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Introduction to Quantum Error Correction

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Introduction to Quantum Error Correction 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 2 αa n =||α = ee0 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.
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