“putting the bell on schrodinger’s cat”

Rob Schoelkopf

Applied Physics, Yale University

PI’s: RS Michel Devoret Luigi Frunzio Steven Girvin Leonid Glazman Liang Jiang Mazyar Mirrahimi

“Putting the Bell on Schrodinger’s Cat”

Postdocs & students wanted!

Overview

• A few remarks on quantum circuits, then and now…

Observing quantum jumps of photon parity: Sun, Petrenko et al., Nature 511, 444 (2014).

• Tracking photon parity of a cat state in real time

CHSH for encoded qubit: Petrenko, Vlastakis, et al., submitted & arXiv (2015).

• “Putting the Bell on a cat”: efficient tomography and single-shot violation for a logical qubit

Before we met…

Some early influences:

• Koch, PhD thesis ~ 1985

• MQT: Martinis, Devoret, Clarke,…

• Control/influence of EM environment!

Forerunner of circuit QED!

In AMO language: strong coupling, bad cavity limit

Devoret, Esteve, Martinis, and Urbina, 1989.

Before we met…

This is sufficient to perform interesting manipulations on the quantum state…

… decoherence…is mainly due to the… electromagnetic environment of the circuit. …we estimate that the life-time of a Q-bit …can be longer than 100 µs.

When I began… First Yale/Chalmers spectroscopy of

Cooper-pair box, ca. 2000 -2002, aka Bouchiat + RF-SET

Linewidth? ~ 1 GHz

T1 = 103 µs

T2E = 145 µs

time (µs)

time (µs)

Rea

dout

Vol

tage

(mV

) R

eado

ut V

olta

ge (m

V)

We’ve come a long way together…

2 1

1 1 12T T Tφ

= +

echo2 1~ 1.5T T

A 3D transmon qubit

Coherence of Photon States in Cavity State preparation by SNAP:

Heeres et al., arXiv:1503.01496,

Thy: Krastanov et al., arXiv: 1502.08015

T1c = 1.2 ms T2c = 0.8 ms

Tφc = 1 ms

Dephasing and relaxation actually limited by qubit… (reverse Purcell and qubit thermal population)

Reagor, Pfaff, et al., in preparation ∆f / f = 100 Hz / 4 GHz ~ 25 ppb !!

“Putting the Bell” on Schrodinger’s Cat

Petrenko, Vlastakis, et al., submitted (2015). Gustave Dore, ca. 1868

“But who will volunteer to place it?”

“The mice in council”

Violation of a CHSH inequality for a macroscopic quantum state

courtesy University of Illinois, Urbana-Champaign

Schrodinger’s Cat = an entangled state between a microscopic object (atom or qubit) and a macroscopic object (easily distinguished by “environment”)

g eor

( )alive d1 ead2

e gΨ = +

“meow?”

“ack!”

??

Non-linear electromagnetic

oscillator

Ene

rgy

g

egeω

efωge efω ω≠

Atom: The Transmon Qubit

2† † †

J0

ˆ 2 ˆ2

cos geH E a aQ a a aaC

π ω λ

Φ = − +…= Φ −

~ 5 10

~ 0.25ge

B

ω

ω

− GHz/k K

Superconductor

Superconductor (Al)

Insulating barrier 1 nm

Josephson junction (dissipation-free?)

Other practitioners (many!): UCSB/Google, Berkeley, Princeton, Delft, Zurich, Chicago…

geH e eω≈ Koch et al., PRA, 2007; Houck et al., PRL, 2008

The Cat: A Cavity Oscillator

E

x

x

ω

2ψzpf / 2x mω=

Glauber (coherent) state 2| |

2

0| |

!

n

ne n

n

β ββ∞−

=

⟩ = ⟩∑max zpfx xβ=

if we can only apply classical controls (e.g. laser, force), can only make displacements

a β β β=

0t =

2Enβ ωω

=

=

What’s a Cat State of an Oscillator?

E

x

x

ω

ψCat state of an oscillator (field)

0t =

/ 2x mω∆ = ( )12

ψ β β= + −

2d xβ= ∆

Size of the superposition, d: 22 4 4d nβ= =

2Enβ ωω

=

=

Bigger cats die faster: rate = 2d κ


E

x

x

ω

Cat state of an oscillator (field)

0t =

/ 2x mω∆ =

( )12

ψ β β= + −

This one is EVEN parity!

ψ

2Enβ ωω

=

=

2d xβ= ∆


E

x

x

ω


0t =

/ 2x mω∆ =

( )12

ψ β β= − −

This one is ODD parity!

ψ

2Enβ ωω

=

=

2d xβ= ∆


E

x

x

ω


0t =

/ 2x mω∆ =

( )12

ψ β β= − −

This one is ODD parity!

2ψ

2Enβ ωω

=

=

2d xβ= ∆


E

x

x

ω

2ψSchrödinger cat state

/ 2x mω∆ =

( )12

ψ β β= + −

2t πω=

What happens now, when packets collide?

~ /x β∆fringes

The sign of fringe = “parity”

2Enβ ωω

=

=

Seeing the Interference: Wigner Function }{( ) ( ) ( )2

m m mW D DTr Pα α ρ απ

= −

( )†

1 ni a aP e π= = −Parity

or xΦ

Thy:

Negative fringes = “whiskers”

Expt’l. Wigner tomography: Leibfried et al., 1996 ion traps (NIST) Haroche/Raimond , 2008 Rydberg (ENS) Hofheinz et al., 2009 in circuits (UCSB)

orQ p

Using a Cavity as a Logical Qubit?

Qubits 1..7

High-Q (memory)

Ancilla qubit Readout

Our approach : • Cavity is the memory • One error syndrome

Register as memory: 1. More qubits 2. More decay channels

1. 1 qubit! 2. Single readout channel!

Ancilla qubits Readout

earlier ideas: Gottesman, Kitaev & Preskill, PRA 64 , 012310 (2001)

“Hardware-efficient QEC” Leghtas, Mirrahimi, et al., PRL 111, 120501(2013).

Encoding a Quantum Bit in a Cavity State

2-Level System (e.g. transmon)

g e+ g j e+e

gContinuous Variables (e.g. cavity)

cavity 7.22sf GHz=55s sτ µ=

Dispersive cQED Coupling: 2 Cavities + 1 Qubit + Paramp

( )† †/ q qs sH a a e e a aω χ ω= − +

JBA readout 8.17rf GHz=30r nsτ =

5.94qf GHz= 1 10T sµ=2 10T sµ=

qubit

98%RF =

95%PF =

Readout Fidelity

Parity Readout Fidelity

Dispersive Hamiltonian:

Strong* Dispersive Regime

“doubly-QND” interaction

Allows qubit to control many photons at once (and vice-versa)

( )2~ / s qgχ ω ω−,χ γ κ

† †q sa ae e e eaH aω χω= + −

* Schuster et al., 2007; prev. attained only in Rydberg cQED (ENS-Paris)

n=0

n=1

n=0

n=1

n=2

n=2

χ~ 0n

~ 0.5n

~ 1n

0n =1n =2n =

qubi

t abs

orpt

ion

cavity

qubit

Deterministic Cat Creation: QCMAP Gate

Leghtas et al., Phys. Rev A 87, 042315 (2013) Theory:

0gψ = ⊗

cavity

qubit



( ) 0N g eψ = + ⊗

cavity

qubit



( )N g eψ β= + ⊗

cavity

qubit


Deterministic Cat Creation: QCMAP Gate †

int e eH a aχ=

( ), , i tN g e e χψ β β −= +

cavity

qubit



t π χ=after time:

“Here be kittens!”

( ), ,N g eψ β β= + −

†int e eH a aχ=

Encoding a Quantum Bit in a Cavity State

Continuous Variables (e.g. cavity)

Efficiently Measuring 𝑋𝑋𝑐𝑐 ,𝑌𝑌𝑐𝑐, 𝑍𝑍𝑐𝑐

−2

−2

0 2

0

2

𝑅𝑅𝑅𝑅(𝛼𝛼)

𝐼𝐼𝐼𝐼(𝛼𝛼

)

P

+1

−1

Cavity state along 𝑋𝑋𝑐𝑐

Measured Wigner function of cavity


−2

−2

0 2

0

2



)

cZ cZcX

cY

P

+1

−1



−2

−2

0 2

0

2



)

cZ cZcX

cY

P

+1

−1 0.07cZ P Pβ β−= − =

0 0.76cX P= =

80.14jcY Pπ

β= =



−2

−2

0 2

0

2



)

cZ cZcX

cY

P

+1

−1

Cavity state along 𝑌𝑌𝑐𝑐

A Bell State

( )12B g e e gψ = +

g e+ g j e+

g


g e+ g j e+

g


The Schrodinger Cat or “Bell-Cat”

( )12B g eψ β β= + −

g e+ g j e+

g

Continuous Variables (e.g. cavity)


0P

8jPπ

β

P Pβ β−−

CX

CZCY

X

ZYBψ

Measuring Bell-Cat Correlations

Vlastakis et.al. Science 2013

0P

8jPπ

β

P Pβ β−−

CX

CZCY

X

ZYBψ

Performing a CHSH Measurement

' ' ' ' ' 'ABA B c c c cO AB AB A B A B= + − +

yRθ

/2yRθ π+

0P

P Pβ β−−

CX

CZ

Vlastakis et al. Science 2013

Violating the CHSH Inequality

g

e

( )12

g e+

XZZX c c c cO XZ XX ZX ZZ= + − +

g

( )12

g e+ 135°

g

( )12

g e+

45°

MAX VIOLATION MAX VIOLATION

Bell

Sign

al 𝑂𝑂

𝛽𝛽 = 1

Violating the CHSH Inequality

2.30 ± 0.04

−2.28 ± 0.04

XZZXO

ZXXZO

WE TAKE ALL MEASUREMENTS: No correction for detector inefficiency

Bell

Sign

al 𝑂𝑂

𝛽𝛽 = 1

Continuously Varying Bell-Cat Size

Increasing 𝛽𝛽: Realizing a Schrodinger’s Cat

Experiment

2.14 ± 0.04

Still violating at 𝑑𝑑2 = 16 photons Finite encoding and

measurement fidelity

dead alivege +

𝑑𝑑 = 2𝛽𝛽 Be

ll Si

gnal

𝑂𝑂

see also Brune et al., 1996

What Comes Next?

Cavity as a Correctable Memory?

Store a qubit as a superposition of two cats of same parity

0 ( )L α β β+= = + −C N

1 ( )L i i iα β β+= = + −C N

10e eg Lg Lc cc c⇒↑ ++↓

Leghtas, Mirrahimi, et al., PRL 111, 120501(2013).

Z. Leghtas M. Mirrahimi

Cat States for Hardware-Efficient QEC?

even odd even odd odd even

High-Q (memory)

Ancilla qubit Readout

Leghtas, Mirrahimi, et al., PRL 111, 120501(2013). “Cat codes”: much less hardware required

1st tracking of a parity or error syndrome in real-time:

Sun, Petrenko et al., Nature 511, 444 (2014). parity msmt. ala’ Bertet et al.,

Merci! … and my apologies

Bon anniversaire Quantronics!

Summary

Qubits: T2 ~ 2*T1 ~ 0.0001 sec Cavities: T2 ~ 2*T1 ~ 0.001 sec

• Coherence in circuit QED passing the QEC threshold.

• Tracking the jumps of an error syndrome: photon parity

• Cat-codes: a new shortcut for QEC?

Leghtas, Mirrahimi et al., PRL 2013. Sun, Petrenko et al., Nature 511, 444 (2014). Petrenko, Vlastakis et al., submitted & arXiv (2015)

Next challenge: “breakeven” for error correction!

• Bell violation between qubit and continuous variable system: benchmarking a module

“putting the bell on schrodinger’s cat”

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