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Fault-tolerant Graph-state Quantum-Computation Panos Aliferis & Debbie Leung Institute of Quantum Information, Ca

Fault-tolerant Graph-state Quantum-Computation Panos Aliferis & Debbie Leung Institute of Quantum Information, Caltech. Starting September 1st, 2005: Dept of Combinatorics & Optimization Institute of Quantum Computing University of Waterloo wcleung@iqc.ca wcleung@math.uwaterloo.ca

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Fault-tolerant Graph-state Quantum-Computation Panos Aliferis & Debbie Leung Institute of Quantum Information, Ca

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  1. Fault-tolerant Graph-state Quantum-Computation Panos Aliferis & Debbie LeungInstitute of Quantum Information, Caltech

  2. Starting September 1st, 2005: Dept of Combinatorics & Optimization Institute of Quantum Computing University of Waterloo wcleung@iqc.ca wcleung@math.uwaterloo.ca wcleung@cs.caltech.edu wcleung@caltech.edu *

  3. Fault-tolerant Graph-state Quantum-Computation Panos Aliferis & Debbie LeungInstitute of Quantum Information, Caltech

  4. Fault-tolerant Graph-state Quantum-Computation Panos Aliferis & Debbie LeungInstitute of Quantum Information, CaltechGSQC-FT: 0503130 (AL) Prior & related results: - Raussendorf PhD thesis (2003) - Nielsen & Dawson, 0405134 - Dawson, Haselgrove, & Nielsen (in-prep) Tools GSQC: 0404132 (Childs, L, Nielsen) FT: 0504218 (A, Gottesman, Preskill)

  5. RBB 0301052 N 0402005 Graph State Quantum Computation (GSQC) To simulate a circuit C : - prepare graph state |gCi (e.g. using |+i, CP) - apply single qubit measurements (with feedforward – meas bases can depend on prior meas outcomes) - simulation is “element-wise” FT Qn 1: Can we achieve FT by simulating a FT circuit ? Physical noise ?! noise in simulated circuit FT Qn 2: Will errors propagate via feedforward ? If 1 error corrupts a meas outcome, can it affect the simulation of a subsequent op ... ?

  6. Graph State Quantum Computation (GSQC) FT Qn 1: Can we achieve FT by simulating a FT circuit ? Physical noise ?! noise in simulated circuit FT Qn 2: Will errors propagate via feedforward ? If 1 error corrupts a meas outcome, will it affect the simulation of a subsequent op ... ? Here: answer the above using different tools than prior works (by “robbing” pieces of recent results) - Use the notion of composable simulation (CLN04) (that shows how&why GSQC works) to show why errors don’t propagate and to relate to circuit noise models - FT & threshold then follows by AGP05 R03, NC04, DHN What else do we learn?

  7. x1 x2 x3 General quantum circuit C 0/1 |0i |0i |0i U3 U5 U1 0/1 U4 U2 0/1 p(x1,x2,x3) Abstract representation of state transform of the logical space. Need not correspond to physical implementation.

  8. x1 x2 x3 General quantum circuit C 0/1 |0i |0i |0i U3 U5 U1 0/1 U4 U2 0/1 p(x1,x2,x3) Abstract representation of state transform of the logical space. Need not correspond to physical implementation. e.g. fault-tolerant encoded logical operations.

  9. x2 x3 x1 General quantum circuit C k1 k7 0/1 |0i |0i |0i k5 U3 U5 U1 k2 k3 0/1 k8 U4 k6 U2 0/1 k4 p(x1,x2,x3) Abstract representation of state transform of the logical space. Need not correspond to physical implementation. e.g. in GSQC, the simulations use measurements & teleportation, with extra, uncontrolled, known Pauli operations.

  10. x1 x2 x3 Composable simulation S(U3) S(|0i) S(|0i) S(|0i) 0/1 S(U5) S(U1) 0/1 S(U4) S(U2) 0/1 p(x1,x2,x3) It is lazy (but understandly) to hope that simulating circuit elements one by one automatically simulates the entire circuit. The simulation O ! S(O) is composable if S(O2) ± S(O1) = S(O2± O1)

  11. in U U(in) ein eout S(U) eout ein in x »p(x) |ai S( ) Pein(in) (Peout± U)(in) S( ) in x »p(x) Peout(|ai) Composable simulation in GSQC • The simulation O ! S(O) is composable • if S(O2) ± S(O1) = S(O2± O1) • For GSQC : • If, 8in demand: 8in 8ein 9 eout s.t. • s(u) • i.e. ein­ Pein(in)!eoutpr(eout) eout­(Peout±U)(in) • If, 8in demand: 8in8 ein • For |ai  demand: Explain symbols & interpretation Pein( ) 9 eout s.t. eout pr(eout) eout­Peout (|ai)

  12. x1 x2 x3 Composable simulation in GSQC S(U3) S(|0i) S(|0i) S(|0i) S(U5) S(U1) S(U4) S(U2) 1 2 6 State at various stages of the simulation: 1: eout11,eout12,eout13 Pr(eout11 eout12 eout13) eout11 eout12 eout13­Peout11(|0i) Peout12(|0i) Peout13(|0i) 2: eout11,eout12,eout13eout21 eout22Pr(eout11 eout12 eout13) Pr(eout21eout22|eout11eout12) eout21 eout22 eout13­Peout21 Peout22 ±U1(|0i|0i) Peout13(|0i) p(x1,x2,x3)

  13. x1 x2 x3 Composable simulation in GSQC S(U3) S(|0i) S(|0i) S(|0i) S(U5) S(U1) S(U4) S(U2) 1 2 6 State at various stages of the simulation: 1: eout11,eout12,eout13 Pr(eout11 eout12 eout13) eout11 eout12 eout13­Peout11(|0i) Peout12(|0i) Peout13(|0i) 2: eout11,eout12,eout13eout21 eout22Pr(eout11 eout12 eout13 eout21 eout22) eout21 eout22 eout13­Peout21 Peout22 ±U1(|0i|0i) Peout13(|0i) .... 6:eout11,eout12,eout13... eout61 eout62 eout63Pr(eout11 eout12 eout13 ... eout61 eout62 eout63) eout61 eout62 eout63­Peout61 Peout62 Peout63 ±U5± ... ±U1(|0i|0i|0i) p(x1,x2,x3) S(meas)

  14. Composable simulation in GSQC The state is a mixture over possible each will give the correct output distribution “Pauli-frame history” of the simulation State at various stages of the simulation: 1: eout11,eout12,eout13 Pr(eout11 eout12 eout13) eout11 eout12 eout13­Peout11(|0i) Peout12(|0i) Peout13(|0i) 2: eout11,eout12,eout13eout21 eout22Pr(eout11 eout12 eout13 eout21 eout22) eout21 eout22 eout13­Peout21 Peout22 ±U1(|0i|0i) Peout13(|0i) .... 6:eout11,eout12,eout13... eout61 eout62 eout63Pr(eout11 eout12 eout13 ... eout61 eout62 eout63) eout61 eout62 eout63­Peout61 Peout62 Peout63 ±U5± ... ±U1(|0i|0i|0i) p(x1,x2,x3)

  15. Composable simulation in GSQC “Pauli-frame history” Info of the Pauli-frame at any time: “classical part” of the simulation. State at various stages of the simulation: 1: eout11,eout12,eout13 Pr(eout11 eout12 eout13) eout11 eout12 eout13­Peout11(|0i) Peout12(|0i) Peout13(|0i) 2: eout11,eout12,eout13eout21 eout22Pr(eout11 eout12 eout13 eout21 eout22) eout21 eout22 eout13­Peout21 Peout22 ±U1(|0i|0i) Peout13(|0i) .... 6:eout11,eout12,eout13... eout61 eout62 eout63Pr(eout11 eout12 eout13 ... eout61 eout62 eout63) eout61 eout62 eout63­Peout61 Peout62 Peout63 ±U5± ... ±U1(|0i|0i|0i) p(x1,x2,x3)

  16. Composable simulation in GSQC “Pauli-frame history” “quantum part” “classical part” State at various stages of the simulation: 1: eout11,eout12,eout13 Pr(eout11 eout12 eout13) eout11 eout12 eout13­Peout11(|0i) Peout12(|0i) Peout13(|0i) 2: eout11,eout12,eout13eout21 eout22Pr(eout11 eout12 eout13 eout21 eout22) eout21 eout22 eout13­Peout21 Peout22 ±U1(|0i|0i) Peout13(|0i) .... 6:eout11,eout12,eout13... eout61 eout62 eout63Pr(eout11 eout12 eout13 ... eout61 eout62 eout63) eout61 eout62 eout63­Peout61 Peout62 Peout63 ±U5± ... ±U1(|0i|0i|0i) p(x1,x2,x3)

  17. S(|0i): S(|+i): a a 0 0 0 b Mz b Mx 0 XaZ0|0i X0Zb|+i S( Mx ): S( Mz ): a b a b c XaZb|i Mx c a©c XaZb|i Mz a©c Composable simulation for state prep & meas

  18. S(H): £ a b b©d a £ d XaZb|i Mx |+i Xb©d Za H|i a1 b1 S(CP): a1 b1© a2 S(Z): a2 b2 a b a2 b2© a1 a b©c c XaZb|i Z(-1)a Mx Xa1Zb1 ­ Xa2Zb2 |i Xa1Z b1©a2­ Xa2Z b2©a1(CP|i) |+i Xa Zb©c Z|i H Composable simulation for a universal gate set Aside: 1. Recipe for GSQC Quantum part: - |+i - CP - matching in/out - meas 2. Also for 1-way QC via - deletion principle - optional CP

  19. a’ a b a©a’ b©b’ b’ XaZb|i Xa©a’ Zb©b’ |i Composable simulation for Pauli’s S(Xa’ Zb’) (or S(I)): Note: Known Pauli operations : shifts in the classical parts. Unknown Pauli errors : errors in the classical parts.

  20. ~ O ~ O UF O » » UF O env env What about noise? 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 where UF = I ­ A0 + i Pi­ Ai on sys ­ env Pi = nontrivial Pauli’s

  21. S(Z): a b a b©c c Z(-1)a XaZb|i Mx |+i Xa Zb©c Z|i H       How physical errors affect each simulation? What about noise? 2. Noisy simulation: interperse UFs between ideal operations e.g. : where UFs act • Expand each UF as Isys­ A0 env + i Pi sys­ Ai env • Commute each summand 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 !! • i.e., classical part is always correct, & fault do not propagate • – localization

  22. S(Z): a b a b©c c Z(-1)a XaZb|i Mx |+i Xa Zb©c Z|i H       How physical errors affect each simulation? What about noise? 2. Noisy simulation: interperse UFs between ideal operations e.g. Simulation output: Noiseless term: eineout pr(eineout) eout­ (Peout ± U)(in) Noisy term: eineout pr(eineout) i A’i [·] ­e’i out­ (PiPeout ± U )(in) = eineout pr(eineout) iA’i [·]­e’i out­Pe’i outPe’i out PiPeout± U(in)

  23. S(Z): a b a b©c c Z(-1)a XaZb|i Mx |+i Xa Zb©c Z|i H       How physical errors affect each simulation? What about noise? 2. Noisy simulation: interperse UFs between ideal operations e.g. Simulation output: Noiseless term: eineout pr(eineout) eout­ (Peout ± U)(in) Noisy term: eineout pr(eineout) i A’i [·] ­e’i out­ (PiPeout ± U )(in) = eineout pr(eineout) iA’i [·]­e’i out­Pe’i outPe’i out PiPeout± U(in)

  24. S(Z): a b a b©c c Z(-1)a XaZb|i Mx |+i Xa Zb©c Z|i H       How physical errors affect each simulation? What about noise? 2. Noisy simulation: interperse UFs between ideal operations e.g. Simulation output: Noiseless term: eineout pr(eineout) eout­ (Peout ± U)(in) Noisy term: eineout pr(eineout) i A’i [·] ­e’i out­ (PiPeout ± U )(in) = eineout pr(eineout) iA’i [·]­e’i out­Pe’i outPi’± U(in) subsequent simulations unaffected (composability) U’F(S)

  25. How physical errors affect each simulation? What about noise? 3. Elementary operations in GSQC: - prepare |+i  - CP  each in a unique simulation - single qubit meas  - storage } WLOG error acts in later sim Together with localization -- noise affecting each elementary operation affects only 1 simulated operation Simulation output: Noiseless term: eineout pr(eineout) eout­ (Peout ± U)(in) Noisy term: eineout pr(eineout) i A’i [·] ­e’i out­ (PiPeout ± U )(in) = eineout pr(eineout) iA’i [·]­e’i out­Pe’i outPi’± U(in) subsequent simulations unaffected (composability) U’F(S)

  26. Thus, in GSQC: (1) by interpreting errors in the classical part as unknown Pauli errors + composability, faults of elementary ops in 1 simulation give combined fault the corr simulated op only (2) the joint no-fault term in 1 simulation gives an ideal simulated op E.g. 1. For independent stochastic noise, simulated operation O has fault with prob at most Tot(S(O)) = sum of prob of faults of all elementary ops in S(O). To reliably simulate circuit C , find FT circuit C ’ that handles error Tot(S(O)) , then simulate C ’ with GSQC . Threshold for GSQC ¸ circuit / maxO(#ops in S(O)).

  27. Thus, in GSQC: (1) by interpreting errors in the classical part as unknown Pauli errors + composability, faults of elementary ops in 1 simulation give combined fault the corr simulated op only (2) the joint no-fault term in 1 simulation gives an ideal simulated op E.g. 2. For general noise, consider output of 1 simulation: eineout pr(eineout) iA’i [·]­e’i out­Pe’i outP’i± U(in) U’F(S)

  28. Thus, in GSQC: (1) by interpreting errors in the classical part as unknown Pauli errors + composability, faults of elementary ops in 1 simulation give combined fault the corr simulated op only (2) the joint no-fault term in 1 simulation gives an ideal simulated op E.g. 2. For general noise, consider output of 1 simulation: eineout pr(eineout) i A’i [·]­e’i out­Pe’i outP’i± U(in) .... and compose many simulations: ein...eout pr(ein ...eout) ni,...,1i A’ni[·] ... A’1i[·] ­e’ni out­Pe’i out­P’inUn ... P’1iU1(in) For each Pauli-frame history, the env-sys state is in the form treated in circuit model (e.g. AGP05). Sum of amp of fault terms upper bounds noisy amp of simulated op. U’F(S)

  29. Meditating on this .... ein...eout pr(ein ...eout) ni,...,1i A’ni[·] ... A’1i[·] ­e’ni out­Pe’i out­P’inUn ... P’1iU1(in) GSQC is just QC with constantly changing Pauli-frame. These eout are like the perfect part of error syndromes. Qns: NonDeterministic operations? Insights on using error syndromes or Pauli-encryption/randomization to tame noise? e.g., to reduce the extent of non-Markovian-ness ? Composability of FT analysis?

  30. Thus, in GSQC: (1) by interpreting errors in the classical part as unknown Pauli errors + composability, faults of elementary ops in 1 simulation give combined fault the corr simulated op only (2) the joint no-fault term in 1 simulation gives an ideal simulated op E.g. 2. For general noise, consider output of 1 simulation: eineout pr(eineout) i A’i [·]­e’i out­Pe’i outP’i± U(in) .... and compose many simulations: ein...eout pr(ein ...eout) ni,...,1i A’ni[·] ... A’1i[·] ­e’ni out­Pe’i out­P’inUn ... P’1iU1(in) For each Pauli-frame history, the env-sys state is in the form treated in circuit model (e.g. AGP05). Sum of amp of fault terms upper bounds noisy amp of simulated op. U’F(S)

  31. Composable simulation for a universal gate set S(H): £ a b b©d a £ d XaZb|i Mx |+i Xb©d Za H|i S(Z): a b a b©c c XaZb|i Z(-1)a Mx |+i Xa Zb©c Z|i H

  32. Composable simulation for a universal gate set S(Xa’ Zb’): a’ a b a©a’ b©b’ b’ XaZb|i Xa©a’ Zb©b’ |i a1 b1 S(CP): a1 b1© a2 a2 b2 a2 b2© a1 Xa1Zb1 ­ Xa2Zb2 |i Xa1Z b1©a2­ Xa2Z b2©a1(CP|i)

  33. |+i d1 |+i d2 |+i d1 |+i |+i Y Z |+i d2 Simulating an optional C-Z, summary: simulates To do the C-Z: To skip the C-Z: Do: Skip: also simulates up to Z-rotations

  34. Universal Initial state 3 qubits, 8 cycles Starting from the cluster state measurement in Z basis

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