Qubit Fluxonium Transmon

Emanuel Hubenschmid Seminar: Superconducting quantum hardware for quantum computing 19.06.2020 What is so special?

Google (53 qubits processor Sycamore) IBM (4 qubit prozessor)

F. Arute et al. “Quantum supremacy using a programmable superconducting processor”. Nature 574.7779 (Oct. 2019), 505. J. M. Gambetta et al. “Building logical qubits in a superconducting quantum computing system”. npj Quantum Information 3.1 (Jan. 2017).

2 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid Contents

1 Introduction 1.1 The DiVincenzo criteria 1.2 Some familiar qubits 2 The transmon qubit 2.1 Characterization of the qubit 2.2 Initialization, control and read-out 2.3 Relaxation and decoherence 3 The fluxonium qubit 3.1 Characterization of the qubit 3.2 Initialization, control and read-out 3.3 Relaxation and decoherence 4 Summary

3 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid 1 Introduction - 1.1 The DiVincenzo criteria

1 Introduction 1.1 The DiVincenzo criteria

4 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid | ⟩ ∈ H⊗n → ⊗n | ⟩ 2. Initializability of the qubits Ψ qubit i=0 0

3. Decoherence times much longer than the gate operation time T1,T2 ≫ τ

4. A “universal” set of quantum gates H ,…

0 5. A qubit-specific measurement capability

1 Introduction - 1.1 The DiVincenzo criteria

The DiVincenzo criteria

1. Scalable system with well characterized qubits H = Hqubit ⊗ Hr

D. P. DiVincenzo. “The Physical Implementation of Quantum Computation”. Fortschritte der Physik 48.9-11 (Sept. 2000), 771.

5 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid 3. Decoherence times much longer than the gate operation time T1,T2 ≫ τ

4. A “universal” set of quantum gates H ,…

0 5. A qubit-specific measurement capability

1 Introduction - 1.1 The DiVincenzo criteria

The DiVincenzo criteria

1. Scalable system with well characterized qubits H = Hqubit ⊗ Hr

| ⟩ ∈ H⊗n → ⊗n | ⟩ 2. Initializability of the qubits Ψ qubit i=0 0

D. P. DiVincenzo. “The Physical Implementation of Quantum Computation”. Fortschritte der Physik 48.9-11 (Sept. 2000), 771.

5 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid 4. A “universal” set of quantum gates H ,…

0 5. A qubit-specific measurement capability

1 Introduction - 1.1 The DiVincenzo criteria

The DiVincenzo criteria

1. Scalable system with well characterized qubits H = Hqubit ⊗ Hr

| ⟩ ∈ H⊗n → ⊗n | ⟩ 2. Initializability of the qubits Ψ qubit i=0 0

3. Decoherence times much longer than the gate operation time T1,T2 ≫ τ

D. P. DiVincenzo. “The Physical Implementation of Quantum Computation”. Fortschritte der Physik 48.9-11 (Sept. 2000), 771.

5 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid 0 5. A qubit-specific measurement capability

1 Introduction - 1.1 The DiVincenzo criteria

The DiVincenzo criteria

1. Scalable system with well characterized qubits H = Hqubit ⊗ Hr

| ⟩ ∈ H⊗n → ⊗n | ⟩ 2. Initializability of the qubits Ψ qubit i=0 0

3. Decoherence times much longer than the gate operation time T1,T2 ≫ τ

4. A “universal” set of quantum gates H ,…

D. P. DiVincenzo. “The Physical Implementation of Quantum Computation”. Fortschritte der Physik 48.9-11 (Sept. 2000), 771.

5 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid 1 Introduction - 1.1 The DiVincenzo criteria

The DiVincenzo criteria

1. Scalable system with well characterized qubits H = Hqubit ⊗ Hr

| ⟩ ∈ H⊗n → ⊗n | ⟩ 2. Initializability of the qubits Ψ qubit i=0 0

3. Decoherence times much longer than the gate operation time T1,T2 ≫ τ

4. A “universal” set of quantum gates H ,…

0 5. A qubit-specific measurement capability

D. P. DiVincenzo. “The Physical Implementation of Quantum Computation”. Fortschritte der Physik 48.9-11 (Sept. 2000), 771.

5 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid 1 Introduction - 1.2 Some familiar qubits

1 Introduction 1.2 Some familiar qubits

6 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid 1 Hˆ = 4E (ˆn − n )2 − E cosφ ˆ + E (φ ˆ − 2πϕ/ϕ )2 C g J 2 L 0

EL = 0 EL ≪ EJ, ng = 0

2 3 EJ/EC ∼ 1 EJ/EC ∼ 10 − 10

1 Introduction - 1.2 Some familiar qubits

Comparison of qubits

Charge qubit Flux qubit C g E ,C ng J J U V U g EJ,CJ ϕ

φ φ

G. Catelani et al. “Relaxation and frequency shifts induced by quasiparticles in superconducting qubits”. Physical Review B 84.6 (Aug. 2011).

7 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid 2 3 EJ/EC ∼ 1 EJ/EC ∼ 10 − 10

1 Introduction - 1.2 Some familiar qubits

Comparison of qubits

Charge qubit Flux qubit C g E ,C ng J J U V U g EJ,CJ ϕ

φ φ

EL = 0 EL ≪ EJ, ng = 0

1 Hˆ = 4E (ˆn − n )2 − E cosφ ˆ + E (φ ˆ − 2πϕ/ϕ )2 C g J 2 L 0

G. Catelani et al. “Relaxation and frequency shifts induced by quasiparticles in superconducting qubits”. Physical Review B 84.6 (Aug. 2011).

7 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid 1 Introduction - 1.2 Some familiar qubits

Comparison of qubits

Charge qubit Flux qubit C g E ,C ng J J U V U g EJ,CJ ϕ

φ φ

EL = 0 EL ≪ EJ, ng = 0

2 3 EJ/EC ∼ 1 EJ/EC ∼ 10 − 10 1 Hˆ = 4E (ˆn − n )2 − E cosφ ˆ + E (φ ˆ − 2πϕ/ϕ )2 C g J 2 L 0

G. Catelani et al. “Relaxation and frequency shifts induced by quasiparticles in superconducting qubits”. Physical Review B 84.6 (Aug. 2011).

7 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid Solution: Transmon qubit Fluxonium qubit

Increase EJ/EC with large shunting capacitance Small junction shunted by an JJ array

EJ/EC ∼ 100 EJ/EC ∼ 1 − 10

1 Introduction - 1.2 Some familiar qubits

Comparison of qubits Problem: Single Cooper pair circuits decohere due to low frequency offset charge noise

J. Koch et al. “Charge-insensitive qubit design derived from the Cooper pair box”. Physical Review A 76.4 (Oct. 2007). V. E. Manucharyan et al. “Fluxonium: Single Cooper-Pair Circuit Free of Charge Offsets”. Science 326.5949 (Oct. 2009), 113.

8 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid 1 Introduction - 1.2 Some familiar qubits

Comparison of qubits Problem: Single Cooper pair circuits decohere due to low frequency offset charge noise Solution: Transmon qubit Fluxonium qubit

Increase EJ/EC with large shunting capacitance Small junction shunted by an JJ array

EJ/EC ∼ 100 EJ/EC ∼ 1 − 10

8 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid 2 The transmon qubit - 2.1 Characterization of the qubit

2 The transmon qubit 2.1 Characterization of the qubit

9 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid 2 The transmon qubit - 2.1 Characterization of the qubit

The device

J. Koch et al. “Charge-insensitive qubit design derived from the Cooper pair box”. Physical Review A 76.4 (Oct. 2007).

10 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid 2 The transmon qubit - 2.1 Characterization of the qubit

The capacitance network

J. Koch et al. “Charge-insensitive qubit design derived from the Cooper pair box”. Physical Review A 76.4 (Oct. 2007).

11 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid 2 The transmon qubit - 2.1 Characterization of the qubit

The capacitance network

J. Koch et al. “Charge-insensitive qubit design derived from the Cooper pair box”. Physical Review A 76.4 (Oct. 2007).

11 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid Thévenin-Theorem

2 The transmon qubit - 2.1 Characterization of the qubit

The capacitance network

J. Koch et al. “Charge-insensitive qubit design derived from the Cooper pair box”. Physical Review A 76.4 (Oct. 2007).

12 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid 2 The transmon qubit - 2.1 Characterization of the qubit

The capacitance network

Thévenin-Theorem

J. Koch et al. “Charge-insensitive qubit design derived from the Cooper pair box”. Physical Review A 76.4 (Oct. 2007).

12 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid Effective offset charge: ng = Qr/2e + CgVg/2e

Charging energy: Josephson energy: 2 E = E |cos(πΦ/Φ )| EC = e /2CΣ J J,max 0 CΣ = CJ + CB + Cg

2 The transmon qubit - 2.1 Characterization of the qubit

The capacitance network

2 Hˆ = 4EC(ˆn − ng) − EJ cosφ ˆ

J. Koch et al. “Charge-insensitive qubit design derived from the Cooper pair box”. Physical Review A 76.4 (Oct. 2007).

12 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid Effective offset charge: ng = Qr/2e + CgVg/2e

Josephson energy: EJ = EJ,max |cos(πΦ/Φ0)|

2 The transmon qubit - 2.1 Characterization of the qubit

The capacitance network

2 Hˆ = 4EC(ˆn − ng) − EJ cosφ ˆ Charging energy: 2 EC = e /2CΣ CΣ = CJ + CB + Cg

J. Koch et al. “Charge-insensitive qubit design derived from the Cooper pair box”. Physical Review A 76.4 (Oct. 2007).

12 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid Effective offset charge: ng = Qr/2e + CgVg/2e

2 The transmon qubit - 2.1 Characterization of the qubit

The capacitance network

2 Hˆ = 4EC(ˆn − ng) − EJ cosφ ˆ Charging energy: Josephson energy: 2 E = E |cos(πΦ/Φ )| EC = e /2CΣ J J,max 0 CΣ = CJ + CB + Cg

J. Koch et al. “Charge-insensitive qubit design derived from the Cooper pair box”. Physical Review A 76.4 (Oct. 2007).

12 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid 2 The transmon qubit - 2.1 Characterization of the qubit

The capacitance network

Effective offset charge: ng = Qr/2e + CgVg/2e

2 Hˆ = 4EC(ˆn − ng) − EJ cosφ ˆ Charging energy: Josephson energy: 2 E = E |cos(πΦ/Φ )| EC = e /2CΣ J J,max 0 CΣ = CJ + CB + Cg

J. Koch et al. “Charge-insensitive qubit design derived from the Cooper pair box”. Physical Review A 76.4 (Oct. 2007).

12 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid ⇒ Ψ(φ) = eingφg(φ/2)

Ψ(φ + 2π) = Ψ(φ)

  E E ⇒ g′′(x) +  + J cos(2x) g(x) = 0 EC EC φ/2

2 The transmon qubit - 2.1 Characterization of the qubit

Eigenenergies

   2  d  E (n )Ψ(φ) = 4E  − n  − E cos φ Ψ(φ) m g C dφ g J

J. Koch et al. “Charge-insensitive qubit design derived from the Cooper pair box”. Physical Review A 76.4 (Oct. 2007).

13 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid ⇒ Ψ(φ) = eingφg(φ/2)

  E E ⇒ g′′(x) +  + J cos(2x) g(x) = 0 EC EC φ/2

2 The transmon qubit - 2.1 Characterization of the qubit

Eigenenergies

Ψ(φ + 2π) = Ψ(φ)    2  d  E (n )Ψ(φ) = 4E  − n  − E cos φ Ψ(φ) m g C dφ g J

J. Koch et al. “Charge-insensitive qubit design derived from the Cooper pair box”. Physical Review A 76.4 (Oct. 2007).

13 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid   E E ⇒ g′′(x) +  + J cos(2x) g(x) = 0 EC EC φ/2

2 The transmon qubit - 2.1 Characterization of the qubit

Eigenenergies

Ψ(φ + 2π) = Ψ(φ) ⇒ Ψ(φ) = eingφg(φ/2)    2  d  E (n )Ψ(φ) = 4E  − n  − E cos φ Ψ(φ) m g C dφ g J

J. Koch et al. “Charge-insensitive qubit design derived from the Cooper pair box”. Physical Review A 76.4 (Oct. 2007).

13 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid 2 The transmon qubit - 2.1 Characterization of the qubit

Eigenenergies

Ψ(φ + 2π) = Ψ(φ) ⇒ Ψ(φ) = eingφg(φ/2)    2  d  E (n )Ψ(φ) = 4E  − n  − E cos φ Ψ(φ) m g C dφ g J   E E ⇒ g′′(x) +  + J cos(2x) g(x) = 0 EC EC φ/2

J. Koch et al. “Charge-insensitive qubit design derived from the Cooper pair box”. Physical Review A 76.4 (Oct. 2007).

13 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid 2 The transmon qubit - 2.1 Characterization of the qubit

Eigenenergies

J. Koch et al. “Charge-insensitive qubit design derived from the Cooper pair box”. Physical Review A 76.4 (Oct. 2007).

13 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid −1/2 ≃ −(8EJ/EC)

≃ v u   √ 24m+5 u 2 E m/2+3/4 m t  J  − 8EJ/EC ≃ (−1) EC e m! π 2EC

Relative anharmonicity (charge degeneracy ng = 1/2):

E12 − E01 αr = E01

2 The transmon qubit - 2.1 Characterization of the qubit

Anharmonicity vs. charge dispersion

Peak to peak charge dispersion:

ϵ = Em(ng = 1/2) − Em(ng = 0)

J. Koch et al. “Charge-insensitive qubit design derived from the Cooper pair box”. Physical Review A 76.4 (Oct. 2007).

14 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid −1/2 ≃ −(8EJ/EC)

Relative anharmonicity (charge degeneracy ng = 1/2):

E12 − E01 αr = E01

2 The transmon qubit - 2.1 Characterization of the qubit

Anharmonicity vs. charge dispersion

Peak to peak charge dispersion:

ϵ = Em(ng = 1/2) − Em(ng = 0) ≃ v u   √ 24m+5 u 2 E m/2+3/4 m t  J  − 8EJ/EC ≃ (−1) EC e m! π 2EC

J. Koch et al. “Charge-insensitive qubit design derived from the Cooper pair box”. Physical Review A 76.4 (Oct. 2007).

14 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid −1/2 ≃ −(8EJ/EC)

2 The transmon qubit - 2.1 Characterization of the qubit

Anharmonicity vs. charge dispersion

Peak to peak charge dispersion:

ϵ = Em(ng = 1/2) − Em(ng = 0) ≃ v u   √ 24m+5 u 2 E m/2+3/4 m t  J  − 8EJ/EC ≃ (−1) EC e m! π 2EC

Relative anharmonicity (charge degeneracy ng = 1/2):

E12 − E01 αr = E01

J. Koch et al. “Charge-insensitive qubit design derived from the Cooper pair box”. Physical Review A 76.4 (Oct. 2007).

14 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid 2 The transmon qubit - 2.1 Characterization of the qubit

Anharmonicity vs. charge dispersion

Peak to peak charge dispersion:

ϵ = Em(ng = 1/2) − Em(ng = 0) ≃ v u   √ 24m+5 u 2 E m/2+3/4 m t  J  − 8EJ/EC ≃ (−1) EC e m! π 2EC

Relative anharmonicity (charge degeneracy ng = 1/2): − E12 E01 −1/2 αr = ≃ −(8EJ/EC) E01

J. Koch et al. “Charge-insensitive qubit design derived from the Cooper pair box”. Physical Review A 76.4 (Oct. 2007).

14 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid 2 The transmon qubit - 2.1 Characterization of the qubit

Anharmonicity vs. charge dispersion

Peak to peak charge dispersion:

ϵ = Em(ng = 1/2) − Em(ng = 0) ≃ v u   √ 24m+5 u 2 E m/2+3/4 m t  J  − 8EJ/EC ≃ (−1) EC e m! π 2EC

Relative anharmonicity (charge degeneracy ng = 1/2): − E12 E01 −1/2 αr = ≃ −(8EJ/EC) E01

J. Koch et al. “Charge-insensitive qubit design derived from the Cooper pair box”. Physical Review A 76.4 (Oct. 2007).

14 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid 2 The transmon qubit - 2.2 Initialization, control and read-out

2 The transmon qubit 2.2 Initialization, control and read-out

15 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid Resonator frequency Cg/CΣ 2 † Hˆ=4EC(ˆn − ng) − EJ cosφ ˆ + ℏωraˆ aˆ+ 2βevˆnˆ 0 † Vrms(ˆa +a ˆ ) ”Vg → Vg + v”

2 The transmon qubit - 2.2 Initialization, control and read-out

Circuit QED for the transmon

J. Koch et al. “Charge-insensitive qubit design derived from the Cooper pair box”. Physical Review A 76.4 (Oct. 2007).

16 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid Resonator frequency Cg/CΣ

0 † Vrms(ˆa +a ˆ ) ”Vg → Vg + v”

2 The transmon qubit - 2.2 Initialization, control and read-out

Circuit QED for the transmon

2 † Hˆ=4EC(ˆn − ng) − EJ cosφ ˆ + ℏωraˆ aˆ+ 2βevˆnˆ

J. Koch et al. “Charge-insensitive qubit design derived from the Cooper pair box”. Physical Review A 76.4 (Oct. 2007).

16 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid Resonator frequency Cg/CΣ

0 † Vrms(ˆa +a ˆ )

2 The transmon qubit - 2.2 Initialization, control and read-out

Circuit QED for the transmon

2 † Hˆ=4EC(ˆn − ng) − EJ cosφ ˆ + ℏωraˆ aˆ+ 2βevˆnˆ

”Vg → Vg + v”

J. Koch et al. “Charge-insensitive qubit design derived from the Cooper pair box”. Physical Review A 76.4 (Oct. 2007).

16 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid Cg/CΣ

0 † Vrms(ˆa +a ˆ )

2 The transmon qubit - 2.2 Initialization, control and read-out

Circuit QED for the transmon

Resonator frequency 2 † Hˆ=4EC(ˆn − ng) − EJ cosφ ˆ + ℏωraˆ aˆ+ 2βevˆnˆ

”Vg → Vg + v”

J. Koch et al. “Charge-insensitive qubit design derived from the Cooper pair box”. Physical Review A 76.4 (Oct. 2007).

16 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid 0 † Vrms(ˆa +a ˆ )

2 The transmon qubit - 2.2 Initialization, control and read-out

Circuit QED for the transmon

Resonator frequency Cg/CΣ 2 † Hˆ=4EC(ˆn − ng) − EJ cosφ ˆ + ℏωraˆ aˆ+ 2βevˆnˆ

”Vg → Vg + v”

J. Koch et al. “Charge-insensitive qubit design derived from the Cooper pair box”. Physical Review A 76.4 (Oct. 2007).

16 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid 2 The transmon qubit - 2.2 Initialization, control and read-out

Circuit QED for the transmon

Resonator frequency Cg/CΣ 2 † Hˆ=4EC(ˆn − ng) − EJ cosφ ˆ + ℏωraˆ aˆ+ 2βevˆnˆ 0 † Vrms(ˆa +a ˆ ) ”Vg → Vg + v”

J. Koch et al. “Charge-insensitive qubit design derived from the Cooper pair box”. Physical Review A 76.4 (Oct. 2007).

16 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid P j |j⟩⟨j| = 1, |j⟩ : Transmon eigenvectors

Jaynes Cummings model

X † X † Hˆ = ℏ ωj |j⟩⟨j| + ℏωr aˆ aˆ+ ℏgi,j |i⟩⟨j| (ˆa +a ˆ ) j i,j |g⟩ 0 ⟨ | | ⟩ → ∀ 1/4 2βeVrms i nˆ j 0 i > j + 1 ∝ (EJ/EC) |e⟩

2 The transmon qubit - 2.2 Initialization, control and read-out

Circuit QED for the transmon

ˆ − 2 − ℏ † 0 † H = 4EC(ˆn ng) EJ cosφ ˆ + ωraˆ aˆ+ 2βeVrmsnˆ(ˆa +a ˆ )

J. Koch et al. “Charge-insensitive qubit design derived from the Cooper pair box”. Physical Review A 76.4 (Oct. 2007).

17 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid Jaynes Cummings model

X † X † Hˆ = ℏ ωj |j⟩⟨j| + ℏωr aˆ aˆ+ ℏgi,j |i⟩⟨j| (ˆa +a ˆ ) j i,j |g⟩ 0 ⟨ | | ⟩ → ∀ 1/4 2βeVrms i nˆ j 0 i > j + 1 ∝ (EJ/EC) |e⟩

2 The transmon qubit - 2.2 Initialization, control and read-out

Circuit QED for the transmon P j |j⟩⟨j| = 1, |j⟩ : Transmon eigenvectors

ˆ − 2 − ℏ † 0 † H = 4EC(ˆn ng) EJ cosφ ˆ + ωraˆ aˆ+ 2βeVrmsnˆ(ˆa +a ˆ )

J. Koch et al. “Charge-insensitive qubit design derived from the Cooper pair box”. Physical Review A 76.4 (Oct. 2007).

17 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid Jaynes Cummings model

|g⟩ 1/4 ∝ (EJ/EC) |e⟩

2 The transmon qubit - 2.2 Initialization, control and read-out

Circuit QED for the transmon P j |j⟩⟨j| = 1, |j⟩ : Transmon eigenvectors

ˆ − 2 − ℏ † 0 † H = 4EC(ˆn ng) EJ cosφ ˆ + ωraˆ aˆ+ 2βeVrmsnˆ(ˆa +a ˆ )

X † X † Hˆ = ℏ ωj |j⟩⟨j| + ℏωr aˆ aˆ+ ℏgi,j |i⟩⟨j| (ˆa +a ˆ ) j i,j 0 ⟨ | | ⟩ → ∀ 2βeVrms i nˆ j 0 i > j + 1

J. Koch et al. “Charge-insensitive qubit design derived from the Cooper pair box”. Physical Review A 76.4 (Oct. 2007).

17 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid Jaynes Cummings model

|g⟩ 1/4 ∝ (EJ/EC) |e⟩

2 The transmon qubit - 2.2 Initialization, control and read-out

Circuit QED for the transmon P j |j⟩⟨j| = 1, |j⟩ : Transmon eigenvectors

ˆ − 2 − ℏ † 0 † H = 4EC(ˆn ng) EJ cosφ ˆ + ωraˆ aˆ+ 2βeVrmsnˆ(ˆa +a ˆ )

X † X † Hˆ = ℏ ωj |j⟩⟨j| + ℏωr aˆ aˆ+ ℏgi,j |i⟩⟨j| (ˆa +a ˆ ) j i,j 0 ⟨ | | ⟩ → ∀ 2βeVrms i nˆ j 0 i > j + 1   X X †  †  Hˆ ≈ ℏ ωj |j⟩⟨j|+ℏωr aˆ aˆ+ ℏ gi,i+1 (|i⟩⟨i + 1| + |i + 1⟩⟨i|)a ˆ + h.c. j i

J. Koch et al. “Charge-insensitive qubit design derived from the Cooper pair box”. Physical Review A 76.4 (Oct. 2007).

17 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid Jaynes Cummings model

|g⟩

|e⟩

1/4 ∝ (EJ/EC)

2 The transmon qubit - 2.2 Initialization, control and read-out

Circuit QED for the transmon P j |j⟩⟨j| = 1, |j⟩ : Transmon eigenvectors

ˆ − 2 − ℏ † 0 † H = 4EC(ˆn ng) EJ cosφ ˆ + ωraˆ aˆ+ 2βeVrmsnˆ(ˆa +a ˆ )

X † X † Hˆ = ℏ ωj |j⟩⟨j| + ℏωr aˆ aˆ+ ℏgi,j |i⟩⟨j| (ˆa +a ˆ ) j i,j 0 ⟨ | | ⟩ → ∀ 2βeVrms i nˆ j 0 i > j + 1   X X †  †  Hˆ ≈ ℏ ωj |j⟩⟨j| + ℏωr aˆ aˆ+ ℏ gi,i+1 |i⟩⟨i + 1| aˆ + h.c. j i

J. Koch et al. “Charge-insensitive qubit design derived from the Cooper pair box”. Physical Review A 76.4 (Oct. 2007).

17 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid 1/4 ∝ (EJ/EC)

2 The transmon qubit - 2.2 Initialization, control and read-out

Circuit QED for the transmon P j |j⟩⟨j| = 1, |j⟩ : Transmon eigenvectors

2 † 0 † Hˆ = 4EC(ˆn − ng) − EJ cosφ ˆ + ℏωraˆ aˆ+ 2βeV nˆ(ˆa +a ˆ ) rms Jaynes Cummings model

X † X † Hˆ = ℏ ωj |j⟩⟨j| + ℏωr aˆ aˆ+ ℏgi,j |i⟩⟨j| (ˆa +a ˆ ) j i,j |g⟩ 0 ⟨ | | ⟩ → ∀ 2βeVrms i nˆ j 0 i > j + 1   X X |e⟩ †  †  Hˆ ≈ ℏ ωj |j⟩⟨j| + ℏωr aˆ aˆ+ ℏ gi,i+1 |i⟩⟨i + 1| aˆ + h.c. j i

J. Koch et al. “Charge-insensitive qubit design derived from the Cooper pair box”. Physical Review A 76.4 (Oct. 2007).

17 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid 2 The transmon qubit - 2.2 Initialization, control and read-out

Circuit QED for the transmon P j |j⟩⟨j| = 1, |j⟩ : Transmon eigenvectors

2 † 0 † Hˆ = 4EC(ˆn − ng) − EJ cosφ ˆ + ℏωraˆ aˆ+ 2βeV nˆ(ˆa +a ˆ ) rms Jaynes Cummings model

X † X † Hˆ = ℏ ωj |j⟩⟨j| + ℏωr aˆ aˆ+ ℏgi,j |i⟩⟨j| (ˆa +a ˆ ) j i,j |g⟩ 0 ⟨ | | ⟩ → ∀ 2βeVrms i nˆ j 0 i > j + 1   X X |e⟩ †  †  Hˆ ≈ ℏ ωj |j⟩⟨j| + ℏωr aˆ aˆ+ ℏ gi,i+1 |i⟩⟨i + 1| aˆ + h.c. j i 1/4 ∝ (EJ/EC)

J. Koch et al. “Charge-insensitive qubit design derived from the Cooper pair box”. Physical Review A 76.4 (Oct. 2007).

17 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid Effective desperive shift 2 gij χ = χ01 − χ12/2, χij = ωij−ωr

P † gi,i+1 Dˆ = exp Sˆ − Sˆ ; Sˆ = i aˆ|i + 1⟩⟨i| ∆i

2 The transmon qubit - 2.2 Initialization, control and read-out

The dispersive limit

  ′ 2 † ℏω ′ †  gij  ˆ ˆ ˆ ≈ 01 ℏ ℏ O   DHD σˆz + ( ωr + χσˆz )ˆa aˆ+ 2 2 ∆i

J. Koch et al. “Charge-insensitive qubit design derived from the Cooper pair box”. Physical Review A 76.4 (Oct. 2007).

18 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid Effective desperive shift 2 gij χ = χ01 − χ12/2, χij = ωij−ωr

2 The transmon qubit - 2.2 Initialization, control and read-out

The dispersive limit

  ′ 2 † ℏω ′ †  gij  ˆ ˆ ˆ ≈ 01 ℏ ℏ O   DHD σˆz + ( ωr + χσˆz )ˆa aˆ+ 2 2 ∆i

P † gi,i+1 Dˆ = exp Sˆ − Sˆ ; Sˆ = i aˆ|i + 1⟩⟨i| ∆i

J. Koch et al. “Charge-insensitive qubit design derived from the Cooper pair box”. Physical Review A 76.4 (Oct. 2007).

18 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid 2 The transmon qubit - 2.2 Initialization, control and read-out

The dispersive limit

Effective desperive shift 2 gij χ = χ01 − χ12/2, χij = ωij−ωr

  ′ 2 † ℏω ′ †  gij  ˆ ˆ ˆ ≈ 01 ℏ ℏ O   DHD σˆz + ( ωr + χσˆz )ˆa aˆ+ 2 2 ∆i

P † gi,i+1 Dˆ = exp Sˆ − Sˆ ; Sˆ = i aˆ|i + 1⟩⟨i| ∆i

J. Koch et al. “Charge-insensitive qubit design derived from the Cooper pair box”. Physical Review A 76.4 (Oct. 2007).

18 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid 2 The transmon qubit - 2.3 Relaxation and decoherence

2 The transmon qubit 2.3 Relaxation and decoherence

19 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid = 2 pur = κ |⟨f |aˆ|i⟩| ⇒ T1 ∼ 16 µs

(1) Spontaneous emission 2 4 1 d ω rad ℏω01 P = 3 ⇒ T = ∼ 0.3 ms 4πϵ0 3c 1 P

(2) Purcell effect 2 2π ⟨ |ℏ P ˆ† | ⟩ Γi→f = ℏ ρ(ωk ) 1, f k λk (bk aˆ+ h.c.) 0, i

qp (3) Quasiparticle tunneling T1 ∼ 1 s flux ∼ (4) Flux coupling T1 70 ms (5) Dielectric losses   1 1 1 1 −1 ⇒   ∼ T1 = rad + pur + qp + flux 15 µs T1 T1 T1 T1

2 The transmon qubit - 2.3 Relaxation and decoherence

Relaxation time T1

J. Koch et al. “Charge-insensitive qubit design derived from the Cooper pair box”. Physical Review A 76.4 (Oct. 2007).

20 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid = 2 pur = κ |⟨f |aˆ|i⟩| ⇒ T1 ∼ 16 µs

(2) Purcell effect 2 2π ⟨ |ℏ P ˆ† | ⟩ Γi→f = ℏ ρ(ωk ) 1, f k λk (bk aˆ+ h.c.) 0, i

qp (3) Quasiparticle tunneling T1 ∼ 1 s flux ∼ (4) Flux coupling T1 70 ms (5) Dielectric losses   1 1 1 1 −1 ⇒   ∼ T1 = rad + pur + qp + flux 15 µs T1 T1 T1 T1

2 The transmon qubit - 2.3 Relaxation and decoherence

Relaxation time T1

(1) Spontaneous emission 2 4 1 d ω rad ℏω01 P = 3 ⇒ T = ∼ 0.3 ms 4πϵ0 3c 1 P

J. Koch et al. “Charge-insensitive qubit design derived from the Cooper pair box”. Physical Review A 76.4 (Oct. 2007).

20 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid = 2 pur = κ |⟨f |aˆ|i⟩| ⇒ T1 ∼ 16 µs qp (3) Quasiparticle tunneling T1 ∼ 1 s flux ∼ (4) Flux coupling T1 70 ms (5) Dielectric losses   1 1 1 1 −1 ⇒   ∼ T1 = rad + pur + qp + flux 15 µs T1 T1 T1 T1

2 The transmon qubit - 2.3 Relaxation and decoherence

Relaxation time T1

(1) Spontaneous emission 2 4 1 d ω rad ℏω01 P = 3 ⇒ T = ∼ 0.3 ms 4πϵ0 3c 1 P

(2) Purcell effect 2 2π ⟨ |ℏ P ˆ† | ⟩ Γi→f = ℏ ρ(ωk ) 1, f k λk (bk aˆ+ h.c.) 0, i

J. Koch et al. “Charge-insensitive qubit design derived from the Cooper pair box”. Physical Review A 76.4 (Oct. 2007).

20 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid qp (3) Quasiparticle tunneling T1 ∼ 1 s flux ∼ (4) Flux coupling T1 70 ms (5) Dielectric losses   1 1 1 1 −1 ⇒   ∼ T1 = rad + pur + qp + flux 15 µs T1 T1 T1 T1

2 The transmon qubit - 2.3 Relaxation and decoherence

Relaxation time T1

(1) Spontaneous emission 2 4 1 d ω rad ℏω01 P = 3 ⇒ T = ∼ 0.3 ms 4πϵ0 3c 1 P

(2) Purcell effect 2 2π ⟨ |ℏ P ˆ† | ⟩ Γi→f = ℏ ρ(ωk ) 1, f k λk (bk aˆ+ h.c.) 0, i = 2 pur = κ |⟨f |aˆ|i⟩| ⇒ T1 ∼ 16 µs

J. Koch et al. “Charge-insensitive qubit design derived from the Cooper pair box”. Physical Review A 76.4 (Oct. 2007).

20 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid flux ∼ (4) Flux coupling T1 70 ms (5) Dielectric losses   1 1 1 1 −1 ⇒   ∼ T1 = rad + pur + qp + flux 15 µs T1 T1 T1 T1

2 The transmon qubit - 2.3 Relaxation and decoherence

Relaxation time T1

(1) Spontaneous emission 2 4 1 d ω rad ℏω01 P = 3 ⇒ T = ∼ 0.3 ms 4πϵ0 3c 1 P

(2) Purcell effect 2 2π ⟨ |ℏ P ˆ† | ⟩ Γi→f = ℏ ρ(ωk ) 1, f k λk (bk aˆ+ h.c.) 0, i = 2 pur = κ |⟨f |aˆ|i⟩| ⇒ T1 ∼ 16 µs qp (3) Quasiparticle tunneling T1 ∼ 1 s

J. Koch et al. “Charge-insensitive qubit design derived from the Cooper pair box”. Physical Review A 76.4 (Oct. 2007).

20 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid (5) Dielectric losses   1 1 1 1 −1 ⇒   ∼ T1 = rad + pur + qp + flux 15 µs T1 T1 T1 T1

2 The transmon qubit - 2.3 Relaxation and decoherence

Relaxation time T1

(1) Spontaneous emission 2 4 1 d ω rad ℏω01 P = 3 ⇒ T = ∼ 0.3 ms 4πϵ0 3c 1 P

(2) Purcell effect 2 2π ⟨ |ℏ P ˆ† | ⟩ Γi→f = ℏ ρ(ωk ) 1, f k λk (bk aˆ+ h.c.) 0, i = 2 pur = κ |⟨f |aˆ|i⟩| ⇒ T1 ∼ 16 µs qp (3) Quasiparticle tunneling T1 ∼ 1 s flux ∼ (4) Flux coupling T1 70 ms

J. Koch et al. “Charge-insensitive qubit design derived from the Cooper pair box”. Physical Review A 76.4 (Oct. 2007).

20 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid   1 1 1 1 −1 ⇒   ∼ T1 = rad + pur + qp + flux 15 µs T1 T1 T1 T1

2 The transmon qubit - 2.3 Relaxation and decoherence

Relaxation time T1

(1) Spontaneous emission 2 4 1 d ω rad ℏω01 P = 3 ⇒ T = ∼ 0.3 ms 4πϵ0 3c 1 P

(2) Purcell effect 2 2π ⟨ |ℏ P ˆ† | ⟩ Γi→f = ℏ ρ(ωk ) 1, f k λk (bk aˆ+ h.c.) 0, i = 2 pur = κ |⟨f |aˆ|i⟩| ⇒ T1 ∼ 16 µs qp (3) Quasiparticle tunneling T1 ∼ 1 s flux ∼ (4) Flux coupling T1 70 ms (5) Dielectric losses

J. Koch et al. “Charge-insensitive qubit design derived from the Cooper pair box”. Physical Review A 76.4 (Oct. 2007).

20 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid 2 The transmon qubit - 2.3 Relaxation and decoherence

Relaxation time T1

(1) Spontaneous emission 2 4 1 d ω rad ℏω01 P = 3 ⇒ T = ∼ 0.3 ms 4πϵ0 3c 1 P

J. Koch et al. “Charge-insensitive qubit design derived from the Cooper pair box”. Physical Review A 76.4 (Oct. 2007).

20 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid P → ∂H01 H H + λ ∂λ δλ

2πA2 S(ω) = |ω| ng, Φ,Ic

Transmon (EJ/EC = 85) CPB (EJ/EC = 1) Noise source 1/f amplitude A T2 in ns T2 in ns Charge 10−4 − 10−3e 400′000 1′000 −6 −5 ′ ′ ′ ′ Flux 10 − 10 Φ0 3 600 000 1 000 000 −7 −6 ′ ′ Critical current 10 − 10 Ic 35 000 17 000

2 The transmon qubit - 2.3 Relaxation and decoherence

Decoherence time T2

−1 ℏ ∂E01 T ≃ 2 A ∂λ

J. Koch et al. “Charge-insensitive qubit design derived from the Cooper pair box”. Physical Review A 76.4 (Oct. 2007).

21 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid 2πA2 S(ω) = |ω| ng, Φ,Ic

2 The transmon qubit - 2.3 Relaxation and decoherence

Decoherence time T2 P → ∂H01 H H + λ ∂λ δλ

−1 ℏ ∂E01 T ≃ 2 A ∂λ

J. Koch et al. “Charge-insensitive qubit design derived from the Cooper pair box”. Physical Review A 76.4 (Oct. 2007).

21 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid ng, Φ,Ic

2 The transmon qubit - 2.3 Relaxation and decoherence

Decoherence time T2 P → ∂H01 H H + λ ∂λ δλ

−1 ℏ ∂E01 T ≃ 2 A ∂λ 2πA2 S(ω) = |ω|

J. Koch et al. “Charge-insensitive qubit design derived from the Cooper pair box”. Physical Review A 76.4 (Oct. 2007).

21 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid Transmon (EJ/EC = 85) CPB (EJ/EC = 1) Noise source 1/f amplitude A T2 in ns T2 in ns Charge 10−4 − 10−3e 400′000 1′000 −6 −5 ′ ′ ′ ′ Flux 10 − 10 Φ0 3 600 000 1 000 000 −7 −6 ′ ′ Critical current 10 − 10 Ic 35 000 17 000

2 The transmon qubit - 2.3 Relaxation and decoherence

Decoherence time T2 P → ∂H01 H H + λ ∂λ δλ

−1 ℏ ∂E01 T ≃ 2 A ∂λ 2πA2 S(ω) = |ω| ng, Φ,Ic

J. Koch et al. “Charge-insensitive qubit design derived from the Cooper pair box”. Physical Review A 76.4 (Oct. 2007).

21 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid 2 The transmon qubit - 2.3 Relaxation and decoherence

Decoherence time T2 P → ∂H01 H H + λ ∂λ δλ

−1 ℏ ∂E01 T ≃ 2 A ∂λ 2πA2 S(ω) = |ω| ng, Φ,Ic

J. Koch et al. “Charge-insensitive qubit design derived from the Cooper pair box”. Physical Review A 76.4 (Oct. 2007).

21 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid 3 The fluxonium qubit - 3.1 Characterization of the qubit

3 The fluxonium qubit 3.1 Characterization of the qubit

22 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid 3 The fluxonium qubit - 3.1 Characterization of the qubit

The device

V. E. Manucharyan et al. “Fluxonium: Single Cooper-Pair Circuit Free of Charge Offsets”. Science 326.5949 (Oct. 2009), 113.

23 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid Thévenin-Theorem

3 The fluxonium qubit - 3.1 Characterization of the qubit

The device

V. E. Manucharyan et al. “Fluxonium: Single Cooper-Pair Circuit Free of Charge Offsets”. Science 326.5949 (Oct. 2009), 113.

24 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid Thévenin-Theorem

3 The fluxonium qubit - 3.1 Characterization of the qubit

The device

V. E. Manucharyan et al. “Fluxonium: Single Cooper-Pair Circuit Free of Charge Offsets”. Science 326.5949 (Oct. 2009), 113.

24 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid 3 The fluxonium qubit - 3.1 Characterization of the qubit

The device

Thévenin-Theorem

V. E. Manucharyan et al. “Fluxonium: Single Cooper-Pair Circuit Free of Charge Offsets”. Science 326.5949 (Oct. 2009), 113.

24 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid = 2.5 GHz = 9.0 GHz

= 0.52 GHz

2 2 EC = e /2CJ EJ = (Φ0/2π) /LJ

2 EL = (Φ0/2π) /LA

(1) Small charge fluctuations ⇒ Large flux fluctuations ⇒ NLJA ≫ LJ − (2) Reduce offset charge ⇒ e 8RQ/ZJA < ε ≪ 1 − − (3) Suppress quantum phase slips ⇒ Ne 8RQ/ZJA ≪ e 8RQ/ZJ 1/2 (4) Reduce parasitic capacitance to ground ⇒ N < (CJA/Cg)

3 The fluxonium qubit - 3.1 Characterization of the qubit

The Eigenstates

1 H = 4E nˆ2 + E φˆ2 − E cos(φ ˆ − 2πΦ /Φ ) C 2 L J ext 0

V. E. Manucharyan et al. “Fluxonium: Single Cooper-Pair Circuit Free of Charge Offsets”. Science 326.5949 (Oct. 2009), 113.

25 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid = 9.0 GHz

= 0.52 GHz

2 = 2.5 GHz EJ = (Φ0/2π) /LJ

2 EL = (Φ0/2π) /LA

3 The fluxonium qubit - 3.1 Characterization of the qubit

The Eigenstates 2 EC = e /2CJ 1 H = 4E nˆ2 + E φˆ2 − E cos(φ ˆ − 2πΦ /Φ ) C 2 L J ext 0

V. E. Manucharyan et al. “Fluxonium: Single Cooper-Pair Circuit Free of Charge Offsets”. Science 326.5949 (Oct. 2009), 113.

25 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid = 9.0 GHz

2 = 2.5 GHz EJ = (Φ0/2π) /LJ

= 0.52 GHz

3 The fluxonium qubit - 3.1 Characterization of the qubit

The Eigenstates 2 EC = e /2CJ 1 H = 4E nˆ2 + E φˆ2 − E cos(φ ˆ − 2πΦ /Φ ) C 2 L J ext 0

2 EL = (Φ0/2π) /LA

V. E. Manucharyan et al. “Fluxonium: Single Cooper-Pair Circuit Free of Charge Offsets”. Science 326.5949 (Oct. 2009), 113.

25 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid = 2.5 GHz = 9.0 GHz

= 0.52 GHz

3 The fluxonium qubit - 3.1 Characterization of the qubit

The Eigenstates 2 2 EC = e /2CJ EJ = (Φ0/2π) /LJ 1 H = 4E nˆ2 + E φˆ2 − E cos(φ ˆ − 2πΦ /Φ ) C 2 L J ext 0

2 EL = (Φ0/2π) /LA

V. E. Manucharyan et al. “Fluxonium: Single Cooper-Pair Circuit Free of Charge Offsets”. Science 326.5949 (Oct. 2009), 113.

25 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid = 2.5 GHz = 9.0 GHz

= 0.52 GHz

− (2) Reduce offset charge ⇒ e 8RQ/ZJA < ε ≪ 1 − − (3) Suppress quantum phase slips ⇒ Ne 8RQ/ZJA ≪ e 8RQ/ZJ 1/2 (4) Reduce parasitic capacitance to ground ⇒ N < (CJA/Cg)

3 The fluxonium qubit - 3.1 Characterization of the qubit

The Eigenstates 2 2 EC = e /2CJ EJ = (Φ0/2π) /LJ 1 H = 4E nˆ2 + E φˆ2 − E cos(φ ˆ − 2πΦ /Φ ) C 2 L J ext 0

2 EL = (Φ0/2π) /LA

(1) Small charge fluctuations ⇒ Large flux fluctuations ⇒ NLJA ≫ LJ

V. E. Manucharyan et al. “Fluxonium: Single Cooper-Pair Circuit Free of Charge Offsets”. Science 326.5949 (Oct. 2009), 113.

25 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid = 2.5 GHz = 9.0 GHz

= 0.52 GHz

− − (3) Suppress quantum phase slips ⇒ Ne 8RQ/ZJA ≪ e 8RQ/ZJ 1/2 (4) Reduce parasitic capacitance to ground ⇒ N < (CJA/Cg)

3 The fluxonium qubit - 3.1 Characterization of the qubit

The Eigenstates 2 2 EC = e /2CJ EJ = (Φ0/2π) /LJ 1 H = 4E nˆ2 + E φˆ2 − E cos(φ ˆ − 2πΦ /Φ ) C 2 L J ext 0

2 EL = (Φ0/2π) /LA

(1) Small charge fluctuations ⇒ Large flux fluctuations ⇒ NLJA ≫ LJ − (2) Reduce offset charge ⇒ e 8RQ/ZJA < ε ≪ 1

V. E. Manucharyan et al. “Fluxonium: Single Cooper-Pair Circuit Free of Charge Offsets”. Science 326.5949 (Oct. 2009), 113.

25 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid = 2.5 GHz = 9.0 GHz

= 0.52 GHz

1/2 (4) Reduce parasitic capacitance to ground ⇒ N < (CJA/Cg)

3 The fluxonium qubit - 3.1 Characterization of the qubit

The Eigenstates 2 2 EC = e /2CJ EJ = (Φ0/2π) /LJ 1 H = 4E nˆ2 + E φˆ2 − E cos(φ ˆ − 2πΦ /Φ ) C 2 L J ext 0

2 EL = (Φ0/2π) /LA

(1) Small charge fluctuations ⇒ Large flux fluctuations ⇒ NLJA ≫ LJ − (2) Reduce offset charge ⇒ e 8RQ/ZJA < ε ≪ 1 − − (3) Suppress quantum phase slips ⇒ Ne 8RQ/ZJA ≪ e 8RQ/ZJ

V. E. Manucharyan et al. “Fluxonium: Single Cooper-Pair Circuit Free of Charge Offsets”. Science 326.5949 (Oct. 2009), 113.

25 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid = 2.5 GHz = 9.0 GHz

= 0.52 GHz

3 The fluxonium qubit - 3.1 Characterization of the qubit

The Eigenstates 2 2 EC = e /2CJ EJ = (Φ0/2π) /LJ 1 H = 4E nˆ2 + E φˆ2 − E cos(φ ˆ − 2πΦ /Φ ) C 2 L J ext 0

2 EL = (Φ0/2π) /LA

V. E. Manucharyan et al. “Fluxonium: Single Cooper-Pair Circuit Free of Charge Offsets”. Science 326.5949 (Oct. 2009), 113.

25 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid 3 The fluxonium qubit - 3.1 Characterization of the qubit

The Eigenstates 2 2 EC = e /2CJ = 2.5 GHz EJ = (Φ0/2π) /LJ = 9.0 GHz 1 H = 4E nˆ2 + E φˆ2 − E cos(φ ˆ − 2πΦ /Φ ) C 2 L J ext 0

2 EL = (Φ0/2π) /LA = 0.52 GHz

V. E. Manucharyan et al. “Fluxonium: Single Cooper-Pair Circuit Free of Charge Offsets”. Science 326.5949 (Oct. 2009), 113.

25 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid 3 The fluxonium qubit - 3.2 Initialization, control and read-out

3 The fluxonium qubit 3.2 Initialization, control and read-out

26 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid 1 H = 4E nˆ2 + E φˆ2 − E cos(φ ˆ − 2πΦ /Φ ) + gnˆ(ˆa +a ˆ†) + ℏω aˆ†aˆ C 2 L J ext 0 R

3 The fluxonium qubit - 3.2 Initialization, control and read-out

Circuit QED

V. E. Manucharyan et al. “Fluxonium: Single Cooper-Pair Circuit Free of Charge Offsets”. Science 326.5949 (Oct. 2009), 113.

27 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid 3 The fluxonium qubit - 3.2 Initialization, control and read-out

Circuit QED

1 H = 4E nˆ2 + E φˆ2 − E cos(φ ˆ − 2πΦ /Φ ) + gnˆ(ˆa +a ˆ†) + ℏω aˆ†aˆ C 2 L J ext 0 R

V. E. Manucharyan et al. “Fluxonium: Single Cooper-Pair Circuit Free of Charge Offsets”. Science 326.5949 (Oct. 2009), 113.

27 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid 3 The fluxonium qubit - 3.2 Initialization, control and read-out

Microwave spectroscopy

V. E. Manucharyan et al. “Fluxonium: Single Cooper-Pair Circuit Free of Charge Offsets”. Science 326.5949 (Oct. 2009), 113.

28 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid 3 The fluxonium qubit - 3.2 Initialization, control and read-out

Microwave spectroscopy

V. E. Manucharyan et al. “Fluxonium: Single Cooper-Pair Circuit Free of Charge Offsets”. Science 326.5949 (Oct. 2009), 113.

28 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid 3 The fluxonium qubit - 3.2 Initialization, control and read-out

Microwave spectroscopy

V. E. Manucharyan et al. “Fluxonium: Single Cooper-Pair Circuit Free of Charge Offsets”. Science 326.5949 (Oct. 2009), 113.

28 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid 3 The fluxonium qubit - 3.3 Relaxation and decoherence

3 The fluxonium qubit 3.3 Relaxation and decoherence

29 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid 3 The fluxonium qubit - 3.3 Relaxation and decoherence

High coherence device

L. B. Nguyen et al. “High-Coherence Fluxonium Qubit”. Physical Review X 9.4 (Nov. 2019).

30 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid 3 The fluxonium qubit - 3.3 Relaxation and decoherence

High coherence device

L. B. Nguyen et al. “High-Coherence Fluxonium Qubit”. Physical Review X 9.4 (Nov. 2019).

30 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid 3 The fluxonium qubit - 3.3 Relaxation and decoherence

High coherence device

L. B. Nguyen et al. “High-Coherence Fluxonium Qubit”. Physical Review X 9.4 (Nov. 2019).

30 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid 4 Summary

4 Summary

31 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid - CPB with effective shunting√ capacitance − - Short-ciruit offset charge noise with JJ array - Charge dispersion ∝ e 8EJ/EC - Large JJ array protects small JJ from large ⇒ Exponential insentivity to 1/f noise flux fluctuations - Anharmonicity ∝ −(8E /E )−1/2 J C - Unharmonic as the flux qubit but as insesitive - Less sensitive to flux or crit. current noise as to flux noise as the transmon CPB

4 Summary

Summary

The transmon qubit The fluxonium qubit

32 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid √ − - Short-ciruit offset charge noise with JJ array - Charge dispersion ∝ e 8EJ/EC - Large JJ array protects small JJ from large ⇒ Exponential insentivity to 1/f noise flux fluctuations - Anharmonicity ∝ −(8E /E )−1/2 J C - Unharmonic as the flux qubit but as insesitive - Less sensitive to flux or crit. current noise as to flux noise as the transmon CPB

4 Summary

Summary

The transmon qubit The fluxonium qubit

- CPB with effective shunting capacitance

32 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid - Short-ciruit offset charge noise with JJ array - Large JJ array protects small JJ from large flux fluctuations - Anharmonicity ∝ −(8E /E )−1/2 J C - Unharmonic as the flux qubit but as insesitive - Less sensitive to flux or crit. current noise as to flux noise as the transmon CPB

4 Summary

Summary

The transmon qubit The fluxonium qubit

- CPB with effective shunting√ capacitance − - Charge dispersion ∝ e 8EJ/EC ⇒ Exponential insentivity to 1/f noise

32 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid - Short-ciruit offset charge noise with JJ array - Large JJ array protects small JJ from large flux fluctuations - Unharmonic as the flux qubit but as insesitive - Less sensitive to flux or crit. current noise as to flux noise as the transmon CPB

4 Summary

Summary

The transmon qubit The fluxonium qubit

- CPB with effective shunting√ capacitance − - Charge dispersion ∝ e 8EJ/EC ⇒ Exponential insentivity to 1/f noise −1/2 - Anharmonicity ∝ −(8EJ/EC)

32 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid - Short-ciruit offset charge noise with JJ array - Large JJ array protects small JJ from large flux fluctuations - Unharmonic as the flux qubit but as insesitive to flux noise as the transmon

4 Summary

Summary

The transmon qubit The fluxonium qubit

- CPB with effective shunting√ capacitance − - Charge dispersion ∝ e 8EJ/EC ⇒ Exponential insentivity to 1/f noise −1/2 - Anharmonicity ∝ −(8EJ/EC) - Less sensitive to flux or crit. current noise as CPB

32 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid - Large JJ array protects small JJ from large flux fluctuations - Unharmonic as the flux qubit but as insesitive to flux noise as the transmon

4 Summary

Summary

The transmon qubit The fluxonium qubit

- CPB with effective shunting√ capacitance − - Short-ciruit offset charge noise with JJ array - Charge dispersion ∝ e 8EJ/EC ⇒ Exponential insentivity to 1/f noise −1/2 - Anharmonicity ∝ −(8EJ/EC) - Less sensitive to flux or crit. current noise as CPB

32 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid - Unharmonic as the flux qubit but as insesitive to flux noise as the transmon

4 Summary

Summary

The transmon qubit The fluxonium qubit

- CPB with effective shunting√ capacitance − - Short-ciruit offset charge noise with JJ array - Charge dispersion ∝ e 8EJ/EC - Large JJ array protects small JJ from large ⇒ Exponential insentivity to 1/f noise −1/2 flux fluctuations - Anharmonicity ∝ −(8EJ/EC) - Less sensitive to flux or crit. current noise as CPB

32 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid 4 Summary

Summary

The transmon qubit The fluxonium qubit

- CPB with effective shunting√ capacitance − - Short-ciruit offset charge noise with JJ array - Charge dispersion ∝ e 8EJ/EC - Large JJ array protects small JJ from large ⇒ Exponential insentivity to 1/f noise flux fluctuations - Anharmonicity ∝ −(8E /E )−1/2 J C - Unharmonic as the flux qubit but as insesitive - Less sensitive to flux or crit. current noise as to flux noise as the transmon CPB

32 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid Thank You For Your Attention! Do You Have Any Questions? 5 Appendix

Analogy to the rotor

2 Hˆ = 4EC(ˆn − ng) − EJ cosφ ˆ

2 Lˆ H = z − mgl cosφ ˆ rot 2ml 2

J. Koch et al. “Charge-insensitive qubit design derived from the Cooper pair box”. Physical Review A 76.4 (Oct. 2007).

33 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid 5 Appendix

Analogy to the rotor

2 Hˆ = 4EC(ˆn − ng) − EJ cosφ ˆ

(Lˆ + qB l 2/2)2 H = z 0 − mgl cosφ ˆ rot 2ml 2

J. Koch et al. “Charge-insensitive qubit design derived from the Cooper pair box”. Physical Review A 76.4 (Oct. 2007).

33 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid 5 Appendix

Charge number states vs. transmon eigenstates

33 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid 5 Appendix

Ac vs dc noise

J. Koch et al. “Charge-insensitive qubit design derived from the Cooper pair box”. Physical Review A 76.4 (Oct. 2007).

33 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid 5 Appendix

Microwave spectroscopy

V. E. Manucharyan et al. “Fluxonium: Single Cooper-Pair Circuit Free of Charge Offsets”. Science 326.5949 (Oct. 2009), 113.

33 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid 5 Appendix

Measured relaxation and coherence time

L. B. Nguyen et al. “High-Coherence Fluxonium Qubit”. Physical Review X 9.4 (Nov. 2019).

33 / 33 19.06.2020 Transmon and Fluxonium Qubit Emanuel Hubenschmid