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Home , Quantum teleportation

Quantum with SC

David Castells Graells, Ankit Anand

David Castells Graells, Ankit Anand | 26.03.2018 | 1 David Castells Graells, Ankit Anand | 26.03.2018 | 2 Motivation

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

▪ Quantum teleportation: transferring unknown 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

David Castells Graells, Ankit Anand | 26.03.2018 | 3 The Protocol Bell measurement EPR pair 흓푨 흓푩

Input state 흍

흍 ′ time classical 00 01 10 11 R

흍 Output state

• Creation of entangled pair shared between sender and receiver • Joint two 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

Bennett, Charles H., et al. "Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen David Castells Graells, Ankit Anand | 26.03.2018 | 4 channels." 70.13 (1993): 1895. The Protocol

Feed-forward A classical A

B

Hadamard gate

David Castells Graells, Ankit Anand | 26.03.2018 | 5 The Protocol Measurement: Unconditional (Determ.) vs. Post selected

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

Feed-forward A classical A

B

State preparation: Deterministic vs. Probabilistic Technical challenge: finite lifetime qubits

David Castells Graells, Ankit Anand | 26.03.2018 | 6 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

David Castells Graells, Ankit Anand | 26.03.2018 | 7 Historical overview 2012 - Ma, et. al. 1997 - Bouwmeester, et.al. 2000 - Kim, et. al. 2004 - Barrett, et. al. • Single 2014 - Pfaff, et. al. • Single photons • Non-lin. Phot. interact • Atomic qubits • 143 km (free) • Diamond qb • Lab scale • Lab scale • Same trap(~μm) • Probabilistic • 3 meters • Deterministic • Deterministic • Deterministic • 2 BSM - • Deterministic • 2 BSM - Post-select. • 4 BSM - Uncond. • 4 BSM - Uncond Uncnd Fpr = 71% • 4 BSM - Uncond • Proof-of-principle • Low efficiency • F = 78% Post-slct F = 86% • F = 86%

2013 - Steffan, et.al. 1998 - Furusawa, et. al. 2004 - Riebe, et. al. 2009 - Olmschenk, et. al. • Phot. continuous- • Atomic qubits • 2 at. + 2 ph qubits 2012 - Bao, et. al. 1993 variable state • Same trap (~1μm) • 1 m • Atomic qubits Bennet, et. al. • Lab scale • Deterministic • Probabilistic • 150m • Theoretical • Deterministic • 4 BSM - Uncond. • 2 BSM - Uncond. • Probabilistic proposal • 4 BSM - Uncond. • F = 75% • F = 90% • 2 BSM - Post-sel. • F = 58% • F = 84% pr • Fpr = 88% | | Superconducting Qubits: Protocol requirements An ideal protocol requires • Creation of entangled pair shared between sender and receiver • Two qubit measurement identifying all four 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 푝푒푟 sec ) David Castells Graells, Ankit Anand | 26.03.2018 | 9 Superconducting Qubits • A chip designed with four superconducting qubits [1] connected via three resonator (only 3 in use) • Qubits are coupled to each other via (Superconducting coplanar waveguide resonator)(microwave 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. David Castells Graells, Ankit Anand | 26.03.2018 | 10 [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. , 449(7161), p.443. 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. 푥 푦 • 퐸 푡 = 퐸 푡 cos 휔푑푡 + 퐸 sin 휔푑푡 퐻 퐸푥 푡 퐸푦 푡 • 푟 = 휔 − 휔 1 1 + 𝜎 + 𝜎 ℏ 푑 2 푥 2 푦 • E.g if you want rotation about x then you can 푦 푥 have, 휔 = 휔푑 , 퐸 푡 = 0 ⇒ 푅푥 휃 = ∫ 퐸 푡 푑푡 • 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.

David Castells Graells, Ankit Anand | 26.03.2018 | 11 Superconducting Qubit: Driving the autopilot

Feed-forward

Feed-forward

David Castells Graells, Ankit Anand | 26.03.2018 | 12 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.

David Castells Graells, Ankit Anand | 26.03.2018 | 13 Superconducting Qubit: conquering the complications • Dynamic decoupling pulse applied to maintain while measurement was going on. [1] • Reality is not so real-time! (Fig B)

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

David Castells Graells, Ankit Anand | 26.03.2018 | 14 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_decoupling David Castells Graells, Ankit Anand | 26.03.2018 | 15 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

David Castells Graells, Ankit Anand | 26.03.2018 | 16 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

David Castells Graells, Ankit Anand | 26.03.2018 | 17 Success probability joint readout

00 01 10 11 ퟎퟎ 0.88 0.09 0.01 0.02 ퟎퟏ 0.11 0.79 0.08 0.02 ퟏퟎ 0.06 0.10 0.77 0.06 ퟏퟏ 0.02 0.03 0.08 0.87

Post-selection: Discriminate 00 with 91% success

David Castells Graells, Ankit Anand | 26.03.2018 | 18 Process fidelity

3 Input: 00 푓푝푟 = 푇푟 휒푖푑푒푎푙 휒 ; 𝜌표푢푡 = ෍ 휒푙푘𝜎푙𝜌푖푛𝜎푘 푙,푘=0

algorithm 𝝆풊풏 → I, X, Y, …

→ 𝝆풐풖풕

𝜎푖 = {퐼, 𝜎푥, 𝜎푦, 𝜎푧} , 𝜎푙𝜌𝜎푘 = σ푖 훼푖𝜎푙 푖 푖 𝜎푘  rotations

Wire frame  Ideal 휒, with only one non-zero component

David Castells Graells, Ankit Anand | 26.03.2018 | 19 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

David Castells Graells, Ankit Anand | 26.03.2018 | 20 Transfer process matrix Input: 00

Post-selection

Deterministic

David Castells Graells, Ankit Anand | 26.03.2018 | 21 Transfer process matrix

Deterministic

Deterministic + Feed-Forward

David Castells Graells, Ankit Anand | 26.03.2018 | 22 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 푝푒푟 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.

David Castells Graells, Ankit Anand | 26.03.2018 | 23 Questions ??

David Castells Graells, Ankit Anand | 26.03.2018 | 24 David Castells Graells, Ankit Anand | 26.03.2018 | 25 The Protocol A

A

B

Hadamard gate

0 0 흍풊풏 흍ퟐퟑ 2 3 H 푎 0 1 + 푏 1 1

0 2 0 3 + 1 3 / 2

CNOT

0 2 0 3 + 1 2 1 3 / 2

David Castells Graells, Ankit Anand | 26.03.2018 | 26 The Protocol A

A

B

Hadamard gate 푎 0 + 푏 1 0 0 + 1 1 흍ퟏퟐퟑ 1 1 2 3 2 3 CNOT

푎 0 1 0 2 0 3 + 0 1 1 2 1 3 + 푏 1 1 1 2 0 3 + 1 1 0 2 1 3

H

ퟎퟎ ퟏퟐ 푎 0 3 + 푏 1 3 + ퟎퟏ ퟏퟐ 푎 1 3 + 푏 0 3 + ퟏퟎ ퟏퟐ 푎 0 3 − 푏 1 3 + ퟏퟏ ퟏퟐ 푎 1 3 − 푏 0 3

David Castells Graells, Ankit Anand | 26.03.2018 | 27 The Protocol Feed-forward A classical A

B

Hadamard gate Pauli matrices

흍ퟏퟐퟑ 흍풐풖풕 퐈 1 0 푎 푎 ퟎퟎ ⟺ 푎 0 + 푏 1 = ퟏퟐ 3 3 0 1 푏 푏 푿 = 𝝈ෝ 풙 0 1 푏 푎 ퟎퟏ ⟺ 푏 0 + 푎 1 = ퟏퟐ 3 3 푏 풁 = 𝝈ෝ 1 0 푎 풛 1 0 푎 푎 ퟏퟎ ퟏퟐ ⟺ 푎 0 3 − 푏 1 3 = −푏 푏 풀෩ = 풊𝝈ෝ 0 −1 풚 0 1 −푏 푎 ퟏퟏ ퟏퟐ ⟺ −푏 0 3 + 푎 1 3 = −1 0 푎 푏

David Castells Graells, Ankit Anand | 26.03.2018 | 28 Definitions

▪ Fidelity ▪ Average output fidelity (classically: 2/3)

푓 = 휙푖푑푒푎푙 𝜌 휙푖푑푒푎푙

▪ Average process fidelity (classically: 1/2) 3

푓푝푟 = 푇푟 휒푖푑푒푎푙 휒 ; 𝜌 = ෍ 휒푙푘𝜎푙𝜌푖푑푒푎푙𝜎푘 푙,푘=0 Ideal 휒 has only one non-zero component

For teleportation, the ideal state corresponds to the input state

David Castells Graells, Ankit Anand | 26.03.2018 | 29 Process fidelities of the feed-forward pulses

I X Y Z ퟎퟎ ⊗ 흍 0.80 0.67 0.74 0.64 ퟎퟏ ⊗ 푿 흍 0.64 0.69 0.73 0.65 ퟏퟎ ⊗ 풁 흍 0.66 0.63 0.66 0.65 ퟏퟏ ⊗ 풀෩ 흍 0.73 0.68 0.62 0.75

David Castells Graells, Ankit Anand | 26.03.2018 | 30