Department of Applied Physics, Yale University

Michael Hatridge

Remote entanglement of transmon

qubits

Katrina Sliwa Anirudh Narla Shyam Shankar Zaki Leghtas Mazyar Mirrahimi Evan Zalys-Geller

Chen Wang Luigi Frunzio Steven Girvin Robert Schoelkopf Michel Devoret

35

What is remote entanglement and why is it important?

How do we engineer interactions over arbitrary length scales?

Q1

Q2

Q3

Alice

Bob

Monroe, Hanson, Zeilinger

QN+1

QN+2

QN+3

Direct interaction Remote entanglement via msmt of ancilla

34

Remote entanglement with flying qubits

ALICE

BOB

bert

arnie

|𝐺

|𝑔

|𝑔

|𝐺

1/ 2 |𝐺𝑔 + |𝐸𝑒 ⊗

1/ 2 |𝐺𝑔 + |𝐸𝑒

How can we do this with a superconducting system?

Instead of qubits use coherent states

How do we entangle flying qubits?

How well can we transmit our flying qubit?

How do we build efficient detector?

quantum-limited amplification

𝑅𝑥 𝜋/2 ,

𝑅𝑥 𝜋/2 ,

1/ 2 |𝐺𝐺 + |𝐸𝐸 or

1/ 2 |𝐺𝐺 − |𝐸𝐸 or

1/ 2 |𝐺𝐸 + |𝐸𝐺 or

1/ 2 |𝐺𝐸 − |𝐸𝐺

measure X

(sign)

measure Z

(parity)

33

Part 1: measurement with coherent

states

32

microwave cavity

coherent pulse

phase meter

dispersive cavity/pulse interaction

Dispersive measurement: classical version

|𝑔

transmission line

31

microwave cavity

coherent pulse

phase meter

dispersive cavity/pulse interaction

Dispersive measurement: classical version

|𝑔 |𝑒

transmission line

30

microwave cavity

coherent pulse

dispersive cavity/pulse interaction

Now a wrinkle: finite phase uncertainty

|𝑔 |𝑒

transmission line

phase meter

29

microwave cavity

coherent pulse

dispersive cavity/pulse interaction

|𝑔 |𝑒

phase meter

Measurement with bad meter (still classical)

noise added by amp.

AND signal lost in transmission

• Each msmt tells us only a little • State after msmt not pure! • This example optimistic, best

commercial amp adds 20-30x noise • We fix this with quantum-limited

amplification

28

Ideal phase-preserving amplifier • What is power dissipated in Rload due to noise in Rsource, for Rload = Rsource = R?

BTkRR

TRBkR

R

VP B

Brmsload

22

2

4

2

NBload TTBGkP

𝜌𝑖 → 𝜌𝑓 = 𝑇𝑟𝑈𝜌𝑖𝑈†

𝑆 𝜌𝑓 = 𝑆 𝜌𝑖

MATH

Phase-sensitive amps

180° hybrid (beam splitter)

𝜙 = 0

𝜙 = 𝜋2

|0

|𝛼 𝐺 ≫ 1

𝐺 ≫ 1

Signal in

Idler in

180° hybrid (beam splitter)

These ports are often internal degrees of freedom, in our amp they are accessible. We’ll use this for remote entanglement

Signal out

Idler out

• Adds its inputs, outputs 2 copies of combined inputs • Adds minimum fluctuations to signal output*

* 𝜎𝑜𝑢𝑡2 = 2𝜎𝑖𝑛

2 (Caves’ Thm) Caves, Phys Rev D (1982)

27

microwave cavity

coherent pulse

Quantum-limited amplification: projective msmt

|𝑔 |𝑒

phase meter w/ P. P. pre-amp

• state of qubit pure after each msmt • For unknown initial state

𝑐𝑔|𝑔 + 𝑐𝑒|𝑒 , repeat

many times to estimate 𝑐𝑔2

, 𝑐𝑒2

only quantum fluctuations

coherent superposition

26

microwave cavity

WEAK coherent

pulse

Quantum-limited amplification: ‘partial’ msmt

|𝑔 |𝑒

phase meter w/ P. P. pre-amp

• state of qubit pure after each msmt • counter-intuitive, but is achievable

in the laboratory

only quantum fluctuations

coherent superposition

25

Part 2: Partial measurement with

transmon qubit and JPC

24

200 nm

The Josephson tunnel junction

SUPERCONDUCTING TUNNEL JUNCTION

𝐼 = 𝐼0 sin𝜙

𝜑0

1nm

Al/AlOx/Al tunnel junction

nonlinear inductor shunted by capacitor

02e

LJ

CJ

𝐼

𝜙

23

Superconducting transmon qubit

Potential energy

f

Josephson junction with shunting capacitor anharmonic oscillator

lowest two levels form qubit

fge ~ 5.025 GHz, fef ~ 4.805 GHz

Koch et al., Phys. Rev. A (2007)

|𝑔

|𝑒

|𝑓

22

mixer

Measurement configuration

TR

number of inddent signal modes:

cavity ring-down time 1 2

r

Q

/S RM T

𝑎 𝑔 ⊗ 𝛼𝑔, 0 + 𝑏 𝑒 ⊗ 𝛼𝑒 , 0

isolator circulator

Sig Idl Pump

compact resonator

+ qubit pulses

JPC

HEMT

𝐼𝑚 = 𝐼 𝑡 𝑑𝑡𝑇𝑚

0

readout pulse

Ref

transmon

𝑄𝑚 = 𝑄 𝑡 𝑑𝑡𝑇𝑚

0

on qubit state?

𝐼𝑚 = 𝐼 𝑡 𝑑𝑡𝑇𝑚

0

Readout

phase

tan−1 𝑄𝑚𝐼𝑚

𝜋2

width 𝜅

Readout

amplitude

𝐼𝑚2 + 𝑄𝑚

2 𝑓

dispersive shift 𝜒

Qubit +

resonator + qubit

pulses

JPC

|𝑔 |𝑒

|𝑔 |𝑒 𝜗 = 2 tan−1 𝜒

𝜅

readout

pulse at

𝑓𝑑

𝑓

Ref

𝑄𝑚 = 𝑄 𝑡 𝑑𝑡𝑇𝑚

0

Sig Idl

Pump

𝑓𝑑

HEMT

− 𝜋2

vacuum 50Ω

21

Isolating the transmon from the environment

transmon output coupler

Purcell filter

waveguide-SMA adapter

input coupler

Cavity fc,g = 7.4817 GHz 1/ = 30 ns

10 mm

25 mm

Qubit fQ=5.0252 GHz T1 = 30 s T2R = 8 s

20

The 8-junction Josephson Parametric Converter

G=20 dB

NR= 9 dB

BW= 9 MHz

D.R. ~ 10 photons @ 4.5 MHz

Tunability ~100’s of MHz

~88% of output noise is

quantum noise!

→ quantum fluctuations

on an oscilloscope

Idler

Signal

ADD SOME REF?

Bergeal et al Nature (2010)

See also Roch et al PRL (2012)

20

15

10

5

G (

dB

)

7.507.487.467.447.42

x109

F 7.42 7.46 7.50

Frequency (GHz)

Direct G

(dB

)

20

10

not a defect! quantum jumps of connected qubit

10 m

19

𝐼𝑚/𝜎

𝑄𝑚

/𝜎

|𝑔 10

5

0

10 5 0 -5 -10

102

104

1

|𝑒

𝑓 , … 𝑄𝑚

/𝜎

𝐼𝑚/𝜎

6000

0

10

5

0

10 5 0 -5 -10

|𝑔 |𝑒

MA

𝑓 , …

Rotate to z=0 State preparation Confirm state

8.6 σ

Rotate to z=0 State preparation Confirm state

𝑅𝑥 𝜋/2 𝑅𝑥 𝜋 𝐼𝑑 or

𝑛 11 𝑛 11

640 ns

𝑇𝑚 320 ns

Trep = 20 µs

See also Riste et al PRL (2012) Johnson et al PRL (2012)

Preparation by measurement + post-selection 18

Rotate to z=0 State preparation Confirm state

𝑅𝑥 𝜋/2 𝑅𝑥 𝜋 𝐼𝑑 or

𝑛 11 𝑛 11

640 ns

𝑇𝑚 320 ns

|𝑒

102

104

1

MA MB|MA = |𝑔 Trep = 20 µs

𝐼𝑚/𝜎

𝑄𝑚

/𝜎

10

5

0

10 5 0 -5 -10

|𝑔 |𝑒

𝑓 , …

𝑅𝑥 𝜋 “|𝑒 ”

𝑄𝑚

/𝜎

𝐼𝑚/𝜎

10

5

0

10 5 0 -5 -10

|𝑔 |𝑒

𝑓 , …

𝐼𝑑 “|𝑔 ”

Fidelity=0.994! Error budget:

Msmt ~0.003

T1 ~0.001-0.002

rotation <0.002

Now that we have outcomes MA= |𝑔 either do nothing to retain |𝑔 OR rotate qubit by 𝑅𝑥 𝜋 to create |𝑒

Preparation by measurement + post-selection 17

𝑄𝑚

/𝜎

𝐼𝑚/𝜎

10

5

0

10 5 0 -5 -10

|𝑒

𝐼𝑚/𝜎 𝑄

𝑚/𝜎

10

5

0

10 5 0 -5 -10

102

104

1

|𝑔 |𝑒

𝑓 , …

|𝑔 |𝑒

𝑓 , …

𝑅𝑥 𝜋 “|𝑒 ” 𝐼𝑑 “|𝑔 ”

How ideal is this operation?

Fidelity=0.994!

Strong measurements allow rapid, high-fidelity state preparation and tomography

Say : “practically useful, but doesn’t tell us how ideal we are, and how much room for improvement in signal processing”

16

A picture is worth a thousand math symbols * : Mapping (𝐼𝑚, 𝑄𝑚) to the bloch vector

For weak msmt:

𝜕𝑥𝑓

𝜕𝑄𝑚 𝐼𝑚,𝑄𝑚=0

= η𝐼𝑚𝑔

− 𝐼𝑚𝑒

2

• Now calculate the effect of some added classical noise on the density matrix • Efficiency 𝜂 = 𝜎𝑚

𝜎2

• 𝑥𝑓, 𝑦𝑓 provide important information about efficiency

Rotation (𝑸𝒎) Convergence to poles (𝑰𝒎)

Information loss due to 𝜼 < 𝟏

COWBOY HAT

w/ curvy arrow

𝑄𝑚

𝐼𝑚

𝐼𝑚 gives latitude information

𝑄𝑚 gives longitude information

𝑥

𝑦

𝑧

*Gambetta, et al PRA (2008); Korotkov/Girvin, Les Houches (2011); M. Hatridge et al Science (2013)

The equator is a dangerous place: lost information pulls trajectory

towards the z-axis

15

mixer

Back-action characterization protocol

TR

number of inddent signal modes:

cavity ring-down time 1 2

r

Q

/S RM T

𝑛 11 𝑛 11

Tomography

𝑅𝑥 𝜋/2

𝑅𝑥 𝜋/2 ,

𝑅𝑦 𝜋/2 ,

700ns

qubit

cavity

𝑥𝑓, 𝑦𝑓, 𝑧𝑓

variable 𝑛

𝐼𝑑 or

𝑇𝑚 320 ns

Variable strength

measurement

State preparation

X = 1 𝑥

𝑦

𝑧

or

or Y = 1

Z = 1

(𝐼𝑚, 𝑄𝑚)

14

histogram of measurement after p/2 pulse tomography along X, Y and Z after measurement

Measurement with 𝑰 𝒎 𝝈 = 0.4

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0

-1

Counts

Max 0

𝑄𝑚

/𝜎

𝐼𝑚/𝜎

𝑄𝑚

/𝜎

𝐼𝑚/𝜎

6

0

6 0 -6

6

0

6 0 -6

6

0

6 0 -6

𝑋 𝑐 𝑌 𝑐

𝑍 𝑐

6

0

6 0 -6

𝐼 𝑚

𝜎

13

histogram of measurement after p/2 pulse tomography along X, Y and Z after measurement

Measurement with 𝑰 𝒎 𝝈 = 1.0

Pro

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0

-1

Counts

Max 0

𝑄𝑚

/𝜎

𝐼𝑚/𝜎

𝑄𝑚

/𝜎

𝐼𝑚/𝜎

6

0

6 0 -6

6

0

6 0 -6

6

0

6 0 -6

𝑋 𝑐 𝑌 𝑐

𝑍 𝑐

6

0

6 0 -6

𝐼 𝑚

𝜎

12

histogram of measurement after p/2 pulse tomography along X, Y and Z after measurement

Measurement with 𝑰 𝒎 𝝈 = 2.8

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0

-1

Counts

Max 0

𝑄𝑚

/𝜎

𝐼𝑚/𝜎

𝑄𝑚

/𝜎

𝐼𝑚/𝜎

6

0

6 0 -6

6

0

6 0 -6

6

0

6 0 -6

𝑋 𝑐 𝑌 𝑐

𝑍 𝑐

6

0

6 0 -6

𝑓 , … show at ~10-4 contamination

𝐼 𝑚 𝜎

11

𝑛

Measurement strength 𝐼𝑚/𝜎

𝑄𝑚

/𝜎

𝑄𝑚

/𝜎

𝑛 Experiment Theory

x- and y-component along 𝑰𝒎 = 𝟎

0 2 𝐼𝑚/𝜎

0

4

0

4

𝑌 𝑐 along 𝐼𝑚 = 0

𝑋 𝑐 along 𝐼𝑚 = 0

𝑋𝑐,

𝑌𝑐

0 6

0

-1

1

Amplitude determined by one fit parameter: 𝜼 = 0.57 ± 0.02

𝑰 𝒎 𝝈 = 0.82

𝑋 𝑐 = sin𝑄𝑚

𝜎

𝐼 𝑚𝜎

+ 𝜃 exp −𝐼 𝑚𝜎

21 − 𝜂

𝜂

𝑌 𝑐 = cos𝑄𝑚

𝜎

𝐼 𝑚𝜎

+ 𝜃 exp −𝐼 𝑚𝜎

21 − 𝜂

𝜂

-6

𝜼 ≥ 𝟎. 𝟓 → 3 body entanglement (qubit, signal, idler)

𝑄𝑚/𝜎

10

Part 3: remote entanglement

experiment

9

Two qubit readout schematic

𝑇1 = 30 𝜇s 𝑇2𝑅 = 15𝜇s

𝑓𝑄𝑔𝑒

= 4.672 GHz 𝜒

2𝜋 = 2.7 MHz 𝜅 2𝜋 = 6.7 MHz

𝑇1 = 15𝜇s 𝑇2𝑅 = 15 𝜇s

𝑓𝑄𝑔𝑒

= 6.074GHz 𝜒

2𝜋 = 3.5 MHz 𝜅 2𝜋 = 4.1 MHz

𝑓𝑝 = 16.58 GHz

𝑓𝑟𝑑 = 7.464 GHz 𝑓𝑟

𝑑 = 9.116 GHz

50 Ω

8

Simultaneous readout of two qubits

Signal Alone

𝑄𝑚

𝐼𝑚

|𝑒𝑠

|𝑔𝑠

Idler Alone

𝑄𝑚

𝐼𝑚

|𝑒𝑖

|𝑔𝑖

Together

𝑄𝑚

𝐼𝑚

|𝑒𝑒

|𝑔𝑔

|𝑔𝑒

|𝑒𝑔

“Joint Readout”

|𝑒𝑠 |𝑔𝑠

|𝑔𝑖

|𝑒𝑖

𝐼𝑚 encodes Z info

𝑄𝑚 encodes Z info

CHANGE ALL TO FUZZY BALLS

7

|𝑒𝑒

|𝑔𝑔 |𝑔𝑒

|𝑒𝑔

How to perform “entangling readout”

Signal Alone

𝑄𝑚

𝐼𝑚

|𝑒𝑠

|𝑔𝑠

Idler Alone

𝑄𝑚

𝐼𝑚

Together

𝐼𝑚 encodes Z info

𝐼𝑚 encodes Z info

𝑄𝑚 (sign)

𝐼𝑚 (parity)

|𝑒𝑒 |𝑔𝑔

|𝑔𝑒 , |𝑒𝑔

“Entangling Readout”

|𝑒𝑖

|𝑔𝑖

|𝑔𝑖

|𝑒𝑖

• 𝐼𝑚 is now blind to contents of |𝑔𝑒 , |𝑒𝑔

• With/ appropriate initial state, outcome is Bell state w/ 50 % success rate

• 𝑄𝑚 encodes phase of Bell state

6

|𝑒𝑒 |𝑔𝑔

|𝑔𝑒 , |𝑒𝑔

Back action of two qubit msmt creates entanglement

For weak msmt:

𝜕𝑥𝑓

𝜕𝑄𝑚 𝐼𝑚,𝑄𝑚=0

= η𝐼𝑚𝑔

− 𝐼𝑚𝑒

2

• Now calculate the effect of some added classical noise on the density matrix • Efficiency 𝜂 = 𝜎𝑚

𝜎2

• 𝑥𝑓, 𝑦𝑓 provide important information about efficiency

Rotation (𝑸𝒎) Convergence to poles (𝑰𝒎)

Information loss due to 𝜼 < 𝟏

COWBOY HAT

w/ curvy arrow

𝑄𝑚

𝐼𝑚

𝐼𝑚 gives info on even vs. odd

parity (a bit too much, actually)

𝑄𝑚 gives sign info for odd

parity states

Even parity states:

= |𝑔𝑔

= |𝑒𝑒

= |𝑔𝑒 − |𝑒𝑔

= |𝑔𝑒 + 𝑖|𝑒𝑔

= |𝑔𝑒 + |𝑒𝑔

= |𝑔𝑒 − 𝑖|𝑒𝑔

Odd parity states:

Make note about which we keep…which are good ones

The climax… this is my favorite part..

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+1

Tomography of strong entangling msmt

Histogram

𝑄𝑚

𝜎

𝑋𝑋 𝑐 𝑍𝐼 𝑐

𝑍𝑍 𝑐

𝐼𝑚 𝜎

10

0

-10

-10 0 10

𝑄𝑚

𝜎

𝐼𝑚 𝜎

10

0

10 0

-10

-10

𝑄𝑚

𝜎

10 0 -10

10

0

-10

10 0 -10

10

0

-10

4

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Tomography of weak entangling msmt

Histogram

𝑄𝑚

𝜎

𝑋𝑋 𝑐 𝑍𝐼 𝑐

𝑍𝑍 𝑐

𝐼𝑚 𝜎

5

0

-5

-5 0 5

𝑄𝑚

𝜎

𝐼𝑚 𝜎

5

0

5 0

-5

-5

𝑄𝑚

𝜎

5 0 -5

5

0

-5

5 0 -5

5

0

-5

3

XX YY ZZ

Average a strip along 𝑄𝑚 ≃ 0

Blo

ch c

om

p. v

alu

e 5

0

𝑄𝑚

𝜎

𝐼𝑚 𝜎 𝑄𝑚 𝜎

-5 0 5

0

0.5

-0.5

𝑋𝑋 𝑐

5 0

-5

-5

𝑄𝑚

𝜎

Signature of entangling operation

• currently, too much information is lost • correct dependence of qubit correlations • expect to demonstrate entanglement soon (F > 0.5)

2

Evolution of single-qubit readout vs time

Year

1

0

0.5 To

tal e

ffic

ien

cy 𝜂

20

10

20

12

20

14

Conclusions

• Coherent states can be used as flying qubits • Quantum mechanics goes through the amplifier • New tools for building large-scale quantum

entanglement

Compare with optical systems: 𝜂~10−3

due to collector/detector inefficiencies

1