Coupled qubits. Frank K. Wilhelm Department Physik, Center for Nanoscience, and Arnold Sommerfeld Center for theoretical physics

Transcription

1 Coupled qubits Frank K. Wilhelm Department Physik, Center for Nanoscience, and Arnold Sommerfeld Center for theoretical physics Ludwig-Maximilians-Universität München

2 Contents Coupling schemes Efficient gates Decoherence And protection

3 Natural interaction Inductive energy M 12 E ind = M 12 I 1 I 2= 1 x,1 2 x,2 L Pseudospin notation 2 H = K 1 z z J. B. Majer, F. G. Paauw, A. C. J. ter Haar, C. J. P. M. Harmans, and J. E. Mooij Phys. Rev. Lett. 94, 951 (25)

4 Tunable interaction Coupling Damping Tunable coupler: DC-SQUID: Cross-talk, unforgiving off -state Π-SQUID (high-tc, FM): Hard to make, extra noise JoFET/SNS-Transistor: Extra noise, slow M.J. Storcz and F.K. Wilhelm, Appl. Phys. Lett. 83, 2389 (23). T. P. Orlando, J. E. Mooij, C. H. van der Wal, L. Levitov, S. Lloyd, and J. J. Mazo, Phys. Rev. B 6, (1999) 2

5 Bipolar dynamical inductance Bias 1 2 I b =2 I c cos sin 2 Screening 1 2 I s = I c sin cos 2 Extra flux at constant bias Increases screening Reduces Ic,eff : Makes SQUID more symmetric Tunable coupling by current pulse B.L.T. Plourde, J. Zhang, K.B. Whaley, F.K.W., T.L. Robertson, T. Hime, S. Linzen, P.A. Reichardt, C.E. Wu, and J. Clarke PRB 7, 1451(R) (24). C. Hilbert and J. Clarke, J. Low Temp. Phys. 61, 237 (1985).

6 Efficient gates Charge Basis H = n1, n2 E ch, n1, n2 n1, n2 n1, n2 E J,1 n1 n1 1 n1 n1 1 1 n 2 E J,2 n2 n2 1 n2 n2 1 1 n 2 Pseudospin Representation E J1 1 H = [ E m 1 2 n g2 2 E c1 1 2 n g1 ] z 1 x E J2 Em 1 [ E m 1 2 n g1 2 E c2 1 2 n g2 ] 1 z 1 x z z Partially controlled Hamiltonian with crosstalk

7 CNOT quantum logic gate Target flips, iff control =1 1 U= Part of the standard universal set of gates 3 CNOTs + single-qubit rots: Arbitrary 2-qubit gate Infidelity 1 F = U U CNOT 2.2 for showing entangled states at output G. Vidal and G.M. Dawson, Phys. Rev. A 69, 131 (24)

8 The challenge of finding the right pulse Unitary gate: Propagator U =T exp i dt ' H t ' Rotating wave case: Exact: For high symmetry: [ H t, H t ' ] = in some reference frame Approximate: For weak driving Area theorem: Gate given by pulse area H U =cos i sin t = dt ' H t ' H Superconducting qubits: No such symmetry, driving of charge/flux Weak driving not desirable for fast gates

9 Textbook solution Look up: Ising coupling Idealized pulse sequence Flux Tunneling Times set by matrix elements Bang bang control Demands: Sudden rectangular pulses + full control

10 Control theory Restrictions: Fixed tunneling Fixed coupling Finite Rise time Theorem: This is sufficient for full control Analogy: Parallel parking Control theory Application of NMR / Femtochemistry / Enginnering method

11 Gradient optimization Trotterize total gate Minimize U = U t i = e U CNOT U T 2=2 2 ℜ tr U Constraint: T U CNOT i t H u1 t i u n t i Maximize ℜ tr U T U CNOT h U t k =ℜ tr { i t k H u1 t k u n t k U t k } Lagrange adjoint system: t = i H u1 t u n t U t Maximization condition { } h H =ℜ tr i U = ui ui Gradient flow using formalized derivatives computed from forward and backward sweep h t k u i t k u i t k ui t k N. Khaneja, T. Reiss, C. Kehlet, T. Schulte-Herbrüggen, and S. J. Glaser, J. Magn. Reson. 172, 296 (25). T. Schulte-Herbrüggen, A. Spörl, N. Khaneja, and S. J. Glaser, e-print: quant-ph/5214 (25).

12 Test object: The NEC CNOT

13 Experimental status Visibility limted by rise times even in theory even with short rise time not at all perfect phase twist not decoherence-limited T. Yamamoto, Y. A. Pashkin, O. Astaviev, Y. Nakamura, and J. S. Tsai, Nature (London) 425, 941 (23).

14 Optimum control result % Fidelity, no phase twist, T=π/K=55 ps<< T2 Relatively smooth: Bandwith limited (7 harmonics) A.K. Spörl, T. Schulte-Herbrüggen, S.J. Glaser, V. Bergholm, M.J. Storcz, J. Ferber, F.K. Wilhelm, quant-ph/5422

15 Optimum solution: Lessons Control Target palindromic pulse: n(t ) = n(t - t ) -1 = U CNOT H is a real matrix and U CNOT (time reversal inv.) see C. Griesinger, C. Gemperle, O. W. Sørensen, and R. R. Ernst, Molec. Phys. 62, 295 (1987). T=π/K=55 ps : tunnelin π pulse with built-in phase gates

16 Leakage Higher charge states -1,2 not infinitely away Include into simulation: F=99.99 % Number may increase by optimizing WITH leakage

17 Evolution of the reduced spins Reduced Bloch Sphere i =tr i

18 Experimental evolution

19 Generation of shaped pulses? Custom pulse generator [Dan van der Weide et al., Wisconsin] Broadband pulse Fourier decomposition Spectral Equalizer Fourier composition Technique similar to Femtochemistry Superposition of Gaussian Pulses: Stable and compatible to filters

20 Pulse shaping complexity Broadband pulse (1 ps) Four-Pole Cryogenic Filter Sample Laplace space: U 2 s =Z 12 s I 1 s 7 filter stages lead to 94% fidelity

21 Rapid Single Flux Quantum On-chip generation, no filtering SFQ-Pulse Complex circuit Pictures: M. Feldman group, Rochester Compatibility needs to be established

22 Pulse w/ tunable coupler Single qubit Entangler Single qubit SU 2 SU 2 SU 4 / SU 2 SU 2 SU 2 SU 2 Pulse shape simple + separated structure (Khaneja Glaser decomposition)

23 In-situ pulse optimization Pulse optimization Modeling stage Sample design Control design Spectroscopy Pulse adjustment (3 s) Experiment stage Fabrication Quantum computing

24 Two qubit decoherence models One of two uncorrelated Baths H = H Q z 1 X B1 1 z X B2 H B1 H B2 e.g. control electronics One common bath H =H Q z 1 1 z X B H B e.g. noisy coupler; long-range correlated electromagnetic noise

25 Basis representation Appropriate basis for this problem: 1 = 11 2 = = 4 = 1 1 Hop (case of equal qubit paramters): K 1 K H = 2 s K s H B K =s = 1 2 s= X 1 X 2 = 1 2 s= X 1 X 2 s= for one common bath Singlet/triplet basis provides convenient representation

26 Bloch-Redfield rates 4 levels 6 dephasing rates 3 relaxation rates One common bath: One rate only from thermal fluctuations

27 Gate Quality Factors 1 U CNOT = Fidelity 1 F= N ρinitial N j U j=1 j CNOT F U CNOT j ρcnot Simulation of CNOT ρinitial Application of CNOT-1 Overlap + Purity + Quantum degree + Entanglement capability Invented: Poyatos, Cirac, Zoller, PRL 97; Previously studied: Thowart, Hänggi, PRA 1

28 Gate performance CPHASE α=1-3 CNOT CPhase more robust single qubit rotation (Hadamard) limiting factor M.J. Storcz und F.K. Wilhelm, Phys. Rev. A 67, (23) F.K. Wilhelm, M.J. Storcz, C.H. Van der Wal, C.J.P.M. Harmans, and J.E. Mooij, Adv. Sol. St. Phys. 43, 763 (23).

29 Deviations from full symmetry 1 2 G= 1 2 K = M.J. Storcz, F. Hellmann, C. Hrlescu, and F.K. Wilhelm, cond-mat/54787.

30 Rate asymmetry K = 2=E s

31 Modest asymmetry 1% asymmetry, different bath coupling angles

32 XY-coupled flux qubits C Or superconducting charge qubits! C C C 4 i H = i=1 i i z i x 3 i 1 i i 1 i=1 K i i x x y y N* Assumptions Switchable coupling Nice to have:tunnel matrix element tunable L.S. Levitov, T.P. Orlando, J.B. Majer, J.E. Mooij, preprint (21); D.V. Averin and C. Bruder, PRL 91, 573 (23) J. Siewert, R. Fazio, G.M. Palma, and E. Sciacca, J. Low Temp. Phys. 118, 795 (2) C

33 Decoherence free subspace (DFS) Hilbert Space RFS: Error operators F =diag DFS: Error operators F = 1 Errors / noise coupling: RFS Collective: 1 z 2 z 3 z 4 z X Two-qubit: z z X Single-qubit: z X DFS :-) Protected :- Not Protected J. Kempe et al., Phys. Rev. A 63 (21); J. Vala and K.B. Whaley, Phys. Rev. A 66 (22) L 11 1L 11 1 L L 11

34 Universal two-qubit encoding Single-qubit operations: exp i x t =exp i H 12 t e i y t =e i z / 4 i x t e e x, y, z i x / 4 exp i z t =exp i 2 z t RFS DFS (in DFS) Controlled phase shift (two-qubit operation): e i S x / 4 ih 23 t / 2 i 2 ih 23 t / 2 x e e e e i S x / S x = 2 S = x x x x x =e 2 i 1 z z (in RFS) J. Kempe et al., Phys. Rev. A 63 (21); J. Vala and K.B. Whaley, Phys. Rev. A 66 (22) L 11 1L 11 1 L L 11

35 Coupler noise Correlated two-qubit noise from coupling elements (1/f-noise) 1 H = j i iz i e i / 2 ix e i / 2 H qq H B 2 s 1 Power spectrum S = J coth =2 T, T of the noise? 2T c Apply standard unitary transformation that maps the Hamiltonian onto 1 2 H = j i iz i ix 1 z z H qq H B 2 but this also affects the spectral function! J 2 2 J = J = s / c Super-Ohmic! Noise from coupling elements: transformed to super-ohmic flux noise! M.J. Storcz et al., cond-mat/4778

36 Elimination of coupling errors

37 Encoded vs. non-encoded 4 physical qubits encode 2 logical qubits CNOT vs. 2 physical qubits = 2 logical qubits (Ohmic Bath) αsq smaller 4 Qubit DFS, Single-Qubit +Two-Qubit errors of equal strength 2 Qubits, no encoding, performs better! But: Temperature dependence for αsq/α2qb!

38 Summary Fixed and tunable Ising coupling Fast HiFi-pulses with low leakage Protection of logic gates by symmetry, even with fabrication jitter DFS encoding against coupling erros

39 Thanks to My bosses past and present G. Schön J.E. Mooij My sponsors DM $ $ J. von Delft My group including alumni/ae J. Ferber H. Gutmann My collaborators M.C. Goorden U. Hartmann K. Jähne M. Storcz

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