quantum teleportation with sc qubits - eth z · the protocol bennett, charles h., et al....

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David Castells Graells, Ankit Anand

26.03.2018 1

Quantum teleportation with SC Qubits

David Castells Graells, Ankit Anand

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Motivation

▪ Transfer the state of an information carrier essential primitive in both classical and

quantum communication and information processing

▪ Quantum teleportation: transferring unknown quantum state between two parties at two

different physical locations without transferring the physical carrier of information itself

▪ Use of non-local correlations:

▪ Entangled pair shared between sender and receiver

▪ Exchange of classical information

▪ Extended range of quantum communication quantum repeaters

▪ Used to implement logic gates for universal quantum computation

26.03.2018David Castells Graells, Ankit Anand 3

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The Protocol

26.03.2018David Castells Graells, Ankit Anand 4Bennett, Charles H., et al. "Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen

channels." Physical review letters 70.13 (1993): 1895.

tim

e

𝝍

𝝍

𝝓𝑨 𝝓𝑩

Input state

EPR pair

• Creation of entangled pair shared between sender and receiver

• Joint two qubit measurement in bell basis. Identify BS at the sender with 1/4 probability

• Quantum state projected on Bob’s qubit up to a rotation

• Feed forward classical information to perform the final qubit rotation

00 01 10 11

Bell measurement

R

𝝍 ′

Output state

classical

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The Protocol


Feed-forward

classical

Pauli matricesHadamard gate

A

A

B

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The Protocol

Feed-forward

classical

A

A

B


Technical challenge:

finite lifetime qubitsState preparation: Deterministic vs. Probabilistic

Measurement: Unconditional (Determ.) vs. Post selected

00 01 10 11(25%) { I X Z Y }


Outline

1) Historical overview

2) Implementation with SC qubits:

i. Protocol requirements

ii. System description

iii. Gate implementation

iv. Measurement

3) Readout characterization

4) Results

5) Conclusions and outlook

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2012 - Bao, et. al.

• Atomic qubits

• 150m

• Probabilistic

• 2 BSM - Post-sel.

• Fpr = 88%

1993Bennet, et. al.

• Theoretical

proposal

1997 - Bouwmeester, et.al.

• Single photons

• Lab scale

• Deterministic

• 2 BSM - Post-select.

• Proof-of-principle

1998 - Furusawa, et. al.

• Phot. continuous-

variable state

• Lab scale

• Deterministic

• 4 BSM - Uncond.

• F = 58%

2000 - Kim, et. al.

• Non-lin. Phot. interact

• Lab scale

• Deterministic

• 4 BSM - Uncond.

• Low efficiency

2012 - Ma, et. al.

• Single photons

• 143 km (free)

• Probabilistic

• 2 BSM -

Uncnd Fpr = 71%

Post-slct F = 86%

2013 - Steffan, et.al.2004 - Riebe, et. al.

• Atomic qubits

• Same trap (~1μm)

• Deterministic

• 4 BSM - Uncond.

• F = 75%

2004 - Barrett, et. al.

• Atomic qubits

• Same trap(~μm)

• Deterministic

• 4 BSM - Uncond

• F = 78%

2009 - Olmschenk, et. al.

• 2 at. + 2 ph qubits

• 1 m

• Probabilistic

• 2 BSM - Uncond.

• F = 90%

• Fpr = 84%

2014 - Pfaff, et. al.

• Diamond spin qb

• 3 meters

• Deterministic

• 4 BSM - Uncond

• F = 86%

Historical overview

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Superconducting

Qubits: Protocol

requirements An ideal protocol requires

• Creation of entangled pair shared between

sender and receiver

• Two qubit measurement identifying all four bell

state at the sender

• Feed forward classical information to perform the

final qubit rotation

• High rate over high distance to maximize

usefulness.

3 types of experiment done here in single set up

• Post selected teleportation

• Deterministic teleportation (use correlation

between sender’s and receiver’s measurements).

• Deterministic teleportation with feed forward

(measurement of sender is feedforward to

receiver to do rotation and retrieve the exact

input).

Here, all of the above is achieved except spacelike

separation (6mm at the rate of 104 𝑝𝑒𝑟 sec ) 26.03.2018David Castells Graells, Ankit Anand 9

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Superconducting

Qubits• A chip designed with four

superconducting transmon qubits [1] connected via three resonator (only 3 in use)

• Qubits are coupled to each other via Quantum Bus (Superconducting coplanar waveguide resonator)(microwave photon confined in a transmission line cavity). [2]

• Resonators acts as a Quantum Bus which is used to create Bell state distributed between sender and receiver and to perform a deterministic Bell analysis at the sender.

[1] Koch, J., Terri, M.Y., Gambetta, J., Houck, A.A., Schuster, D.I., Majer, J., Blais, A., Devoret, M.H., Girvin, S.M. and Schoelkopf, R.J., 2007. Charge-insensitive qubit design

derived from the Cooper pair box. Physical Review A, 76(4), p.042319.

[2] Majer, J., Chow, J.M., Gambetta, J.M., Koch, J., Johnson, B.R., Schreier, J.A., Frunzio, L., Schuster, D.I., Houck, A.A., Wallraff, A. and Blais, A., 2007. Coupling

superconducting qubits via a cavity bus. Nature, 449(7161), p.443.


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Superconducting

Qubit: Driving the

autopilot

• A rotation on a single Qubit can be performed by sending microwave pulse on the coupled transmission line.

• Length of the pulse determines the angle of rotation while the axis is set by Quadrature Amplitude modulation of the pulse.

• As an example of single Qubit operation.

• 𝐸 𝑡 = 𝐸𝑥 𝑡 cos 𝜔𝑑𝑡 + 𝐸𝑦 sin 𝜔𝑑𝑡

•𝐻𝑟

ℏ= 𝜔 − 𝜔𝑑 1 1 +

𝐸𝑥 𝑡

2𝜎𝑥 +

𝐸𝑦 𝑡

2𝜎𝑦

• E.g if you want rotation about x then you can have, 𝜔 = 𝜔𝑑 , 𝐸

𝑦 𝑡 = 0 ⇒ 𝑅𝑥 𝜃 = ∫ 𝐸𝑥 𝑡 𝑑𝑡

• All gates can be achieved through the control and rotation operation.

• Therefore it is the pulse sequence which at the end determines all the operation you are performing.


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Feed-forward

Feed-forward

Superconducting

Qubit: Driving the

autopilot


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Superconducting

Qubits : Measurement

technique• Resonators acts as a Quantum Bus

which is used to create Bell state

distributed between sender and

receiver and to perform a

deterministic Bell analysis at the

sender.

• In your harmonic oscillator you have

voltage as a variable which you can

read using amplifier. You want to

measure it’s magnitude and phase

as well. That’s why you mix it with LO

(Heterodyne/homodyne detection).

• Quadrature measurement is done by

mixing it with a local oscillator a sin

and a cosine signal.


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Superconducting

Qubit: conquering

the complications • Dynamic decoupling pulse applied to

maintain coherence while

measurement was going on. [1]

• Reality is not so real-time! (Fig B)

[1] https://en.wikipedia.org/wiki/Dynamical_decoupling


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Superconducting

Qubit: conquering

the complications • Fidelity was not good as

distinguishing four state is difficult

than having only 1, this was

overcame in figure A, where only one

is measured it reduces the success

probability to ¼ (so fidelity is traded

with the success probability.)

• Two types of measurements, phase

sensitive and phase preserving are

done while preparing the sample in

given states.

[1] https://en.wikipedia.org/wiki/Dynamical_decoupling26.03.2018David Castells Graells, Ankit Anand 15

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Q1/Q2 readout characterization

Post-selection


Phase-selective mode

▪ Tune parametric amplifier transition

frequency (at which is pumped) in

resonance to readout

maximum gain

▪ Readout at mean value effective

resonator frequency for qubits in state

00 and 01

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Q1/Q2 readout characterization

Deterministic


Phase-preserving mode

▪ Detuned pump from readout

frequency

lower gain at readout freq.

and bandwidth

both quadratures of

transmitted field amplified

▪ Readout at mean resonator frequency

for qubits in state 01 and 10

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Success probability joint readout

00 01 10 11

𝟎𝟎 0.88 0.09 0.01 0.02

𝟎𝟏 0.11 0.79 0.08 0.02

𝟏𝟎 0.06 0.10 0.77 0.06

𝟏𝟏 0.02 0.03 0.08 0.87


Post-selection: Discriminate 00 with 91% success

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Process fidelity

𝑓𝑝𝑟 = 𝑇𝑟 𝜒𝑖𝑑𝑒𝑎𝑙 𝜒 ; 𝜌𝑜𝑢𝑡 =

𝑙,𝑘=0

3

𝜒𝑙𝑘𝜎𝑙𝜌𝑖𝑛𝜎𝑘

𝜎𝑖 = {𝐼, 𝜎𝑥 , 𝜎𝑦, 𝜎𝑧} , 𝜎𝑙𝜌𝜎𝑘 = σ𝑖 𝛼𝑖𝜎𝑙 𝑖 𝑖 𝜎𝑘 rotations

Wire frame Ideal 𝜒, with only one non-zero component


Input: 00

𝝆𝒊𝒏 →

I, X, Y, …

→ 𝝆𝒐𝒖𝒕

algorithm

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Error sources & limitations

• Short lived microwave photon limits the physical separation

• Mesoscopic scale Short lived qubits

• Measurement in all four basis together is difficult

• Limited fidelity:

• fidelities of single-qubit and two-qubit operations

• readout fidelities

• time required for the feed-forward in relation to the coherence times of the qubits used


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Transfer process matrix


Post-selection

Deterministic

Input: 00

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Transfer process matrix


Deterministic

Deterministic +

Feed-Forward

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Conclusion and Outlook

▪ The teleportation protocol has been implemented on a Superconducting Qubit

by three different method on a same set-up

▪ Rate of teleportation achieved was 104 𝑝𝑒𝑟 sec between two macroscopic

system separated by 6 mm

▪ Larger distance and higher rate would be required to achieve spacelike

separation and should be done in future to claim it is really a teleportation not

some spooky leakage.


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Questions ??


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The Protocol

0 2 0 3

0 2 0 3 + 1 3 / 2

0 2 0 3 + 1 2 1 3 / 2

Hadamard gate

A

A

B

H

𝝍𝟐𝟑

CNOT

𝑎 0 1 + 𝑏 1 1

𝝍𝒊𝒏


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The Protocol

Hadamard gate

A

A

B

𝑎 0 1 + 𝑏 1 1 0 2 0 3 + 1 2 1 3

𝑎 0 1 0 2 0 3 + 0 1 1 2 1 3 + 𝑏 1 1 1 2 0 3 + 1 1 0 2 1 3

𝟎𝟎 𝟏𝟐 𝑎 0 3 + 𝑏 1 3 + 𝟎𝟏 𝟏𝟐 𝑎 1 3 + 𝑏 0 3

+ 𝟏𝟎 𝟏𝟐 𝑎 0 3 − 𝑏 1 3 + 𝟏𝟏 𝟏𝟐 𝑎 1 3 − 𝑏 0 3

CNOT

𝝍𝟏𝟐𝟑

H


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The Protocol Feed-forward

classical

Pauli matricesHadamard gate

A

A

B

𝟎𝟎 𝟏𝟐 ⟺ 𝑎 0 3 + 𝑏 1 3

𝟎𝟏 𝟏𝟐 ⟺ 𝑏 0 3 + 𝑎 1 3

𝟏𝟎 𝟏𝟐 ⟺ 𝑎 0 3 − 𝑏 1 3

𝟏𝟏 𝟏𝟐 ⟺ −𝑏 0 3 + 𝑎 1 3

𝝍𝟏𝟐𝟑𝐈 1 0

0 1

𝑎𝑏

=𝑎𝑏

𝝍𝒐𝒖𝒕

𝑿 = ෝ𝝈𝒙 0 11 0

𝑏𝑎

=𝑎𝑏

𝒁 = ෝ𝝈𝒛1 00 −1

𝑎−𝑏

=𝑎𝑏෩𝒀 = 𝒊ෝ𝝈𝒚

0 1−1 0

−𝑏𝑎

=𝑎𝑏


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Definitions

▪ Fidelity

▪ Average output fidelity (classically: 2/3)

𝑓 = 𝜙𝑖𝑑𝑒𝑎𝑙 𝜌 𝜙𝑖𝑑𝑒𝑎𝑙

▪ Average process fidelity (classically: 1/2)

𝑓𝑝𝑟 = 𝑇𝑟 𝜒𝑖𝑑𝑒𝑎𝑙 𝜒 ; 𝜌 =

𝑙,𝑘=0

3

𝜒𝑙𝑘𝜎𝑙𝜌𝑖𝑑𝑒𝑎𝑙𝜎𝑘

Ideal 𝜒 has only one non-zero component

For teleportation, the ideal state corresponds to the input state


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Process fidelities of the feed-forward pulses


I X Y Z

𝟎𝟎 ⊗ 𝝍 0.80 0.67 0.74 0.64

𝟎𝟏 ⊗ 𝑿 𝝍 0.64 0.69 0.73 0.65

𝟏𝟎 ⊗ 𝒁 𝝍 0.66 0.63 0.66 0.65

𝟏𝟏 ⊗ ෩𝒀 𝝍 0.73 0.68 0.62 0.75

quantum teleportation with sc qubits - eth z · the protocol bennett, charles h., et al....

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