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Home , Quantum error correction

Schrödinger Cats, Maxwell’s Demon and

Experiment Theory SMG Luigi Frunzio Liang Jiang Rob Schoelkopf Leonid Glazman M. Mirrahimi ** Andrei Petrenko Nissim Ofek Shruti Puri Reinier Heeres Yaxing Zhang Philip Reinhold Victor Albert** Yehan Liu Kjungjoo Noh** Zaki Leghtas Richard Brierley Brian Vlastakis Claudia De Grandi +….. Zaki Leghtas Juha Salmilehto Matti Silveri Uri Vool Huaixui Zheng Marios Michael +….. QuantumInstitute.yale.edu

1 is a New Paradigm

• Quantum computing: a completely new way to store and process information.

• Superposition: each quantum bit can be BOTH a zero and a one.

• Entanglement: the quantum computer can explore ALL possible outcomes. (Einstein: ‘Spooky action at a distance.’)

• Massive parallelism: carry out computations that are impossible Daily routine engineering and calibration test: Carry out on spooky ANY conventional operations that computer. Einstein said were impossible!

2 Register of N conventional bits can be in 1 of 2N states.

000, 001, 010, 011, 100, 101, 110, 111

The Power of N Register of N quantum bits has ~22 states! (crude estimate)

Ψ~ 000±±±±±±± 001 010 011 100 101 110 111

Even a small quantum computer of 30 will be so powerful its operation is difficult to simulate on a conventional supercomputer.

Quantum computers are good for problems that have simple input and simple output but must explore a large space of states at intermediate stages of the calculation.

3 Applications of Quantum Computing

Quantum Quantum materials chemistry

Machine learning* Cryptography and *read the fine print Privacy Enhancement

4 Storing information in quantum states sounds great…,

but how on earth do you build a quantum computer?

5 ATOM vs CIRCUIT

Hydrogen atom Superconducting circuit oscillator

L C

10−7 mm 0.1 - 1 mm (Not to scale!) 1 ~ 1012 ‘Artificial atom’

6 How to Build a with an Artificial Atom…

Superconducting T = 0.01 K integrated circuits are a promising technology for scalable quantum computing

Josephson junction: 200 nm Aluminum/AlOx/Aluminum The “transistor of quantum computing”

Provides anharmonic structure AlOx tunnel barrier

7 Qubit Antenna pads are capacitor plates ωω01≠ 12 Josephson ω12 1 = e tunnel junction

Energy ω01 0 = g ~1 mm

ω01 ~5− 10GHz

1012 mobile electrons Superconductivity gaps out single-particle excitations Quantized energy level spectrum is simpler than hydrogen

Quality factor QT = ω 1 comparable to that of hydrogen 1s-2p Enormous transition dipole moment 8 The first electronic quantum processor (2009)

Executed Deutsch-Josza and Grover search algorithms

Lithographically produced integrated circuit with semiconductors replaced by superconductors.

Michel Rob Devoret Schoelkopf

DiCarlo et al., Nature 460, 240 (2009) 9 The huge information content of quantum superpositions

comes with a price:

Great sensitivity to noise perturbations and dissipation.

The quantum phase of superposition states is

well-defined only for a finite ‘ time’ T 2

10 Cat Code Exponential Growth in QEC “Moore’s Law” for T 3D multi-mode SC Qubit Coherence 2 cavity several groups 100-200 us (Delft, IBM, MIT, Yale, …)

lowest thresholds for quantum error correction NIST/IBM, Yale, ...

MIT-LL Nb Trilayer

Oliver & Welander, MRS Bulletin (2013) R. Schoelkopf and M. Devoret 11 Girvin’s Law:

There is no such thing as too much coherence.

We need quantum error correction!

12 The Quantum Error Correction Problem

I am going to give you an unknown .

If you measure it, it will change randomly due to state collapse (‘back action’).

If it develops an error, please fix it.

Mirable dictu: It can be done!

13 Quantum Error Correction for an unknown state requires storing the quantum information non-locally in (non- classical) correlations over multiple physical qubits. ‘Logical’ qubit ‘Physical’ qubits ‘Physical’ N

Non-locality: No single physical qubit can “know” the state of the logical qubit. 14 Quantum Error Correction

‘Logical’ qubit Cold bath

Entropy

‘Physical’ qubits ‘Physical’ Maxwell N Demon

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) 15 All previous attempts to overcome the factor of N and reach the ‘break even’ point of QEC have failed. With major technological advances a 49-transmon ‘surface code’ might conceivably break even at

T 2 = 40 µ s . [O’Brien et al. arXiv:1703.04136] However to date, good repeatable (QND) (weight 4) error syndrome measurements have not been demonstrated [Takita et al., Phys. Rev. Lett. 117, 210505 (2016)] ‘Scale up and then error correct’ seems almost hopeless.

We need a simpler and better idea...

‘Error correct and then scale up!’

16 “Hardware-Efficienct Bosonic Encoding” Leghtas, Mirrahimi, et al., PRL 111, 120501(2013). Replace ‘Logical’ qubit with this: High-Q (memory) Ancilla Readout qubit

• Cavity has longer lifetime (~ms)

‘Physical’qubits • Large Hilbert space

N • Single dominant error channel photon loss: Γ = κ nˆ • Single readout channel

earlier ideas: Gottesman, Kitaev & Preskill, PRA 64, 012310 (2001) Chuang, Leung, Yamamoto, PRA 56, 1114 (1997) 17 Photonic Code States

Can we find novel (multi-photon) code words that can store quantum information even if some photons are lost?

High-Q (memory) Ancilla Ancilla transmon coupled to Readout qubit resonator gives us universal control to make ‘any’ code word states we want.

18 Quick review of microwave resonators and photonic states

19 Coherent state is closest thing to a classical sinusoidal RF signal

ψψα()Φ=0 (Φ− )

20 Photon number distribution in a coherent state (measured via quantized light shift of qubit transition frequency) V = 2χ e e a†a [2χ : 3,000 (κ ,γ )]

N.B. power broadened 100X … 4 3 2 1

2χ

New low-noise way to do axion dark matter detection? Microwaves (arXiv:1607.02529) are particles! We will use Schrödinger cat states of cavity photons

(normalization is only approximate) 22 Parity of Cat States

iπa†a n n P = e = (−1) → P = ∑ pn (−1) n Photon number 10 8 6 4 2 0 Coherent state: ψ ==α 2 Mean photon number: 4 Readoutsignal Even parity cat state: ψ =+αα− Only photon numbers: 0, 2, 4, … Pˆ ψψ=+

Odd parity cat state: ψ =αα−−

Only photon numbers: 1, 3, 5, … Spectroscopy frequency (GHz) ˆ P ψψ=− Schoelkopf Lab

23 Key enabling technology: ability to make nearly ideal measurement of photon number parity (without measuring photon number!)

∞ † Pˆ =(1)− aa=∑ n (1)− n n n=0

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

† Use dispersive ee−itV →−iaπ ee a =+g geePˆ coupling:

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

If we can measure parity, we can perform complete state tomography (measure Wigner function) 24 Wigner Funcon of a Cat State Vlastakis, Kirchmair, et al., Science (2013) Interference fringes prove cat is coherent (even for sizes > 100 photons)

-4 0 4 4 (Yes...this is data.)

0 “P ” even parity 1 vacuum noise ⎣⎦⎡−αα++⎤ -4 2 Rapid parity oscillations With small displacements “X ” 25 Using Schrödinger cat states to store and correct quantum information

26 QUANTUM ERROR CORRECTION Keeping your Cat Alive

Courtesy of Mitra Farmand Encode information in two orthogonal even-parity code “words”

code word Wigner functions: ψψ=+↑↓WW1 ψ 2

W2 W1 =+αα− W1

Wi2 =+αα− i + =

Store a qubit as a superposition of two cats of same parity

Photon loss flips the parity which is the error syndrome we can measure (and repeat hundreds of times).

28 Coherent states are eigenstates of photon destruction operator. a ααα=

Effect of photon loss on code words:

aW1 = a( αα+− ) → ( αα−−) (if α real) 22 aW1 =+ a( αα−) → ( αα+−) = W1

aW2 =+ a( iiαα− i) → i( αα −−i ) 22 2 aW22=+ a( iαα− i) =()i ( i αα +−i ) =− W

After loss of 4 photons cycle repeats: 4 aW(ξξ11++22 W) → ( ξξ1 W 1 2 W2 )

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

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

Wt() = ee−−κκtt/2αα± − /2

Maxwell Demon takes this into account ‘in software.’ 2016: First true Error Correcon Engine that works

Analog Inputs

Analog Outputs

MAXWELL’S DEMON

• Commercial FPGA with custom software A prototype quantum developed at Yale computer being prepared for cooling close to • Single system performs all measurement, absolute zero. 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 31 computer to the controller and back! Implemenng a Full QEC System: Debugger View

(This is all real, raw data.) Ofek, et al., Nature 536, 441–445 (2016). 32 Process Fidelity: Uncorrected Transmon

τ≈15µs

33 System’s Best Component

ψψ==+=g nn01 ψe

τ≈ 290µs

τ≈15µs

34 Process Fidelity: Cats without QEC

α = 2 + =

τ≈ 290µs

τ≈130µs

τ≈15µs

35 Process Fidelity: Cats with QEC

α = 2 QEC – NO POST-SELECTION.

τ≈ 290µs τ≈ 320µs τ≈130µs

τ≈15µs

36 Only High-Confidence Trajectories

Still keep Exclude results with ~80% of α = 2 heralded errors data τ≈ 560µs τ≈ 290µs τ≈ 320µs τ≈130µs

τ≈15µs

37 We are on the way!

“Age of Qu. Error Correcon.”

“Age of Quantum Feedback”

“Age of Measurement”

“Age of Entanglement”

“Age of Coherence”

Achieved goal of reaching “break-even” point for error correcon. Now need to surpass by 10x or more.

M. Devoret and RS, Science (2013) 38 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).

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

40 Cat in Two Boxes Qubit measures joint parity! iπ ()nnˆˆ12+ (two-legged cat only) PP12== 1P 2 e

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

1 Ψ=⎡++±αα−− αα ⎤ ± 2 ⎣⎦ 41 Qubit measures joint parity!

Cat in Two Boxes iπ ()nnˆˆ12+ PP12== 1P 2 e

- Universal controllability Experiment by Yale group: - 3-level qubit can measure Science 352, 1087 (2016)

PP1,P 2 , and 12

42 Theory

Entanglement of two logical cat states

1 Ψ=⎡++±αα−− αα ⎤ ± 2 ⎣⎦

9 sigma violation of Bell inequality

Experiment

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

43 Entanglement of Two Logical Cat-Qubits CHSH: (Milman et al.: evaluate Wigner at 4 points in 4D phase space)

B = W (0 ,0 )+W ( ,0 )+W (0 , ) W ( , ) 1 2 1 2 1 2 1 2

W (10 ,20 ) W (1,20 ) W (10 ,2) W (1,2) B =2.18 0.02 ±

CHSH Bell: 2≤≤B 2 2 44 ↓ → W Encoding qubits in cavity photon states 1

↑ → W2 Minimal encoding W 0 cannot correct 1 =

errors but has W2 = 1 minimal loss rate:

04+ W = aW1 = 2 3 Minimal ‘binomial code’ 1 2 corrects 1 loss: aW2 = 2 1 W2 = 2

General binomial code corrects L losses, G gains and D dephasing events: M. Michael et al., Phys. Rev. X 6, 031006 (2016) ‘New class of error correction codes for a bosonic mode’ 45 Strong Dispersive Hamiltonian ω Haa=++ω††q σzz χσ aaH + χκ>>, Γ r2 damping resonator qubit dispersive coupling

z cavity frequency =+ωr χσ

g ‘strong-dispersive’ limit e 2χκ ~ 2× 103 κ ω g 2 ωχ− χκ~ ? r Δ ωχr + 46 Future work:

Cat pumping, non-linear driving and damping

ρρκαρ&=−iH[,]+ Da (44− )

4-photon damping 4-photon drive

Stabilize a manifold of 4 coherent states to keep cats alive!

Use parity monitoring to correct for single photon loss. 47 Photon number parity ∞ † Pˆ =(1)− aa=∑ n (1)− n n n=0

Remarkably easy to measure using our quantum engineering toolbox

and

Measurement is 99.8% QND 48 Measuring Photon Number Parity

- use quantized light shift of qubit frequency

ωχ+ 2 aa† q σ z 2

σσzz −−i2χπ ntˆˆ i n e 22= e z nˆ =1,3,5,... x nˆ = 0,2,4,...

Gleyzes, S. et al. Nature 446, 297 (2007) Sun, Petrenko et al., Nature 511, 444 (2014) 49 Previous Results: Tracking Photon Number Parity Jumps Pˆ =(1)− nˆ +1 Parity -1

odd even odd even odd even

Syndrome msmt time: As photons leave one by Map: ~ 200 ns one, the sign of the parity Detect: ~ 500 ns changes between +1 and -1 Fidelity: > 98%, QND ~99.8% Can perform 100’s of QND syndrome measurements per jump time Sun, Petrenko et al., Nature 511, 444 (2014). nˆ Using photon number parity Pˆ =(−1) to do cavity state tomography

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

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.

Handy identity (Luterbach and Davidovitch):

ˆ W (βββ))()=ΨDD(+Pˆ − Ψ Pˆ =(− 1)N = parity

52 1 even =+( αα− ) 2 1 (normalization approx. only) odd =( αα−− ) 2

a +++ααα= a even= α odd How cats die: a −ααα=− − a odd= α even

Γϕ =2nκ κ = 2κα | |2 = (4n ) 2

53 Transmon Qubit in 3D Cavity

Josephson junction

~ mm sp in

Spin flip 50 mm r r r 7 x † dE• de=2× 1 mm ≈ 10 Debye!! Vgaadipole =+σ () g = rms h Huge dipole moment: strong coupling g ∼ 100 MHz 54 Information is Physical Store Information in Quantum States?

Quantum bit: single atom? State State “0” “1”

Quantum superposition Principle:

a quantum bit can be “0” and “1” at the same time,

01+ 01−

55 Quantum Computing is a New Paradigm

• Quantum computing: a completely new way to store and process information.

• Superposition: each quantum bit can be BOTH a zero and a one.

• Entanglement: the quantum computer can explore ALL possible outcomes. (Einstein: ‘Spooky action at a distance.’)

• Massive parallelism: computations that are impossible Daily routine engineering and calibration test: Carry out on spooky ANY conventional operations that computer. Einstein said were impossible!

56 The Hardware

57 QEC Setup: 2 Cavies + 1 Transmon Ancilla

8.2 ��� 6.5 ��� �=​�↓� ​�↑† �+​�↓� ​|�⟩⟨�| − ​�↓�� ​�↑† �​|�⟩⟨�| − ​�↓�� (​�↑† �​)↑2

Input Storage Transmon Readout JPC

Input Output To outside Ancilla (transmon) readout fidelity ~ 99.5% in 400 ns Thank you58 QLab The Experiment: Implementing the cat code with superconducting circuits

1. Map qubit state into photonic cat code words 2. Monitor number parity jumps M 3. Conditioned on M, map cavity state back to qubit 4. Perform process tomography to determine fidelity 5. Compare error-corrected logical qubit performance to ‘best component’: (0,1 photon encoding vs. cats)

(Many slides courtesy R. Schoelkopf) 59 60 at the single photon level Large dipole coupling of transmon qubit to cavity permits:

• Quantum engineer’s toolbox to make arbitrary states:

‘Dispersive’ Hamiltonian: qubit detuned from cavity -qubit can only virtually absorb/emit photons

(DOUBLY QND)

††ωq zz Haa=++ω σ χσ aaH + ωωrq≠ r2 damping resonator qubit Dispersive coupling

61 Dispersive Hamiltonian ω Haa=++ω††q σzz χσ aaH + r2 damping resonator qubit dispersive coupling

z cavity frequency =+ωr χσ

g ‘strong-dispersive’ limit e 2χκ ~ 2× 103 κ ω

ωχr − ωχr + 62 Strong-Dispersive Limit yields a powerful toolbox g e Cavity frequency depends on qubit state ω

ωχr − ωχr +

Microwave pulse at this Microwave pulse at this frequency excites cavity frequency excites cavity only if qubit is in only if qubit is in

g Engineer’s tool #1: Dα Conditional displacement of cavity

63 Photon number parity ∞ † Pˆ =(1)− aa=∑ n (1)− n n n=0 Remarkably easy to measure using strong-dispersive interaction between qubit and cavity

† V = 2χ e e a a [2χ : 3,000 (κ ,γ )] (cavity and qubit linewidths) † ee−itV →−iaπ ee a =+g geePˆ

64 Photon number parity ∞ † Pˆ =(1)− aa=∑ n (1)− n n n=0 Remarkably easy to measure using strong-dispersive interaction between qubit and cavity

† V = 2χ e e a a [2χ : 3,000 (κ ,γ )] (cavity and qubit linewidths) † ee−itV →−iaπ ee a =+g geePˆ

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

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

65 Quantum Information Science Control theory, Coding theory, Computational Complexity theory, Networks, Systems, Information theory Ultra-high-speed digital, analog, Programming languages, Compilers, microwave electronics, FPGA Quantum algorithms Electrical Computer Engineering Science QIS Physics Applied Quantum Physics Chemistry , Quantum optics, Circuit QED, Materials Science

quantuminstitute.yale.edu 66 Quantum Error Correction

‘Logical’ qubit Cold bath

Entropy

‘Physical’ qubits ‘Physical’ Maxwell N Demon

QEC is an emergent collective phenomenon: adding N-1 worse qubits to the 1 best qubit gives an improvement! 67