quantum control of superconducting...

Quantum Control of Superconducting Circuits

Liang Jiang Yale University, Applied Physics

Victor Albert, Stefan Krastanov, Chao Shen, Changling Zou

Brian Vlastakis, Matt Reagor, Andrei Petrenko, Steven Touzard, Zaki Leghtas, Reinier Heeres, Wolfgang Pfaff

Mazyar Mirrahimi, Michel Devoret, Rob Schoelkopf

City Tech Physics Seminar 2014.11.13

Supported by: DARPA, ARO, AFOSR, Sloan, Packard.

45mm

Superconduc*ng Cavity-‐Qubit System

storage cavity

transmon qubit

coupler port

Effec*ve Hamiltonian:

−χ sa†a e eH =ω sa

†a +ω q e e

readout cavity

χ s ≫κ s ,γ q

Interac9on strength dominates photon loss & qubit decoherence

κ s : photon decay rate γ q : qubit decoh. rate

•  QND readout of the qubit has been demonstrated in many groups (e.g., F>99.5% in 300 ns)

•  Long lived SC cavity (T>10 ms >> Tqubit~100us)

storage cavity Manipulate the cavity

H = ω s − χ s e e( )a†a +ω q e e

, gβ0,gD gβ

Weak Drive: Condi9onal cavity displacement

0,e0,gD eβ

β,g

0,e

in-‐phase

quadrature

Phase-‐space diagram

Quantum circuit

cavity

qubit

gDβ

Strong Drive: Uncondi9onal cavity displacement

βDβ 0indep. of qubit state

in-‐phase

quadrature

Phase-‐space diagram

θ

β

Quantum circuit

cavity Dβ

Create superposi4on of cavity state depending on the qubit.

+ε t( )a† eiω st + h.c.

storage cavity Manipulate the qubit

H =ω sa†a + ω q − χ sa

†a( ) e e

Query the qubit: ‘Are there m photons in the cavity?’

Kirchmair G. et al. Nature 495 205-‐209 2013

Johnson B.R. et al. Nature Phys. 6, 663-‐667 2010

Integrated Signal

Spectroscopy frequency (GHz)

Qubit Spectroscopy

χ s

cavity

qubit Yπm

m,eYπm m,g

Yπm n,g n,gn ≠ m

Weak Drive: Condi9onal qubit rota9on:

Yπ n,g n,efor all n

Strong Drive: Uncondi9onal qubit rota9on:

qubit Yπ

Quantum circuit

+Ωm t( ) e g ei ωq−mχs( )t + h.c.

Determinis)c qubit-‐cavity mapping

Leghtas et. al PRA 87, 042315 (2013) Theory:

Transfer arbitrary state from qubit to cavity

Storage

0

Yπ /2 Yπ0

Dβ Cπ Dβ†Dβ

qubit a g b e+

a bβ β+ −

g

Vlastakis et. al Science 342, 6158 (2013) Experiment:

0.0

-‐0.6

0.6

Measure Wigner func*on (based on QND parity meas)

-‐ Can we prepare the cavity in arbitrary superposiTon of photon number states?

( ) ( ) ( ) ( )† †1 a aW Tr D Dα α ρ α⎡ ⎤= ⎢ ⎥⎣ ⎦

−

-‐ Can we robustly encode quantum informaTon in cavity?

Theory of Cat Codes

Mo*va*ons: 1.  How to construct robust quantum memory?

a)  Overcome cavity dephasing errors b)  Overcome cavity loss errors

2.  How to control such robust quantum memory? a)  Gates over single logical qubits b)  Gates between two logical qubits

† † †1 1 1

1 12 2loss a a a a a aκ ρ κ ρ ρ ρ−

⎛ ⎞= − −⎜ ⎟⎝ ⎠D

Dominant Errors in Cavity Quantum Memory

2 21 12 2dephase n n n nφ φκ ρ κ ρ ρ ρ⎛ ⎞= − −⎜ ⎟⎝ ⎠

D

Dephasing Error Photon Loss Error

No Change in photon number! Reduce photon number by one!

(Env. is probing photon number) (Env. is stealing our photons one by one)

Strategy: quickly shuffle photon number … Strategy: monitor photon parity …

generator

Dissipation to Transmission line

LC oscillator

I

Q

D a⎡⎣ ⎤⎦ρ = aρa† − 1

2a†aρ − 1

2ρa†a

Steady state:

ρ∞ = α∞ α∞ ,α∞ = 2ε /κ

Driven damped harmonic oscillator

Driving + DissipaTon à  Pure Steady State à  Suppress dephasing noise!

ddt

ρ = [εa† − ε * a,ρ]+κ1D a⎡⎣ ⎤⎦ρ

ddt

ρ =κ1D a −α∞⎡⎣ ⎤⎦ρ

† *d d[ ( ) , ]d d d

dd a a adt

ρ ε ε ρ κ ρ⎡ ⎤= − + ⎣ ⎦D

Case d=2 : 2-dim steady state subspace!

{ }2 2, 2 /cρ α α ε κ∞ ± ∞ ∞∈ ± =∑

Case d=4 : 4-dim steady state subspace!

{ }1/4( ) 4 4( ) , (2 / )ic iρ α α ε κ∞ ± ∞ ∞∈ ± =∑

Mul*-‐photon Driven & Mul*-‐photon Damped Oscillator

I

Q

I

Q

I

Q

I

Q

I

Q

I

Q

I

QI

Q

I

Q

I

Q

I

Q

45mm

Pump & Readout Cavity

Storage Cavity

Vertical Transmon Qubit

Coupler Port

Mirrahimi, Leghtas, Albert, et al., NJP 16, 045014 (2014)

ddt

ρ =κ d D ad −α∞d⎡⎣ ⎤⎦ρ

α∞ = 2εd /κ d( )1/d

Ongoing experiment in Devoret’s group

𝑄

𝐼

𝑄

𝐼

Steady states dependent on ini*al states Case d=2 : Two dimensional steady state subspace!

{ }2 2, 2 /cρ α α ε κ∞ ± ∞ ∞∈ ± =∑

2( ) 2nN c nα α∞ ∞+ − =∑0n =ie δφα∞ α∞

0

1Z

Z

n

n

+ = =

− = =

Choice of memory basis (|n=0> and |n=1>)

|+X⟩ ≈ |α⟩|−X⟩ ≈ |−α⟩

|+Z⟩ = |C+α⟩

|−Z⟩ = |C−

α⟩

Y

|+Z⟩ = |C(0mod4)α

⟩

|−Z⟩ = |C(2mod4)α

⟩

|+X⟩ ≈ |C+α⟩|−X⟩ ≈ |C+

iα⟩

Y

(b)

1

|+𝑧⟩= |0⟩

|−𝑧⟩= |1⟩

|+𝑥⟩= |0⟩+ |1⟩ |−𝑥⟩= |0⟩− |1⟩

Vulnerable to both -‐ Dephasing Errors -‐ Photon Loss Errors

|+X⟩ ≈ |α⟩|−X⟩ ≈ |−α⟩

|+Z⟩ = |C+α⟩

|−Z⟩ = |C−

α⟩

Y

|+Z⟩ = |C(0mod4)α

⟩

|−Z⟩ = |C(2mod4)α

⟩

|+X⟩ ≈ |C+α⟩|−X⟩ ≈ |C+

iα⟩

Y

(b)

1

2

2 1

( ) 2

( ) 2 1

Z n

Z n

C N c n

C N c n

α

α

α α

α α +

++ = = + − =−− = = − − = +

∑∑

Choice of qubit basis (2-‐photon process) |+𝒛⟩= |𝜶⟩+ |−𝜶⟩

|−𝒛⟩= |𝜶⟩− |−𝜶⟩

|−𝒙⟩= |−𝜶⟩ |+𝒙⟩= |+𝜶⟩

2| |2

!

n

nc en

α α−=

+1

-1

Par

ity

|+X⟩ ≈ |α⟩|−X⟩ ≈ |−α⟩

|+Z⟩ = |C+α⟩

|−Z⟩ = |C−

α⟩

Y

|+Z⟩ = |C(0mod4)α

⟩

|−Z⟩ = |C(2mod4)α

⟩

|+X⟩ ≈ |C+α⟩|−X⟩ ≈ |C+

iα⟩

Y

(b)

1

Choice of qubit basis (2-‐photon process)

( )0

0

| |

| 1 |

nn

nn

n

x c n

x c n

∞

=

∞

=

+ 〉 = 〉

− 〉 = − 〉

∑

∑with

|−𝒙⟩= |−𝜶⟩ |+𝒙⟩= |+𝜶⟩

2| |2

!

n

nc en

α α−=

|+X⟩ ≈ |α⟩|−X⟩ ≈ |−α⟩

|+Z⟩ = |C+α⟩

|−Z⟩ = |C−

α⟩

Y

|+Z⟩ = |C(0mod4)α

⟩

|−Z⟩ = |C(2mod4)α

⟩

|+X⟩ ≈ |C+α⟩|−X⟩ ≈ |C+

iα⟩

Y

(b)

1


|−𝒙⟩= |−𝜶⟩ |+𝒙⟩= |+𝜶⟩

Key property: Difference in average photon number <n> is exponen-ally small, for any superposiTon state of

e−α2

Cα+ and Cα

−


+Z = Cα (0mod 4) = N α + −α( ) + iα + −iα( )( ) = c4n∑ 4n

−Z = Cα (2mod 4) = N α + −α( )− iα + −iα( )( ) = c4n+2∑ 4n+ 2

|+X⟩ ≈ |α⟩|−X⟩ ≈ |−α⟩

|+Z⟩ = |C+α⟩

|−Z⟩ = |C−

α⟩

Y

(a) |+Z⟩ = |C(0mod4)α

⟩

|−Z⟩ = |C(2mod4)α

⟩

|+X⟩ ≈ |C+α⟩|−X⟩ ≈ |C+

iα⟩

Y

1

+1

-1

Par

ity

Key property: Difference in average photon number <n> is exponen-ally small, for any superposiTon state of

e−α2 /2

Cα0mod 4( ) and Cα

2mod 4( )

Zaki Leghtas

Leghtas, et al., PRL 111, 120501 (2012).

No change in photon number (modulo 2 or 4) è

Constant populaTon of è

No logical bit-‐flip errors

±Z

Effect of photon dephasing (in presence of driven dissipa*ve process)

[ ] 2 21 12 2

n n n n nφ φκ ρ κ ρ ρ ρ⎛ ⎞= − −⎜ ⎟⎝ ⎠D

|𝛼|

(a) Two-photon ƉƌŽĐĞƐƐ (b) Four-photon ƉƌŽĐĞƐƐ

𝛾 ��

/2𝜅

2𝛾��

/𝜅

|𝛼|

Suppression of cavity dephasing error -‐-‐ by mul*-‐photon driven dissipa*ve process

[ ]† *d d[ ( ) , ]d d d

dd a a a ndt φρ ε ε ρ κ ρ κ ρ⎡ ⎤= − + +⎣ ⎦D D

MulT-‐photon driven dissipaTon Dephasing

/ dφκ κ

( )1/d2 / ddα ε κ∞ =

Two Dimensionless Para:

Induced phase-‐flip rate: ExponenTally suppressed with the cat size . α∞

2

Out[133]=

0 5 10 1510-4

0.001

0.01

0.1

1

»a• 2

-2g

bit-flipêk f

Num: kfêk2 ph=1ê200Num: kfêk2 ph=1ê20

Out[133]=

0 5 10 1510-4

0.001

0.01

0.1

1

»a• 2

-2g

bit-flipêk f


0 2 4 6 810-610-510-40.0010.010.11

»a• 2

-2g

bit-flipêk f

Pert. Thry.

Num: kfêk2 ph=1ê200Num: kfêk2 ph=1ê200 2 4 6 8

10-610-510-40.0010.010.11

»a• 2

-2g

bit-flipêk f

Pert. Thry.


Victor Albert

† † †1 1 1

1 12 2loss a a a a a aκ ρ κ ρ ρ ρ−

⎛ ⎞= − −⎜ ⎟⎝ ⎠D

Dominant Errors in Cavity Quantum Memory

2 21 12 2dephase n n n nφ φκ ρ κ ρ ρ ρ⎛ ⎞= − −⎜ ⎟⎝ ⎠

D

Dephasing Error Photon Loss Error

No Change in photon number! Reduce photon number by one!

(Env. is probing photon number) (Env. is stealing our photons one by one)

Strategy: quickly shuffle photon number … Strategy: monitor photon parity …

QND Measurement of Qubit!

0

5

-5

0 50 100 150 Time (ms)

Results from Devoret group, Yale: Hatridge et al., Science 2013*

dispersive circuit QED readout + JJ paramp

Readout fidelity > 99.5% in ~ 300 nsec

Many groups now working with JJ paramps & feedback, including: Berkeley, Delft, JILA, ENS/Paris, IBM, Wisc., Saclay, UCSB, …

*First jumps: R. Vijay et al., (UCB)

QND Measurement of Photon Number Parity

H = −χqsa†a e e U T = π

χqs

⎛

⎝⎜⎞

⎠⎟=

g g + e e for even n

g g − e e for odd n

⎧⎨⎪

⎩⎪

Sun, Petrenko, et al, Nature 511, 444 (2014)

Effect of photon loss (in presence of driven dissipa*ve process)

a Cα

+ → Cα − and a Cα

− → Cα +

a Cα (0mod 4) → Cα

(3mod 4) = N α − −α( ) + i iα − −iα( )( )a Cα

(2mod 4) → Cα (1mod 4) = N α − −α( )− i iα − −iα( )( )

For two-‐photon process: photon loss à logical bit-‐flip error

For four-‐photon process: photon loss à tractable by parity measurement

Summary on Quantum Memory

Two-‐photon process: 1.  Logical qubit basis of 2.  Photon dephasing induces phase-‐flip errors whose rate

is exp. suppressed by the cat size. 3.  Photon loss induces bit-‐flip errors.

Four-‐photon process: 1.  Logical qubit basis of 2.  Photon dephasing induces phase-‐flip errors whose rate is

exp. suppressed by the cat size. 3.  Photon loss induces errors that are tractable by photon-‐

number parity measurements.

How to achieve universal gates on encoded quantum memory?

( )†2 2 22- 2-

†[ [ , ] [, ]] phX phid a aat

a ad

ρ ε ρ κε ρρ= + − +− + D

Quantum Zeno dynamics for 1.  Resonant drive è Small displacement

2.  Two-‐photon process è ProjecTon on to logical space

D(iε ) Cα

+ = N e− iεα −α + iε + eiεα α + iε( )Cα

+ , Cα −{ }

N e−iεα −α + iε + eiεα α + iε( )→ cos(εα) Cα + + isin(εα) Cα

−

ε XΠ Cα

+ , Cα − (a + a† )Π

Cα + , Cα

− = (α +α∗)ε X Cα + Cα

− + Cα − Cα

+( )∝ X L

2-X phε κ<<Steady State Subspace

gates (2-‐ph process) Xθ

Two-photon process:

Four-photon process:

Decay operator κ2-‐pha2 Driving Hamiltonian iε2-‐ph (a*2-‐a2) Arbitrary rot. around X εX(a*+a) π/2-‐rotaTon around Z -‐χKerr(a*a)2 Two-‐qubit entanglement εXX(a1*a2+a2*a1)

Decay operator κ4-‐pha4 Driving Hamiltonian iε4-‐ph (a*4-‐a4) Arbitrary rot. around X εX(a*2+a2) π/2-‐rotaTon around Z -‐χKerr(a*a)2 Two-‐qubit entanglement εXX(a1*2a22+a2*2a12)

Universal Gates on Quantum Memory

Mirrahimi, Leghtas, Albert, et al., NJP 16, 045014 (2014)

Mazyar Mirrahimi

Summary of Cat Codes

1. How to construct robust quantum memory? a)  Overcome cavity dephasing errors b)  Overcome cavity loss errors

2. How to control such robust quantum memory? a)  Single qubit gates b)  Two qubit gates

Autonomous QEC of cat code

Quantum Zeno Dynamics Steady State Subspace

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Q

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Symmetry & Conserved Quan**es in Lindblad Systems

Q: What informaTon from iniTal state is preserved as infinite Tme?

Albert, L J, PRA 89,022118 (2014); Mirrahimi, et al., NJP 16, 045014 (2014)

For open system ddt

ρ = Lρ =κ 2D a2 −α∞2⎡⎣ ⎤⎦ρ, initial state ρinit involves into ρ∞ = eLtρinit t→∞

,

which is c++ c+−

c−+ c−−

⎡

⎣⎢⎢

⎤

⎦⎥⎥

in the steady state basis Cα+ , Cα

−{ }.

Each degree cjk is associated with a conserved quantity J jk , such that cjk = Tr J jk† ρinit

⎡⎣ ⎤⎦.

The corresponding quantities can be calculated:

which satisfies d

dtJ jk = L†J jk ≡ 0 with no evolution (conserved).

Key result: #(conserved quanTTes) = #(degrees in steady state density matrix) Conserved quantities: (1) efficient tool to compute steady state, (2) extract stored quantum information, (3) perturbative calculation for other decoherences,

Victor Albert

Outlook: Two-‐Qubit Quantum Modules

Design of Two-‐Qubit Quantum Modules

1.  Memory Qubit (m) •  Long coherence Tme

2.  CommunicaTon Qubit (c)

•  IniTalizaTon •  Measurement •  Entanglement generaTon

3.  Local Two-‐Qubit Unitary Gates

𝑚 𝑐

Outlook: Scalable QC

DiVincenzo’s Criteria •  A scalable physical system with well characterized qubits •  The ability to ini*alize the state of the qubits •  The ability to measure specific single qubits •  A universal set of quantum gates •  Long relevant decoherence *me, much longer than the gate opera*on *mes

…

Use cavity mode for long-‐lived quantum

memory!

Outlook: Scalable QC

DiVincenzo’s Criteria •  A scalable physical system with well characterized qubits •  The ability to ini*alize the state of the qubits •  The ability to measure specific single qubits •  A universal set of quantum gates (esp., focusing on the remote gates) •  Long relevant decoherence *me, much longer than the gate opera*on *mes

…

… Switchable

Router

Acknowledgement

Victor Albert

Stefan Krastanov

Chao Shen

Brian Vlastakis

Zaki Leghtas Rob Schoelkopf

Michel Devoret

Mazyar Mirrahimi Reinier Heeres

quantum control of superconducting...

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