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Coherent feedback formulation of a continuous quantum error correction protocol
Coherent feedback formulation of a continuous quantum error correction protocol
Coherent feedback formulation of a continuous quantum error correction protocol
Coherent feedback formulation of a continuous quantum error correction protocol
Coherent feedback formulation of a continuous quantum error correction protocol
Coherent feedback formulation of a continuous quantum error correction protocol
Coherent feedback formulation of a continuous quantum error correction protocol
Coherent feedback formulation of a continuous quantum error correction protocol
Coherent feedback formulation of a continuous quantum error correction protocol
Coherent feedback formulation of a continuous quantum error correction protocol
Coherent feedback formulation of a continuous quantum error correction protocol
Coherent feedback formulation of a continuous quantum error correction protocol
Coherent feedback formulation of a continuous quantum error correction protocol
Coherent feedback formulation of a continuous quantum error correction protocol
Coherent feedback formulation of a continuous quantum error correction protocol
Coherent feedback formulation of a continuous quantum error correction protocol
Coherent feedback formulation of a continuous quantum error correction protocol
Coherent feedback formulation of a continuous quantum error correction protocol
Coherent feedback formulation of a continuous quantum error correction protocol
Coherent feedback formulation of a continuous quantum error correction protocol
Coherent feedback formulation of a continuous quantum error correction protocol
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Coherent feedback formulation of a continuous quantum error correction protocol

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Presentation at the "Stochastic processes at the quantum level" workshop/meeting at Aberystwyth University, Wales, UK, October 21-22, 2009.

Presentation at the "Stochastic processes at the quantum level" workshop/meeting at Aberystwyth University, Wales, UK, October 21-22, 2009.

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  • 1. Coherent-Feedback Formulation of a Continuous Quantum Error Correction Protocol arXiv:0907.0236 Hendra Nurdin (The Australian National University) Joint work with: Joe Kerckhoff Dmitri Pavlichin Hideo Mabuchi (Stanford University)
  • 2. Talk plan   Reminder of quantum error correction (QEC).   Continuous QEC.   Description of a scheme for coherent-feedback QEC for a simple 3 qubit bit flip code.   Concluding remarks.
  • 3. Quantum error correction (QEC)   Quantum error correction is essential for quantum information processing since qubits are susceptible to decoherence.   When decoherence alters the state of a qubit, a QEC algorithm acts to restore it to the state prior to decoherence.
  • 4. QEC principles   Main ingredient is to introduce reduncancy by encoding a logical qubit into a number of physical qubits.   Simple 3 qubit code: A logical qubit is encoded by 3 physical qubits. Logical qubit |0 >L encoded by 3 physical qubits |000 > and |1 >L encoded by |111 >.   A logical state a|0 >L + b|1 >L is encoded as a|000 > + b| 111 >. The 3 qubit codespace is C = span{|000 >,|111>}.
  • 5. The 3 qubit bit flip code   This simple code belongs to a class of QEC codes called stabilizer codes (Gottesman, PRA 54, 1862)   C = span{| 000 >,|111 >}   Correctable errors are single qubit bit flips X1, X2 or X3   Error is determined by measuring the parity Z1Z2 between qubits 1 and 2, and Z2Z3 between qubits 2 and 3. Measurement results called the error syndrome.   Error syndromes are: {1,1} (no error), {-1, 1} (qubit 1 has flipped), {1,-1} (qubit 3 has flipped) and {-1,-1} (qubit 2 has flipped)
  • 6. Continuous QEC   Most proposed QEC schemes are discrete. Error detection and recovery operations are done periodically with a sufficiently small period.   In continuous QEC, the idea is to detect and correct errors continuously as they occur. Suitable for low level continuous time differential equation based models.   Continuous QEC using continuous monitoring and measurement-feedback: Ahn, Doherty & Landahl et al, PRA 65, 042301; Ahn, Wiseman & Milburn, PRA 65, 042301; Chase, Landahl & Geremia, PRA 77, 032304.
  • 7. Why coherent-feedback?   Not necessary to go up to the “macroscopic” level and have interfaces to electronic circuits for measurements. Not limited by bandwidth of electronic devices.   No classical processing required and avoids challenges imposed by the requirement of such processing; e.g., numerical integration of nonlinear quantum filtering equations in real-time.   Entirely “on-chip” implementation; a controller can be on the same hardware platform as the controlled quantum system. In particular, in solid state monolithic circuit QED.
  • 8. Quantum network notation and operations   Open Markov quantum system G = (S,L,H).   Concatenation product S2 0 L2 G2 G1 = (S2 , L2 , H2 ) (S1 , L1 , H1 ) = , , H1 + H2 0 S1 L1   Series product G2 G 1 = (S2 , L2 , H2 ) (S1 , L1 , H1 ) = (S2 S1 , S2 L1 + L2 , H1 + H2 + {L† S2 L1 }) 2 Gough & James, IEEE-TAC (to appear), 2009, arXiv:0708.4483
  • 9. error correction and trogen vacancy center in diamond, etc. , are interrogated se- architectures are es- quentially by a coherent optical probe with amplitude of quantum informa- similar arrangements have previously been considered in the research in these ar- context of quantum information science 10 . A qubit is en- models for quantum coded in the ground states and of the intracavity Continuous parity measurement erhaps more familiar atom; an optical transition between and the excited state theorists, there has e is coupled strongly to a quantized cavity mode with ey ideas 2 to the vacuum Rabi frequency g. For simplicity we assume atomic quation-based mod- selection rules such that e decays only to , with excited- context of ab initio ntribute to a line of √ √ rs 3–6 and broad- dU (t) = ( κb1 + κb2 + α)dA∗ (t) − h.c. hich focuses on con- √ √ √ √ abilizer coding and ( κb1 + κb2 + α)∗ ( κb1 + κb2 + α) s approach is attrac- − dt fundamentally con- 2 2 uppression with for- √ γ ∗ ontrol theory 7 . It + γσ (i) dB ∗ (t) − h.c. − σ σdt mentation advantage 2 k=1 s in that continuous out the need for ex- 2 κ this of course relies + (b∗ b2 − h.c.)dt + g (σ (i)∗ bi − h.c.)dt emolition syndrome 2 1 i=1 rimentally favorable √ aightforward imple- ¯ α κ y measurement suf- + (b1 + b2 ) − h.c. dt U (t) he quantum bit-flip 2 ectrodynamics cav- of our scheme both ¯ dU (t) = ((Π12 − I)dΛ(t) + αΠ12 dA∗ (t) − αdA(t) ¯ ical simulation and a adiabatic elimina- |α|2 ¯ − dt U (t) oherent-state optical 2 qubits and exploits ace of clocked quan- lim ¯ lim (U (t) − U (t))ψ = 0 for all ψ . The strength of the g→∞ α, κ → ∞ s be modulated eas- √ = const α stment of the power FIG. 1. Color online Schematic depiction of two cavities κ Kerckhoff, Bouten, Silberfarb & driven sequentially by a resonant laser beam. A three-level atom is in the reduced Hilbert space. ementation is shown trapped inside each cavity, and identical atom-cavity dynamics ap- aining a single three- Mabuchi, PRA 79, 024305 ply in each. After probing both cavities, the laser light is directed to kali-metal atom, ni- a homodyne receiver. But cannot include bit flip errors! 024305-1 ©2009 The American Physical Society
  • 10. Modified parity measurement model   Full single atom-cavity-field model with bit flip errors: √ 1 √ √ dU (t) = ( κb + α)dA1 (t) − h.c. − ( κb + α) ( κb + α)dt ∗ ∗ 2 √ Γ ∗ + ΓσX dA∗ (t) − h.c. − σX σX dt 2 2 √ γ ∗ + γσdA3 (t) − h.c. − σ σdt + g(σ ∗ b − h.c.)dt U (t) ∗ 2   Reduced atom-field model with bit flip errors: ¯ dU (t) = (Z − I)dΛ11 (t) + αZdA∗ (t) − αdA1 (t) 1 ¯ √ 1 ¯ + ΓXdA∗ (t) − h.c. − (|α|2 + γ)dt U (t) 2 2 lim ¯ (U (t) − U (t))ψ for all ψ in the reduced Hilbert space. g, κ → ∞ g κ = const
  • 11. Modified parity measurement model   The reduced model can be written as: Z 0 , √0 ,0 I, α ,0 0 1 ΓX 0   Reduced model for two coherently driven cavities:               Z2 0 0 0 Z1 0 0 √0 α √0  0 1 0  ,   , 0  0 1 0  ,  ΓX1  , 0 I,  0  , 0 0 0 1 ΓX2 0 0 1 0 0      Z2 Z1 0 0 αZ2 Z1 √ =  0 1 0  ,  √ΓX1  , 0 0 0 1 ΓX2
  • 12. The “bare bones” of it " = a 0 A 0 B 0C ! + b 1A1B1C ! A |e ! g Z AZ C B Z Z ! A " – – – – |1 r La se B + C se La |0 r + A M R XA XB XC 1 ZAZB – – + + ZAZC – + – + . L. O B C O L. . •  Coherent lasers drive atomic Raman transitions between the two ground states of the atom to correct bit flips. •  To make it work, need more than this …
  • 13. n are replaced by unitary processing of the probe beams and coherent feedback to its. We exploit a limit theorem for quantum stochastic differential equations to an- eedback networks based on the bit-flip/phase-flip code, obtaining simple closed-loop s with only four Hilbert-space dimensions required for the controller. Our approach to physical implementations that feature solid-state qubits embedded in planar circuits. The actual scheme 03.67.Pp,02.30.Yy,42.50.-p,03.65.Yz quantum error correction !" and measurement of syn- ntral to the modern field of e. Although substantial work nsions and refinements of ab- orts [2, 3, 4] have begun to fo- !# task of developing implemen- ! " !!!" $# ly the fundamental principles ally accommodate the struc- !"" models. Such new approaches $% quantum memories that make physical resources and intro- ted abstractions for quantum !"" plement those we have inher- r science. !% recently proposed [5] a cavity cavity QED) implementation ty measurement required for FIG. 1: Schematic diagram of a coherent-feedback quantum 9, 10, 11, 12] versionQuantummemory showing qubits-in-cavities (Q1,facilitate switching   of the switches R1, R2 inserted to Q2 and Q3), circula- to higher or phase-flip codes, which be- amplitude bit flip correcting Raman lasers. (R1 and R2). tors, beam-splitters, steering mirrors and relays el in a strong coupling limit. The calculation we present is based on a modified arrange- further step of describing co- ment that leads to the same closed-loop master equation but hat realize quantum memories factorizes into four simple sub-networks. e encoded qubit is suppressed
  • 14. Qubit atomic level scheme ! !"" !!" ! !#" ! !$" !&" !%"
  • 15. Raman transitions with ac Stark shift compensation ac Stark shift compensation   Given by the terms: √ 1 1, γ (σgr + σgG ) , ∆ Πr − ΠG 2 Raman transitions √ 1 1, γ (σhr + σhH ) , ∆ Πr − ΠH 2   Together with coupling to probe laser becomes: Z 0 √ 1 , 0, 0 1, γ (σgr + σgG ) , ∆ Πr − ΠG 0 1 2 √ 1 1, γ (σhr + σhH ) , ∆ Πr − ΠH 2
  • 16. QSDE model for switch/relay (+* (,* $%#!&' !"# ()* !!" -%$%#!&' !"# -%$%#!&' !"" - - $%#!&' $%# &' !"# 34516 012 .!/%- .!/%- &' &' !"# !$" !#" •  Simple QSDE model: Πg −Πh βΠg Πh −σhg , ,0 , 0, 0 −Πh Πg −βΠh −σgh Πg Detailed physical model: H. Mabuchi, arXiv:0907.2720
  • 17. QEC network description !"   (Modified) half network diagram 3 (+* (,* $%#!&' !"# ()* !!" ;88 !# -88 9: ! " !!!" $# -%$%#!&' !"# -%$%#!&' !"" - - $%#!&' $%# &' 9< -87 ;8< !"# 34516 !"" 012 $% .!/%- .!/%- &' &' !"# !$" !#" ;77 !"" ( * ( * ( * ;78 &' !!! " FIG. 3: Details of the ‘set-reset flip-flop cross-over relay’ com- 98 !% ponent model [22]: (a) input and output ports, (b) coupling of input/output fields to resonant modes of two cavities, and !"#7 ;<7 & (c) relay internal level diagram. Half it would be possible to f (p=probe path, f=feedback path): but in a practical implementation network Gp G FIG. 4: Signal-flow diagram of the half-network Gp Gf . utilize double-pole double-throw relays that switch not Gp = R12 B3 ((Q we Q21 ) (1, 0, 0)) B1 , only the Raman beam itself but also an auxiliary beam 13 proceed to assemble the full network√model N = Gp Gf G Gf G . Here GΓ = ( ΓX , 0, 0) Gf = (Q11 chosen soQ22√ΓX (B0) p 2√ΓX0,0,Γ0) describes bit-flip 1decoher- whose frequency, polarization and amplitude is Q 32 ( ) , 0, 5 ( (1, , 0)) that it provides equivalent compensation. 2 3 Some details of our relay model are displayed in (1, 0, 0)) (R11 Fig. 3 ence of the register qubits, and the component connec- [22]. In electrical engineering parlance, the devices we tions for Gp Gf are shown in Fig. 4 (note that as the utilize correspond to open quantum systems versions of a signal routing shown in Fig. 1 yields rather unwieldy cross-over relay driven by a set-reset flip-flop. Each relay network calculations, we are here adopting a modified
  • 18. QEC network master equation   The QEC network master equation in the limit that Δ, β  ∞ with β2/ Δ constant is: 7 1 ρt = −i[H, ρt ] + ˙ Li ρt L∗ − {L∗ Li , ρt } , i=1 i 2 i where (Ω ≡ β 2 /γ∆), √ (R2 ) √ (R1 ) H = 2ΩΠ(R1 ) Πh g X1 + 2ΩΠh Π(R2 ) X3 g −ΩΠ(R1 ) Π(R2 ) X2 , g g α L1 = √ σhg 1 ) (1 + Z1 Z2 ) + Π(R1 ) (1 − Z1 Z2 ) (R h , 2 α L2 = √ σgh 1 ) (1 − Z1 Z2 ) + Π(R1 ) (1 + Z1 Z2 ) (R g , 2 α L3 = √ σhg 2 ) (1 + Z3 Z2 ) + Π(R2 ) (1 − Z3 Z2 ) (R h , 2 α L4 = √ σgh 2 ) (1 − Z3 Z2 ) + Π(R2 ) (1 + Z3 Z2 ) (R g , 2 √ √ √ L5 = ΓX1 , L6 = ΓX2 , L7 = ΓX3 .
  • 19. Ideal QEC network performance 1 0.995 0.99 Fidelity 0.985 0.98 "=0 "=10 "=30 0.975 "=60 "=90 0.97 0 0.002 0.004 0.006 0.008 0.01 !t Fidelity = < ψ0 |ρ(t) |ψ0 >; Simulation parameters: Γ = 0.01, α = 10, |ψ0 > = (| ggg > - |hhh >)/√2; ρ(0) = |ψ0 > < ψ0 |.
  • 20. Ideal QEC network performance -./01234,2540167819:4/28;<4!=>! *+,-./0-1213,.41156)/78.!9:! "#&' "#'$!$ "#&+ "#'$!% "#&* "#&$ "#'$!% "#&% "#&) "#'$!( ; ? "#&& "#'$!( "#&! "#& "#'$!& "#!( "#!' "#'$!& % % & ! & ! "#$ "#$ " " " " !& !& !"#$ !"#$ " !% !! !% !! , " ) Fidelity = < ψ0 |ρ(T) |ψ0 >; Simulation parameters: Γ = 0.01, α = 10,β = 30, |ψ0 > = a |ggg > + √(1-a2)eiθ|hhh >, a in [-1,1], θ in [-π,π]; ρ(0) = |ψ0 > < ψ0 |.
  • 21. Concluding remarks   We have proposed a coherent-feedback QEC scheme for a simple 3 qubit QEC code that protects against single bit flip errors; can be easily adapted to a 3 qubit phase flip code.   Simulations of the QEC network master equation indicates the scheme can slow down decoherence due to single bit flips.   Ideas for the future: Adaptation to more complex stabilizer codes, but necessarily also with more complex quantum circuits. Perhaps also to non-stabilizer codes (more challenging?).

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