Quantum Computing - MIT OpenCourseWare · PDF fileQC1: Quantum Computing with Superconductors 1. Introduction to Quantum Computation 1.The Unparalleled Power of a Quantum Computer

QC1: Quantum Computing with Superconductors 1. Introduction to Quantum Computation

1.The Unparalleled Power of a Quantum Computer

2.Two state systems: qubits

3.Types of Qubits

2. Quantum Circuits

3. Building a Quantum Computer with Superconductors

1.Quantizing Superconducting Josephson Circuits

2.Dynamics of Two-Level Quantum Systems

3.Types of superconducting qubits

4.Experiments on superconducting qubits

1. Charge qubits

2. Phase/Flux qubits

3. Hybrid qubits

4. Advantages of superconductors as qubits

Massachusetts Institute of Technology

November 29, 2005 6.763 2005 Lecture QC1

Quantum Computing

Qubits are two level systems a) Spin states can be true two level systems, or b) Any two quantum energy levels can also be used

0We will call the lower energy state and the higher energy state 1

In general, the wave function can be in a superposition of these two states

ψ = a 0 + b 1

Massachusetts Institute of Technology 6.763 2005 Lecture QC1

1

Computing with Quantum States • Consider two qubits, each in superposition states

1 +ψ A

0 + = A A B

= ψ B

0 B

1

• We can re-write these states a single state of the 2-e- system

ψ =

0

ψ ψ A B

=( 1 )⊗( 0 )+ +A A B

1 B

= 0 + 0 +0 +A B A

1 B

1 A

0 B

1 A 1

B

• All four “numbers” exist simultaneously • Algorithm designed so that states interfere to give one “number”

with high probability Massachusetts Institute of Technology

6.763 2005 Lecture QC1

The Promise of a Quantum Computer

A Quantum Computer … • Offers exponential improvement in

speed and memory over existing computers

• Capable of reversible computation • e.g. Can factorize a 250-digit number in

seconds while an ordinary computer will take 800 000 years!


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1. Quantum Computing Roadmap Overview

2. Nuclear Magnetic Resonance Approaches

3. Ion Trap Approaches

4. Neutral Atom Approaches

5. Optical Approaches

6. Solid State Approaches

7. Superconducting Approaches

8. "Unique" Qubit Approaches

9. The Theory Component of the Quantum Information Processing and Quantum Computing Roadmap

http://qist.lanl.gov Massachusetts Institute of Technology


Outline 1. Introduction to Quantum Computation

1.The Unparallel Power of a Quantum Computer


3.Types of Qubits

2. Quantum Circuits




3.Types of Superconducting qubits


1. Charge qubits


3. Hybrid qubits



3

Circuits for QubitsCircuits for Qubits


• Need to find circuits (dissipationless) which have two “good” energy levels

• Need to be able to “manipulate” qubits and couple them together


Harmonic Oscillator LC Circuit

1 2 1 2H =

1 mv 2

1 mω 2 2 H = C V + L Ix

2 +

2 2 2

Φdx v = d Φ and I = v= d t Ldt

dx 2 1 2 2 1 ⎛ d Φ

⎟⎞2

+1 C 1

Φ 2C1 ⎛ ⎞ { ⎜H = m⎜ + mω xH =

2 M ⎝ d t ⎠ 2 { L C dt

{2 ⎝ ⎠ 2 M

ω 2

dx d Φp = mdt p = C = C V

dt Quantum Mechanically Quantum Mechanically

∆ ∆ ≥ h / 2 L C ∆ I ∆ V ≥ h / 2x p

E = h ω ( n +E = h ω ( n + 12

) Massachusetts Institute of Technology

12

)


4

Outline

1. Introduction to Quantum Computation



3.Types of Qubits

2. Quantum Circuits

Building a Quantum Computer with Superconductors 3.





1. Charge qubits


3. Hybrid qubits



Quantization of Circuits

1. Find the energy of the circuit

2. Change the energy into the Hamiltonian of the circuit by identifying the canonical variables

3. Quantize the Hamiltonian

¾ Usually we can make it look like a familiar quantum system


5

Quantization of a Josephson JunctionQuantization of a Josephson Junction

Charging Energy Josephson Energy cUC =

1 Q2 =

1 CV 2 U J =Φ 0 I (1− cos ϕ )

2π

1 ⎛⎜ Φ 0 ⎞

2 ⎛ ϕ∂ ⎞

2

2 C 2

= ⎟ C⎜ ⎟2 ⎝ 2π ⎠ ⎝ ∂ t ⎠

2 EC =

2C EJ

Φ

20

π

Ie = c

Hamiltonian: H = 1 ⎛⎜ Φ 0 ⎞⎟

2

C⎛⎜ϕ∂ ⎟⎞

2

+Φ 0 Ic (1− cos ϕ )

2 ⎝ 2π ⎠ ⎝ ∂ t ⎠ 2π

Circuit behaves just like a physical pendulum.

For Al-Al2O3-Al junction with an area of 100x100 nm2

C = 1fF and Ic=300 nA, which gives EC=10µ eV and EJ=600µ eV

To see quantization, Temperature < 300 mK Massachusetts Institute of Technology


6

7

8

Outline 1. Introduction to Quantum Computation



3.Types of Qubits

2. Quantum Circuits






1. Charge qubits


3. Hybrid qubits



Dynamics Two-Level Quantum Systems

ψ α 2⎛ −F −V ⎞ ⎛ ⎞ Eigenenergies E= F 2 +V , = ⎜ ⎟ H = ⎜

⎝−V * F ⎠⎟ β⎝ ⎠

1⎛ ⎞ At F=0, let ψ ( t = 0 ) = ⎜ ⎟ 0⎝ ⎠ E+

E-

F

1 0

⎛ ⎜⎝

01

⎛ ⎞⎜ ⎟⎝ ⎠

+V

-V

⎞⎛11

⎟⎟⎠

⎞⎜⎜⎝

⎛ −1 1

2 1

⎞ ⎟ V V⎠ 1 ⎛ ⎞ i t 1 ⎛ 1 ⎞ − i t

ψ ( ) 1

t = ⎜ ⎟12 ⎝ ⎠ e h +

2 ⎜⎝ −1 ⎟⎠ e h

1 0V t ⎛ ⎞ V t ⎛ ⎞ = co s ⎜ ⎟ + i s in ⎜ ⎟0 1h ⎝ ⎠ h ⎝ ⎠

1⎛ ⎞ System oscillates between ⎜ ⎟ 0⎝ ⎠ 0⎛ ⎞ and ⎜ ⎟ with period1

2 ⎜⎝⎜ 1⎟⎠⎟ ⎝ ⎠


9

Rabi Oscillations

Drive the system with V(t)=V0 eiωt at the resonant frequency ω = E+ − E−

1If ψ (t = 0) = ⎛ ⎞ then⎜ ⎟0,

⎝ ⎠ 1 0V t ⎛ ⎞ o ω ⎛ ⎞0( )ψ t = cos ⎜ ⎟ + i sin

V tei t

⎜ ⎟0 1h ⎝ ⎠ h ⎝ ⎠

Oscillations between states can be controlled by V0 and the time of AC drive, with period

hΤ =

2V 0


Charge-State Superconducting Qubit

Images removed for copyright reasons.

Please see: Nakamura, Y., Yu A. Pashkin, and J. S. Tsai. Nature 398, 786 (1999).


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Charge qubit a Cooper-pair box E EJ / C 3 . 0 ~

Coherence up to ~ 5 ns, presently limited by background charge noise (dephasing) and by readout process (relaxation)


Please see: Nakamura, Y., Yu A. Pashkin, and J. S. Tsai. Nature 398, 786 (1999).


Types of Superconducting Qubits

• Charge-state Qubits (voltage-controlled) – Cooper pair boxes

• Flux/Phase-state qubits (flux-current control) – Persistent Current Qubits – RF SQUID Qubits – Phase Qubits (single junction)

• Hybrid Charge-Phase Qubits


11


Three-Junction Loop Measurements

j

µ µm µm2

LIc µA EJ/Ec

µm µm2

LSQUID Ic µA

Jc2

0≡

1≡

20 µm

1pF 0.45µm

1.1µm 0.55µm

1.1µm

I V

I+ V+Image removed for copyright reasons.

6.763 2005 Lecture QC1 Persistent current qubits require high-quality sub-micron unctions with low current density, and only MIT Lincoln has demonstrated this capability in Nb.

Three-junction Loop Jct. Size ~ 0.45 m, 0.55Loop size ~16x16

3-junction~ 30pH ~1 & 2

~ 350 & 550 DC SQUID Shunt capacitors ~ 1pF Jct. Size ~ 1.1Loop size ~20x20

~ 50pH ~10 & 20

M~35pH ~350 & 730A/cm

1-20 GHz


Energy Band Diagram of MIT-LL PC-Qubit:

12

Observation of Coherent Superposition of Macroscopic States Jonathan Friedman, Vijay Patel, Wei Chen, Sergey Tolpygo and James Lukens


Please see: Friedman, J., et al. Nature 406, 43 (July, 2000); and Van der Wal, C., et al. Science 290, 773 (Oct. 2000).


Advantages of Superconductors for Quantum Computing

• Employs lithographic technology • Scalable to large circuits • Combined with on-chip, ultra-fast

control electronics – Microwave Oscillators – Single Flux Quantum classical electronics


13



Circuits Fabricated at Lincoln Laboratory

Quantum Computation with Superconducting Quantum Devices T.P. Orlando, Ken Segall, D. Crankshaw, D.Nakada, S. Lloyd, L. Levitov,- MIT

M. Tinkham , Nina Markovic, Segio Vanenzula– Harvard; K. Berggren, Lincoln Laboratory

1/27/03

SQUID on-chip oscillator and qubit


On-chip oscillator couples to qubit: No spectroscopy yet due to high temperature


14

Image removed for copyright reasons.


Feasibility of Superconductive Control Electronics Fabrication

On-chip Control for an RF-SQUID M.J. Feldman, M.F. Bocko, Univ. of Rochester

www.ece.rochester.edu/~sde/



15

Quantum Computation Quantum

/

Dilution Refrigerator Insert


with Superconducting Devices T.P. Orlando, Yang Yu, D.Nakada, B. Singh, J. Lee, D. Berns,

Ken Segall, D. Crankshaw, B Cord- MIT

1/27 03

Installed and to begin dc data taking in February and ac data taking in April

Sample Holder

DC Measurement on Nb Persistent Current Qubit

IR sw

VVg

IL sw

Isw

Isw

ISW

FLUX

Isw

FLUX

|1>

|0>


SQUID Detector

16

Cum

ulative Distribution Function

Weakly-Driven limit, 3.25 GHz

) T= 25 mK

Cumulative Distribution Function:

Magnetic Flux

Switc

hing

Cur

rent

of S

QU

ID

Resonance peaks at 3.25 GHz

= |0>

= |1>

Yang Yu (MIT) and Will Oliver (MIT-LL

Observation of Coherent Superposition of Macroscopic States Jonathan Friedman, Vijay Patel, Wei Chen, Sergey Tolpygo and James Lukens

Please see: Friedman, J., 406 ; and Van der Wal, C., Science .



et al. Nature , 43 (July, 2000)et al. 290, 773 (Oct. 2000)

17

switc

hing

pro

babi

lity

(%)

time between pulses ∆t (µs)

νRF = 16409.50 MHz



CHARGE-FLUX QUBIT

Quantronics Group CEA-Saclay

France

M. Devoret (now at Yale) D. Esteve, C. Urbina D. Vion, H. Pothier P. Joyez, A. Cottet

Coherence time measured by Ramsey fringes : 500ns Qubit transition frequency: 16.5 GHz; coherence quality factor: 25 000

RAMSEY FRINGES

Image removed for copyright reasons.

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Quantum Computing - MIT OpenCourseWare · PDF fileQC1: Quantum Computing with Superconductors 1. Introduction to Quantum Computation 1.The Unparalleled Power of a Quantum Computer