Quantum State Protection and Transfer using Superconducting Qubits

Quantum StateProtection and Transfer

using Superconducting Qubits

Dissertation Defense of

Kyle Michael Keane

Department of Physics & Astronomy Committee:

Alexander Korotkov

Leonid Pryadko

Vivek Aji

June 29, 2012

Journal Articles1. A. N. Korotkov and K. Keane, “Decoherence suppression by quantum measurement

reversal,” Phys. Rev. A, 81, 040103(R), April 2010.

2. K. Keane and A. N. Korotkov, “Simple quantum error detection and correction for superconducting qubits,” arxiv:1205.1836, May 2012 (submitted to Phys. Rev. A).

1. “Decoherence suppression of a solid by uncollapsing,” Portland, OR, March 15-19, 2010), Z33.00011.

2. “Currently realizable quantum error correction/detection algorithms for superconducting qubits” Dallas, TX, March 21-25, 2011), Z33.00011.

3. “Modeling of a flying microwave qubit”Boston, MA, Feb. 27-March 2, 2012, Y29.00010

APS March Meeting Presentations

Posters1. “Theoretical analysis of phase qubits,” Quantum Computing Program Review (Minneapolis,

MN, 2009)

2. “Suppression of -type decoherence of phase qubits using uncollapsing and quantum error detection/correction,” Coherence in Superconducting qubits (San Diego, CA, 2010)

Outline

1. Introduction2. Decoherence by uncollapsing

Korotkov and Keane, PRA 2010

3. Repetitive N-qubit codes and energy relaxationKeane and Korotkov, arxiv:1205.1836, submitted to PRA, 2012

4. Two-qubit quantum error “correction” and detectionKeane and Korotkov, arxiv:1205.1836, submitted to PRA, 2012

5. Qubit state transferKeane and Korotkov, APS March Meeting, 2012

6. Summary

INTRODUCTIONLet’s begin with a basic

U

22

0 0 0cos2 2 2IU

L

δ δ

quantum variable

Superconducting Phase Qubitsstate

control

Iμw

flux bias

Ib

meas. pulse

Imeas

SQUID readout

Isq Vsq

25 mKSQUID

flux bias

qubit

C I0 L

microwavesX-, Y-rotations

flux biasZ-rotations

operation

UΔU

|0|1

State MeasurementSQUID-based Measurement:

lower barrierfor time t

U

relaxes

|1

|0

readout w/ SQUID

Tunneling Detected = state has been projected onto |1 and destroyedTunneling Not Detected = state has been projected onto |0

tunnels with rate

Γ 𝑡≫1

Weak Measurement

lower barrier for short time t

U

relax

|1

|0

readout w/ SQUID

Tunneling Not Detected = state projected onto|0 ORstate was |1 and didn’t have enough time to tunnel

tunnels with rate

There is a small change to the energy spacing during the lowering of the barrier

2 2

0 1 10 1

1

ie p

p

Γ 𝑡 ≈1

1 ( )te p t

Tunneling Detected = state has been projected onto |1 and destroyed

Uncollapsing

If tunneling does not occur, the qubit state is recoveredIn experiment, only data for cases where tunneling does not occur is kept

State Prepared

Doesn’t Tunnel Doesn’t Tunnel

Partial Measurement

Projects state toward 0 (was 1)

Partial Measurement

Projects state toward 0

π-pulse π-pulse

Zero-Temperature Energy Relaxation

11

1

1

/ 22 2 /

2 2 /

2 /

0 1 with probability

0 1

0 with probability 1

t Tt T

t T

t T

ee

e

e

This can be “unravelled” into discrete outcomes with probabilities

|0

|1

The population of the excited state moves into the ground state

DECOHERENCE SUPPRESSION BY UNCOLLAPSING

Project One


Protection from Energy Relaxation

• Quantum Error Correction (Shor/Steane/Calderbank circa 1995)• Requires larger Hilbert space and controllable entanglement)

• Decoherence-Free Subspaces (Lidar 1998)• Requires larger Hilbert space and specfic subspaces

• Dynamical Decoupling (Lloyd and Viola 1998)• Does Not Protect Against Markovian Processes (Pryadko 2008)

Standard methods to protect against decoherence:

Our proposed method• Simple modification of uncollapsing procedure

• Our proposal was demonstrated in another system

• Requires selection of only certain cases• Similar to probabilistic QEC and linear optics QC

Ideal Procedure

storage period

11

Prepared

π-rotationPartial meas. (p

u )

Partial meas. (p)

π-rotation

𝑒−𝑡 /𝑇1

time

axis of π-rotation

Initial value

Returned toinitial value

Similar protection for all density matrix elements


Results

Yields a state arbitrarily close to initial

Some improvement even with naive uncollapsing strength


Fide

lity

Measurement Strength (p)

Process with Decoherence

storage period t

11

Prepared

π-rotationPartial meas. (p

u )

Partial meas. (p)

π-rotation

κ1

κ 𝑖≡𝑒−𝑡 𝑖 /𝑇 1

time

κ2κ3κ4

axis of π-rotation

Initial value

Pure dephasing and energy relaxation during entire process

Returned toinitial value


Results

Pure dephasing uniformly decreases fidelity

Explains phase qubit uncollapsing experiment (Katz, 2008)

Still works with relaxation during operations

Perfect suppression requires small prob. of success


Fide

lity

and

Prob

abili

ty

Measurement Strength (p)

Experimental Demonstration

Weak Measurementpolarization beam splitter, half wave

plate, and dark port

Optical Circuit Results

Nearly exact match to theory

Jong-Chan Lee, et. al., Opt. Express 19, 16309-16316 (2011)

Relaxationsimilar components, (except no dark port)

Protecting EntanglementInitially

entangledstate

Q1

Q2

WM π𝑻 𝟏 WM π

Entanglement is recovered

Q1

Q2

WM π𝑻 𝟏 WM π

Circumvents Entanglement Sudden Death

Same optics group did this extension experimentYong-Su Kim, et. al., Nature Physics, 8, 117-120 (2012)

Summary

• Does not require a larger Hilbert space• Modification of existing experiments in superconducting phase qubits• Demonstrated using photonic polarization qubit• Extended to protect entanglement

REPETITIVE CODING AND ENERGY RELAXATION

Project Two

Keane and Korotov, arxiv 2012

Motivation

|0 ⟩ |1 ⟩Bit Flip

|1 ⟩ |0 ⟩

A bit flip looks like a more difficult error

process than T1|0 ⟩ |0 ⟩

T1

|1 ⟩ |0 ⟩AND

Repetitive coding protects against bit

flips

PROT

ECTS

????

????

?

THEREFORE…

Repetitive Quantum Codes and Energy Relaxation

|

|0N-1

tomography

T1(i)X X

All “N-1” are 0: good Any in 1: either discard (detection) or try to correct (correction)

Encoding by N c-X gates

|

|0N-1 X

||0|0|0

||0

c-X gate (cNOT)

cNOT|

cNOT|

α | 0 |0 N −1 + β | 1 |0 N−1 →α | 0 N + β | 1 N

Syndrome Result

FAILS

Two-Qubit Encoding

syndrome

𝒔𝒊𝒏𝒈𝒍𝒆𝒒𝒖𝒃𝒊𝒕

|ψ ⟩|0 ⟩

Two qubitsEqual decoherence strength

(𝑝=1−𝑒−𝑡 /𝑇 1 )

𝐹 (ψ )=⟨ψ|ρ 𝑓𝑖𝑛𝑎𝑙|ψ ⟩

T1(i)


Fide

lity

Decoherence Strength (p)

N-Qubit Error Detection|

|0N-1

tomography

T1(i)X X All “N-1” are 0: keep

Any in 1: discard

p

optimal, but 2 qubits are sufficient

Since the procedure works

𝐹 𝑎𝑣=∫ ⟨ |ρ| ⟩𝑃𝑆

d |

~𝐹 𝑎𝑣=∫ ⟨|ρ|⟩ d |

∫ ⟨|𝑃𝑆| ⟩d |


Fide

lity


ignore

detect

single

N-Qubit Error Correction|

|0N-1

tomography

T1(i)X X All “N-1” are 0: keep

Any in 1: cannot correct!

p

QEC is impossibleIn our paper we show

that no unitary operation can improve the fidelity

for p<0.5


Fide

lity


ignore

correct

single

Summary

• Can be used for QED, but not for QEC of energy relaxation

• 3 qubits are optimal, but 2 qubits are sufficient

TWO-QUBIT QUANTUM ERROR DETECTION/CORRECTION

Project Three


Two-Qubit Error “Correction”/Detection

0: good 1: either discard (only detection) or correct (if know which error)

Y/2 -Y/2

||0

tomography

X-correction needed

Y-correction needed

Z-correction needed

no correction needed (insensitive)

( )2 2YY R

Notations:

= c-Z E1

E2

( | 0 | 1 ) | 0 cos ( | 0 | 1 ) sin ( || 1 | 0 ) |0 1i E1 = X-rotation of main qubit by arbitrary angle 2:

E1 = Y-rotation of main qubit:

E2 = Z-rotation of ancilla qubit:

( | 0 | 1 ) | 0 cos ( | 0 | | 01 ) sin ( | 1 | 0 1) |

( | 0 | 1 ) | 0 cos ( | 0 | 1 ) sin ( | 0 | 1| 0 ) | 1 E2 = Y-rotation of ancilla qubit:

( | 0 | 1 ) | 0 |( | 0 | 1 ) ( )|0 1ie

(| 00 | 01 ) (| 10 | 11 )

good

good

good


Two-Qubit Error “Correction”/Detection

0: good 1: either discard (only detection) or correct (if know which error)

Y/2 -Y/2

||0

tomography( )

2 2YY R

Notations:

= c-Z E1

E2

(| 00 | 01 ) (| 10 | 11 )

Various Decoherence Strengths

Fide

lity

Rotation Strength (2θ/π)

corrdet

ign

All Four Errors

Fide

lity

Rotation Strength (2θ/π)

corrdet

ign


QED for Energy Relaxationstore in resonators

0: good 1: discard

Y/2 -Y/2

||0

tomography ( )2 2YY R

Notations:

= c-Z T1

T1Y/2-Y/2

1100

QED of real decoherenceThe fidelity is improved

by selection of measurement result 0Fide

lity

Relaxation Strength

detect

ignore


Almost “repetitive”

Summary

• QEC is possible for intentional errors• QED is possible for energy relaxation• Experiments can be done with superconducting

phase qubits

QUANTUM STATE TRANSFERProject Four

Keane and Korotov, APS 2012

System

Resonatoror

Phase Qubit

Transmission Line

Tunable CouplersHigh-Q Storage

Initiallyhere

Senthere

SuperconductingWaveguide

Tunable Inductance

Example from UCSB

(𝑪𝑫)=(𝑟 𝐿 𝑡𝑡 𝑟 𝑅)(𝑨𝑩)

𝑨 𝑩𝑪 𝑫

Tunable Parameter

1

𝑡×𝐴𝐴

𝑟 𝐿×𝐴 𝑀𝐿 𝐽

and are tuned byvarying

𝑟 𝐿

𝑡0

Korotkov, PRB 2011

Ideal ProcedureTr

ansm

issi

onCo

effici

ents

Qubitinitially is here

Qubittransferredto here

𝒕𝟏(𝒕<𝒕𝒎)=𝒕𝟏𝒎𝒂𝒙

√𝟐𝒆− Δt / τ 𝐛𝐮−𝟏𝒕𝟐(𝒕<𝒕𝒎)=𝒕𝟐𝒎𝒂𝒙

𝒕𝟏(𝒕>𝒕𝒎)=𝒕𝟏𝒎𝒂𝒙

𝒕𝟐(𝒕>𝒕𝒎)=𝒕𝟐𝒎𝒂𝒙

√𝟐𝒆+Δ t / τ 𝐛𝐮−𝟏

𝒕𝟏 𝒕𝟐Time (t)𝐭𝐦

𝒕𝒎𝒂𝒙

𝑡𝑚𝑎𝑥 ≈0.05 resonators,

𝜼=𝟎 .𝟗𝟗𝟗

Typical parameters

(UCSB)

Duration ns

ON/OFF

Desired Efficiency

Korotkov, PRB 2011

Main idea

AB

Transmission line Receiving resonator

PERFECT TRANSFER

( 𝑟1 𝑨+𝑡𝑩𝑡 𝑨+𝒓𝟐𝑩)=(

𝑟1 𝑡𝑡 𝑟2)(

𝑨¿¿𝑩)

“into line”

“into resonator”A B

𝒓𝟐𝑩

𝑡 𝑨𝑡𝑩𝑟1 𝑨

τ 𝒃𝒖=τ𝒓𝒕 /|𝒕|𝟐= buildup time

resonator round trip time transmission coefficient

Korotkov, PRB 2011

Procedural RobustnessTr

ansm

issi

onCo

effici

ents

𝐭𝐦 Time (t)

𝒕𝟐𝒕𝟏

�̇�=− |𝑡|2

2 τ𝑟𝑡𝐵+ 𝑡

τ𝑟𝑡𝐴

𝒕 𝑩𝑨

𝒕𝟏(𝒕<𝒕𝒎)=𝒕𝟏𝒎𝒂𝒙

√𝟐𝒆− Δt / τ 𝐛𝐮−𝟏𝒕𝟐(𝒕<𝒕𝒎)=𝒕𝟐𝒎𝒂𝒙

𝒕𝟏(𝒕>𝒕𝒎)=𝒕𝟏𝒎𝒂𝒙

𝒕𝟐(𝒕>𝒕𝒎)=𝒕𝟐𝒎𝒂𝒙

√𝟐𝒆+Δ t / τ 𝐛𝐮−𝟏

𝒎𝒂𝒙

Keane and Korotkov, APS 2012

We vary the parameters in the above equations: , , and

η=𝐸 𝑓𝑖𝑛

𝐸𝑖𝑛

Shaping of ControlTr

ansm

issi

onCo

effici

ents

𝐭𝐦Time (t)

𝑡 2

𝑡1

shaping error efficiency loss

Robustness

No Problem! of 33.3 ns is 2.5 ns)

𝒎𝒂𝒙


Switching TimeTr

ansm

issi

onCo

effici

ents

𝐭𝐦𝟐 Time (t)

𝑡 2

𝑡1

𝐭𝐦

Vary and together Vary only

timing error efficiency loss

Robustness

No Problem!

ns

of 230 ns is 11.5 ns)

𝒎𝒂𝒙


Maximum Transmission CoefficientTr

ansm

issi

onCo

effici

ents

𝐭𝐦 Time (t)

𝒎𝒂𝒙

𝑡 2𝑡1

Vary only

Vary and together

amplitude error efficiency loss

Robustness

No Problem!(experiments have good control of tunable coupler)


Frequency Mismatch

ω1 ω2

ω1≠ω2

�̇�=− |𝑡|2

2 τ𝑟𝑡𝐵+ 𝑡

τ𝑟𝑡𝐴𝑒𝑖(𝜔1−ω2 )𝑡

resonator frequency

Frequency Mismatch

frequency error efficiency loss

Robustness

Requires Attention(resonator frequencies should be kept nearly equal throughout procedure)


Summary

• Robust to procedural errors (timing, shaping, maximum transmission coefficient)

• Requires active maintenance of nearly equal resonator frequencies

The second conclusion is very important for experiments — For the current solid-state tunable couplers

there is an effective frequency shift during modulation of the transmission coefficient

CLOSING REMARKSrecapitulation

Summary• Decoherence suppression by uncollapsing

– Probabilistically suppresses Markovian energy relaxation– After our proposal, it was demonstrated by another group– Extended in another experiment to entangled qubits

• N-qubit repetitive codes and relaxation– Can be used for QED, but not for QEC (2 qubits are sufficient)

• Two-qubit “QEC”/QED experiments– Can be performed with current technology

• Quantum state transfer– Robust against procedural errors– Requires resonator frequencies to be kept nearly equal

THANK YOU!

APPENDICESJust in case

Representations of Errors-Example: Energy Relaxation

11 01

10 11

1 1 1-

1- 1

p pD

p p

11 111

1t tt T

00 11t tt t

01 011

12

t tt T

10 101

12

t tt T

00 0R

pK

1 0

0 1DRKp

2 2

2 2

2

0 1 1, with probability 1

1

0 , with probability

pp

p

p

10

1

1T

0 1

From the normalization requirement

Need to derive this from commutator!!!!!

Need to derive this from somewhere!!!!!

Solving these equations and combining into an operation

Choosing a specific operator sum decomposition

If you initially have a pure state, the classical mixture created by this process becomes explicit

† †R R DR DR R R R DR DR DRt K K K K P P LINK

1/1 t Tp e

This can be done for any operation however only some give physically meaningful interpretations

† †R R DR DRD K K K K

Master Equation RETURN

Documents

Quantum State Protection and Transfer using Superconducting Qubits