Schrödinger Cats, Maxwell s Demon and Quantum Error Correction

Transcription

1 Schrödinger Cats, Maxwell s Demon and Quantum Error Correction Experiment Michel Devoret Luigi Frunzio Rob Schoelkopf Andrei Petrenko Nissim Ofek Reinier Heeres Philip Reinhold Yehan Liu Zaki Leghtas Brian Vlastakis +.. QuantumInstitute.yale.edu Theory SMG Liang Jiang Leonid Glazman M. Mirrahimi ** Shruti Puri Yaxing Zhang Victor Albert** Kjungjoo Noh** Richard Brierley Claudia De Grandi Zaki Leghtas Juha Salmilehto Matti Silveri Uri Vool Huaixui Zheng Marios Michael +.. 1

2 In the first lecture we learned: An arbitrary quantum channel (CPTP map) is the most general possible operation on a quantum system. Therefore if quantum error correction is possible, it can be performed via a quantum channel. ρʹ = ρ sys sys = d 2 2 k = 1 E k ρ sys E k error map d Rkρʹsys Rk recovery map k = 1 Under what conditions is recovery possible? 2

3 Let the system ( sys ) be N physical qubits. A logical qubit encoded in sys consists of two orthogonal words in the Hilbert of sys Knill-Laflamme condition A recovery map for a set of errors exists if where α Equivalently { W0 W1 } code = span, P = W W + W W code P E E P = α P code i j code ij code is a Hermitian matrix. µ i j ν ij µν { E E E },,, N 1 2 where α is state-independent (i.e. ind ependent o f µ, ν). ij W E E W = αδ In learning the error, we learn nothing about the stored information. 3

4 Knill-Laflamme condition A recovery map for a set of errors exists if where Proof: α P E E P = α P code i j code ij code is a Hermitian matrix. { E E E },,, N 1 2 Let SαS = d diagonalize α. Let K = SE. P KKP = dp code i j code ij code Different error states K j W µ are orthogonal and hence identifiable by measurement of the projector (which does not tell us about the stored quantum information) K P K Π Π = Π j code j 2 j =, ( j) j d jj 4

5 Given knowledge of which error occurred, there exists a unitary map from the error state back to the original state in the code space. Errors can be non-unitary (increase entropy) But Knill-Laflamme condition says we can correct them with a unitary, if the choice of unitary is conditioned on measurement result. K P K Π Π = Π j code j 2 j =, ( j) j d jj 5

6 Next up: Quantum Error Correction Codes for Bosonic Modes (microwave photons) Photons are excitations of harmonic oscillators (e.g. one mode of a microwave resonator) We will use superpositions of photon Fock states (number states) as quantum code words. Example: Binomial Code (aka kitten code ) Code states Error states W W 0 1 = 2 1 = aw 0 = 2 1 aw1 = 2 3 M. Michael et al., PRX 6, (2016) 6

7 H = ω nˆ + nˆ = a d nˆ dt H 0 damping a = κ nˆ Experimentally realistic error model: 1. Amplitude damping of harmonic oscillator 2. No intrinsic dephasing (frequency noise) Quantum channel for damped oscillator: 0 photon loss 1 photon loss ρ( t+ δt) = E ρ(0) E + E ρ(0) E + E E 1 = κδ ta E 1 1 = κδ t nˆ Sanity check: probability of 1 photon loss: Tr{ E ρ } p = E t n ˆ = κδ 7

8 Quantum channel for damped oscillator: 0 photon loss 1 photon loss ρ( t+ δt) = E ρ(0) E + E ρ(0) E + E E 1 = κδ ta E 1 1 = κδ t nˆ Sanity check: probability of 1 photon loss: p = Tr E ρe = κδt n { } ˆ What is Guess: E0? Completeness says: E0 E0 = 1 E1 E κ E0 = U 1 E1 E1 1 δ t a a; U = I E ˆ 0 exp t n 2 1 Exact answer: κ = δ 2 1 ρ( t δt) ρ( t) κδt Lindblad master + = + aρ( t) a [ a aρ() t + ρ() t a a] 2 equation 8

9 0 photon loss 1 photon loss ρ( t+ δt) = E ρ(0) E + E ρ(0) E [Master Equation] Quantum trajectory interpretation: Toss a coin to randomly select E0ρ(0) E0 ρ( t δt) p + = with probability p { 0 = Tr E0ρ(0) E0} or select 0 E ρ(0) E δt = with probability p1 = Tr { E1ρ (0) E1 } p1 ρ( t ) Each Monte Carlo run simulates an actual experimental history (and gives a pure state). Ensemble averaging gives same result as master equation evolution density matrix becomes impure. 9

10 What if, instead of Monte Carlo data, we had real data? Resonator PMT no click click no click click c E0ρ(0) E0 + δt = with probability p { 0 = Tr E0ρ(0) E0} ρ ( t ) c p 0 E ρ(0) E δt = with probability p1 = Tr { E1ρ (0) E1 } p1 ρ ( t ) The measurement record tells us exactly the state of the environment. Hence the resonator cannot be entangled with it. Conditional density matrix ρ ( t+ δt) c must be in a pure state(!) 10

11 What if, instead of Monte Carlo data, we had real data? no click click Resonator PMT no click click c E0ρ(0) E0 + δt = with probability p { 0 = Tr E0ρ(0) E0} ρ ( t ) c p 0 E ρ(0) E δt = with probability p1 = Tr { E1ρ (0) E1 } p1 ρ ( t ) Conditional density matrix is pure: ρc( t+ δt) = ψc ψc If detector clicks: ψ c E ψ aψ = = ψ0 E1E1 ψ0 ψ0 a aψ0 11

12 But what happens if the detector does not click? κ E ˆ 0 ψ δ tn 0 2 ψ c = ; E 0 = e ψ E E ψ Example 1 (Fock state): ψ 0 = n (eigenstate of E0) ψ = n (no click) ψ = n 1 (click) Example 2: ψ0 = ; ψ ˆ 0 n ψ0 = 1 Click: ψ = 99 ψ nˆ ψ = 99 (!!!) c c c No click: κ ψc exp δ t State changes even though no click!! ψ c n ψ c <1. 12

13 But what happens if the detector does not click? Sir Arthur Conan Doyle: Silver Blaze Scotland Yard Detective: Is there any other point to which you wish to draw my attention? Sherlock Holmes: To the curious incident of the dog during the night. Detective: The dog did nothing in the night-time. Holmes: That was the curious incident. The quantum state changes even if the dog does not bark. 13

14 If we keep track of when the dog barks and doesn t bark, the system state remains pure. We will use this for QEC. One scheme involves some remarkable properties of coherent states and Schrödinger cat states. (Mazyar Mirrahimi) 14

15 Quick review of microwave resonators and photonic states 15

16 Coherent state is closest thing to a classical sinusoidal RF signal ψ( Φ ) = ψ ( Φ α) 0 16

17 Experimental setup: superconducting qubit (artificial twolevel atom) in a superconducting microwave resonator Microwave resonator microwave drive artificial atom with states g, e H = ω a a + ω e e + 2χa a e e r q resonator qubit dispersive coupling 17

18 Photon number distribution in a coherent state (measured via quantized light shift of qubit transition frequency) V = 2χ e e a a χ : 3,000 (, [ 2 κ γ )] N.B. power broadened 100X χ New low-noise Microwaves way to do axion are dark particles! matter detection? (arxiv: )

19 We will use Schrödinger cat states of cavity photons (normalization is only approximate) 19

20 Parity of Cat States P = e iπaa = ( 1) n P = p n ( 1) n Coherent state: Mean photon number: 4 Even parity cat state: ψ n = α = 2 ψ = α + α Readout signal 10 8 Photon number Only photon numbers: 0, 2, 4, ˆP ψ =+ ψ Odd parity cat state: Only photon numbers: 1, 3, 5, ˆP ψ = ψ ψ = α α Spectroscopy frequency (GHz) Schoelkopf Lab 20

21 Key enabling technology: ability to make nearly ideal measurement of photon number parity (without measuring photon number!) ˆ aa ( 1) = n ( 1) P = n= 0 n n We learn whether n is even or odd without learning the value of n. Use dispersive coupling: e e a a ˆ ivt iπ e e = g g + e e P 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) 21

22 Wigner Function of a Cat State Vlastakis, Kirchmair, et al., Science (2013) Interference fringes prove cat is coherent (even for sizes > 100 photons) (Yes...this is data.) P 0-4 even parity 1 2 α + + α X vacuum noise Rapid parity oscillations With small displacements 22

23 Using Schrödinger cat states to store and correct quantum information 23

24 The magic of coherent states: Lose a photon and stay in same state! The damped oscillator does NOT entangle with the environment and stays in a pure state! 1 2 n α α 2 α = exp( a a) 0 = e n n= 0 n! a α = α α E α α E p = α α 2 α nˆ α = α but a a α α* α α κ κδ t t ˆ 2 Nevertheless E0 α = exp{ n} α = Z() t e α 2 Hence, damping comes not from loss of photons but from the no click events (when the dog does not bark)! 24

25 Driven damped oscillator: ( ) V = i[ú^ a Úa d dt 1 ρ() t = iv [, ρ()] t + κ aρ() t a [ a aρ() t + ρ() t a 2 a] Lindblad master equation Define: b 2 = a Ú κ d 1 dt ρ κ ρ 2 ρ ρ 2Ú ρ( ) α α, α κ () t = b () t b [ b b () t + ( t) b b] Energy is being stochastically dumped into the bath, yet the dissipation stabilizes a pure state (coherent state)! 25

26 Can we extend this idea to use dissipation to autonomously stabilize a manifold of 2 states? Two-photon pumping: Two-photon damping: V = iλ a a E2 = κδ 2 t a a 2 2 Bath engineering b= a α 2 2 ( ) d dt 1 ρ κ ρ ρ ρ 2 () t = b () t b [ b b () t + () t b b] Stable manifold: (can be used as a qubit) span + α, α { } 26

27 Devoret group: arxiv: Stable manifold using two-photon pumping and two-photon damping 1 even parity 2 { + α + α } Expt l Wigner functions of cat states 1 2 { + α + i α } 1 odd parity 2 { + α α } Problem: still suffers from occasional single-photon loss. 1 2 { + α i α } 27

28 Recall that if we can keep track of the state of the bath the system state remains pure. Resonator PMT no click click no click click c E0ρ(0) E0 + δt = with probability p { 0 = Tr E0ρ(0) E0} ρ ( t ) c p 0 E ρ(0) E δt = with probability p1 = Tr { E1ρ (0) E1 } p1 ρ ( t ) The measurement record tells us exactly the state of the environment. Hence the resonator cannot be entangled with it. Conditional density matrix ρ ( t+ δt) c must be in a pure state(!) 28

29 Problem: We don t actually have a PMT for microwave photons sitting outside the cavity. (Also some of the photons are lost internally.) Solution: We are able to measure the photon number parity. We said before that photon loss leaves coherent state invariant, but actually there is an important phase: a + α =+ α + α, a α = α α This is necessary to achieve: even odd a α α 2 α α { + + } = { + } code word error word 29

30 even odd a α α 2 α α { + + } = { + } code word error word We therefore need a second code word with even parity. 30

31 Encode information in two (nearly) orthogonal even-parity code words ψ = ψ W + ψ W 1 2 code word Wigner functions: W1 = α + α W = iα + iα 2 W 1 W 2 + = 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). 31

32 Coherent states are eigenstates of photon destruction operator. a α = α α Effect of photon loss on code words: ( + ) ( ) aw a α α α α α 1 = (if r ( α α ) ( α α ) 1 a W = a + + = W 2 ( iα α ) ( iα iα ) aw = a + i i ( ) 2 iα α () i ( iα iα ) After loss of 4 photons cycle repeats: ( ξ + ξ ) ( ξ + ξ ) 4 a 1 W1 2 W2 1 W1 2 W2 eal) a W = a + i = + = W We can recover the state if we know: N mod 4 (via monitoring parity jumps) Loss

33 We can recover the state if we know: N mod 4 (via monitoring parity jumps) Loss Amplitude damping is deterministic (independent of the number of parity jumps!) Wt () = e α ± e κt/2 κt/2 α Maxwell Demon takes this into account in software. (We don t yet have 4-photon pumping/damping working.)

34 QUANTUM ERROR CORRECTION Keeping your Cat Alive Courtesy of Mitra Farmand

35 2016: First true Error Correction Engine that works Analog Inputs Analog Outputs MAXWELL S DEMON 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 A prototype quantum computer being prepared for cooling close to absolute zero. 35

36 Implementing a Full QEC System: Debugger View (This is all real, raw data.) Ofek, et al., Nature 536, (2016). 36

37 Process Fidelity: Uncorrected Transmon τ 15µ s 37

38 System s Best Component ψ = ψ n= g 0 + ψ n= e 1 τ 290µ s τ 15µ s 38

39 Process Fidelity: Cats without QEC α = 2 + = τ 290µ s τ 130µ s τ 15µ s 39

40 Process Fidelity: Cats with QEC α = 2 QEC NO POST-SELECTION. τ 290µ s τ 320µ s τ 130µ s τ 15µ s 40

41 Only High-Confidence Trajectories α = 2 Exclude results with heralded errors Still keep ~80% of data τ 560µ s τ 290µ s τ 320µ s τ 130µ s τ 15µ s 41

42 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. Now need to surpass by 10x or more. M. Devoret and RS, Science (2013) 42

43 Experiment: Extending the lifetime of a quantum bit with error correction in superconducting circuits, Ofek, et al., Nature 536, (2016). Theory: cat codes Leghtas, Mirrahimi, et al., PRL 111, (2013). kitten codes M. Michael et al., Phys. Rev. X 6, (2016). New class of error correction codes for a bosonic mode

44 Thanks for listening! QuantumInstitute.yale.edu 44

45 Extra slides 45

46 Storing information in quantum states sounds great, but how on earth do you build a quantum computer? 46

47 ATOM vs CIRCUIT Hydrogen atom Superconducting circuit oscillator L C 7 10 mm (Not to scale!) mm 12 1 electron ~ 10 electrons Artificial atom 47

48 How to Build a Qubit with an Artificial Atom T = 0.01 K Superconducting integrated circuits are a promising technology for scalable quantum computing Aluminum/AlOx/Aluminum 200 nm AlOx tunnel barrier Josephson junction: The transistor of quantum computing Provides anharmonic energy level structure 48

49 Transmon Qubit ω ω Antenna pads are capacitor plates Energy ω 12 ω 01 1 = e 0 = g ~1 mm Josephson tunnel junction ω 01 ~5 10GHz Superconductivity gaps out single-particle excitations Quantized energy level spectrum is simpler than hydrogen Quality factor mobile electrons Q= ωt 1 comparable to that of hydrogen 1s-2p Enormous transition dipole moment 49

50 The first electronic quantum processor (2009) Executed Deutsch-Josza and Grover search algorithms Lithographically produced integrated circuit with semiconductors replaced by superconductors. Michel Devoret Rob Schoelkopf DiCarlo et al., Nature 460, 240 (2009) 50

51 Scale then correct Correct then scale Surface/Toric Code (readout wires not shown) Large, complex: o Non-universal (Clifford gates only) o Measurement via many wires o Difficult process tomography Large part count Fixed encoding Cat Code Photonic Qubit hardware shortcut (readout wire shown) Precision: o Universal control (all possible gates) o Measurement via single wire o Easy process tomography o Long-lived cavities o Fault-tolerant QEC Reduced part count Flexible encoding 51

52 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 2 = 40 s. [O Brien et al. arxiv: ] T µ However to date, good repeatable (QND) (weight 4) error syndrome measurements have not been demonstrated [Takita et al., Phys. Rev. Lett. 117, (2016)] Scale up and then error correct seems almost hopeless. We need a simpler and better idea... Error correct and then scale up! 52

53 Hardware-Efficienct Bosonic Encoding Leghtas, Mirrahimi, et al., PRL 111, (2013). Replace Logical qubit with this: Ancilla Readout High-Q (memory) N Physical qubits Cavity has longer lifetime (~ms) Large Hilbert space Single dominant error channel photon loss: Γ=κ ˆn Single readout channel earlier ideas: Gottesman, Kitaev & Preskill, PRA 64, (2001) Chuang, Leung, Yamamoto, PRA 56, 1114 (1997) 53

54 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 control to make any code word states we want. Readout Ancilla qubit High-Q (memory) 54