Parity-Protected Josephson Qubits

Transcription

1 Parity-Protected Josephson Qubits Matthew Bell 1,2, Wenyuan Zhang 1, Lev Ioffe 1,3, and Michael Gershenson 1 1 Department of Physics and Astronomy, Rutgers University, New Jersey 2 Department of Electrical Engineering, University of Massachusetts, Boston 3 LPTHE, CNRS UMR 7589, 4 place Jussieu, Paris, France CUNY, Sept. 29, 2016

2 Outline Qubits: state of the art Idea of Parity-Based Protection Charge-Pairing Qubits Flux-Pairing Qubits Superinductors 2

3 Main Idea Physical (faulty) qubit: a two-level quantum system that satisfies DiVincenzo criteria. Logical (fault-tolerant) qubit: a collection of N physical qubits that can correct for arbitrary errors in a single qubit by running error correction codes. error correction codes faulty physical qubits fault-tolerant logical qubit The fault tolerance theorem: once a sufficient low error rate of physical qubits is attained, there is a strategy for correcting errors so that one can carry out indefinitely long computations. 3

4 Physical Qubits: error rate The error rate : τ 0 ε τ d τ 0 - the longest time for oneand two-qubit gates τ d min, ( ) T1 T ϕ - the coherence time Threshold values of ε for different ECCs: Earlier codes (Steane, Bacon-Shor): ε 11 5 Surface correction code (Kitaev et al.): ε 11 2

5 Physical Qubits: speed Van Meter & Horsman, Communications of the ACM 56, 84 (2013) QC speed below 1MHz is not practical.

6 Qubit Implementations (Isolation Speed) Trapped 43 Ca + ions Single-shot readout fidelity 99.93% Single-qubit gate fidelity % Harty et al., PRL 113, (2014) Trapped ions, neutral atoms, quantum dots, superconducting qubits, etc. Quantum dots Single-qubit gate fidelity > 99.5% Kim et. al., Nat.QI 1, (2015) Single-qubit gate fidelity > 99.95% Two-qubit gate fidelity 99.5% Trapped Cs atoms Single-qubit gate fidelity 99.83% Xia et al., PRL 114, (2015) State-of-the-art: Gambetta et al., arxiv: Superconducting qubits

7 Figure of Merit = 1/ε 7

8 Superconducting Qubits: State of the Art Single-qubit gates: τ 0 ε ~ 10 T 2 4 Two-qubit gates: T 2 40 µs τ 0 40 ns Martinis Group, UCSB/Google Devoret & Schoelkopf, Science 2013 ε =

9 Physical Qubit vs. Logical Qubit How many physical qubits does it take to build a single logical qubit? The number of physical qubits that form a single logical qubit ε tt ε The threshold error rate for the surface error codes is ~1%. If the error rate is 1/10 of ε tt, «a reasonably fault-tolerant logical qubit... takes physical qubits». Physical qubits are macroscopic ( 100 μμ), areal density ~10 6 times less than computer chips - enormous impact on the large scale architecture. Martinis (UCSB and Google): We re somewhere between the invention of the transistor and the invention of the integrated circuit not yet 9

10 Main Message Simple design of physical qubit (e.g. transmon) does not necessarily eliminate the enormous complexity of logical qubits. Any design that may help to reduce the complexity of the final product (i.e. the logical qubit) is worth considering. 10

11 Classical Dynamics: JC Model and Washboard Potential I = I C sin φ dd dd + I C sin φ = I ħc 2e d 2 φ dt 2 + I C sin φ = I V = ħ 2e dφ dd ω p = 2eI C ħc = 1 L J C - Josephson plasma frequency E J = ħi C 2e small oscillations at ω p If time is normalized to 1/ω J φ + sin φ = I I C I = 0 Motion of a particle with coordinate ϕ and mass C in a washboard potential: 0 < I < I C U φ = E J 1 cos φ I I C φ 11

12 Josephson Junction as a Non-Linear Oscillator The Josephson junction is a non-dissipative (at T 0) nonlinear inductor shunted by a capacitor a non-linear non-dissipative oscillator. Josephson energy E J = ħi C 2e = Φ 0 2π 2 1 Charging E L J energy C = 2e 2 2C J E = E J ccc φ Mag. flux quantum Φ 0 = h 2e 20G μμ2 ω P = 1 L J C = 2E JE C ħ Josephson junction impedance: the Josephson plasma frequency (typically ~80GGG~4K) - depends on oxide transparency, not on JJ area Z J = L J C 1kΩ 2E C E J - tunable, E J /E C ~ (JJ area) 2

13 Josephson Junction as a Non-Linear Oscillator Josephson junctions: non-dissipative inductors, their non-linearity allows addressing only the quantum states involved in computation without exciting the rest of the spectrum. Josephson energy E J = ħi C 2e = Φ 0 2π 2 1 Charging E L J energy C = 2e 2 2C J L = ħ 2ee C cos φ = L J cos φ Estimates for Small Junctions Area: 0.06 µm 2 Normal-state resistance: 7 kω E J 1K E C 1.2K Micro energy controls a macro system!

14 Qubits: state of the art Idea of Parity-Based Protection Charge-Pairing Qubits Flux-Pairing Qubits Superinductors 14

15 Parity Protected Qubits The goal: to engineer two quantum states indistinguishable by the environment. H = K X n + X n + V k 2 g the number of discrete components of the lowest-energy even (odd) states k discrete variable n=2 - parity protection X ±2 k = k ± 2 The two-level approximation Hamiltonian H = E 01 2 σ z + E 2 σ x the rate of parity violation Two almost degenerate low-energy states: E 01 eee g 15

16 Parity Protected Qubits (cont d) g the number of discrete components of the lowest-energy even (odd) states Decay is suppressed if parity is protected (E = 0). Coupling to k noises is suppressed if the envelope decays slowly (large g). Some fault tolerant rotations can be realized by fast changes of V(t) protection from the gate pulse noise. T 1 = llll T 2 16

17 Charge Pairing Qubit Correlated tunneling of TWO Cooper pairs Discrete variable: number of Cooper pairs on the central island Parity protection (cancellation of single-particle tunneling) due to destructive AB interference Aharonov-Bohm phase Y. Aharonov & D. Bohm, PR 115, 485 (1959) The wave function of a charge q moving around a magnetic flux Φ B acquires a topological phase shift Gladchenko et al. Nat. Phys. 5, 48 (2009) Bell et al. PRL 112, (2014) φ AA = q ħ A dd = 2e ħ Φ B

18 Flux(on) Pairing Qubit Correlated tunneling of TWO fluxons Discrete variable: number of flux quanta in the loop Parity protection (cancellation of single-particle tunneling) due to destructive AC interference Bell et al. PRL 109, (2012) Bell et al. PRL 116, (2016) Aharonov-Casher phase Y. Aharonov & A. Casher, PRL 53, 319 (1984) The wave function of a particle with magnetic dipole moment µ moving in 2D around a line charge acquires a topological phase shift proportional to the line charge density. 18

19 The toolbox for protected qubits Novel Josephson elements: ccc 2φ ccc φ/2 Superinductors 19

20 Qubits: state of the art Idea of Parity-Based Protection Charge-Pairing Qubits Flux-Pairing Qubits Superinductors 20

21 Charge-Pairing Qubits (discrete variable - Cooper pairs number) Ioffe, Doucot, Feigel man et. al. (2002 present) H = 2E 2 cos 2φ 2E 1 cos φ + E C n n g 2 The two-level approximation Hamiltonian H = E 01 2 σ z + E 2 σ x Perfect symmetry (E 1 = 0): E 01 ω p eee g cos πn g g = 4 E 2 /E C ω p = 4 E 2 E C - plasma frequency Required: E 2 E C 21

22 Decoupling from noises Perfect symmetry (E 1 = 0): Energy relaxation is suppressed (very long T 1 ) Coupling to the charge noise is exp. small eee g sss πn g Parity violation: single Cooper pair hopping δe 1 (t) 0. Sources of parity violation: rhombus asymmetry, flux noise. To suppress coupling to the flux noise, the rhombi chain should be longer. 22

23 Fabrication Multi-angle deposition of Al through a shadow mask. 23

24 Microwave Measurements ω 1 ω 2 ω 1 ω 1 ω 2 super-l CPB the device is coupled to the read-out LC resonator Multiplexing: several devices with systematically varied parameters. 24

25 Device and Readout Small JJ: ~ μμ 2 Large JJ: ~ μμ 2 Global magnetic field: Φ R Φ 0 /2 φ = 2π Φ Φ 0 E phase difference across the chain Two experimental knobs : - the gate voltage controls n g on the central island; - the flux in the phase loop controls φ across the chain. 25

26 Time-Domain Measurements Optimal regime: max E 2, min E 1 φ = nn ( min E 01 ) Q ω 00 T Factor-of-100 suppression of energy relaxation M. Bell et al. PRL 112, (2014) 26

27 Summary The toolbox for protected qubits: ccc 2φ, ccc φ/2, super-l Charge- (flux-) pairing qubits offer the possibility of coherence protection and fault-tolerant operations. Observed: - suppression of energy relaxation in a minimalistic rhombi chain; - spectroscopic evidence of Aharonov-Casher effect in flux-pairing devices. Current work: - optimization of the parameters of the flux-pairing qubits (smaller E L and larger E dps ); - development of better superinductors (for applied projects as well as a novel tool to study 1D QPTs). 27