Schrödinger Cats, Maxwell’s Demon and Quantum Error Correction
Experiment Theory Michel Devoret 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 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) 2 Full Steane Code – Arbitrary Errors
Single round of error correction
6 ancillae
7 qubits All previous attempts to overcome the factor of N and reach the ‘break even’ point of QEC have failed.
Current industrial approach (IBM, Google, Intel, Rigetti):
Scale up, then error correct
• Large, complex: o Non-universal (Clifford gates only) o Measurement via many wires o Difficult process tomography • Large part count • Fixed encoding ‘Surface Code’ (readout wires not shown)
O’Brien et al. arXiv:1703.04136 predict ‘break-even’
will be difficult even at the 50 qubit scale. 4 All previous attempts to overcome the factor of N and reach the ‘break even’ point of QEC have failed.
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.
5 Scale then correct Correct then scale
Surface Code Cat Code Photonic Qubit (readout wires not shown) hardware shortcut (readout wire shown) • Large, complex: • Precision: o Non-universal (Clifford o Universal control gates only) (all possible gates) o Measurement via o Measurement via many wires single wire o Difficult process o Easy process tomography tomography • Large part count o Long-lived cavities • Fixed encoding o Fault-tolerant QEC • Reduced part count
• Flexible encoding 6 “Hardware-Efficienct Bosonic Encoding” Leghtas, Mirrahimi, et al., PRL 111, 120501(2013). High-Q Replace ‘Logical’ qubit with this: (memory) Ancilla Readout
• Cavity has long lifetime (~ms) • Single dominant error channel ‘Physical’ qubits photon loss: Γ=κ nˆ N makes QEC easier
earlier ideas: Gottesman, Kitaev & Preskill, PRA 64, 012310 (2001) Chuang, Leung, Yamamoto, PRA 56, 1114 (1997) 7 Photonic Code States
Can we find novel (multi-photon) code words that can store quantum information even if some photons are lost?
Ancilla transmon coupled to resonator gives us universal High-Q (memory) control to make ‘any’ code Ancilla word states we want. Readout
| Ψ〉 =ψψ0|0 L 〉 + 1L |1 〉
Logical code words quantum (superpositions of information photon Fock states) 8 Encoding qubits in cavity photon states
Minimal encoding 00= 0 photons cannot correct L = errors but has 11L 1 photon minimal loss rate: nn = −κ 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. 9 Quick review of microwave resonators and photonic states
10 Coherent state α is closest thing to a classical sinusoidal RF signal
ψ()Φ ≡〈Φ | αψ 〉 =0 ( Φ − α )
Can displace in both position and momentum αα=||eiθ
11 Coherent state = displaced vacuum Poisson distribution of photon number † α = e[ααaa− *] 0 Pn 1 2 − ||α † 2 = ee2 αa 0 n =||α
1 ∞ n − ||α 2 α = en2 ∑ n=0 n! n
This is the only kind of state that can be created with a classical drive applied to a cavity.
12 Dispersive coupling of two-level ancilla to high-Q cavity yields universal control
High-Q cavity (memory) Ancilla Readout Cavity and ancilla are detuned
ωc ≠ ωq
High-Q cavity (memory) Ancilla ω H =++ω aa††q σ zzχ σ aa [2χ 3,000 (κ ,γ )] c 2
13 Quantum optics at the single-photon level
• Universal control enables: photon state engineering
Goal: arbitrary photon Fock state superpositions
ψ =a001 ++ aa 12 2 + a 3 3 +
Use the coupling between the cavity (harmonic oscillator) and the two-level qubit (anharmonic oscillator) to achieve this goal.
14 Previous State of the Art for Complex Oscillator States Expt’l. Wigner tomography: Leibfried et al., 1996 ion traps (NIST – Wineland group)
Rydberg atom cavity QED Phase qubit circuit QED Haroche/Raimond, 2008 Rydberg (ENS) Hofheinz et al., 2009 (UCSB – Martinis/Cleland) ~ 10 photons ~ 10 photons
Q Φ Dispersive Hamiltonian ω H =++ω aa††q σ zzχ σ aa c 2 resonator qubit dispersive coupling
z cavity frequency =ωr + χσ
g ‘strong-dispersive’ limit e 2χκ ~ 2× 103 κ ω
ωχr − ωχr + 16 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 ground state only if qubit is in excited state
g Engineer’s tool #1: Dα Conditional displacement of cavity
17 ω H =++ω aa††q σ zzχ σ aa c 2 resonator qubit dispersive coupling
Reinterpret dispersive term: - quantized light shift of qubit frequency
ωχ+ 2 aa† q σ z 2
18 Microwave photon number distribution in a coherent state (measured via quantized light shift of qubit transition frequency) ω H = ω aa††+ q σ z + χ σ z aa [2χ 3,000 (κ ,γ )] c 2
… N.B. power broadened 100X 2χ
New low-noise way to do axion dark matter detection? (arXiv:1607.02529) Microwaves are particles! π Engineer’s tool #2: n Conditional flip of qubit if exactly n photons
ω H =++ω aa††q σ zzχ σ aa c 2 resonator qubit dispersive coupling
Reinterpret dispersive term: - quantized light shift of qubit frequency
ωχ+ 2 aa† q σ z 2 20 strong dispersive coupling
† z VDISPERSIVE ≈ χσ aa
Qubit Spectroscopy
Coherent state in the cavity
2χ Conditional bit flip π n
21 Strong Dispersive Coupling Gives Powerful Tool Set
Cavity-conditioned bit flip π n g Qubit-conditioned cavity displacement Dα
• multi-qubit geometric entangling phase gates (Paik et al.) • Schrödinger cats are now ‘easy’ (Kirchmair et al.)
Photon Schrödinger cat states on demand experiment theory G. Kirchmair M. Mirrahimi B. Vlastakis Z. Leghtas
22 Paradoxically, we will use code words made of ‘delicate’ Schrödinger cat states of cavity photons
(normalization is only approximate) 23 Parity of Cat States
Photon number
10 108 9 86 7 64 5 42 3 02 1 0 Coherent state: ψ =α = 2 Mean photon number: 4
Readoutsignal 4 2 Even parity cat state: ψ =αα +− 6 8 Only photon numbers: 0, 2, 4, … 0 Pˆ ψψ= + 5 3 7 1 9 Odd parity cat state: ψ =αα −−
Only photon numbers: 1, 3, 5, … Spectroscopy frequency (GHz) ˆ P ψψ= − Schoelkopf Lab
24 Key enabling technology: ability to make nearly ideal measurement of photon number parity (without measuring photon number!) Photon number
Coherent state
∞ † Odd cat Pˆ =( − 1) aa =∑ n ( − 1) n n = Readoutsignal n 0 Even cat
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 complete state tomography (measure Wigner function) 25 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) 99.8% QND(!) 26 nˆ Using photon number parity Pˆ =( −1) to do cavity state tomography
Wigner function: --quasi-probability distribution in phase space equivalent to the full density matrix for state tomography.
27 Wigner Function = “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 (β)=Ψ+DD( ββ )Pˆ () −Ψ Pˆ =−=( 1)N parity
28 State Tomography: Wigner Function 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 Fringes prove this is a coherent cat, not a mixture “X ” 29 Using Schrödinger cat states to store and correct quantum information
Courtesy of Mitra Farmand 30 Encode information in two orthogonal even-parity logical code “words”
code word Wigner functions: Ψ =ψψ0101LL + 1 L X L 0L =αα +− 0L
1L =iiαα +−
a αα= α Store a qubit as a Magic property: coherent states superposition are invariant under photon loss! of two cats of same parity Pˆ |Ψ〉 = ( + 1) | Ψ〉 a 0L =α { αα −− }
Photon loss flips the parity which is the error syndrome we can measure (99.8% QND). 31 Coherent states are eigenstates of photon destruction operator. a α= αα
Effect of photon loss on code words:
aa0L =( αα +−) → ( αα −− ) (if α real)
22 aa00LL=( αα +− ) → ( αα+ −) =
aa1L =( iiαα +− i) → i( αα−−i )
22 2 a 1LL=a( iαα +−i ) =() ii( i αα +−) =−1
After loss of 4 photons cycle repeats: 4 a (ψψ001 L++ 1L) → ( ψψ 0 01 L 1L)
We can recover the state if we know: (via monitoring parity jumps) NLoss mod 4 2016: First true Error Correction Engine that works
Analog Inputs
Analog Outputs
MAXWELL’S DEMON A prototype quantum • Commercial FPGA with custom software computer being prepared developed at Yale 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 Schoelkopf-Devoret lab computer to the controller and back! 33 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 Process Fidelity: Uncorrected Transmon
τµ≈15 s
36 System’s Best Component
ψψ=g nn =+=01 ψe
τµ≈ 290 s
τµ≈15 s
37 Process Fidelity: Cats without QEC
α = 2 + =
τµ≈ 290 s
τµ≈130 s
τµ≈15 s
38 Process Fidelity: Cats with QEC
α = 2 QEC – NO POST-SELECTION.
τµ≈ 290 s τµ≈ 320 s τµ≈130 s
τµ≈15 s
39 Only High-Confidence Trajectories
Still keep Exclude results with ~80% of data α = 2 heralded errors τµ≈ 560 s τµ≈ 290 s τµ≈ 320 s τµ≈130 s
τµ≈15 s
40 Cavity is not just a quantum Universal Gate Set on a Logical memory, it is a qubit. Qubit Encoded in an Oscillator (‘cat code’)
Heeres et al., Nature Communications 8, 94 (2017) 41 Encoding qubits in cavity photon states: ‘kitten code’
04+ 0,4 photons 0L = aka ‘binomial code’ 2
1L = 2 2 photons
initial state: ψα= 01LL+ β single photon loss: a ψ=2[ αβ3 + 1]
‘kitten code’ can correct single photon loss
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’ 42 Kitten code QEC: arXiv:1805.09072
43 Kitten code QEC approaching breakeven: arXiv:1805.09072
44 Repeated gates on logical qubit
Logical qubit Ramsey fringes with QEC
Kitten code QEC: arXiv:1805.09072 45 We are on the way!
“Age of Qu. Error Correction.”
“Age of Quantum Feedback”
“Age of Measurement”
“Age of Entanglement”
“Age of Coherence”
Achieved goal of reaching “break-even” point for error correction with cat code and (almost) with kitten code. Now need to surpass by 10x or more.
M. Devoret and RS, Science (2013) 46 ‘Circuit QED’
In (microwave) light there is truth…..
47 Extra Slides
48 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.’ 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 50 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
51 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.
52 Entanglement of Two Logical Cat-Qubits CHSH: (Milman et al.: evaluate Wigner at 4 points in 4D phase space)
CHSH Bell: 2≤≤B 2 2 53