Schrödinger Cats, Maxwell’s Demon and Quantum Error Correction

QuantumInstitute.yale.edu

ExperimentMichel DevoretLuigi FrunzioRob Schoelkopf

Andrei Petrenko Nissim OfekReinier HeeresPhilip ReinholdYehan LiuZaki LeghtasBrian Vlastakis+…..

TheorySMGLiang JiangLeonid GlazmanM. Mirrahimi **

Shruti PuriYaxing ZhangVictor Albert**Kjungjoo Noh**Richard BrierleyClaudia De GrandiZaki LeghtasJuha SalmilehtoMatti SilveriUri VoolHuaixui ZhengMarios Michael+…..

Quantum Error Correction

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‘Logical’ qubit

N ‘

Phys

ical

’ qub

itsCold bath

MaxwellDemon

Entropy

N qubits have errors N times faster. Maxwell demon must overcome this factor of N – and not introduce errors of its own! (or at least not uncorrectable errors)

Full Steane Code – Arbitrary Errors

Single round of error correction

6 ancillae

7 qubits

4

‘Surface Code’(readout wires not shown)

Current industrial approach (IBM, Google, Intel, Rigetti):

Scale up, then error correct

• Large, complex: o Non-universal (Clifford gates only)o Measurement via many wireso Difficult process tomography

• Large part count • Fixed encoding

All previous attempts to overcome the factor of N and reach the ‘break even’ point of QEC have failed.

O’Brien et al. arXiv:1703.04136 predict ‘break-even’ will be difficult even at the 50 qubit scale.

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We need a simpler and better idea...

‘Error correct and then scale up!’

Don’t use material objects as qubits.

Use microwave photon states stored in high-Q SC resonators.

All previous attempts to overcome the factor of N and reach the ‘break even’ point of QEC have failed.

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Surface Code(readout wires not shown)

Cat Code Photonic Qubithardware shortcut

(readout wire shown)

Scale then correct Correct then scale

• Precision: o Universal control

(all possible gates)o Measurement via

single wireo Easy process

tomographyo Long-lived cavitieso Fault-tolerant QEC

• Reduced part count • Flexible encoding

• Large, complex: o Non-universal (Clifford

gates only)o Measurement via

many wireso Difficult process

tomography• Large part count • Fixed encoding

“Hardware-Efficienct Bosonic Encoding”

earlier ideas: Gottesman, Kitaev & Preskill, PRA 64, 012310 (2001)Chuang, Leung, Yamamoto, PRA 56, 1114 (1997)

High-Q(memory)

Ancilla Readout

Leghtas, Mirrahimi, et al., PRL 111, 120501(2013).

Replace ‘Logical’ qubit with this:

N ‘

Phys

ical

’ qub

its

• Cavity has long lifetime (~ms)• Single dominant error channel

photon loss:makes QEC easier

n̂κΓ =

7

Photonic Code States

8

Can we find novel (multi-photon) code words that can store quantum information even if

some photons are lost?

High-Q(memory)

Ancilla Readout

Ancilla transmon coupled to resonator gives us universal control to make ‘any’ code word states we want.

0 L 1 L| | 0 |1ψ ψΨ⟩ = ⟩ + ⟩

quantum information

Logical code words (superpositions of photon Fock states)

Encoding qubits in cavity photon states

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We will use more complicated states with more photons (e.g. Schrödinger cat states)

More photons means higher loss (error) rate

This is the analog of N physical qubits forming a logical qubit. QEC Maxwell demon has to overcome the higher error rate.

n nκ= −

L

L

0 0

1 1

=

=

Minimal encoding cannot correct errors but has minimal loss rate:

0 photons

1 photon

10

Quick review of microwave resonators and photonic states

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0( ) | ( )ψ α ψ αΦ ≡⟨Φ ⟩ = Φ −

| | ie θα α=

Can displace in both position and momentum

Coherent state is closest thingto a classical sinusoidal RF signal

α

12

Coherent state = displaced vacuum

†

2†

2

[ * ]

1| |2

1| |2

0

0

0

!

a a

a

n

n

e

e e

e nn

α α

α α

α

α

α

−

−

−

∞

=

=

=

= ∑

Poisson distributionof photon number

nP

n

2| |n α=

This is the only kind of state that can be created with a classical drive applied to a cavity.

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High-Q cavity(memory)

Ancilla Readout

† †

2cq z za a a aH

ωσ χ σω= + + 3,000 ( ,[ )]2 κ γχ

Ancilla High-Q cavity

(memory)

Dispersive coupling of two-level ancillato high-Q cavity yields universal control

Cavity and ancillaare detuned

qcω ω≠

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Quantum optics at the single-photon level

0 1 2 30 1 2 3a a a aψ = + + + +

• Universal control enables: photon state engineering

Goal: arbitrary photon Fock state superpositions

Use the coupling between the cavity (harmonic oscillator)and the two-level qubit (anharmonic oscillator) to achieve this goal.

Previous State of the Art for Complex Oscillator States

Haroche/Raimond, 2008 Rydberg (ENS)

Expt’l. Wigner tomography: Leibfried et al., 1996 ion traps (NIST – Wineland group)

Hofheinz et al., 2009 (UCSB – Martinis/Cleland)

Rydberg atom cavity QED Phase qubit circuit QED

~ 10 photons ~ 10 photons

ΦQ

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Dispersive Hamiltonian

resonator qubit dispersivecoupling

rcavity frequency zω χσ= +

ω

eg

rω χ+rω χ−

κ‘strong-dispersive’ limit

32 ~ 2 10χ κ×

† †

2cq z za a a aH

ωσ χ σω= + +

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Strong-Dispersive Limit yields a powerful toolbox

ω

eg

rω χ+rω χ−

Microwave pulse at this frequency excites cavityonly if qubit is in ground state

Microwave pulse at this frequency excites cavityonly if qubit is in excited state

gDα Conditional displacement of cavityEngineer’s tool #1:

Cavity frequency depends onqubit state

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resonator qubit dispersivecoupling

Reinterpret dispersive term:- quantized light shift of qubit frequency

†q 2

2za aω χ

σ+

† †

2cq z za a a aH

ωσ χ σω= + +

…

2χ

N.B. power broadened100X

Microwaves are particles!

Microwave photon number distribution in a coherent state(measured via quantized light shift of qubit transition frequency)

† †

2q z

cza a aH a

ωσ χ σω ++= 3,000 ( ,[ )]2 κ γχ

New low-noise way to do axion dark matter detection?(arXiv:1607.02529)

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nπ Conditional flip of qubit if exactly n photonsEngineer’s tool #2:

resonator qubit dispersivecoupling

Reinterpret dispersive term:- quantized light shift of qubit frequency

†q 2

2za aω χ

σ+

† †

2cq z za a a aH

ωσ χ σω= + +

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†DISPERSIVE

zV a aχ σ≈

strong dispersive coupling

2χ

Qubit Spectroscopy

Coherent state in the cavity

Conditional bit flip nπ

Strong Dispersive Coupling Gives Powerful Tool Set

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Cavity-conditioned bit flip

Qubit-conditioned cavity displacement gDα

nπ

• multi-qubit geometric entangling phase gates (Paik et al.)• Schrödinger cats are now ‘easy’ (Kirchmair et al.)

experiment theoryG. Kirchmair M. MirrahimiB. Vlastakis Z. Leghtas

Photon Schrödinger cat states on demand

23(normalization is only approximate)

Paradoxically, we will use code words made of‘delicate’ Schrödinger cat states of cavity photons

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0246810

Spectroscopy frequency (GHz)

Coherent state:

Mean photon number: 4

Even parity cat state:

Odd parity cat state:

Only photon numbers: 0, 2, 4, …

Only photon numbers: 1, 3, 5, …

Parity of Cat States

P̂ ψ ψ= +

P̂ ψ ψ= −

2ψ α= =

ψ α α= + −

ψ α α= − −

Photon number

Rea

dout

sig

nal

Schoelkopf Lab

012345678910

246

8

135

79

0

Photon number

Rea

dout

sig

nal

Odd cat

Even cat

Coherent state

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Key enabling technology: ability to make nearly ideal measurement of photon number parity

(without measuring photon number!)

†

0

ˆ ( 1) ( 1)a a n

nnP n

∞

=

= −= − ∑

We learn whether n is even or odd without learning the value of n.

(analogous to measuring Z1Z2 without measuring Z1, Z2)

Measurement is 99.8% QND. (Can be repeated hundreds of times.)

If we can measure parity, we can perform completestate tomography (measure Wigner function)

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- use quantized light shift of qubit frequency

†q 2

2za aω χ

σ+

Measuring Photon Number Parity

ˆ ˆ22 2e

z zi nt i n

eσ σχ π− −

=

ˆ 0, 2, 4,...n =ˆ 1,3,5,...n = x

z

Gleyzes, S. et al. Nature 446, 297 (2007)

Sun, Petrenko et al., Nature 511, 444 (2014) 99.8% QND(!)

Using photon number parity to do cavity state tomography

( ) ˆˆ 1 nP = −

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Wigner function: --quasi-probability distribution in phase space

equivalent to the full density matrix for state tomography.

Wigner Function = “Displaced Parity”Vlastakis, Kirchmair, et al., Science (2013)

(( ˆ) ) ( )D DW Pβ β β= Ψ + − Ψ

Handy identity (Luterbach and Davidovitch):

ˆˆ ( 1) parityNP = − =28

Full state tomography on large dim. Hilbert space can be done very simply over a single input-output wire.

Simple Recipe: 1. Apply microwave tone to displace

oscillator in phase space.2. Measure mean parity.

0

0

4

4

-4

-4

State Tomography:Wigner Function of a Cat State

Vlastakis, Kirchmair, et al., Science (2013)

12

α α− + + Fringes prove this is acoherent cat, not a mixture

Interference fringes prove cat is coherent (even for sizes > 100 photons)

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(Yes...this is data.)

“X ”

“P ”vacuum noise

even parity

30

Using Schrödinger cat statesto store and correct quantum information

Courtesy of Mitra Farmand

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L L0 10 1ψ ψΨ = +

Encode information in two orthogonaleven-parity logical code “words”

L0 α α= + −

L1 i iα α= + −

Store a qubit as a superposition

of two cats of same parity

code word Wigner functions:

L0L1

Photon loss flips the parity which is the error syndromewe can measure (99.8% QND).

{ }L0a α αα= − − ˆ | ( 1) |P Ψ⟩ = + Ψ⟩

LX

a α αα=Magic property: coherent states are invariant under photon loss!

( ) ( )L )0 (if reala a α α α α α→= + − − −

Coherent states are eigenstates of photon destruction operator.

After loss of 4 photons cycle repeats:

( ) ( )40 L 1 L 0 L 1 L0 1 0 1a ψ ψ ψ ψ+ +→

We can recover the state if we know: (via monitoring parity jumps) Loss 4modN

( ) ( )2 2L L0 0a a α α α α= + − − =→ +

( ) ( )L1a a i ii i iα α α α→= + − − −

( ) ( )2 2 2L L1 ( ) 1a i i ia i iα α α α= + − = + − = −

a α α α=

Effect of photon loss on code words:

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2016: First true Error Correction Engine that works

AnalogOutputs

AnalogInputs

• Commercial FPGA with custom software developed at Yale

• Single system performs all measurement, control, & feedback (latency ~200 nanoseconds)

• ~15% of the latency is the time it takes signals to move at the speed of light from the quantum computer to the controller and back!

A prototype quantum computer being prepared

for cooling close to absolute zero.

MAXWELL’S DEMON

Schoelkopf-Devoret lab

Experiment:

‘Extending the lifetime of a quantum bit with error correction in superconducting circuits,’

Ofek, et al., Nature 536, 441–445 (2016).

Theory:

‘cat codes’Leghtas, Mirrahimi, et al., PRL 111, 120501(2013).

Implementing a Full QEC System: Debugger View

(This is all real, raw data.) Ofek, et al., Nature 536, 441–445 (2016). 35

15 sτ µ≈

Process Fidelity: Uncorrected Transmon

36

290 sτ µ≈

15 sτ µ≈

System’s Best Component

37

0 1eg n nψ ψ ψ= = + =

290 sτ µ≈

130 sτ µ≈

15 sτ µ≈

2α =

Process Fidelity: Cats without QEC

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+ =

290 sτ µ≈320 sτ µ≈

130 sτ µ≈

15 sτ µ≈

2α =

Process Fidelity: Cats with QEC

QEC – NO POST-SELECTION.

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290 sτ µ≈560 sτ µ≈

320 sτ µ≈

130 sτ µ≈

15 sτ µ≈

2α =

Only High-Confidence TrajectoriesStill keep~80% of dataExclude results with

heralded errors

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Universal Gate Set on a Logical Qubit Encoded in an Oscillator

(‘cat code’)

Heeres et al., Nature Communications 8, 94 (2017)

Cavity is not just a quantum memory, it is a qubit.

Encoding qubits in cavity photon states: ‘kitten code’

aka ‘binomial code’ L

L

0 40

22

1

+=

=

M. Michael et al., Phys. Rev. X 6, 031006 (2016)‘New class of error correction codes for a bosonic mode’

General binomial code corrects L losses, G gains and D dephasing events:

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L L

[ 3 1 ]

0 1

2a

ψ α β

ψ α β

+=

= +

initial state:

single photon loss:

‘kitten code’ can correct single photon loss

0,4 photons

2 photons

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Kitten code QEC: arXiv:1805.09072

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Kitten code QEC approaching breakeven: arXiv:1805.09072

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Repeated gates on logical qubit

Logical qubit Ramsey fringes with QEC

Kitten code QEC: arXiv:1805.09072

We are on the way!

“Age of Coherence”

“Age of Entanglement”

“Age of Measurement”

“Age of Quantum Feedback”

“Age of Qu. Error Correction.”

M. Devoret and RS, Science (2013)

Achieved goal of reaching “break-even” point for error correctionwith cat code and (almost) with kitten code.Now need to surpass by 10x or more.

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In (microwave) light there is truth…..

‘Circuit QED’

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Extra Slides

We can recover the state if we know: (via monitoring parity jumps) Loss 4modN

Amplitude damping is deterministic (independent of the number of parity jumps!)

/2 /2( ) t te eW t κ κα α− −−±=

Maxwell Demon takes this into account ‘in software.’

Cat in Two Boxes Qubit measures joint parity!1 2ˆ ˆ( )

12 1 2i n nP P eP π += =

50

Theoretical proposal by Paris group:Eur. Phys. J. D 32, 233–239 (2005)

12

α α α α±Ψ + + ± − −=

(two-legged cat only)

Cat in Two Boxes Qubit measures joint parity!

1 2ˆ ˆ( )12 1 2

i n nP P eP π += =

51

Experiment by Yale group:Science 352, 1087 (2016)

- Universal controllability- 3-level qubit can measure

1 2 12, , and P PP

Two-cavities:4-dimensional phase space and Wigner functions.

Theory

Experiment

12

α α α α±Ψ + + ± − −=

52

Entanglement of two logical cat states

9 sigma violation of Bell inequality

Entanglement of Two Logical Cat-Qubits

CHSH Bell: 2 2 2B≤ ≤

CHSH: (Milman et al.: evaluate Wigner at 4 points in 4D phase space)

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