Design and analysis of autonomous quantum memories based on coherent feedback control Hideo Mabuchi, Stanford University

R SET in

RESET in

POWER in

OUT

OUT

R

OUT

OUT

POWER in

SET in

RESET in

jei

jsi

jgijhi

set

pow

er

ARO, NSF, (DARPA-MTO)

_½t = ¡i[H; ½t] +

7X

i=1

µLi½tL

¤i ¡

1

2fL¤i Li; ½tg

¶

Continuous syndrome detection Coherent feedback

+ (S,L,H) modeling

Autonomous/Embedded

Quantum error correction “circuits”

http://www.rfdesignline.com/howto/209400216

J. Vuckovic

Continuous syndrome detection Coherent feedback

+ (S,L,H) modeling

Autonomous/Embedded

• 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)

j®i 7! j ¡ ®i

j®i 7! j®i

j®i 7! j®i

!c

j+i

!c

j¡i

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

Continuous syndrome detection Coherent feedback

+ (S,L,H) modeling

Autonomous/Embedded

w y

u z

PLA

NT

CO

NTR

OLL

ER

Coherent-feedback control

w y

u z

PLA

NT

¯

laser ¢A ¢'

CO

NTR

OLL

ER

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

Q1

Q3

j2®i

R1

R2

j¯i

j¯i

Continuous syndrome detection Coherent feedback

+ (S,L,H) modeling

Autonomous/Embedded

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)

R SET in

RESET in

POWER in

OUT

OUT R

OUT

OUT POWER

in

SET in

RESET in

jei

jsi

jgijhi

set

pow

er

¢jri

jei

jgijhi

¢jri

jei

jgijhi

j®i 7! j ¡ ®i

j®i 7! j®i j+i

j®i 7! j ¡ ®ij¡i

Probe interaction: Z- (Duan-Kimble/Nielsen) or X-parity (Kerckhoff)

jgijhi

jei jri

R SET in

RESET in

POWER in

OUT

OUT

OUT

OUT POWER

in

SET in

RESET in

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

Modeling propagation losses G. Sarma (unpublished)

J. Vuckovic

Master equation simulations with losses G. Sarma (unpublished)

PZT2 HWP PBS

HWP PBS

w y

u z

PLAN

T

CON

TRO

LLER

PD

PZT1

G

z w

K

u

y

• optical feedback • dynamic compensator

9 MHz 14 cm

7 MHz 49 cm

des

ign

°c

=°p¡

2(k

1+

k4)

Coherent-feedback control James, Nurdin and Petersen, IEEE-TAC 53, 1787 (2008); HM, Phys. Rev. A 78, 032323 (2008)

β

ϕ( ) ϕ0

Nonlinear dynamic controller

H. Mabuchi, Appl. Phys. Lett. 98, 193109 (2011) Coherent feedback control of optical bistability

In1

In2 Out1

w x

¯

¯ ’

Out2

µ

µ ’

¼/4

¼/4

0 100 200 300 400 500 600 700 800 900 10000

5

10

15

20

25

30

35

40

45

50

-5 -4 -3 -2 -1 0 1 2 3 4 5-5

-4

-3

-2

-1

0

1

2

3

4

5

0 1 2 3 4 5 6 7 8 9 10

0

1

2

3

4

5

6

7

8

9

10

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

In1

In2 Out1

w x

¯

¯ ’

Out2

µ

µ ’

¼/4

¼/4

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

Continuous syndrome detection Coherent feedback

+ (S,L,H) modeling

Autonomous/Embedded

www.rfdesignline.com