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

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

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1/19  For How Long is it Possible to Quantum Compute? Eduardo Mucciolo (University of Central Florida, USA)  Eduardo Novais (University of ABC, Brazil)  Harold Baranger (Duke University, USA)  QAMF WORKSHOP Vancouver, UBC - July 23-25, 2010 Sunday, July 25, 2010  2/19  Overview  ● Motivation and previous work ● Hamiltonian formulation of a fault-tolerant quantum processor ● Results  Sunday, July 25, 2010  Motivation:  3/19  Protect Quantum Information from Decoherence  Some standard strategies: 1) Decoherence free-subspaces 2) Topological systems 3) Dynamical decoupling likely the most versatile and universal approach  4) Quantum error correction  - encode information - measure ancilla qubits - correct for errors  Major result: The “threshold theorem” Provided Providedthat thatthe thenoise noiselevel levelisisbelow belowaacertain certaincritical critical value, can bebe protected value,quantum quantuminformation computation can proceedindefinitely “indefinitely” resilient QC (”low-T” regime)  noisy QC (”high-T” regime)  εc Sunday, July 25, 2010  ε  (= single-qubit error probability)  4/19  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.  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  5/19  Novais, EM, Baranger [PRL 98, 040501 (2007); PRA 78 012314 (2008)]  Hamiltonian  qubits  H = Hcomputer + Hbath + V  qubit-bath interaction:  V =  λα fα (x) σα (x) x,α  environmentʼs “free” Hamiltonian (gapless)  environmental degrees of freedom  Three important parameters (critical environment) • spatial dimension D • bath mode velocity c • dynamical exponent z  fα (x1 , t1 ) fα (x2 , t2 )  env  ∼O  1 1 , 2δ α |x1 − x2 | |t1 − t2 |2δα /z  Main hypothesis only one qubit per hypercube  - defines a single-qubit error probability  time  ∆ space  Sunday, July 25, 2010  - maps into an impurity problem for short times.  ξ = (c∆)1/z  ∆ = period of QEC cycle  6/19  Main result of previous work:  Novais, EM, Baranger [PRL 98, 040501 (2007); PRA 78 012314 (2008)]  D + z < 2δα  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.  local error probability ( ε )  “temperature”  qubits + environment strongly entangled; noisy quantum computer (efficiently simulated by a Turing machine) weak entanglement among qubits "upper critical dimension" "lower critical dimension"  not possible to compute  ✓  ✓  ?  threshold theorem needs ( unkown ) new derivation  ?  ✓  traditional threshold theorem  D+z 2  qubits + environment strongly entangled Sunday, July 25, 2010  qubits + environment weakly entangled; strong entanglement among qubits  “correlations” strong  δ  weak  Current Work:  7/19  Novais, EM, Baranger [arXiv: 1004.3247]  1) Change the question: Given a certain desired error tolerance, for long can we compute using QEC? 2) Avoid the hypercube assumption.  - standard quantum error correction code (circuit model QC).  “friendly” assumptions  - 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  8/19  Time-Evolution under QEC: start  end QEC  0 syndrome extraction  QEC  QEC  QEC  QEC  Δ  2Δ  3Δ  4Δ  s1  s2  s3  s4  (Ν−2)Δ (Ν−1)Δ  sN-2  Syndrome history tree  non-error syndrome path  bosonic  Sunday, July 25, 2010  QEC  sN-1  t ΝΔ  sN  9/19  Taking into account the QEC code structure: Example: 5-qubit code (corrects for one-qubit errors) 1 ¯ |0 = [|00000 + |10010 + |01001 + |10100 + |01010 − |11011 − |00110 − |11000 4 −|11101 − |00011 − |11110 − |01111 − |10001 − |01100 − |10111 + |00101 ]  logical qubit  1 ¯ |1 = [|11111 + |01101 + |10110 + |01011 + |10101 − |00100 − |11001 − |00111 4 −|00010 − |11100 − |00001 − |10000 − |01110 − |10011 − |01000 + |11010 ] stabilizers and logical operations  g1 = xzzx1 g2 = 1xzzx  45 = 1024 error operators 16 possible syndromes  g3 = x1xzz g4 = zx1xz ¯ = xxxxx X Y¯ = yyyyy Z¯ = zzzzz  Sunday, July 25, 2010  but only 64 error operators can lead to a non-error syndrome  ¯ 1 g1 g2 g3 g4 h1 h2 h3 h4 h5 h6 h7 h8 h9 h10 h11  ¯ 1 11111 xzzx1 1xzzx x1xzz zx1xz xy1yx 1zyyz yyz1z xxy1y z1zyy yxxy1 1yxxy zyyz1 y1yxx yz1zy zzx1x  ¯ X xxxxx 1yy1x x1yy1 1x1yy y1x1y 1zxz1 xyzzy zzyxy 11zxz yxyzz z11zx xz11z yzzyx zxz11 zyxyz yy1x1  Y¯ yyyyy zxxzy yzxxz zyzxx xzyzx z1y1z yx11x 11xyx zz1y1 xyx11 1zz1y y1zz1 x11xy 1y1zz 1xyx1 xxzyz  Z¯ zzzzz y11yz zy11y yzy11 1yzy1 yxzxy z1xx1 xx1z1 yyxzx 1z1xx xyyxz zxyyx 1xx1z xzxyy x1z1x 11yzy  10/19  Model for the critical environment: physical qubits  HI =  λα f α (x) σ α (x)  qubit-bath interaction  x α={x,z}  ∗ e−ik·x aα,k gα,|k| eik·x a†α,k + gα,|k|  f α (x) = bosonic bath  k=0  ωα,|k| a†α,k aα,k  HB =  aα,k , a†β,k  = δαβ δk,k  α={x,z} k=0  form factor: dispersion relation:  2  2sα  |gα,|k| | ∼ |k|  zα  ωα,|k| ∼ |k|  spectral function  Jα (ω) ∼  2 λα  J(ω) = k  bosonic  Sunday, July 25, 2010  ω  (D+2sα −zα )/zα  |gk |2 δ(ω − ωk )  11/19  Time evolution: First QEC cycle system+bath evolution operator  U (∆, 0) = Tt e−i  R∆ 0  dt HI (t)  =1−i  ∆ 0  1 dt HI (t) − 2  ∆  t  dt 0  dt HI (t) HI (t ) + . . .  0  lowest order perturbation theory (short times / small coupling)  U (∆, 0) ≈ 1 − i  λα ∆ f α (x, 0) σxα α=x,z x  λα ∆  1  expansion parameter  syndrome extraction and error correction operation:  U (∆, 0) −→ vs (∆, 0) evolution operator for logical qubits after a non-error syndrome  v¯0 (∆, 0) ≈ ¯1 + i∆3  αβ ηijk λα λ2β f α (xi , 0) f β (xj , 0) f β (xk , 0) σ ¯xα + O(λ5 ) x α,β={x,z} i,j,k  numerical coefficients O(λ2) renormalization dropped Sunday, July 25, 2010  12/19  Determining the expansion coefficients:  operators leading to a non-error syndrome ¯ 1 g1 g2 g3 g4 h1 h2 h3 h4 h5 h6 h7 h8 h9 h10 h11  ¯ 1 11111 xzzx1 1xzzx x1xzz zx1xz xy1yx 1zyyz yyz1z xxy1y z1zyy yxxy1 1yxxy zyyz1 y1yxx yz1zy zzx1x  ¯ X xxxxx 1yy1x x1yy1 1x1yy y1x1y 1zxz1 xyzzy zzyxy 11zxz yxyzz z11zx xz11z yzzyx zxz11 zyxyz yy1x1  Y¯ yyyyy zxxzy yzxxz zyzxx xzyzx z1y1z yx11x 11xyx zz1y1 xyx11 1zz1y y1zz1 x11xy 1y1zz 1xyx1 xxzyz  Z¯ zzzzz y11yz zy11y yzy11 1yzy1 yxzxy z1xx1 xx1z1 yyxzx 1z1xx xyyxz zxyyx 1xx1z xzxyy x1z1x 11yzy  Notice: 3 or 5 insertions of the interaction only (one per physical qubit, at most)  Sunday, July 25, 2010  xy xy xy xy xy η1524 = η1235 = η2314 = η3425 = η4513 xz xz xz xz xz η1423 = η2534 = η1345 = η2415 = η3512 yz yz yz yz yz η3425 = η1235 = η4513 = η2314 = η1524  (all other coefficients vanish)  = 1 = 1 = 1  13/19  Time evolution (Cont.): rearrangement (separating intra versus inter logical qubit correlations)  v¯0 (∆, 0) ≈ ¯1 + i∆  (λ∗α + Γα )f β (x, 0) σ ¯xα x α={x,z} αβ ηijk (λβ ∆)2  λ∗α = λα  effective coupling constant  x β={x,z} i,j,k  k=0  |gα,|k| |2 e−ik·(xi −xj )  repeat for subsequent cycles  ¯ (N ∆, 0) = v¯0 (N ∆, (N − 1)∆) v¯0 ((N − 1)∆, (N − 2)∆) . . . v¯0 (∆, 0) U  quantum evolution of logical qubit after N QEC cycles:  ¯ (T = N ∆, 0) ≈ Tt e U  i  RT 0  dt  P P x  ∗ α α λ f (x,t) σ ¯ α x α={x,z}  Same functional form as for physical (unprotected) qubits! Main effect of QEC: renormalization of coupling constant Sunday, July 25, 2010  Δ 2Δ  3Δ  14/19  Upper bound to computation time - Trace distance as a measure of information degradation  D ρR (T ) − ρideal  1 = Tr ρR (T ) − ρideal 2  A =  √  A† A  - Information lost by a single logical qubit  D ρR (T ) − ρideal =  |δσ + (T )|2  1 z + [δσ (T )]2 4  δσ α (T ) = σ α (T ) − σ α  - Given a certain Dthresh, one can compute the available time Tmax by solving D ρR (Tmax ) − ρideal = Dthresh  Sunday, July 25, 2010  15/19  Example: isolated logical qubit, no transversal coupling (λx =0)  Long-time behavior controlled by the exponent  D(t) 1  ζz ≡ 2(zz − sz ) − D  Dthresh  ζz> 0  ζz= 0  Dc  ζz< 0  bath spatial dimension  0  Tmax ≈ ∆   ∞            exp C Dthresh      ω0 λ∗ z  2   zz /ζz 2zz /ζz  Dthresh  ω 0   ∗ ω ∆ λ  0 z        (ζz −2zz ) √  Dthresh  2π k0 L  Sunday, July 25, 2010  ζz < 0  λ∗ z∆  ζz = 0  0 < ζz < 2zz ζz > 2zz  Tmax  “superohmic” “ohmic”  “subohmic”  t  16/19  Another example: logical qubit array (M qubits, array dimension Dq)  ζα ≡ 2(zα − sα ) − D  (sparse if intra-logical qubit correlations dominate array)  ζα ≡ 2(zα − sα ) − D + Dq  if inter-logical qubit correlations dominate (dense  Tmax ≈ ∆ ×   ∞          exp B                   1 ω0 ∆  2π k0 L  array)  Dthresh M (λ∗ α /ω0 )  Dthresh M (λ∗ α /ω) ζα −zα  zα /ζα  Dthresh M (λ∗ α ∆)  ζα < 0  best case  ζα = 0  marginal case  0 < ζα < zα ζα > zα  worst case  compromise between array size, code effectiveness, and amplitude of coupling to environment  M large  λ* small  Notice: Different layers of concatenation only alter λα* (higher power in the bare λα) Sunday, July 25, 2010  17/19  Case 1: 2D sparse logical qubit array; level-0 concatenation GaAs double-quantum dot charge physical qubits bath: piezoelectric acoustic phonons (3D) sz = 1/2 J(ω) ∼ ω  3  zz = 1  ζz = 0  D=3  marginal case  Tmax  Dthresh ≈ ∆ exp B M (λ∗z /ω0 )  Case 2: 1D dense logical qubit array; level-0 concatenation superconductor charge qubits bath: electromagnetic (gate voltage) fluctuations (effectively 1D) sz = 1/2 zz = 1 J(ω) ∼ ω  D=1 Dq = 1  Sunday, July 25, 2010  ζz = 1 “bad” case  Tmax ≈ ω0  Dthresh M (λ∗z /ω0 )  18/19  Summary ● 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  19/19  Funding:  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! Sunday, July 25, 2010  

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