Gate Set Tomography Kenneth Rudinger - PowerPoint Presentation

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Gate Set Tomography

Kenneth Rudinger

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Outline

Motivation for quantum tomography

Framework for self-consistent, calibration-free tomography:

Gate set tomographyAchieving Heisenberg accuracy scaling with gate set tomographyUsing gate set tomography to build a better trapped ion qubit at Sandia National Laboratories

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Towards true QIP

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Towards true QIP

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Towards true QIP

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Towards true QIP

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Goal of tomography:

Make

ε

ij

as small as possible as cheaply as possible.

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The problem with tomography

Critical problem: relies on

precalibrated

reference frames that

don’t

really exist in hardware!

Goal: Calibration-free tomography.

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“Black box picture” of quantum information processor

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“Black box picture” of quantum information processor

prepare

do experiments

measure

outcome

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“Black box picture” of quantum information processor

prepare

do experiments

measure

outcome

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“Black box picture” of quantum information processor

prepare

do experiments

measure

outcome

Markovian

model:

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Perform collection of experiments.

Compute:

Gate Set Tomography Framework

Many choices for gate strings, estimator

F

.

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Gate Set Tomography

Simplest algorithm: Linear Inversion (LGST)

“Process

tomographywithout calibration”.

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Gate Set Tomography

Simplest algorithm: Linear Inversion (LGST)

“Process

tomographywithout calibration”.

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Linear gate set tomography

Use unknown gates as uncalibrated “fiducials”.

Run “process tomography” on each gate,

and on empty gate string.

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Linear algebra

 gate set.

arXiv:1310.4492

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How does LGST perform?

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“RMS Frobenius distance”:

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LGST on simulated data

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LGST review

N increases:

ε

0 No self-calibration problem

Experimentally demonstrated

i

ε

decreases slowly. (N

-0.5

)

Can we do better?

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Want to be sensitive to small errors.

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Push :

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Need

Ο(θ

-2

)

measurements to distinguish from

I.

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Push L times:

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Can amplify coherent errors!

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Can each experiment just be a different gate repeated many times?

e.g. G

x

2

, G

x

4

,..., G

y

2

, G

y

4

...

Not sufficient. Need to amplify other errors as well.

e.g. Tilt error

Also want sequences like

G

x

G

y,

(G

x

G

y

)

2

, (G

x

G

y

)

4

...

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Call these short sequences

germs.

Germs chosen to amplify errors. (E.g. tilt, over-rotation, dephasing.)

Do LGST on successively longer “powers”.

We call this

extended linear gate set tomography (

eLGST)

.

Can instead minimize

χ

2

:

Least Squares gate set tomography (LSGST)

.

Long-sequence GST

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Minimize total

χ

2

at each step.

Does estimate fit data?

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Algorithm summary

Start with experimental gate set, eg. {

Gi

, Gx

,

Gy

}

From knowledge of target gate set,

determine set of germs e.g.

{

Gx

,

Gy

,

Gi

,

GxGy

,

GxGyGi

,

GxGiGy

,

GxGiGi

,

GyGiGi

,

GxGxGiGy

,

GxGyGyGi, GxGxGyGxGyGy}

For varying maximum sequence length (L=1,2,…,512), perform “process tomography” experiments on each “extended germ”

Using

least-squares

, iteratively find gate set estimates that minimize

χ

2

.

Compare to target gate set.

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How do we measure success?

Can we find accurate estimate cheaply? (Can we beat N

-0.5

?)

Can we diagnose

and improve

experimental qubits?

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How do we measure success?

Can we find accurate estimate cheaply? (Can we beat N

-0.5

?)

Can we diagnose

and improve

experimental qubits?

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How do we measure success?

Can we find accurate estimate cheaply? (Can we beat N

-0.5

?)

Can we diagnose

and improve

experimental qubits?

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GST on real systems –

c

2

analysisEach box = 1 gate string

Color =

c

2

for that string

Blue

boxes = fits well

Red

boxes = fits poorly

Line-fitting analogy:

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(

GxGy

)

4

d

ata points

b

est fit line

Length of gate string

Germ (to repeat)

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April 2014:

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4/14:

BB1 pulses

May, 2014

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4/14:

5/14

Drift control

December,

2014

BB1

pulses

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4/14:

5/14

Drift control

12/14

BB1

pulses

Improved Gi compensation

February,

2015

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How good are these gates?

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GST

High-fidelity gates

Markovian behavior

Enabled by GST

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Conclusions

GST yields reliable, highly accurate estimates far more cheaply than standard tomography.

GST can diagnose the presence of non-Markovian noise.

GST is being used in the construction of reliable, high-fidelity experimental gates.

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Future directions

Multi-qubit systems

Randomized benchmarking predictions

Non-Markovian analysisDrift control.

More experimental implenetation

Contact me!

kmrudin@sandia.gov

Thank you!

Gnome image courtesy of

http://

sweetclipart.com

/friendly-garden-gnome-1464

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Bonus slides!

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The gauge

GST predictions are

gauge-invariant

:Given a target gate set, can

gauge-optimize

estimated gate set, yielding an “easy-to-read” interpretation.

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New end user infrastructure

Automated report generation

Explains GST to end user

Provides best GST estimate, along with relevant scoring parametersFidelity, trace distance, rotation axes and anglesc2 plots

Diagnostic “whac-a-mole” plots

Website interface for end user to generate own reports.

https://prod.sandia.gov/gst/index.html

Generation takes ~1 minute, depending on data set.

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Experimental error quantification

Can’t use ||.||

F

error for experimental data (don’t know G_true)c2 gives goodness-of-fit; what about error bars on individual gate set parameters?

Tackled experimental error bars in three ways:

Hessian of

c

2

function (in progress)

Parametric bootstrapping

Non-parametric bootstrapping

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Bootstrapping error bars

Non-parametric bootstrapping:

Randomly take subsamples of experimental dataset many times to generate new datasets

Run GST on each new dataset; generate ensemble of gate sets.Compute spread (or other statistics) of new ensemble of gate sets.

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Bootstrapping error bars

Parametric bootstrapping:

Compute GST estimate of experimental dataset.

Use GST estimate to generate many new datasets.Run GST on each new dataset; generate ensemble of gate sets.Compute spread (or other statistics) of new ensemble of gate sets.

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Bootstrapping results

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Bootstrapping error bars

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Bootstrapping error bars

Parametric marginally “better” than non-parametric.

Error bars about 10

-5

to 10

-4

in size for gate elements (larger for SPAM parameters)

This information can also be included in automated GST reports.

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In a method similar to bootstrapping, we can simulate RB experiments given GST data. (Below is SNL ion data.)

Experimental decay rate: 4.9

.

10

-5

GST-predicted decay rate: 4.0

.

10

-5

Can we obviate the need for RB experiments?

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GST vs. RB

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What about germ selection?

How do we choose germs which amplify

all

(non-SPAM) parameters?

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What about germ selection?

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If {

s

i

}

is incomplete,

Q

diverges. Then ||.||

F

behaves poorly.

If

Q

does not diverge, then ||.||

F

behaves well (L

-1

scaling).

We’ve written both integer and convex programs to find “good” germ sets.

This has allowed us to find relatively small germ sets for large gate sets (e.g. 40 germs for gate set with 9 gates).

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What about germ selection?

We

thought that minimizing Q (or finite-L

variants thereof) should correlate to minimizing ||.||

F

.

Found the error in derivation;

need to work out correction.

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Current 1-qubit GST analysis requires ~10

3

different gate sequences to perform.

Without any changes to GST paradigm, 2-qubit GST would require ~105 unique gate sequences.(162

fiducials, ~80 germs, 10 length sequences)

…This is a lot. Can we somehow reduce number of sequences?

Lots of redundancy.

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Future work: resource reduction

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Future work: resource reduction

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Reduce number L’s used?

Here we see a 3-fold reduction in number of 1-qubit experiments at a cost of only a factor of 3 in accuracy.

Can we similarly throw out various sequences (L values, fiducials, germs) for 2-qubit GST to get dramatic reduction in sequence requirement?

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Can we use germ selection techniques to pick out various long gate sequences that always amplify errors?

“Derandomized benchmarking?”

Can we use model selection techniques with GST to diagnose quantum devices of unknown a priori dimension?

Thanks!59Additional future work