Observation of topological phenomena in a programmable lattice of 1800 superconducting qubits

Transcription

1 Observation of topological phenomena in a programmable lattice of 18 superconducting qubits Andrew D. King Qubits North America 218 Nature , 218

2 Interdisciplinary teamwork Theory Simulation QA experiments Processor dev. Fabrication Design 1 / 2

3 Applications of D-Wave processor I Discrete optimization (e.g. SAT problems) I Sampling (e.g. Boltzmann machine) Quantum simulation (e.g. 2D and 3D frustrated magnets) I I I I Using advanced annealing controls Pause, reverse anneal, and quench Performing experiments that were infeasible before 2 / 2

4 The 2D case: Kosterlitz-Thouless phase transition 216 Nobel for theoretical discoveries of topological phase transitions and topological phases of matter Vadim Berezinskii J. Michael Kosterlitz David Thouless Most easily described in 2D XY model. 3 / 2

5 2D XY model Classical 2D spin: XY-Hamiltonian H = J XY i,j Si Sj = J XY cos (θ i θ j ) i,j Ground state: all spins aligned Continuous rotational symmetry: O(2) or U(1) 4 / 2

6 Topological excitations vortex antivortex Defects appear in vortex/antivortex pairs (Stokes Theorem) But when are these pairs tightly bound? Below the KT phase transition T c = J XY π 2 5 / 2

7 Experimental Observation of KT Phase Transition As observed physically in... superfluid 4 He films, 1978 thin film superconductors, 1979 Bose-Einstein condensates, 26 graphene-tin hybrid JJ arrays, / 2

8 Experimental Observation of KT Phase Transition As theorized/simulated in... As observed physically in... superfluid 4 He films, 1978 thin film superconductors, 1979 Bose-Einstein condensates, 26 graphene-tin hybrid JJ arrays, 214 Triangular AFM transverse field Ising model 2L L 6 / 2

9 AFM triangle: Perturbative transverse field Γ Hamiltonian H = i<j J ij σi z σz j Γ σi x i 6-degenerate frustrated ground state Classical E GS = J Quantum E GS = J Γ / E = J E = J E = J 7 / 2

10 AFM triangle: Perturbative transverse field Γ Hamiltonian H = i<j J ij σi z σz j Γ σi x i degenerate frustrated ground state Classical E GS = J Quantum E GS = J Γ E = J E = J E = J Γ Perturbative picture Floppy spins align with transverse field 7 / 2

11 AFM triangle: Perturbative transverse field Γ Hamiltonian H = i<j J ij σi z σz j Γ σi x i 6-degenerate frustrated ground state Classical E GS = J Quantum E GS = J Γ E = J E = J E = J Γ Perturbative picture Floppy spins align with transverse field 7 / 2

12 AFM triangle: Perturbative transverse field Γ Hamiltonian H = i<j J ij σi z σz j Γ σi x i 6-degenerate frustrated ground state Classical E GS = J Quantum E GS = J Γ pseudospin 2 3 E = J E = J E = J Γ Perturbative picture Floppy spins align with transverse field 7 / 2

13 Pseudospin 6 clock states (in perturbative picture) spin pseudospin / 2

14 Pseudospin phase six-state XY model Spin alignment sublattice ordering ψ = Order parameter ψ = average pseudospin Real order parameter m = ψ 9 / 2

15 Pseudospin phase six-state XY model Spin alignment sublattice ordering ψ = Order parameter ψ = average pseudospin Real order parameter m = ψ 9 / 2

16 Pseudospin phase six-state XY model Spin alignment sublattice ordering ψ = Twisting pseudospin phase triangles with no floppy qubit Exactly like the XY model! Except T c depends on Γ. T c = J XY π 2 = Γ π 12(2 3) Only applies in perturbative regime. 9 / 2

17 Triangular AFM phase diagram paramagnetic T/J KT phase ordered Γ/J 1 / 2

18 Triangular AFM phase diagram paramagnetic Ordered KT PM T/J KT phase ordered Γ/J 1 / 2

19 Triangular AFM phase diagram paramagnetic Ordered KT PM T/J KT phase ordered Γ/J Moessner & Sondhi, 21 1 / 2

20 Triangular AFM phase diagram paramagnetic Ordered KT PM T/J KT phase ordered Γ/J Moessner & Sondhi, 21 Isakov & Moessner, 23 1 / 2

21 Demonstration in D-Wave 2Q 5.5 mm L=15 11 / 2

22 Geometrically frustrated lattices Fully-frustrated square-octagonal triangular AFM Same low-energy theory Statistically very different 2L L AFM couplers have J ij = 1 FM couplers have J ij = / 2

23 Pseudospin field Periodic ψ j vortex Open antivortex Open vortex Periodic Classical output state from DW2Q 13 / 2

24 QA schedule and phase diagram mk Upper transition s = 4 T/J s = 5 s =.3 KT PM Ordered Γ/J Energy scale (unitless) 3 2 Energy s 1 Γ(s)/T J(s)/T Annealing parameter s 14 / 2

25 QA schedule and phase diagram mk Upper transition s = 4 T/J s = 5 s =.3 KT PM Ordered Γ/J Energy scale (unitless) 3 2 Energy s 1 Γ(s)/T J(s)/T Annealing parameter s 1 1 Annealing parameter s forward anneal protocol Annealing parameter s reverse anneal protocol Annealing time t (µs) Annealing time t (µs) 14 / 2

26 Quantum evolution Monte Carlo Order parameter m Input classical state (s = 1) Quantum evolution (s = 6) Read out classical state (s = 1) burn-in measurement Ordered initial state Random initial state 25 5 Order parameter m PM critical PM QMC QA 5.3 Quantum evolution steps Annealing parameter s Repeat reverse annealing protocol 5 times Estimate statistics from last 25 anneals Start from random and ordered state. Converge to same place? 15 / 2

27 Quantum evolution Monte Carlo Order parameter m PM critical PM QMC QA 5.3 Complex order parameter ψ = me iθ QA s = QA s = QMC s = QMC s = 6 Annealing parameter s Good agreement between QA and QMC Peak in m and U(1) symmetry in ψ 16 / 2

28 Quantum evolution Monte Carlo Order parameter m PM critical PM T/J Ordered QMC QA Γ/J Annealing parameter s mk Upper transition s =.3 s = 5 KT s = PM Good agreement between QA and QMC Peak in m and U(1) symmetry in ψ 16 / 2

29 Onset of power-law correlation decay b =.366(21) 1 b =.369(11) Periodic ψ j Phase correlation Cij s = 6 QMC 8.4 mk QMC 21.4 mk QA 8.4 mk QA 21.4 mk Open vortex antivortex vortex Open Distance x ij Periodic HOT PM region: Exponential decay COLD KT region: Power-law decay 17 / 2

30 Onset of power-law correlation decay Phase correlation Cij b =.366(21) 1 b =.369(11) s = 6 QMC 8.4 mk QMC 21.4 mk QA 8.4 mk QA 21.4 mk Order parameter m s = 6 b/2 = 68(11) b/2 = 47(4) QMC 8.4 mk QMC 21.4 mk QA 8.4 mk QA 21.4 mk Distance x ij Lattice width L 18 / 2

31 Relaxation dynamics Order parameter m s = 6 QA clock QA striped QMC clock QMC striped Evolution length QMC (sweeps) QA (µs) Evolution length (QA µs, QMC sweeps) Annealing parameter s How fast do they get within.3 of each other? QMC scaling in s is steeper Advantage grows as long-range correlations increase 19 / 2

32 Quantum simulation with D-Wave New features, new possibilities Anneal features allow previously unreachable experiments Simulation of TFIM, not just classical Ising problem Programmable magnetic material Quantitatively accurate, no fitting parameters TFIM accurately models flux qubits at large scale 2 / 2

33 Extras: Embedding a b c 2 / 2

34 Extras: Calibration refinement a.3 Without shim With shim b.3 Without shim With shim Frequency Frequency Average qubit magnetization Average AFM spin-spin correlation Use lattice symmetries to refine calibration Each qubit has average magnetization. Coupler frustration probabilities obey rotational symmetry. 2 / 2

35 Extras: Markov chain convergence Order parameter m s =, T = 8.4 mk.3 Ordered init. Random init Order parameter m s = 6, T = 8.4 mk.3 Ordered init. Random init Order parameter m s =.3, T = 8.4 mk.3 Ordered init. Random init Quantum evolution steps Quantum evolution steps Quantum evolution steps Order parameter m s = 6, T = 15.8 mk.3 Ordered init. Random init Order parameter m s = 6, T = 18.2 mk.3 Ordered init. Random init Order parameter m s = 6, T = 21.4 mk.3 Ordered init. Random init Quantum evolution steps Quantum evolution steps Quantum evolution steps 2 / 2

36 Extras: Phase diagram 1 1 Γ/J = 1.2 a b c Γ/J = L = 15 L = 18 L = 21.9 η/2 =.556 ±.4 T1 =.85 ±.6 χl c e at 1/2 /L L = 3 L = 6 L = 9 L = 12 L = 15 L = 18 L = 21 ml b e at 1/2 /L m.8.7 η/2 = 25 ±.4 T2 = 824 ±.12 Γ/J = 1.2 log-log L d Binder cumulant U U L = 3 L = 6 L = 9 L = 12 L = 15 T/J e T2 from collapse T2 from η T1 from collapse T1 from η QCP L 1/ν (Γ Γc)/Γc Γ/J Γ/J 2 / 2

37 Extras: Quench Frequency a b c.3 QA QMC projected Frequency.3 QA QMC quenched ψ, QMC quenched Residual classical energy per spin Residual classical energy per spin ψ, QMC projected QA evolves during 1 µs quench. Huge difference between QMC and QA classical energies. Classical quench erases the difference ψ mostly unchanged. 2 / 2