design and analysis of autonomous quantum memories based on
TRANSCRIPT
Design and analysis of autonomous quantum memories based on coherent feedback control Hideo Mabuchi, Stanford University
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ARO, NSF, (DARPA-MTO)
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• Idealization applies in the limit of large g, κ with g/κ fixed (QSDE limit theorem)
• Dispersive version with smaller phase shifts, A. B. Nielsen, PRA 81, 012307 (2010)
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Physical model of continuous Z measurement L-M. Duan and H. J. Kimble, Phys. Rev. Lett. 92, 127902 (2004)
A. B. Nielsen, PRA 81, 012307 (2010)
g=104, κ=23.4 MHz
jª0i / (ju1i+ jd1i) (ju2i+ jd2i) / (ju1u2i+ jd1d2i) + (ju1d2i+ jd1u2i)
J. Kerckhoff, L. Bouten, A. Silberfarb and HM, Phys. Rev. A 79, 024305 (2009)
Parity measurement via sequential scattering
Discrete analogy: an odd but valid bit-flip circuit
Syndrome measurement “rate” τ -1 limited by time required for CCNOT gates
Further decrease in τ requires CCNOT! CCROT, statistical syndrome measurement
Continuous limit obtained by taking τ ! 0 with fixed “information rate” κ
τ τ
Continuous syndrome measurement:
• ancilla qubit stream → laser beam • gates → Hamiltonian couplings • time interval → laser intensity
→ reduced complexity ← ions/Q-dots/NV-C’s/circuit QED, …
‘Error-state graph’ for the bit-flip code
• Continuous QND syndrome measurement ) Markov jump dynamics for error state • Mapping of error state to syndrome is degenerate
Assertion (numerically testable via comparison to SME): optimal filter for the error state can be derived as a Wonham filter (Wonham, 1965) for the induced Markov jump process of the error state
Error-state tracking with a Wonham filter
• nonlinear filter, much studied in “hybrid stochastic” control theory Filter stability results: P. Chigansky and R. van Handel, “Model robustness of finite state nonlinear filtering over the infinite time horizon,” Ann. Appl. Probab. 17, 688 (2007).
error-state observer
plant
M1, M2
Ramon van Handel and HM, quant-ph/0511221
Jump dynamics of the error state
Continuous syndrome measurement localizes the error state; bit-flip decoherence induces jump-like transitions
Finite measurement strength/sensitivity gives rise to detection delay and quiescent fluctuations
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Coherent-feedback control
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Measurement-feedback control
Feedback “signal” is quantum Feedback “signal” is classical
• We are generally interested in (semi-)coherent quantum plant dynamics in both cases
• We are generally interested in real-time feedback, i.e., faster than open-loop T1
Coherent feedback vs. measurement feedback
Coherent-feedback quantum memory “schematic” J. Kerckhoff, H. Nurdin, D. Pavlichin and HM, PRL 105, 040502 (2010)
Q2
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Q3
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R1
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Network component models J. Kerckhoff, H. Nurdin, D. Pavlichin and HM, PRL 105, 040502 (2010)
J. Kerckhoff, L. Bouten, A. Silberfarb and HM, Phys. Rev. A 79, 024305 (2009) H. Mabuchi, Phys. Rev. A 80, 045802 (2009)
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Probe interaction: Z- (Duan-Kimble/Nielsen) or X-parity (Kerckhoff)
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Simplified four-state relay model
Idealized/abstracted component model, obtained rigorously in the small-volume limit:
B3
B1
R12
R11 B5
Q32
Q11
Q13
Q22
Q21
Coherent-feedback network “wiring diagram” J. Kerckhoff, H. Nurdin, D. Pavlichin and HM, PRL 105, 040502 (2010)
J. Gough and M. R. James, to appear in IEEE Trans. Automat. Contr. (2009); arXiv:0707.0048v3 L. Bouten, R. van Handel and A. Silberfarb, Journal of Functional Analysis 254, 3123 (2008)
Gp = R12 / B3 / ((Q13 / Q21) (1; 0; 0)) / B1
Gf = (Q11 Q32 Q22) / (B5 2 (1; 0; 0)) / (R11 (1; 0; 0))
N = Gp Gf Gp0 Gf
0 G¡
Closed-loop master equation; simulations J. Kerckhoff, H. Nurdin, D. Pavlichin and HM, PRL 105, 040502 (2010)
J. Kerckhoff, D. S. Pavlichin, H. Chalabi and HM, New J. Phys.13, 055022 (2011)
_½t = ¡i[H; ½t] +
7X
i=1
µLi½tL
¤i ¡
1
2fL¤iLi; ½tg
¶
H =p
2¦(R )g ¦
(R )h X1 +
p2¦
(R )h ¦(R )
g X3 ¡ ¦(R )g ¦(R )
g X2
L1 =®p2f¾(R )
hg (1 + Z1Z2)
+¦(R )h (1¡ Z1Z2)g
L2 =®p2f¾(R )
gh (1¡ Z1Z2)
+¦(R )g (1 + Z1Z2)g
L3 =®p2f¾(R )
hg (1 + Z3Z2)
+¦(R )h (1¡ Z3Z2)g
L4 =®p2f¾(R )
gh (1¡ Z3Z2)
+¦(R )g (1 + Z3Z2)g
PZT2 HWP PBS
HWP PBS
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• optical feedback • dynamic compensator
9 MHz 14 cm
7 MHz 49 cm
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Coherent-feedback control James, Nurdin and Petersen, IEEE-TAC 53, 1787 (2008); HM, Phys. Rev. A 78, 032323 (2008)
β
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Nonlinear dynamic controller
H. Mabuchi, Appl. Phys. Lett. 98, 193109 (2011) Coherent feedback control of optical bistability
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Coherent feedback control in nanophotonic circuits PLINC: Photonic Logic via Interferometry with Nonlinear Components
PLINC exploits cavity-enhanced nonlinearity and circuit-scale
optical coherence to implement attojoule photonic logic
PLINC is a natural scheme for near-future integrated
nanophotonics, testable today using single-atom cavity QED
PLINC circuit theory = coherent-feedback quantum control
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1. Develop QHDL, a subset of industry-standard VHDL for the specification of PLINC circuits
2. Develop software for compiling QHDL into rigorous quantum optical models
3. Use QHDL toolbox + high-performance numerical simulation for analysis and design of functional circuits
4. Validate key coherent feedback concepts in single-atom cavity QED experiments
Schematic Capture
Quantum Hardware Description Language
Quantum Network Model
Net
wor
k O
ptim
izatio
n
Comprehensive software package for automated network model analysis (based on custom code and open source software)
Visual circuit design allows for creation of large, complex networks.
Plain text representation of circuit model. Can easily be published / shared among collaborators.
Compile QHDL circuit file into mathematical model. Analyze/reduce model. Set up and run numerical simulations.
Quantum optical circuit modeling workflow N. Tezak, A. Niederberger, D. S. Pavlichin, G. Sarma and HM, arXiv:1111.3081
Hierarchical Design
SR NAND latch
NAND gate
Schematic Capture Export to QHDL Compute Model
Visual circuit design
• Subset of VHDL (VHSIC Hardware Description Language)
• Specification of a circuit in terms of:
input/output ports, model parameters
circuit schematic/netlist (structural VHDL)
subcomponents as black boxes (Hierarchical Design Principle)
Schematic Capture Export to QHDL Compute Model
A Quantum Hardware Description Language
Schematic Capture Export to QHDL Compute Model
Completely determines model
• Computer algebra approach for analytic model analysis/reduction/ optimization
• Automatically set up numerical simulation of model dynamics
Visual representation of algebraic network expression (automatically generated)
Quantum optical circuit model