Canadian Summer School on Quantum Information (CSSQI) (10th : 2010)

For How Long Is It Possible To Quantum Compute? Mucciolo, Eduardo 2010-07-25

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For How Long is it Possible to Quantum Compute?QAMF WORKSHOPVancouver, UBC - July 23-25, 20101/19Eduardo Mucciolo (University of Central Florida, USA)Harold Baranger (Duke University, USA)Eduardo Novais (University of ABC, Brazil)Sunday, July 25, 2010Overview●  Motivation and previous work●  Hamiltonian formulation of a fault-tolerant quantum processor●  Results2/19Sunday, July 25, 2010Motivation:Protect Quantum Information from DecoherenceMajor result:  The “threshold theorem”resilient QC(”low-T” regime)noisy QC(”high-T” regime)εεc(= single-qubit error probability)likely the most versatileand universal approach- encode information- measure ancilla qubits- correct for errors3/19Provided that the noise level is below a certain critical value, quantum information can be protected indefinitelyProvided that the noise level is below a certain critical value, quantum computation can be proceed “indefinitely”Some standard strategies: 1) Decoherence free-subspaces 3) Dynamical decoupling4) Quantum error correction 2) Topological systems Sunday, July 25, 2010Possible Pitfalls of the Standard Fault-Tolerant QEC Theory:1) The theorem is derived using stochastic error models (not Hamiltonians).2) It implicitly assumes that perturbation theory works and converges.3) Correlated environments and memory effects are not considered.4/19We avoided some of theseproblems in our previous work:- Phys. Rev. Lett. 97, 040501 (2006); 98, 040501 (2007).- Phys. Rev. A 78, 012314 (2008).- Hamiltonian, microscopic error models.- Probabilistic calculations taking into account correlations.- Description of how correlated environments affect the threshold theorem.Sunday, July 25, 2010Quick review: Correlated environments and the Threshold Theoremenvironmentʼs “free”Hamiltonian (gapless)qubit-bath interaction:V =summationdisplayx ,αλαfα( x ) σα( x )environmental degrees of freedomqubitsH = Hcom puter+ Hbat h+ VHamiltonian5/19Three important parameters (critical environment)• spatial dimension  D• bath mode velocity  c• dynamical exponent  z〈 fα( x1,t1) fα( x2,t2) 〉en v∼ Oparenleftbigg1|x1− x2|2 δα,1| t1− t2|2 δα/zparenrightbigg- defines a single-qubit error probability   - maps into an impurity problem for short times.epsilon1Main hypothesisonly one qubitper hypercube∆ξ =( c∆ )1 /ztimespace∆= period of QEC cycleNovais, EM, Baranger [PRL 98, 040501 (2007); PRA 78 012314 (2008)]Sunday, July 25, 20106/19(efficiently simulated by a Turing machine)D+z2noisy quantum computerδ"upper critical dimension""lower critical  dimension"( ε )local error probabilitythreshold theoremneeds ( unkown )new derivationthreshold theoremto computenot possibletraditional“temperature”“correlations”weakstrongMain result of previous work:the effect of long-range correlations are small;QEC and resilient quantum computing are ok.D + z< 2 δαcorrelations grow unbounded; threshold theorem derivation breaks down.D + z> 2 δαqubits + environmentstrongly entangledqubits + environment weakly entangled;strong entanglement among qubitsqubits + environment strongly entangled;weak entanglement among qubits✓✓✓??Novais, EM, Baranger [PRL 98, 040501 (2007); PRA 78 012314 (2008)]Sunday, July 25, 20107/19Current Work:1) Change the question:  Given a certain desired error tolerance, for long     can we compute using QEC?Novais, EM, Baranger [arXiv: 1004.3247]2) Avoid the hypercube assumption.“friendly” assumptions- standard quantum error correction code (circuit model QC).- state preparation, gates, measurements all done perfectly.- quantum evolution with only non-error syndromes.“unfriendly” assumptions- gapless bath: power-law correlations (non exponential)(phonons, EM fluctuations, etc.)Sunday, July 25, 20108/19Time-Evolution under QEC:QEC0 Δ 2 Δ 3 Δ 4 Δ ( Ν− 2) Δ ( Ν− 1) Δ ΝΔtQEC QEC QEC QEC QECstartendbosonicbathsyndromeextractions1s2s3s4sN-2sN-1sNnon-errorsyndromepathSyndrome history treeSunday, July 25, 2010Taking into account the QEC code structure:logical qubitExample: 5-qubit code (corrects for one-qubit errors)|¯0 〉 =14[|00000 〉 + |10010 〉 + |01001 〉 + |10100 〉 + |01010 〉− |11011 〉− |00110 〉− |11000 〉− |11101 〉− |00011 〉− |11110 〉− |01111 〉− |10001 〉− |01100 〉− |10111 〉 + |00101 〉]|¯1 〉 =14[|11111 〉 + |01101 〉 + |10110 〉 + |01011 〉 + |10101 〉− |00100 〉− |11001 〉− |00111 〉− |00010 〉− |11100 〉− |00001 〉− |10000 〉− |01110 〉− |10011 〉− |01000 〉 + |11010 〉]g1= xz z x 1g2=1 xz z xg3= x 1 xz zg4= zx 1 xz¯X = xxxxx¯Y = y y y y y¯Z = zzzzzstabilizersand logical operations45 = 1024 error operators16 possible syndromesbut only 64 error operatorscan lead to a non-errorsyndrome¯1¯X¯Y¯Z¯1 11111 xxxxx yyyyy zzzzzg1xz zx1 1yy 1x zxxz y y 11 yzg21xz zx x1yy 1 yzxxz zy 11 yg3x1xz z 1x1yy zyzxx yzy 11g4zx 1xz y 1x1y xz yzx 1yzy 1h1xy 1yx 1zxz 1 z1y 1z yxz xyh21zyyz xy zzy yx 11 x z1xx 1h3yyz 1z zzyxy 11 xy x xx 1z1h4xxy 1y 11 zxz zz 1y 1 yyxz xh5z1zyy yxy zz xy x11 1z1xxh6yxxy 1 z11 zx 1zz 1y xy yxzh71yxxy xz 11 z y 1zz 1 zxy yxh8zyyz1 yzzyx x11 xy 1xx 1zh9y 1yxx zxz 11 1y 1zz xz xy yh10yz 1zy zyxy z 1xy x1 x1z1xh11zzx 1x yy 1x1 xxz yz 11 yzy9/19Sunday, July 25, 201010/19Model for the critical environment:bosonicbathqubit-bath interactionphysical qubitsHI=summationdisplayxsummationdisplayα = { x,z }λαfα( x ) σα( x )bracketleftBigaα ,k,a†β ,kprimebracketrightBig= δαβδk ,kprimefα(x )=summationdisplayk negationslash=0parenleftBiggα , |k |ei k ·xa†α ,k+ g∗α , |k |e− i k ·xaα ,kparenrightBigbosonicbathHB=summationdisplayα = { x,z }summationdisplayk negationslash=0ωα ,|k |a†α ,kaα ,k| gα , | k ||2∼ |k |2 sαform factor:dispersion relation:ωα , |k |∼ |k |zαspectral functionJ ( ω )=summationdisplayk| gk|2δ ( ω − ωk)Jα( ω ) ∼ λ2αω( D +2 sα− zα) /zαSunday, July 25, 2010U (∆ , 0) = Tte− iR∆0dt HI( t )=1 − iintegraldisplay∆0dt HI( t ) −12integraldisplay∆0dtintegraldisplayt0dtprimeHI( t ) HI( tprime)+ . . .system+bath evolution operator11/19Time evolution: First QEC cycleU (∆ , 0) ≈ 1 − isummationdisplayα = x,zsummationdisplayxλα∆ fα( x , 0) σαxlowest order perturbation theory (short times / small coupling)λα∆ lessmuch 1expansionparametersyndrome extraction and error correction operation: U (∆ , 0) −→ vs(∆ , 0)numerical coefficientsO(λ2) renormalization droppedevolution operator for logical qubits after a non-error syndrome¯v0(∆ , 0) ≈¯1+ i∆3summationdisplayxsummationdisplayα ,β = { x,z }summationdisplayi,j ,kηαβij kλαλ2βfα( xi, 0) fβ( xj, 0) fβ( xk, 0) ¯σαx+ O ( λ5)Sunday, July 25, 2010Determining the expansion coefficients:¯1¯X¯Y¯Z¯1 11111 xxxxx yyyyy zzzzzg1xz zx 1 1yy 1x zxxz y y 11 yzg21xz zx x 1yy 1 yzxxz zy 11 yg3x 1xz z 1x 1yy zyzxx yzy 11g4zx 1xz y 1x 1y xz yzx 1yzy 1h1xy 1yx 1zxz 1 z 1y 1z yxz xyh21zyyz xy zzy yx 11 x z 1xx 1h3yyz 1z zzyxy 11 xy x xx 1z 1h4xxy 1y 11 zxz zz 1y 1 yyxz xh5z 1zyy yxy zz xy x 11 1z 1xxh6yxxy 1 z 11 zx 1zz 1y xy yxzh71yxxy xz 11 z y 1zz 1 zxy yxh8zyyz 1 yzzyx x 11 xy 1xx 1zh9y 1yxx zxz 11 1y 1zz xz xy yh10yz 1zy zyxy z 1xy x 1 x 1z 1xh11zzx 1x yy 1x 1 xxz yz 11 yzyoperators leading to a non-error syndromeNotice: 3 or 5 insertions of the interaction only             (one per physical qubit, at most)ηxy15 24= ηxy12 35= ηxy23 14= ηxy34 25= ηxy45 13=1ηxz14 23= ηxz25 34= ηxz13 45= ηxz24 15= ηxz35 12=1ηyz34 25= ηyz12 35= ηyz45 13= ηyz23 14= ηyz15 24=1(all other coefficients vanish)12/19Sunday, July 25, 201013/19Same functional form as for physical (unprotected) qubits!Main effect of QEC: renormalization of coupling constantrepeat for subsequent cycles¯U ( N ∆ , 0) = ¯v0( N ∆ , ( N − 1) ∆ )¯v0(( N − 1) ∆ , ( N − 2) ∆ ) . . . ¯v0(∆ , 0)rearrangement (separating intra versus inter logical qubit correlations)λ∗α= λαsummationdisplayxsummationdisplayβ = { x,z }summationdisplayi,j ,kηαβij k( λβ∆ )2summationdisplayk negationslash=0| gα , |k ||2e− ik ·( xi− xj)effective coupling constant¯v0(∆ , 0) ≈¯1+ i∆summationdisplayxsummationdisplayα = { x,z }( λ∗α+ Γα) fβ( x , 0) ¯σαxquantum evolution of logical qubit after N QEC cycles:¯U ( T = N ∆ , 0) ≈ TteiRT0dtPxPα = { x,z }λ∗αfα( x ,t )¯σαxΔ2Δ3ΔTime evolution (Cont.):Sunday, July 25, 2010Upper bound to computation time- Information lost by a single logical qubit DparenleftbigρR(T ) − ρidea lparenrightbig=radicalbigg| δσ+(T ) |2+14[δσz(T )]2δσα( T )= 〈 σα( T ) 〉−〈 σα〉- Trace distance as a measure of information degradation DparenleftbigρR(T ) − ρidea lparenrightbig=12Tr bardbl ρR(T ) − ρidea lbardblbardbl A bardbl =√A†A- Given a certain Dthresh, one can compute the available time Tmax by solvingDparenleftbigρR(Tm ax) − ρidea lparenrightbig= Dthr esh14/19Sunday, July 25, 2010Example: isolated logical qubit, no transversal coupling (λx =0)zTmaxDthreshDct10ζ   > 0ζ   = 0ζ   < 0zzD(t)15/19ζz≡ 2( zz− sz) − Dbath spatial dimensionLong-time behavior controlled by the exponent“superohmic”“ohmic”“subohmic”Tm ax≈ ∆∞ ζz< 0expbracketleftbiggCDthr eshparenleftBigω0λ∗zparenrightBig2bracketrightbiggζz=0Dzz/ ζzthreshω0∆parenleftBigω0λ∗zparenrightBig2 zz/ ζz0 < ζz< 2 zzparenleftBig2 pik0LparenrightBig( ζz− 2 zz)√Dthreshλ∗z∆ζz> 2 zzSunday, July 25, 2010Another example: logical qubit array (M qubits, array dimension Dq)16/19Notice: Different layers of concatenation only alter λα* (higher power in the bare λα)ζα≡ 2( zα− sα) − Dζα≡ 2( zα− sα) − D+Dqif intra-logical qubit correlations dominateif inter-logical qubit correlations dominate(sparsearray)(densearray)compromise between array size, code effectiveness, and amplitude of coupling to environmentM large <{  λ* smallbest caseworst casemarginal caseTm ax≈ ∆ ×∞ ζα< 0expbracketleftBigBDthreshM ( λ∗α/ ω0)bracketrightBigζα=01ω0∆bracketleftBigDthreshM ( λ∗α/ ω )bracketrightBigzα/ ζα0 < ζα<zαparenleftBig2 pik0LparenrightBigζα− zαDthreshM ( λ∗α∆ )ζα>zαSunday, July 25, 2010Case 1:   2D sparse logical qubit array; level-0 concatenation               GaAs double-quantum dot charge physical qubits               bath: piezoelectric acoustic phonons (3D)Case 2:   1D dense logical qubit array; level-0 concatenation               superconductor charge qubits               bath: electromagnetic (gate voltage) fluctuations (effectively 1D)zz=1“bad” casesz=1 / 2Dq=1D =1ζz=1Tm ax≈ ω0bracketleftbiggDthr eshM ( λ∗z/ ω0)bracketrightbiggJ ( ω ) ∼ ω17/19zz=1D =3ζz=0marginal caseJ ( ω ) ∼ ω3Tm ax≈ ∆ expbracketleftbiggBDthr eshM ( λ∗z/ ω0)bracketrightbiggsz=1 / 2Sunday, July 25, 2010Summary18/19● We developed a Hamiltonian formulation of QEC in the presence of critical    environments. Details of error code and concatenation level are incorporated.     The non-Markovian dynamics of the bath is taken into account.● The effective, long-time evolution operator of the logical qubits has the     same functional form of the original operator for physical qubits, but with a     renormalized coupling constant.● Given a certain final state tolerance, we developed an estimate of the maximum     computational time for a non-error syndrome history; it is straightforward to       estimate for other histories (will yield shorter computational times). Sunday, July 25, 2010Resources:- Phys. Rev. Lett. 97, 040501 (2006); 98, 040501 (2007).- Phys. Rev. A 78, 012314 (2008).- arXiv:1004.3247- links to publications in http://www.physics.ucf.edu/~mucciolo/THANKS!19/19Funding:Sunday, July 25, 2010

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