arbitrary control of entanglement between two ... › fws1 › files › 2011 › 11 ›...

Arbitrary Control of Entanglement Between Two Superconducting ResonatorsFrederick W. Strauch, Williams College

withKurt Jacobs, UMass Boston

Ray Simmonds, NIST (Boulder)

Research Corp., NSF

Phys. Rev. Lett. 105, 050501 (2010)

Tuesday, November 22, 2011

Outline! Superconducting Qubits & Resonators! NOON States! State-synthesis Algorithm

" JC Ladder => Fock-state Diagram" Stark-shifted Rabi Oscillations" NOON State Synthesis

! Quantum Routing of Entanglement on Oscillator Networks

2


3

C Ib

• Quantum oscillator involving the superconducting current of billions of Cooper pairs.• Spectroscopic transitions between energy levels can be probed by microwaves. • States are metastable, will tunnel through barrier.

|0!

|1!

|2!

|3!

Josephson Junction

"

100 µm

Supercurrent oscillates like a

pendulum!

Phase Qubit


4Sweep of bias current allows experimental control of energy levels.

Tunable Oscillator


5

IsolationQubit

50 µm

Sudeep Dutta et al. (Univ. Maryland)

Hypres Inc.

I (µ A)

f (GHz)

Each microwave transition is an excitation of the junction with an increased tunneling rate. Bright indicates a large number of tunneling events, dark a small number of events.

1 # 2

0 # 1

2 # 33 # 4

0 $ 21 $ 32 $ 4

Multi-Photon Spectroscopy


5

IsolationQubit

50 µm

Sudeep Dutta et al. (Univ. Maryland)

Hypres Inc.

I (µ A)

f (GHz)

Each microwave transition is an excitation of the junction with an increased tunneling rate. Bright indicates a large number of tunneling events, dark a small number of events.

1 # 2

0 # 1

2 # 33 # 4

0 $ 21 $ 32 $ 4Single photon

Two photon

0 # 1

0 $ 2

Multi-Photon Spectroscopy


Coupling Qubits by Cavities

Qubit A

Qubit B

Resonator

“Coherent quantum state storage and transfer between two phase qubits via a resonant cavity”, M. Sillanpaa, J. I. Park, and R. W. Simmonds, Nature 449, 438 (2007)


Coupling Qubits by Cavities

Qubit A

Qubit B

Qubit

%0!

%1! Resonator

%0! %1! %2! %3!

Resonator

“Coherent quantum state storage and transfer between two phase qubits via a resonant cavity”, M. Sillanpaa, J. I. Park, and R. W. Simmonds, Nature 449, 438 (2007)

Qubit

%0!

%1!


Arbitrary Control of Resonator! Martinis Group, UC Santa Barbara

7


State-Synthesis Algorithm

! Law & Eberly developed a scheme to program an arbitrary state of a single harmonic oscillator mode by coupling to a two-level system.

! Climbing Jaynes-Cummings Ladder one rung at a time. 8


State-Synthesis Experiments

9

Hofheinz et al., Nature 454 310 (2008)

Hofheinz et al., Nature 459 546 (2009)


Big Question: Qubits or Resonators! Is it better to use harmonic oscillators

(resonators) or qubits for quantum computing?

! Common wisdom: It depends..." Use qubit nonlinearity to encode

information." Use oscillator coherence to store

information." Use resonators to couple qubits." Use ? to readout information." Use ? to process information.

10


Little Question:

Qubit Coupler

Resonator A

Resonator B

SchematicWhat control sequence is required to generate an

arbitrary entangled state? (e.g. NOON states)

1

|Ψ =Na

na=0

Nb

nb=0

cna,nb |na ⊗ |nb. (1)

|Ψ =1√

N + 1

N

k=0

|k, k, (2)

|Ψ =1√2

(|Na, 0+ |0, Nb) . (3)

H = ωq(t)|11| + 1

2(Ω(t)|10| + Ω∗(t)|01|)

+ωaa†a + ωbb

†b

+ga

σ+a + σ−a

† + gb

σ+b + σ−b

†.

(4)

ω = ωq +g2a

ωq − ωa(2na + 1) +

g2

b

ωq − ωb(2nb + 1), (5)

For Fig. ?? we have set ∆ω = 2g2a/(ωq − ωa) =

−2g2

b/(ωq −ωb). This can always be achieved for a qubitwith a tunable frequency such as the phase or transmonqubit. This allows us to simplify our Rabi pulses to fre-quencies

ωk = ωq + k∆ω (6)

which selects those states with na − nb = k, where k

is an integer (k = 0 is shown in Fig. ??). Note thatthis choice requires ωa < ωq < ωb and |Ω| < g

2a/(ωq −

ωb). By choosing different values of k (or frequencies ωk)one can address each of the “diagonals” of the Fock-statediagram.

U =

Nb

j=1

Ub,j

Ua (7)

with

Ua =Na

j=1

AjRa,j , and Ub,j = Bj

Na

k=1

Rb,jk. (8)

|Ψ = U |0, 0, 0 = |0 ⊗Na

na=1

Nb

nb=1

cna,nb |na, nb, (9)

U†|Ψ = U

†a

1

j=Nb

U†b,j |Ψ = |0, 0, 0. (10)

Tmax = Na(Nb + 1)π

Ω+

Na

j=1

π

ga√

j+

Nb

j=1

π

gb√

j. (11)

TNOON = (Na +Nb−12)π

Ω+

Na

j=1

π

ga√

j+

Nb

j=1

π

gb√

j. (12)

Using this result, and a coupling of ga/(2π) = gb/(2π) =200 MHz for a qubit with ωq/(2π) = 6.5 GHz and res-onators with ωa/(2π) = 6 GHz, ωb/(2π) = 7 GHz (simi-lar to recent experiments [? ]), we find Ω/(2π) ∼ 40 MHzfor the relevant Rabi frequency. Thus, we estimate thatNOON state generation using this protocol will take only270 ns. This compares quite favorably to the coherencetimes of both qubits and resonators, which are now con-sistently greater than 0.5µs [? ? ? ].

TABLE I: NOON State Synthesis Procedure

Step Parameters State

Ra,1 Ωtqa,1 = π/2, ωd = ω0 |0, 0, 0 − i|1, 0, 0A1 gata,1 = π |0, 0, 0 − |0, 1, 0Ra,2 Ωtqa,2 = π, ωd = ω1 |0, 0, 0+ i|1, 1, 0A2 gata,2 = π/

√2 |0, 0, 0+ |0, 2, 0

Ra,3 Ωtqa,3 = π, ωd = ω2 |0, 0, 0 − i|1, 2, 0A3 gata,3 = π/

√3 |0, 0, 0 − |0, 3, 0

Rb,1 Ωtqb,1 = π, ωd = ω0 −i|1, 0, 0 − |0, 3, 0B1 gbtb,1 = π −|0, 0, 1 − |0, 3, 0Rb,2 Ωtqb,2 = π, ωd = ω−1 i|1, 0, 1 − |0, 3, 0B2 gbtb,2 = π/

√2 |0, 0, 2 − |0, 3, 0

Rb,3 Ωtqb,3 = π, ωd = ω−2 −i|1, 0, 2 − |0, 3, 0B3 gbtb,3 = π/

√3 −|0, 0, 3 − |0, 3, 0

1

|Ψ =Na

na=0

Nb

nb=0


|Ψ =1√

N + 1

N

k=0

|k, k, (2)

|Ψ =1√2

(|N, 0+ |0, N) . (3)

H = ωq(t)|11| + 1

2(Ω(t)|10| + Ω∗(t)|01|)

+ωaa†a + ga

σ+a + σ−a

† (4)

H = ωq(t)|11| + 1

2(Ω(t)|10| + Ω∗(t)|01|)

+ωaa†a + ωbb

†b

+ga

σ+a + σ−a

† + gb

σ+b + σ−b

†.

(5)

ω = ωq +g2a

ωq − ωa(2na + 1) +

g2

b

ωq − ωb(2nb + 1), (6)


−2g2


ωk = ωq + k∆ω (7)



2a/(ωq −


U =

Nb

j=1

Ub,j

Ua (8)

with

Ua =Na

j=1

AjRa,j , and Ub,j =Na

k=1

BjkRb,jk. (9)

Aj = exp(−iHata,j), andBjk = exp(−iHbtb,jk). (10)

|Ψ = U |0, 0, 0 = |0 ⊗Na

na=1

Nb

nb=1

cna,nb |na, nb, (11)

U†|Ψ = U

†a

1

j=Nb

U†b,j |Ψ = |0, 0, 0. (12)

Tmax = Na(Nb + 1)π

Ω+

Na

j=1

π

ga√

j+ Na

Nb

j=1

π

gb√

j. (13)


Ω+

Na

j=1

π

ga√

j+

Nb

j=1

π

gb√

j. (14)





√2 |0, 0, 0+ |0, 2, 0


√3 |0, 0, 0 − |0, 3, 0


√2 |0, 0, 2 − |0, 3, 0


√3 −|0, 0, 3 − |0, 3, 0


High NOON and Beyond

! Schrödinger Cat State! Generalizes Bell/GHZ states! Useful for metrology:

! May have applications for quantum state preparation, entanglement transfer and distribution, ...

12

1

|Ψ =Na

na=0

Nb

nb=0


|Ψ =1√

N + 1

N

k=0

|k, k, (2)

|Ψ =1√2

(|N, 0+ |0, N) . (3)

H = ωq(t)|11| + 1

2(Ω(t)|10| + Ω∗(t)|01|)

+ωaa†a + ga

σ+a + σ−a

† (4)

H = ωq(t)|11| + 1

2(Ω(t)|10| + Ω∗(t)|01|)

+ωaa†a + ωbb

†b

+ga

σ+a + σ−a

† + gb

σ+b + σ−b

†.

(5)

ω = ωq +g2a

ωq − ωa(2na + 1) +

g2

b

ωq − ωb(2nb + 1), (6)


−2g2


ωk = ωq + k∆ω (7)



2a/(ωq −


U =

Nb

j=1

Ub,j

Ua (8)

with

Ua =Na

j=1


k=1

BjkRb,jk. (9)


|Ψ = U |0, 0, 0 = |0 ⊗Na

na=1

Nb

nb=1


U†|Ψ = U

†a

1

j=Nb

U†b,j |Ψ = |0, 0, 0. (12)

Tmax = Na(Nb + 1)π

Ω+

Na

j=1

π

ga√

j+ Na

Nb

j=1

π

gb√

j. (13)


Ω+

Na

j=1

π

ga√

j+

Nb

j=1

π

gb√

j. (14)





√2 |0, 0, 0+ |0, 2, 0


√3 |0, 0, 0 − |0, 3, 0


√2 |0, 0, 2 − |0, 3, 0


√3 −|0, 0, 3 − |0, 3, 0

BRIEF ARTICLE

THE AUTHOR

(1) |Ψ|eiφa†a|Ψ| = | cos(Nφ/2)|

1


Recent NOON Linear Optics

13

Science 328, 879 (2010)

N=5


Jaynes-Cummings Ladder

1

|Ψ =Na

na=0

Nb

nb=0


|Ψ =1√

N + 1

N

k=0

|k, k, (2)

|Ψ =1√2

(|Na, 0+ |0, Nb) . (3)

H = ωq(t)|11| + 1

2(Ω(t)|10| + Ω∗(t)|01|)

+ωaa†a + ga

σ+a + σ−a

† (4)

H = ωq(t)|11| + 1

2(Ω(t)|10| + Ω∗(t)|01|)

+ωaa†a + ωbb

†b

+ga

σ+a + σ−a

† + gb

σ+b + σ−b

†.

(5)

ω = ωq +g2a

ωq − ωa(2na + 1) +

g2

b

ωq − ωb(2nb + 1), (6)


−2g2


ωk = ωq + k∆ω (7)



2a/(ωq −


U =

Nb

j=1

Ub,j

Ua (8)

with

Ua =Na

j=1


Na

k=1

Rb,jk. (9)

|Ψ = U |0, 0, 0 = |0 ⊗Na

na=1

Nb

nb=1


U†|Ψ = U

†a

1

j=Nb

U†b,j |Ψ = |0, 0, 0. (11)

Tmax = Na(Nb + 1)π

Ω+

Na

j=1

π

ga√

j+

Nb

j=1

π

gb√

j. (12)


Ω+

Na

j=1

π

ga√

j+

Nb

j=1

π

gb√

j. (13)





√2 |0, 0, 0+ |0, 2, 0


√3 |0, 0, 0 − |0, 3, 0


√2 |0, 0, 2 − |0, 3, 0


√3 −|0, 0, 3 − |0, 3, 0

Rabi pulses (R) drive qubit transitions (q=0 → 1)

Shift pulses (S) transfer quanta between qubit and oscillator.

q =0 q =1

n = 0

n = 1

n = 2 n = 1

n = 2

n = 0

R

Sn = 3

q =0

q =1

R

S

n = 0

n = 1

n = 2

n = 3

Used to generate arbitrary superpositions of Fock states:

Hofheinz et al., Nature 459, 456 (2009)


Two-Mode Jaynes Cummings

Qubit Coupler

Resonator A

Resonator B

Schematic

1

|Ψ =Na

na=0

Nb

nb=0


|Ψ =1√

N + 1

N

k=0

|k, k, (2)

|Ψ =1√2

(|Na, 0+ |0, Nb) . (3)

H = ωq(t)|11| + 1

2(Ω(t)|10| + Ω∗(t)|01|)

+ωaa†a + ga

σ+a + σ−a

† (4)

H = ωq(t)|11| + 1

2(Ω(t)|10| + Ω∗(t)|01|)

+ωaa†a + ωbb

†b

+ga

σ+a + σ−a

† + gb

σ+b + σ−b

†.

(5)

ω = ωq +g2a

ωq − ωa(2na + 1) +

g2

b

ωq − ωb(2nb + 1), (6)


−2g2


ωk = ωq + k∆ω (7)



2a/(ωq −


U =

Nb

j=1

Ub,j

Ua (8)

with

Ua =Na

j=1


Na

k=1

Rb,jk. (9)

|Ψ = U |0, 0, 0 = |0 ⊗Na

na=1

Nb

nb=1


U†|Ψ = U

†a

1

j=Nb

U†b,j |Ψ = |0, 0, 0. (11)

Tmax = Na(Nb + 1)π

Ω+

Na

j=1

π

ga√

j+

Nb

j=1

π

gb√

j. (12)


Ω+

Na

j=1

π

ga√

j+

Nb

j=1

π

gb√

j. (13)





√2 |0, 0, 0+ |0, 2, 0


√3 |0, 0, 0 − |0, 3, 0


√2 |0, 0, 2 − |0, 3, 0


√3 −|0, 0, 3 − |0, 3, 0

ωa < ωq < ωb

Minimal Schemeone qubit (tunable frequency)two resonators (different frequencies)


Two-Mode Jaynes-Cummings Diagram

1

|Ψ =Na

na=0

Nb

nb=0


|Ψ =1√

N + 1

N

k=0

|k, k, (2)

|Ψ =1√2

(|Na, 0+ |0, Nb) . (3)

H = ωq(t)|11| + 1

2(Ω(t)|10| + Ω∗(t)|01|)

+ωaa†a + ga

σ+a + σ−a

† (4)

H = ωq(t)|11| + 1

2(Ω(t)|10| + Ω∗(t)|01|)

+ωaa†a + ωbb

†b

+ga

σ+a + σ−a

† + gb

σ+b + σ−b

†.

(5)

ω = ωq +g2a

ωq − ωa(2na + 1) +

g2

b

ωq − ωb(2nb + 1), (6)


−2g2


ωk = ωq + k∆ω (7)



2a/(ωq −


U =

Nb

j=1

Ub,j

Ua (8)

with

Ua =Na

j=1


Na

k=1

Rb,jk. (9)

|Ψ = U |0, 0, 0 = |0 ⊗Na

na=1

Nb

nb=1


U†|Ψ = U

†a

1

j=Nb

U†b,j |Ψ = |0, 0, 0. (11)

Tmax = Na(Nb + 1)π

Ω+

Na

j=1

π

ga√

j+

Nb

j=1

π

gb√

j. (12)


Ω+

Na

j=1

π

ga√

j+

Nb

j=1

π

gb√

j. (13)





√2 |0, 0, 0+ |0, 2, 0


√3 |0, 0, 0 − |0, 3, 0


√2 |0, 0, 2 − |0, 3, 0


√3 −|0, 0, 3 − |0, 3, 0

Rabi pulses (R) drive qubit transitions (q=0 → 1)Shift pulses (A + B) transfer quanta between qubit and oscillators (a + b).

Need Fock-state-selective interaction!q =0 q =1

n = 0

n = 1

n = 2 n = 1

n = 2

n = 0

R

Sn = 3

q =0

q =1

R

S

n = 0

n = 1

n = 2

n = 3

ωa < ωq < ωb



1

|Ψ =Na

na=0

Nb

nb=0


|Ψ =1√

N + 1

N

k=0

|k, k, (2)

|Ψ =1√2

(|Na, 0+ |0, Nb) . (3)

H = ωq(t)|11| + 1

2(Ω(t)|10| + Ω∗(t)|01|)

+ωaa†a + ga

σ+a + σ−a

† (4)

H = ωq(t)|11| + 1

2(Ω(t)|10| + Ω∗(t)|01|)

+ωaa†a + ωbb

†b

+ga

σ+a + σ−a

† + gb

σ+b + σ−b

†.

(5)

ω = ωq +g2a

ωq − ωa(2na + 1) +

g2

b

ωq − ωb(2nb + 1), (6)


−2g2


ωk = ωq + k∆ω (7)



2a/(ωq −


U =

Nb

j=1

Ub,j

Ua (8)

with

Ua =Na

j=1


Na

k=1

Rb,jk. (9)

|Ψ = U |0, 0, 0 = |0 ⊗Na

na=1

Nb

nb=1


U†|Ψ = U

†a

1

j=Nb

U†b,j |Ψ = |0, 0, 0. (11)

Tmax = Na(Nb + 1)π

Ω+

Na

j=1

π

ga√

j+

Nb

j=1

π

gb√

j. (12)


Ω+

Na

j=1

π

ga√

j+

Nb

j=1

π

gb√

j. (13)





√2 |0, 0, 0+ |0, 2, 0


√3 |0, 0, 0 − |0, 3, 0


√2 |0, 0, 2 − |0, 3, 0


√3 −|0, 0, 3 − |0, 3, 0



n = 0

n = 1

n = 2 n = 1

n = 2

n = 0

R

Sn = 3

q =0

q =1

R

S

n = 0

n = 1

n = 2

n = 3

q =0 q =1

n = 0

n = 1

n = 2 n = 1

n = 2

n = 0

R

Sn = 3

q =0

q =1

R

S

n = 0

n = 1

n = 2

n = 3

ωa < ωq < ωb



1

|Ψ =Na

na=0

Nb

nb=0


|Ψ =1√

N + 1

N

k=0

|k, k, (2)

|Ψ =1√2

(|Na, 0+ |0, Nb) . (3)

H = ωq(t)|11| + 1

2(Ω(t)|10| + Ω∗(t)|01|)

+ωaa†a + ga

σ+a + σ−a

† (4)

H = ωq(t)|11| + 1

2(Ω(t)|10| + Ω∗(t)|01|)

+ωaa†a + ωbb

†b

+ga

σ+a + σ−a

† + gb

σ+b + σ−b

†.

(5)

ω = ωq +g2a

ωq − ωa(2na + 1) +

g2

b

ωq − ωb(2nb + 1), (6)


−2g2


ωk = ωq + k∆ω (7)



2a/(ωq −


U =

Nb

j=1

Ub,j

Ua (8)

with

Ua =Na

j=1


Na

k=1

Rb,jk. (9)

|Ψ = U |0, 0, 0 = |0 ⊗Na

na=1

Nb

nb=1


U†|Ψ = U

†a

1

j=Nb

U†b,j |Ψ = |0, 0, 0. (11)

Tmax = Na(Nb + 1)π

Ω+

Na

j=1

π

ga√

j+

Nb

j=1

π

gb√

j. (12)


Ω+

Na

j=1

π

ga√

j+

Nb

j=1

π

gb√

j. (13)





√2 |0, 0, 0+ |0, 2, 0


√3 |0, 0, 0 − |0, 3, 0


√2 |0, 0, 2 − |0, 3, 0


√3 −|0, 0, 3 − |0, 3, 0



n = 0

n = 1

n = 2 n = 1

n = 2

n = 0

R

Sn = 3

q =0

q =1

R

S

n = 0

n = 1

n = 2

n = 3

q =0 q =1

n = 0

n = 1

n = 2 n = 1

n = 2

n = 0

R

Sn = 3

q =0

q =1

R

S

n = 0

n = 1

n = 2

n = 3

Transition Probability

ωa < ωq < ωb


Stark-shifted Rabi Oscillations

17

Stark-shifted Rabi transitions yield Fock-state-selective qubit rotations!Frequency chosen

to select nb = na + 2

Special condition:ga2 / (ωq - ωa) = -gb2 / (ωq - ωb)

cf. Schuster et al., Nature 445, 515 (2007)


na (photons in mode a)

n b (p

hoto

ns in

mod

e b)

1

|Ψ =Na

na=0

Nb

nb=0


|Ψ =1√

N + 1

N

k=0

|k, k, (2)

|Ψ =1√2

(|Na, 0+ |0, Nb) . (3)

H = ωq(t)|11| + 1

2(Ω(t)|10| + Ω∗(t)|01|)

+ωaa†a + ga

σ+a + σ−a

† (4)

H = ωq(t)|11| + 1

2(Ω(t)|10| + Ω∗(t)|01|)

+ωaa†a + ωbb

†b

+ga

σ+a + σ−a

† + gb

σ+b + σ−b

†.

(5)

ω = ωq +g2a

ωq − ωa(2na + 1) +

g2

b

ωq − ωb(2nb + 1), (6)


−2g2


ωk = ωq + k∆ω (7)



2a/(ωq −


U =

Nb

j=1

Ub,j

Ua (8)

with

Ua =Na

j=1


Na

k=1

Rb,jk. (9)

|Ψ = U |0, 0, 0 = |0 ⊗Na

na=1

Nb

nb=1


U†|Ψ = U

†a

1

j=Nb

U†b,j |Ψ = |0, 0, 0. (11)

Tmax = Na(Nb + 1)π

Ω+

Na

j=1

π

ga√

j+

Nb

j=1

π

gb√

j. (12)


Ω+

Na

j=1

π

ga√

j+

Nb

j=1

π

gb√

j. (13)





√2 |0, 0, 0+ |0, 2, 0


√3 |0, 0, 0 − |0, 3, 0


√2 |0, 0, 2 − |0, 3, 0


√3 −|0, 0, 3 − |0, 3, 0


State-Synthesis Algorithm

18

1

|Ψ =Na

na=0

Nb

nb=0


|Ψ =1√

N + 1

N

k=0

|k, k, (2)

|Ψ =1√2

(|Na, 0+ |0, Nb) . (3)

H = ωq(t)|11| + 1

2(Ω(t)|10| + Ω∗(t)|01|)

+ωaa†a + ga

σ+a + σ−a

† (4)

H = ωq(t)|11| + 1

2(Ω(t)|10| + Ω∗(t)|01|)

+ωaa†a + ωbb

†b

+ga

σ+a + σ−a

† + gb

σ+b + σ−b

†.

(5)

ω = ωq +g2a

ωq − ωa(2na + 1) +

g2

b

ωq − ωb(2nb + 1), (6)


−2g2


ωk = ωq + k∆ω (7)



2a/(ωq −


U =

Nb

j=1

Ub,j

Ua (8)

with

Ua =Na

j=1


Na

k=1

Rb,jk. (9)

Aj = exp(−iHata,j), andBj = exp(−iHbtb,j). (10)

|Ψ = U |0, 0, 0 = |0 ⊗Na

na=1

Nb

nb=1


U†|Ψ = U

†a

1

j=Nb

U†b,j |Ψ = |0, 0, 0. (12)

Tmax = Na(Nb + 1)π

Ω+

Na

j=1

π

ga√

j+

Nb

j=1

π

gb√

j. (13)


Ω+

Na

j=1

π

ga√

j+

Nb

j=1

π

gb√

j. (14)





√2 |0, 0, 0+ |0, 2, 0


√3 |0, 0, 0 − |0, 3, 0


√2 |0, 0, 2 − |0, 3, 0


√3 −|0, 0, 3 − |0, 3, 0

1

|Ψ =Na

na=0

Nb

nb=0


|Ψ =1√

N + 1

N

k=0

|k, k, (2)

|Ψ =1√2

(|Na, 0+ |0, Nb) . (3)

H = ωq(t)|11| + 1

2(Ω(t)|10| + Ω∗(t)|01|)

+ωaa†a + ga

σ+a + σ−a

† (4)

H = ωq(t)|11| + 1

2(Ω(t)|10| + Ω∗(t)|01|)

+ωaa†a + ωbb

†b

+ga

σ+a + σ−a

† + gb

σ+b + σ−b

†.

(5)

ω = ωq +g2a

ωq − ωa(2na + 1) +

g2

b

ωq − ωb(2nb + 1), (6)


−2g2


ωk = ωq + k∆ω (7)



2a/(ωq −


U =

Nb

j=1

Ub,j

Ua (8)

with

Ua =Na

j=1


Na

k=1

Rb,jk. (9)

Aj = exp(−iHata,j), andBj = exp(−iHbtb,j). (10)

|Ψ = U |0, 0, 0 = |0 ⊗Na

na=1

Nb

nb=1


U†|Ψ = U

†a

1

j=Nb

U†b,j |Ψ = |0, 0, 0. (12)

Tmax = Na(Nb + 1)π

Ω+

Na

j=1

π

ga√

j+

Nb

j=1

π

gb√

j. (13)


Ω+

Na

j=1

π

ga√

j+

Nb

j=1

π

gb√

j. (14)





√2 |0, 0, 0+ |0, 2, 0


√3 |0, 0, 0 − |0, 3, 0


√2 |0, 0, 2 − |0, 3, 0


√3 −|0, 0, 3 − |0, 3, 0Interaction times + qubit rotations chosen to satisfy


Solution requires solving for the inverse, clearing amplitudes row-by-row, column-by-column, performing two-state oscillations at each step.

1

|Ψ =Na

na=0

Nb

nb=0


|Ψ =1√

N + 1

N

k=0

|k, k, (2)

|Ψ =1√2

(|Na, 0+ |0, Nb) . (3)

H = ωq(t)|11| + 1

2(Ω(t)|10| + Ω∗(t)|01|)

+ωaa†a + ga

σ+a + σ−a

† (4)

H = ωq(t)|11| + 1

2(Ω(t)|10| + Ω∗(t)|01|)

+ωaa†a + ωbb

†b

+ga

σ+a + σ−a

† + gb

σ+b + σ−b

†.

(5)

ω = ωq +g2a

ωq − ωa(2na + 1) +

g2

b

ωq − ωb(2nb + 1), (6)


−2g2


ωk = ωq + k∆ω (7)



2a/(ωq −


U =

Nb

j=1

Ub,j

Ua (8)

with

Ua =Na

j=1


k=1

BjkRb,jk. (9)


|Ψ = U |0, 0, 0 = |0 ⊗Na

na=1

Nb

nb=1


U†|Ψ = U

†a

1

j=Nb

U†b,j |Ψ = |0, 0, 0. (12)

Tmax = Na(Nb + 1)π

Ω+

Na

j=1

π

ga√

j+ Na

Nb

j=1

π

gb√

j. (13)


Ω+

Na

j=1

π

ga√

j+

Nb

j=1

π

gb√

j. (14)





√2 |0, 0, 0+ |0, 2, 0


√3 |0, 0, 0 − |0, 3, 0


√2 |0, 0, 2 − |0, 3, 0


√3 −|0, 0, 3 − |0, 3, 0

1

|Ψ =Na

na=0

Nb

nb=0


|Ψ =1√

N + 1

N

k=0

|k, k, (2)

|Ψ =1√2

(|Na, 0+ |0, Nb) . (3)

H = ωq(t)|11| + 1

2(Ω(t)|10| + Ω∗(t)|01|)

+ωaa†a + ga

σ+a + σ−a

† (4)

H = ωq(t)|11| + 1

2(Ω(t)|10| + Ω∗(t)|01|)

+ωaa†a + ωbb

†b

+ga

σ+a + σ−a

† + gb

σ+b + σ−b

†.

(5)

ω = ωq +g2a

ωq − ωa(2na + 1) +

g2

b

ωq − ωb(2nb + 1), (6)


−2g2


ωk = ωq + k∆ω (7)



2a/(ωq −


U =

Nb

j=1

Ub,j

Ua (8)

with

Ua =Na

j=1


k=1

BjkRb,jk. (9)


|Ψ = U |0, 0, 0 = |0 ⊗Na

na=1

Nb

nb=1


U†|Ψ = U

†a

1

j=Nb

U†b,j |Ψ = |0, 0, 0. (12)

Tmax = Na(Nb + 1)π

Ω+

Na

j=1

π

ga√

j+ Na

Nb

j=1

π

gb√

j. (13)


Ω+

Na

j=1

π

ga√

j+

Nb

j=1

π

gb√

j. (14)





√2 |0, 0, 0+ |0, 2, 0


√3 |0, 0, 0 − |0, 3, 0


√2 |0, 0, 2 − |0, 3, 0


√3 −|0, 0, 3 − |0, 3, 0


19

1

|Ψ =Na

na=0

Nb

nb=0


|Ψ =1√

N + 1

N

k=0

|k, k, (2)

|Ψ =1√2

(|Na, 0+ |0, Nb) . (3)

H = ωq(t)|11| + 1

2(Ω(t)|10| + Ω∗(t)|01|)

+ωaa†a + ga

σ+a + σ−a

† (4)

H = ωq(t)|11| + 1

2(Ω(t)|10| + Ω∗(t)|01|)

+ωaa†a + ωbb

†b

+ga

σ+a + σ−a

† + gb

σ+b + σ−b

†.

(5)

ω = ωq +g2a

ωq − ωa(2na + 1) +

g2

b

ωq − ωb(2nb + 1), (6)


−2g2


ωk = ωq + k∆ω (7)



2a/(ωq −


U =

Nb

j=1

Ub,j

Ua (8)

with

Ua =Na

j=1


k=1

BjkRb,jk. (9)


|Ψ = U |0, 0, 0 = |0 ⊗Na

na=1

Nb

nb=1


U†|Ψ = U

†a

1

j=Nb

U†b,j |Ψ = |0, 0, 0. (12)

Tmax = Na(Nb + 1)π

Ω+

Na

j=1

π

ga√

j+ Na

Nb

j=1

π

gb√

j. (13)


Ω+

Na

j=1

π

ga√

j+

Nb

j=1

π

gb√

j. (14)





√2 |0, 0, 0+ |0, 2, 0


√3 |0, 0, 0 − |0, 3, 0


√2 |0, 0, 2 − |0, 3, 0


√3 −|0, 0, 3 − |0, 3, 0

0

0

nb

3q =0

q =1

2

1

na1 2 3

(B32)† (B33)

† (B31)† (B30)

†

(B20)† (B21)

† (B22)† (B23)

†

(B10)† (B11)

† (B12)† (B13)

†

(Rb33)†

(Ub3)†

(Ub2)†

(Ub1)†

(Ua)†

(Rb23)†

(Rb13)†

(A3)† (A2)

† (A1)†

Algorithm Details

Clear amplitudes row-by-row, column-by-column. (Stark shift is key!)

Time ~ N2

1

|Ψ =Na

na=0

Nb

nb=0


|Ψ =1√

N + 1

N

k=0

|k, k, (2)

|Ψ =1√2

(|Na, 0+ |0, Nb) . (3)

H = ωq(t)|11| + 1

2(Ω(t)|10| + Ω∗(t)|01|)

+ωaa†a + ga

σ+a + σ−a

† (4)

H = ωq(t)|11| + 1

2(Ω(t)|10| + Ω∗(t)|01|)

+ωaa†a + ωbb

†b

+ga

σ+a + σ−a

† + gb

σ+b + σ−b

†.

(5)

ω = ωq +g2a

ωq − ωa(2na + 1) +

g2

b

ωq − ωb(2nb + 1), (6)


−2g2


ωk = ωq + k∆ω (7)



2a/(ωq −


U =

Nb

j=1

Ub,j

Ua (8)

with

Ua =Na

j=1


k=1

BjkRb,jk. (9)


|Ψ = U |0, 0, 0 = |0 ⊗Na

na=1

Nb

nb=1


U†|Ψ = U

†a

1

j=Nb

U†b,j |Ψ = |0, 0, 0. (12)

Tmax = Na(Nb + 1)π

Ω+

Na

j=1

π

ga√

j+ Na

Nb

j=1

π

gb√

j. (13)


Ω+

Na

j=1

π

ga√

j+

Nb

j=1

π

gb√

j. (14)





√2 |0, 0, 0+ |0, 2, 0


√3 |0, 0, 0 − |0, 3, 0


√2 |0, 0, 2 − |0, 3, 0


√3 −|0, 0, 3 − |0, 3, 0

1

|Ψ =Na

na=0

Nb

nb=0


|Ψ =1√

N + 1

N

k=0

|k, k, (2)

|Ψ =1√2

(|Na, 0+ |0, Nb) . (3)

H = ωq(t)|11| + 1

2(Ω(t)|10| + Ω∗(t)|01|)

+ωaa†a + ga

σ+a + σ−a

† (4)

H = ωq(t)|11| + 1

2(Ω(t)|10| + Ω∗(t)|01|)

+ωaa†a + ωbb

†b

+ga

σ+a + σ−a

† + gb

σ+b + σ−b

†.

(5)

ω = ωq +g2a

ωq − ωa(2na + 1) +

g2

b

ωq − ωb(2nb + 1), (6)


−2g2


ωk = ωq + k∆ω (7)



2a/(ωq −


U =

Nb

j=1

Ub,j

Ua (8)

with

Ua =Na

j=1


k=1

BjkRb,jk. (9)


|Ψ = U |0, 0, 0 = |0 ⊗Na

na=1

Nb

nb=1


U†|Ψ = U

†a

1

j=Nb

U†b,j |Ψ = |0, 0, 0. (12)

Tmax = Na(Nb + 1)π

Ω+

Na

j=1

π

ga√

j+ Na

Nb

j=1

π

gb√

j. (13)


Ω+

Na

j=1

π

ga√

j+

Nb

j=1

π

gb√

j. (14)





√2 |0, 0, 0+ |0, 2, 0


√3 |0, 0, 0 − |0, 3, 0


√2 |0, 0, 2 − |0, 3, 0


√3 −|0, 0, 3 − |0, 3, 0

1

|Ψ =Na

na=0

Nb

nb=0


|Ψ =1√

N + 1

N

k=0

|k, k, (2)

|Ψ =1√2

(|Na, 0+ |0, Nb) . (3)

H = ωq(t)|11| + 1

2(Ω(t)|10| + Ω∗(t)|01|)

+ωaa†a + ga

σ+a + σ−a

† (4)

H = ωq(t)|11| + 1

2(Ω(t)|10| + Ω∗(t)|01|)

+ωaa†a + ωbb

†b

+ga

σ+a + σ−a

† + gb

σ+b + σ−b

†.

(5)

ω = ωq +g2a

ωq − ωa(2na + 1) +

g2

b

ωq − ωb(2nb + 1), (6)


−2g2


ωk = ωq + k∆ω (7)



2a/(ωq −


U =

Nb

j=1

Ub,j

Ua (8)

with

Ua =Na

j=1


k=1

BjkRb,jk. (9)


|Ψ = U |0, 0, 0 = |0 ⊗Na

na=1

Nb

nb=1


U†|Ψ = U

†a

1

j=Nb

U†b,j |Ψ = |0, 0, 0. (12)

Tmax = Na(Nb + 1)π

Ω+

Na

j=1

π

ga√

j+ Na

Nb

j=1

π

gb√

j. (13)


Ω+

Na

j=1

π

ga√

j+

Nb

j=1

π

gb√

j. (14)





√2 |0, 0, 0+ |0, 2, 0


√3 |0, 0, 0 − |0, 3, 0


√2 |0, 0, 2 − |0, 3, 0


√3 −|0, 0, 3 − |0, 3, 0


NOON State Example

!

" =123,0 + 0,3( )High NOON state:

J.P. Dowling, Contemp. Phys. 49, 125 (2008)


NOON State Synthesis

High NOON states are accessible using existing technology!

Main limitation is dueto Rabi frequency, to remain in Stark-shifted regime (can be optimized using pulse shapes).

!

TN = (2N" 12)#$

+2#g

1jj=1

N

%

1

|Ψ =Na

na=0

Nb

nb=0


|Ψ =1√

N + 1

N

k=0

|k, k, (2)

|Ψ =1√2

(|N, 0+ |0, N) . (3)

H = ωq(t)|11| + 1

2(Ω(t)|10| + Ω∗(t)|01|)

+ωaa†a + ga

σ+a + σ−a

† (4)

H = ωq(t)|11| + 1

2(Ω(t)|10| + Ω∗(t)|01|)

+ωaa†a + ωbb

†b

+ga

σ+a + σ−a

† + gb

σ+b + σ−b

†.

(5)

ω = ωq +g2a

ωq − ωa(2na + 1) +

g2

b

ωq − ωb(2nb + 1), (6)


−2g2


ωk = ωq + k∆ω (7)



2a/(ωq −


U =

Nb

j=1

Ub,j

Ua (8)

with

Ua =Na

j=1


k=1

BjkRb,jk. (9)


|Ψ = U |0, 0, 0 = |0 ⊗Na

na=1

Nb

nb=1


U†|Ψ = U

†a

1

j=Nb

U†b,j |Ψ = |0, 0, 0. (12)

Tmax = Na(Nb + 1)π

Ω+

Na

j=1

π

ga√

j+ Na

Nb

j=1

π

gb√

j. (13)


Ω+

Na

j=1

π

ga√

j+

Nb

j=1

π

gb√

j. (14)





√2 |0, 0, 0+ |0, 2, 0


√3 |0, 0, 0 − |0, 3, 0


√2 |0, 0, 2 − |0, 3, 0


√3 −|0, 0, 3 − |0, 3, 0

ωa /(2!) = 6 GHzωq /(2!) = 6.5 GHzωb /(2!) = 7 GHz


Other Methods: Sidebands! New transitions, new paths

! Probably slow... !/2" ~ 10 MHz22

Na = 0

Nb = 0

Na = 1 Na = 2 Na = 3

Nb = 1

Nb = 2

Nb = 3 q =0

q =1

Ared

Ablue

Bred

Bblue

Red + Blue Sidebands for each qubit

Flips qubit + photon

Requires two-photon process for transmons?


Other Methods: Higher Levels! |0>, |1>, |2>, ... to mediate interactions

! Less coherent? T12 < T01

23

Na = 0

Nb = 0

Na = 1 Na = 2 Na = 3

Nb = 1

Nb = 2

Nb = 3 q =0

q =1

q =2

A01

A12

B12

B01

R01

R12

R02

Two types of qutrit-resonator interactions.

Three types of Rabi oscillations.

More complex Stark shifts.


Another Scheme

24

Qubit A Qubit B

Coupling Resonator

Resonator A Resonator B

Control Lines

Two qubits, three resonators.Entanglement transferred from qubits to resonators.

Parallel Operations! General State Synthesis?

Martinis & ClelandMerkel & Wilhelm

arXiv: 1006.1336


Many Applications! High-dimensional states provide more efficient

means to distribute entanglement (quantum routing)

! Higher-dimensional Bell inequalities (more settings, less locality?)

! State preparation of qubits by entangled resonators (ancillas, etc.)

! Quantum logic between resonators?! Measuring entangled photon states?

! These and other issues under investigation25


Quantum Routing! Goal: Use a network of elements (qubits or

resonators) to transfer quantum information.! Programmable---send information between

any two nodes.! Parallel---information between different

pairs of nodes can be sent at the same time.

! Ideally suited for entanglement distribution between distinct registers for teleportation, error detection, ancilla preparation, and other steps toward fault tolerance.

26


Quantum State Transfer

d = 1d = 2

d = 3 d = 4

One can transfer the state of a single qubit from site A to site B using a set of permanently coupled qubits with Hamiltonian:

Dynamics of a single excitation (with ω = 0) maps onto a tight-bonding model with H = ħΩ , where the coupling matrix Ω is proportional to the adjacency matrix of the coupling graph. Certain coupling schemes such as the hypercube (M. Christandl et al., Phys. Rev. Lett. 92, 187902 (2004)) lead to perfect state transfer:

27


! Each vertex represents a qubit. Quantum states travel along all paths simultaneously in superposition with full constructive interference, yielding perfect state transfer.

Hypercube State Transfer



|&! = ' |0! +( |1!



•Circuits do not need to be simple two-dimensional layouts.

•Multi-layer interconnects allow many crossovers and complex couplings.

1 2 3Phase Qubit Cube


•Circuits do not need to be simple two-dimensional layouts.

•Multi-layer interconnects allow many crossovers and complex couplings.

Courtesy Ray Simmonds, NIST Boulder

1 2 3Phase Qubit Cube


Entanglement Distribution by Parallel Quantum State Transfer

! Study tunable resonators: exactly solvable!! Split cube into M subcubes by frequency

detuning.! Send quantum states in parallel.

30

with Chris Chudzicki ‘10, Williams College

M = 1 M = 2 M = 4


Parallel Transfer Fidelity

31

Text

M = # of “messages”

" = 2!0 /##

!0

M = 2


Entanglement Distribution Rate! Distribute entanglement between every pair

of nodes (N total nodes).! Send states one at a time on each subcube.! Keep resonators in a fixed bandwidth! The entanglement distribution rate scales as:

32


Massively Parallel Distribution! Send oscillators states between all

corners simultaneously!

! Excitations are noninteracting bosons: multiple photons just pass through each other.

33


Massively Parallel Rate

34

Various Bandwidths


Conclusions! Resonators are a powerful

tool for quantum information: let’s use them!

! Developed state-synthesis algorithm for arbitrary entangled states " (NOON states very efficient)

! Quantum Routing with oscillator networks" Massively Parallel using

multiple excitations35

Phys. Rev. Lett. 105, 050501 (2010)

ArXiv: 1008.1806


arbitrary control of entanglement between two ... › fws1 › files › 2011 › 11 ›...

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