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Alexandre Blais Université de Sherbrooke, Québec, Canada

Superconducting qubits

Équipe de recherche en Physique de l’Information Quantique

piq

Which quantum computer is right for you?

Quantum information processing: the challenge

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Qubits: Two-level systems Single-qubit control Two-qubit entangling gates

1

Qubit readout

Conflicting requirements: long-lived quantum effects, fast control and readout

• D. DiVincenzo, Fortschritte der Physik 48, 771 (2000)

• Artificial atoms
• Physics 101: Harmonic oscillators and basic electrical circuits
• Superconductivity and Josephson junctions
• Circuit QED: a possible QC architecture
• Recent realizations and challenges

Outline

‘Atomic atoms’

Control by shining laser tuned at the desired transition frequency Hyperfine levels of 9Be+ have long relaxation and dephasing times

T1 ∼ a few years

T2 & 10 seconds

• T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien, Nature 464, 45 (2010)

Energy

|0 |1 |2

Good «two-level atom» approximation

E01 = E1 − E0 = ~ω01

• ω01

Relaxation and dephasing times

|0 |1 |0 |1

≠ bit flip channel T1: Relaxation = amplitude damping channel T2: Dephasing = phase damping channel = phase flip channel

(Energy is conserved)

• Probability of qubit

having relaxed at time t:

e−t/T1 = e−γ1t

|ψi = c0|0i + c1|1i → ρ = ✓ |c0|2 c0c∗

1e−t/T2

c∗

0c1e−t/T2

|c1|2 ◆

Probability of phase decay at time t:

e−t/T2 = e−γ2t

‘Atomic atoms’

Control by shining laser tuned at the desired transition frequency Hyperfine levels of 9Be+ have long relaxation and dephasing times

T1 ∼ a few years

T2 & 10 seconds

Reasonably short gate time

Tnot ∼ 5 µs

Low error per gates: ~ 0.48%

• T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien, Nature 464, 45 (2010)

Energy

|0 |1 |2

Good «two-level atom» approximation

E01 = E1 − E0 = ~ω01

Artificial atoms

Resistor Capacitor Inductor

Standard toolkit

• Based on microfabricated circuit elements
• Well defined energy levels
• Nonlinear distribution of energy levels
• Maximize numbers of thumbs up!

Avoiding dissipation: superconductivity

Normal metals dissipate energy

Temperature Resistance

Sydney Tc of Aluminum ~ -272oC (~1K) Sherbrooke Light bulb filament ~ 3000oC

Normal metal (Copper, Gold, …) Superconductor (Aluminium, Niobium, …)

Superconductivity is a (macroscopic) quantum effect No resistance in superconducting state ⇒ no dissipation

A good starting point for a quantum device…

Dilution fridge temperature ~ 10 mK

Basic circuit elements (classical version)

Resistor (R) Capacitor (C) Inductor (L)

Standard toolkit

✗

Capacitor:

• Two metal plates separated

by an insulator

• Relates voltage to charge

Inductor:

• A non-resistive wire
• Relates voltage to change of

current

Q = CV

V

V

I

Q

V = LdI dt

Φ = LI Φ = Z t

1

dt0 V (t0)

Basic circuit elements (classical version)

Resistor (R) Capacitor (C) Inductor (L)

Standard toolkit

✗

Capacitor:

• Two metal plates separated

by an insulator

• Relates voltage to charge

Inductor:

• A non-resistive wire
• Relates voltage to change of

current

Q = CV

V

V

I

Q

V = LdI dt

I = dQ dt

Current: Change of charge in time

Basic circuit elements (classical version)

Resistor (R) Capacitor (C) Inductor (L)

Standard toolkit

✗

Capacitor:

• Two metal plates separated

by an insulator

• Relates voltage to charge

Inductor:

• A non-resistive wire
• Relates voltage to change of

current

Q = CV

V

V

I

Q

LC oscillator

V = LdI dt

V = LdI dt

V = Q C Q C = Ld2Q dt2 Q(t) = Q(0) cos(ωLCt) ωLC = 1 √ LC

Voltage is the same across L and C:

Time Charge

+Qo

• Qo

Basic circuit elements (classical version)

Q(t) = Q(0) cos(ωLCt) ωLC = 1 √ LC

Oscillations of the charge: One out of countless examples of harmonic oscillator

Time Height

Mass at rest

Classical harmonic oscillator

Mass at rest Maximal velocity

(p = mv)

Energy at arbitrary x:

H = p2 2m + 1 2kx2 E = 1 2kx2

max

xmax zmax

Height, z

E = 1 2mv2

max = p2 max

2m

= Hamiltonian

ω = p k/m

Frequency of oscillation:

Position (x) Time

Momentum

Quantum harmonic oscillator

Energy at arbitrary x:

Height, z

= Hamiltonian

ˆ H = ˆ p2 2m + 1 2kˆ x2

Heisenberg uncertainty principle: Impossible to know precisely both x and p

[ˆ x, ˆ p] = i~ x → ˆ x

p → ˆ p

Classical variables are promoted to hermitian

• perator acting on Hilbert space

Position, x

Quantum harmonic oscillator

Energy at arbitrary x:

Height, z

ˆ H = ˆ p2 2m + 1 2kˆ x2 [ˆ x, ˆ p] = i~ x → ˆ x p → ˆ p

Classical variables are promoted to hermitian

• perator acting on Hilbert space

Position, x

ˆ a = ✓mk 4~2 ◆1/4 ✓ ˆ x + i ˆ p √ mk ◆ ˆ a† = ✓mk 4~2 ◆1/4 ✓ ˆ x − i ˆ p √ mk ◆

Useful to introduce:

= ~ωˆ n

Commutation relation:

[ˆ x, ˆ p] = i~ [ˆ a, ˆ a†] = 1

ˆ H = ~ωˆ a†ˆ a

(ω = p k/m)

ˆ n(ˆ a|ni) = ˆ aˆ n|ni ˆ a|ni = (n 1)ˆ a|ni

Quantum harmonic oscillator

[ˆ a, ˆ a†] = 1

ˆ H = ~ωˆ a†ˆ a = ~ωˆ

n ˆ n|ni = n|ni ||ˆ a|ni||2 = hn|ˆ a†ˆ a|ni = hn|ˆ n|ni = n ˆ a|ni = pn|n 1i ˆ a†|ni = p n + 1|n + 1i

What is the action of and on the eigenstates of ?

ˆ n ˆ a ˆ a†

ˆ a|ni =? ˆ a†|ni =?

First observation: [ˆ

n, ˆ a] = −ˆ a [ˆ n, ˆ a†] = ˆ a†

and

⇒

Second observation:

⇒

n ∈ N0

ˆ a|ni / |n 1i

Quantum harmonic oscillator

ˆ H = ~ωˆ a†ˆ a = ~ωˆ

n ˆ n|ni = n|ni ˆ a|ni = pn|n 1i ˆ a†|ni = p n + 1|n + 1i

n ≥ 0

|0 |1 |2 |3 ˆ a†

ˆ a

ˆ a†

ˆ a

ˆ a†

ˆ a

~ω ~ω

~ω

Energy

ˆ H = ˆ p2 2m + 1 2kˆ x2

ω = p k/m ωLC = 1 √ LC

ˆ H = ˆ Q2 2C + ˆ Φ2 2L

Φ = Z dtV (t)

Flux, !

Flux:

ˆ a† = ✓ C 4L~2 ◆1/4 ˆ Φ − i ˆ Q p C/L !

adds a photon to the LC circuit n = number of photons stored in the LC circuit

ˆ a†

(Electric field) (Magnetic field)

Artificial atom

|0 |1 |2 |3

Energy

ωLC = 1 √ LC

Flux, !

V (t)

= V0 cos ω01t

Initialization to ground state is simple

∼ 10 GHz

∼ 0.5 K

ω01 = 1/ √ LC

Not a good «two-level» atom, not a qubit…

1 m K 3 K

Aluminum on Sapphire

Josephson junction

Resistor (R) Capacitor (C) Inductor (L)

Standard toolkit

✗

Josephson junctions

Superconductor (Al) Superconductor (Al) Insulator (AlOx)

Josephson junction S I S

EJ

100 nm Resistor (R) Capacitor (C) Inductor (L)

Standard toolkit

✗

Josephson junctions

Artificial atom toolkit

Josephson junction Capacitor (C) Inductor (L)

Artificial atom toolkit

Capacitor:

• Two metal plates separated

by an insulator

• Relates voltage to charge

Inductor:

• A non-resistive wire
• Relates current to flux

Q = CV

V

V

I

Q

V = LdI dt

Φ = LI Φ = Z t

1

dt0 V (t0)

Josephson junction:

• Two superconductors separated by an insulator
• Relates current to flux

V

I

I = I0 sin(2πΦ/Φ0)

Superconducting artificial atom

|0 |1 |2 |3

• Energy

Flux, !

V (t) = V0 cos ω01t

Big improvements in relaxation and dephasing times in last 10 years Error per gates of 0.2%, similar to trapped ion results

Low error per gates: E. Magesan et al, Phys. Rev. Lett. 109, 080505 (2012) Long T1 and T2: H. Paik et al, Phys. Rev. Lett. 107, 240501 (2011)

Very short π-pulse time

Tπ ∼ 4 − 20 ns

T2 [µs] Year T1 [µs]

0,0 0,1 1,0 10,0 100,0 1 000,0 0,00 0,01 0,10 1,00 10,00 100,00 1999 2002 2007 2012

~ 300 µm

V(t)

Superconducting transmon qubits

• J. Koch et al. Phys. Rev. A 76, 042319 (2007)

|0 |1 |2 |3

Superconducting qubits, a family tree

Charge

NEC, Saclay, 1999

Quantronium

Saclay, 2002

Transmon

Yale, 2007

Phase

NIST 2002

xmon

=

UCSB, 2013

Fluxonium

Yale, 2009

Flux

Delft, 1999

Circuit QED

V V (t) = V0 cos ω01t |0 |1 |2 |3 |0 |1 |2 |3

ˆ p

Circuit QED

V (t) = V0 cos ω01t

|0 |1 |2 |3 |0 |1 |2 |3

V (t)

Circuit QED: Multi-qubit architecture

V (t) = V0 cos ω01t V (t) V (t) = V0 cos ω01t

Circuit QED: Resonant and dispersive regimes

|0 |1 |2 |3 |0 |1 |2 |3 |0 |1 |2 |3 |0 |1 |2 |3

Resonant regime:

• Identical 0-1 transition frequencies
• Energy exchange between qubits

and oscillator Dispersive regime:

• Different 0-1 transition frequencies
• No energy exchange between qubits

and oscillator

• Qubit-state dependent oscillator

frequency leads allows qubit readout

• Oscillator acts as quantum bus for

entangling qubits

Circuit QED: Multi-qubit architecture

Single-qubit control

1

Quantum bus: entangling gates and readout

|0i |1i |0i |1i |0i |1i

Circuit QED: ‘1D’ realization

Proposal: Blais, Huang, Wallraff, Girvin & Schoelkopf, Phys. Rev. A 69, 062320 (2004) First realization: Wallraff, Schuster, Blais, Frunzio, Huang, Majer, Kumar, Girvin & Schoelkopf. Nature 431, 162 (2004)

Aluminum microwave resonator on Sapphire Transmon qubit a) b) c) 100 µm 2 µm 1 mm

| | | |n | d)

Circuit QED: ‘1D’ realization

Proposal: Blais, Huang, Wallraff, Girvin & Schoelkopf, Phys. Rev. A 69, 062320 (2004) First realization: Wallraff, Schuster, Blais, Frunzio, Huang, Majer, Kumar, Girvin & Schoelkopf. Nature 431, 162 (2004)

Circuit QED: scaling up

• F. Helmer et al, Europhys. Lett. 85, 50007 (2009)

Circuit QED: Multi-qubit architecture

V (t) = V0 cos ω01t V (t) V (t) = V0 cos ω01t

Long-range qubit-qubit interactions

Circuit QED: alternative architecture

V (t) = V0 cos ω01t V (t) V (t) = V0 cos ω01t V (t)

Individual qubit readout Individual qubit readout Two-qubit gates Two-qubit gates

Short-range qubit-qubit interactions

• A. Wallraff’s lab @ ETH Zurich

Circuit QED: recent realizations and challenges

10 years of circuit QED

Yale: Wallraff et al., Nature 431, 162 (2004) Delft: Chiorescu et al., Nature 431, 159 (2004)

Quantum optics on a chip with artificial atoms Quantum information processing

Past, present and future

• M. H. Devoret and R. J. Schoelkopf, Science 339, 1169 (2013)

Operation on single physical qubits Algorithms on multiple physical qubits QND measurement for error correction and control Logical memory with longer lifetime than physical qubits Operations on single logical qubits Algorithms on multiple logical qubits Fault-tolerant quantum computation Time Complexity

Superconducting qubits Trapped ions Rydberg atoms Spin qubits

High-fidelity gates and readout

Gates

Average fidelity: > 99.92%

UCSB: Nature 508, 500 (2014)

Error when simultaneously operating neighbour qubits: < 10-4 Single-qubit gate

IBM: PRL 109, 060501 (2012)

Average fidelity: > 96.75% Two-qubit gate (via bus) Average fidelity: up to 99.4% Two-qubit gate (direct)

UCSB: Nature 508, 500 (2014)

Readout

Fidelity: up to 99.8% in 140 ns

UCSB: PRL 112, 190504 (2014)

Multiplexed readout of 4 qubits Single-qubit readout Two-qubit readout (logical basis) Fidelity: > 90%

ETH Zurich: Nature 500, 319 (2013)

Two-qubit readout (Bell basis) Bell state concurrence: ~ 35%

UCB: PRL 112, 170501 (2014) Delft: Nature 502, 350 (2013)

Complexity

Max number of qubits and resonators

UCSB: 1411.7403

Direct: 9 qubits and 10 resonators Bus: 5 qubits and 7 resonators

Delft: 1411.5542

Recent realizations

Simple algorithms Protocols Quantum error correction

UCBS: Nature Physics 8, 719 (2012)

Shor (15; compiled) Deutsch–Jozsa

Yale: Nature 460, 240 (2009)

Grover (N=4)

Yale: Nature 460, 240 (2009) Saclay: PRB 85, 140503 (2012)

QFT

UCBS: Science 334, 61 (2011)

Deterministic teleportation

ETH Zurich: Nature 500, 319 (2013)

7 mm

3-qubit code

Yale: Nature 482, 382 (2011)

|0i |1i

|ψi

Recent realizations

Quantum error correction Simple algorithms Protocols

UCBS: Nature Physics 8, 719 (2012)

Shor (15; compiled) Deutsch–Jozsa

Yale: Nature 460, 240 (2009)

Grover (N=4)

Yale: Nature 460, 240 (2009) Saclay: PRB 85, 140503 (2012)

QFT

UCBS: Science 334, 61 (2011)

Deterministic teleportation

ETH Zurich: Nature 500, 319 (2013)

7 mm

3-qubit code

Yale: Nature 482, 382 (2011)

|0i |1i

|ψi

Error detection via parity meas.

IBM: 1410.6419 Delt: 1411.5542

« … using a two-by-two lattice of superconducting qubits to perform syndrome extraction and arbitrary error detection via simultaneous quantum non- demolition stabilizer measurements. This lattice represents a primitive tile for the surface code … »

Recent realizations

Simple algorithms

UCBS: Nature Physics 8, 719 (2012)

Shor (15; compiled)

Protocols Quantum error correction

Deutsch–Jozsa

Yale: Nature 460, 240 (2009)

Grover (N=4)

Yale: Nature 460, 240 (2009) Saclay: PRB 85, 140503 (2012)

QFT

UCBS: Science 334, 61 (2011)

Deterministic teleportation

ETH Zurich: Nature 500, 319 (2013)

7 mm

Error detection via parity meas.

« … we report the protection of classical states from environmental bit-flip errors and demonstrate the suppression of these errors with increasing system size. »

UCSB: 1411.7403

T = 29 μs

Data qubit avg. 5 qubit R.C. 9 qubit R.C. 2.7x 8.5x 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Time (µs) 1.0 0.9 0.8 1 Probability , Fidelity 1 2 3 4 5 6 7 8 Total repetition code cycles - k

Past, present and future

• M. H. Devoret and R. J. Schoelkopf, Science 339, 1169 (2013)

Operation on single physical qubits Algorithms on multiple physical qubits QND measurement for error correction and control Logical memory with longer lifetime than physical qubits Operations on single logical qubits Algorithms on multiple logical qubits Fault-tolerant quantum computation Time Complexity

Quantum!

Under the rug…

( )

Krytar 104020020 pump cavity drive 30 dB 30 dB 300 K ADwin AlazarTech ATS9870 qubit drives Miteq AFS3 (x2)

S 1 2

SR 445A

• Q

I trigger

Agilent E8257D

Q

Bias DAC

S 1 2

I

S 1 2

trigger gate

transmons in 3D copper cavity 30 dB 20 dB 20 dB 10 dB

JPA

3 K 15 mK 50 20 dB coil LNF LNC4_8A HEMT PAMTECH CTH0408KCS XMA attenuators 2082-6418 FPGA Tektronix AWG5014 Home-built amplifier 30 dB homodyne signal

S 1 2

I Q

Agilent E8257D Tektronix AWG520

I

S 1 2

50 Anritsu K251 bias tee

trigger

voltage-delayed trigger converter cVint

S 1 2

ChA ChB

MC ZVE-3W- 183 ATM 1507-28 ATM P1607D

• MC

VBFZ- 6260

Summary

• Artificial atoms based on Josephson junctions
• Low error per gate
• Steady improvement
• Circuit QED
• Resonator acts as bus for entangling gates
• Dispersive regime: high-fidelity qubit readout
• Basic protocols being implemented
• J. Mlynek et al., Quantum Device Lab, ETH Zurich (2012)

Postdoc and PhD positions available!

Visit us @ http://epiq.physique.usherbrooke.ca/

✤Superconducting qubits and circuit QED ✤Quantum error correction, quantum algorithms, … ✤Quantum dots, NV centers, … ✤ Full counting statistics and quantum noise