Workshop on Quantum Algorithms, Computational Models and Foundations of Quantum Mechanics

For How Long Is It Possible To Quantum Compute? 2011

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For How Long is it Possible to Quantum Compute? QAMF WORKSHOP Vancouver, UBC - July 23-25, 2010 1/19 Eduardo Mucciolo (University of Central Florida, USA) Harold Baranger (Duke University, USA) Eduardo Novais (University of ABC, Brazil) Sunday, July 25, 2010 Overview ●  Motivation and previous work ●  Hamiltonian formulation of a fault-tolerant quantum processor ●  Results 2/19 Sunday, July 25, 2010 Motivation: Protect Quantum Information from Decoherence Major result:  The “threshold theorem” resilient QC (”low-T” regime) noisy QC (”high-T” regime) ε εc (= single-qubit error probability) likely the most versatile and universal approach - encode information - measure ancilla qubits - correct for errors 3/19 Provided that the noise level is below a certain critical value, quantum information can be protected indefinitely Provided 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 decoupling 4) Quantum error correction 2) Topological systems Sunday, July 25, 2010 Possible 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. 4/19 We avoided some of these problems 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, 2010 Quick review: Correlated environments and the Threshold Theorem environmentʼs “free” Hamiltonian (gapless) qubit-bath interaction: V = ∑ x,α λα fα(x)σα(x) environmental degrees of freedom qubits H = Hcomputer +Hbath + V Hamiltonian 5/19 Three important parameters (critical environment) • spatial dimension  D • bath mode velocity  c • dynamical exponent  z 〈fα(x1, t1) fα(x2, t2)〉env ∼ O ( 1 |x1 − x2|2δα , 1 |t1 − t2|2δα/z ) - defines a single-qubit error probability - maps into an impurity problem for short times. ! Main hypothesis only one qubit per hypercube ∆ ξ = (c∆)1/z time space ∆ = period of QEC cycle Novais, EM, Baranger [PRL 98, 040501 (2007); PRA 78 012314 (2008)] Sunday, July 25, 2010 6/19 (efficiently simulated by a Turing machine) D+z 2 noisy quantum computer δ "upper critical dimension" "lower critical  dimension" ( ε ) l o c a l e r r o r p r o b a b i l i t y threshold theoremneeds ( unkown ) new derivation threshold theorem to compute not possible traditional “temperature” “correlations” weak strong Main result of previous work: the effect of long-range correlations are small; QEC and resilient quantum computing are ok. D + z < 2δα correlations grow unbounded; threshold theorem derivation breaks down. D + z > 2δα qubits + environment strongly entangled qubits + environment weakly entangled; strong entanglement among qubits qubits + environment strongly entangled; w ak entanglement among qubits ✓ ✓ ✓ ? ? Novais, EM, Baranger [PRL 98, 040501 (2007); PRA 78 012314 (2008)] Sunday, July 25, 2010 7/19 Current 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, 2010 8/19Time-Evolution under QEC: QEC 0 Δ 2Δ 3Δ 4Δ (Ν−2)Δ (Ν−1)Δ ΝΔ t QEC QEC QEC QEC QEC start end bosonic bath syndrome extraction s1 s2 s3 s4 sN-2 sN-1 sN non-error syndrome path Syndrome history tree Sunday, July 25, 2010 Taking into account the QEC code structure: logical qubit Example: 5-qubit code (corrects for one-qubit errors) |0̄〉 = 1 4 [|00000〉+ |10010〉+ |01001〉+ |10100〉+ |01010〉 − |11011〉 − |00110〉 − |11000〉 −|11101〉 − |00011〉 − |11110〉 − |01111〉 − |10001〉 − |01100〉 − |10111〉+ |00101〉] |1̄〉 = 1 4 [|11111〉+ |01101〉+ |10110〉+ |01011〉+ |10101〉 − |00100〉 − |11001〉 − |00111〉 −|00010〉 − |11100〉 − |00001〉 − |10000〉 − |01110〉 − |10011〉 − |01000〉+ |11010〉] g1 = xzzx1 g2 = 1xzzx g3 = x1xzz g4 = zx1xz X̄ = xxxxx Ȳ = yyyyy Z̄ = zzzzz stabilizers and logical operations 45 = 1024 error operators 16 possible syndromes but only 64 error operators can lead to a non-error syndrome 1̄ X̄ Ȳ Z̄ 1̄ 11111 xxxxx yyyyy zzzzz g1 xzzx1 1yy1x zxxzy y11yz g2 1xzzx x1yy1 yzxxz zy11y g3 x1xzz 1x1yy zyzxx yzy11 g4 zx1xz y1x1y xzyzx 1yzy1 h1 xy1yx 1zxz1 z1y1z yxzxy h2 1zyyz xyzzy yx11x z1xx1 h3 yyz1z zzyxy 11xyx xx1z1 h4 xxy1y 11zxz zz1y1 yyxzx h5 z1zyy yxyzz xyx11 1z1xx h6 yxxy1 z11zx 1zz1y xyyxz h7 1yxxy xz11z y1zz1 zxyyx h8 zyyz1 yzzyx x11xy 1xx1z h9 y1yxx zxz11 1y1zz xzxyy h10 yz1zy zyxyz 1xyx1 x1z1x h11 zzx1x yy1x1 xxzyz 11yzy 9/19 Sunday, July 25, 2010 10/19Model for the critical environment: bosonic bath qubit-bath interaction physical qubits HI = ∑ x ∑ α={x,z} λα f α(x)σα(x) [ aα,k, a † β,k′ ] = δαβ δk,k′ fα(x) = ∑ k!=0 ( gα,|k|eik·xa † α,k + g ∗ α,|k|e −ik·xaα,k ) bosonic bath HB = ∑ α={x,z} ∑ k!=0 ωα,|k| a † α,kaα,k |gα,|k||2 ∼ |k|2sαform factor: dispersion relation: ωα,|k| ∼ |k|zα spectral function J(ω) = ∑ k |gk|2 δ(ω − ωk) Jα(ω) ∼ λ2α ω(D+2sα−zα)/zα Sunday, July 25, 2010 U(∆, 0) = Tt e−i R∆ 0 dtHI(t) = 1− i ∫ ∆ 0 dtHI(t)− 12 ∫ ∆ 0 dt ∫ t 0 dt′HI(t)HI(t′) + . . . system+bath evolution operator 11/19Time evolution: First QEC cycle U(∆, 0) ≈ 1− i ∑ α=x,z ∑ x λα∆ fα(x, 0)σαx lowest order perturbation theory (short times / small coupling) λα∆! 1 expansionparameter syndrome extraction and error correction operation: U(∆, 0) −→ vs(∆, 0) numerical coefficients O(λ2) renormalization dropped evolution operator for logical qubits after a non-error syndrome v̄0(∆, 0) ≈ 1̄ + i∆3 ∑ x ∑ α,β={x,z} ∑ i,j,k ηαβijk λαλ 2 β f α(xi, 0) fβ(xj , 0) fβ(xk, 0) σ̄αx +O(λ 5) Sunday, July 25, 2010 Determining the expansion coefficients: 1̄ X̄ Ȳ Z̄ 1̄ 11111 xxxxx yyyyy zzzzz g1 xzzx1 1yy1x zxxzy y11yz g2 1xzzx x1yy1 yzxxz zy11y g3 x1xzz 1x1yy zyzxx yzy11 g4 zx1xz y1x1y xzyzx 1yzy1 h1 xy1yx 1zxz1 z1y1z yxzxy h2 1zyyz xyzzy yx11x z1xx1 h3 yyz1z zzyxy 11xyx xx1z1 h4 xxy1y 11zxz zz1y1 yyxzx h5 z1zyy yxyzz xyx11 1z1xx h6 yxxy1 z11zx 1zz1y xyyxz h7 1yxxy xz11z y1zz1 zxyyx h8 zyyz1 yzzyx x11xy 1xx1z h9 y1yxx zxz11 1y1zz xzxyy h10 yz1zy zyxyz 1xyx1 x1z1x h11 zzx1x yy1x1 xxzyz 11yzy operators leading to a non-error syndrome Notice: 3 or 5 insertions of the interaction only             (one per physical qubit, at most) ηxy1524 = η xy 1235 = η xy 2314 = η xy 3425 = η xy 4513 = 1 ηxz1423 = η xz 2534 = η xz 1345 = η xz 2415 = η xz 3512 = 1 ηyz3425 = η yz 1235 = η yz 4513 = η yz 2314 = η yz 1524 = 1 (all other coefficients vanish) 12/19 Sunday, July 25, 2010 13/19 Same functional form as for physical (unprotected) qubits! Main effect of QEC: renormalization of coupling constant repeat for subsequent cycles Ū(N∆, 0) = v̄0(N∆, (N − 1)∆) v̄0((N − 1)∆, (N − 2)∆) . . . v̄0(∆, 0) rearrangement (separating intra versus inter logical qubit correlations) λ∗α = λα ∑ x ∑ β={x,z} ∑ i,j,k ηαβijk (λβ∆) 2 ∑ k"=0 |gα,|k||2 e−ik·(xi−xj)effective coupling constant v̄0(∆, 0) ≈ 1̄ + i∆ ∑ x ∑ α={x,z} (λ∗α + Γα)f β(x, 0) σ̄αx quantum evolution of logical qubit after N QEC cycles: Ū(T = N∆, 0) ≈ Tt ei R T 0 dt P x P α={x,z} λ ∗ α f α(x,t) σ̄αx Δ 2Δ 3Δ Time evolution (Cont.): Sunday, July 25, 2010 Upper bound to computation time - Information lost by a single logical qubit D ( ρR(T )− ρideal ) = √ |δσ+(T )|2 + 1 4 [δσz(T )]2 δσα(T ) = 〈σα(T )〉 − 〈σα〉 - Trace distance as a measure of information degradation D ( ρR(T )− ρideal ) = 1 2 Tr ‖ρR(T )− ρideal‖ ‖A‖ = √ A†A - Given a certain Dthresh, one can compute the available time Tmax by solving D ( ρR(Tmax)− ρideal ) = Dthresh 14/19 Sunday, July 25, 2010 Example: isolated logical qubit, no transversal coupling (λx =0) z Tmax Dthresh Dc t 1 0 ζ  > 0 ζ  = 0 ζ  < 0 z z D(t) 15/19 ζz ≡ 2(zz − sz)−D bath spatial dimension Long-time behavior controlled by the exponent “superohmic” “ohmic” “subohmic” Tmax ≈ ∆  ∞ ζz < 0 exp [ C Dthresh ( ω0 λ∗z )2] ζz = 0 Dzz/ζzthresh ω0∆ ( ω0 λ∗z )2zz/ζz 0 < ζz < 2zz ( 2pi k0L )(ζz−2zz) √Dthresh λ∗z∆ ζz > 2zz Sunday, July 25, 2010 Another example: logical qubit array (M qubits, array dimension Dq) 16/19 Notice: Different layers of concatenation only alter λα* (higher power in the bare λα) ζα ≡ 2(zα − sα)−D ζα ≡ 2(zα − sα)−D+Dq if intra-logical qubit correlations dominate if inter-logical qubit correlations dominate (sparse array) (dense array) compromise between array size, code effectiveness, and amplitude of coupling to environment M large ⇔ λ* small best case worst case marginal case Tmax ≈ ∆×  ∞ ζα < 0 exp [ B DthreshM(λ∗α/ω0) ] ζα = 0 1 ω0∆ [ Dthresh M(λ∗α/ω) ]zα/ζα 0 < ζα < zα ( 2pi k0L )ζα−zα Dthresh M(λ∗α∆) ζα > zα Sunday, July 25, 2010 Case 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” case sz = 1/2 Dq = 1 D = 1 ζz = 1 Tmax ≈ ω0 [ Dthresh M(λ∗z/ω0) ] J(ω) ∼ ω 17/19 zz = 1 D = 3 ζz = 0 marginal case J(ω) ∼ ω3 Tmax ≈ ∆ exp [ B Dthresh M(λ∗z/ω0) ]sz = 1/2 Sunday, July 25, 2010 Summary 18/19 ● 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, 2010 Resources: - 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! 19/19 Funding: Sunday, July 25, 2010

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