e iZ c Z c e i(-1) a Z X a Z b | i XaZb|iXaZb|i H |0 i e i(-1) a Z = X a Z c+b e iZ | i Z-Telep (ZT) Xa eiZXa eiZ = § X a Z c+b e iZ | i

11. H c |0 i |i|i Zc|iZc|i Goal: perform Z rotation e iZ c Z c e i(-1) a Z X a Z b | i XaZb|iXaZb|i H |0 i e i(-1) a Z = X a Z c+b e iZ | i Z-Telep (ZT) Xa eiZXa eiZ = § X a Z c+b e iZ | i |+ i MxMx abab ab©cab©c XaZb|iXaZb|i X a Z b © c Z  | i c S(Z  ): Z (-1) a  H

12. Composable simulation for state prep & meas S(|0 i ): MzMz a 0 X a Z 0 |0 i a0a0 0 S(|+ i ): MxMx b X 0 Z b |+ i 0b0b MzMz abab a©ca©c XaZb|iXaZb|i c S( M z ): MxMx abab a©ca©c XaZb|iXaZb|i c S( M x ): |+ i given

13. General quantum circuit C |0 i U1U1 U2U2 U3U3 U4U4 U5U5 0/1 y1y1 y2y2 y3y3

14. Composing simulations simulates the circuit S |0 i 0/1 y1y1 y2y2 y3y3 S O k... S O 2 S O 1 (P init (|0 ih 0|)) = P final O k... O 2 O 1 (|0 ih 0|) S SU1SU1 SU2SU2 SU3SU3 SU4SU4 SU5SU5 S S S O (e in ­ P ein ( in )) = (e out ­ P eout O( in ))

15. FT Qn 1: Can we achieve FT simply by simulating a FT circuit ? Physical noise ! effective noise in simulated circuit ? FT Qn 2: Will an error propagate via feedforward ? Say, 1 error corrupts a meas outcome, which may induce further errors as it is used to condition subsequent op... ? AL05: offer an interpretation in which measurement outcomes are error free... yes to FT R03, ND04 -- with much work in designing schemes and full-blown nonMarkovian treatment of meas outcome errors

16. S( X a’ Z b’ ) (or S(I)): abab a © a’ b © b’ XaZb|iXaZb|i a’ b’ X a © a’ Z b © b’ | i Composable simulation for Pauli’s Note: Known Pauli operations : shifts in the classical parts. Unknown Pauli errors : errors in the classical parts.

17. 0. Add each |+ i to the graph state only slightly before it’s being measured ~ 1. Model noisy elementary operations O  storage, gate, or state prep  meas  env UFUF O ~O~O » UFUF O ~O~O » where U F = I ­ A 0 +  i P i ­ A i on sys ­ env P i = nontrivial Pauli’s Now working out how physical noise transformed to dealing the noise

18.  : where U F s act 2. Noisy simulation: interperse U F s between ideal operations e.g. |+ i MxMx abab ab©cab©c XaZb|iXaZb|i X a Z b © c Z  | i c S(Z  ): Z (-1) a  H       Sub U F = I sys ­ A 0 env +  i P i sys ­ A i env in above, expand For @ summand, commute P i 's towards end of simulation - “Joint I term” : ideal simulation - Else:(1) classical part may flip (2) quantum part may suffer unknown Pauli error But they’re equivalent !! So classical part is always correct & faults do not propagate but are localized How physical errors affect each simulation?

19. Thus, in GSQC: No error in classical part (they're unknown Pauli errors) Each elementary faults is also localized within a simulation Faults of all elementary operations in 1 simulation give a combined fault in that simulated op, and no where else, with effective noise strength · max # locations in a sim £ max strength per elementary op All FT results for circuits thus apply (including the code design, threshold analysis, and size+depth bounds) In fact, the composable graph state simulation is like a lowest level encoding in FTQC Qn: can GS/other teleportation based implementation decohere different fault paths? Need to understand meas better.

20. * * * * * * Measurement problem 2: * NonMarkovian noise is modeled as a system-bath Hamiltonian evolution. Its sup-norm should be sub-threshold for FT method to work. Will a sharp measurement, with an obligatorily large amplitude for the system-bath Hamiltonian, spoil the result? Remember fast measurement & classical components should help FT but how to model these?

21.  Modelling sharp measurements: Q C B M |x i Q ! |x i Q ­ |x i C ­ |x i M need to give a copy of the measurement outcome x to the environment to ensure its classicality g g    H = H Q + H C + H M + H B + H QC + H CM + H BM + H QM + H QB + H BC || H QC || 1 = || H CM || 1 = g signifies how fast the meas is || H BM || 1 =  quantities how fast adversarial noise can use the meas outcome || H QB || 1 = || H QM || 1 =  the usual QC noise amplitude || H CB || 1 =   ¿  Classical Comp noise amplitude g large if meas fast

22. Modelling sharp measurements:  Q C B M g g    H = H Q + H C + H M + H B + H QC + H CM + H BM + H QM + H QB + H BC || H QC || 1 = || H CM || 1 = g signifies how fast the meas is || H BM || 1 =  quantities how fast adversarial noise can use the meas outcome || H QB || 1 = || H QM || 1 =  the usual QC noise amplitude || H CB || 1 =   ¿  Classical Comp noise amplitude || e -iHt - V || 1 · t ( + 2 +   ) Let t be the meas time (tg ¼ 1), V be ideal meas Let t O be the time for a gate, noise strength ¼  t O (t O À t) Putting in the separation of scale, as long as  ·  t O /t the threshold is unspoiled. Qns: do we need to include  in the noise amplitude for gate alone? What to expect for  in various systems? Decohering fault paths by (un)intentional "random" Pauli's? esp in FT designs heavily relying on gate teleportation...