A Superconducting Quantum Simulator for Topological Order and the Toric Code. Michael J. Hartmann Heriot-Watt University, Edinburgh qlightcrete 2016

Transcription

1 A Superconducting Quantum Simulator for Topological Order and the Toric Code Michael J. Hartmann Heriot-Watt University, Edinburgh qlightcrete 2016

2 Topological Order (in 2D) A 2-dimensional physical system has topological order iff: It has (almost) degenerate ground states ji that are separated by a gap from excited states The degeneracy of the grand state manifold only depends on the topology No local operator can distinguish between two ground states energy gap h l Ôloc ji / j,l

3 Topological Order and Quantum Information Processing Use (almost) degenerate ground states as qubit h l Ôloc ji / j,l ) Ôloc ji not in ground state manyfold energy gap } qubit energy penalty for errors: need a lot of energy to excite out of qubit subspace avoid errors within qubit subspace: no transformation between qubit states by local force

4 The Toric Code spin star A s = Y x j 1/2 j2star(s) B p plaquette B p = Y y j A s j2plaq(p) H = J s X A s J p X B p s p A s,b p : stabilizers ground state: [A j,a l ]=[B j,b l ]=[A j,b l ]=0 A j 0i =+1 0i B j 0i =+1 0i Kitaev, Ann. Phys. (2003)

5 Ground State Degeneracy N =2L 2 physical spins Hilbert space dim: 2 N A s B p L N stabilizers: each condition A j 0i =+1 0i B j 0i =+1 0i divides number of states by 2 Only one ground state? L Periodic boundary conditions: not all stabilizers independent! Y Y A s =1, B p =1 2 N N+2 =4 ground states s p topology determines the number of ground states

6 Excitations A s = Y j x j B p = Y j y j reducible loops are identities - each spin belongs to 2 stars (plaquettes) y creates pair of excitations in stars: electric charges x creates pair of excitations in plaquettes: magnetic vortices strings y j y j+1 irreducible loops transform between ground states move excitations move el. charge around mag. vortex apart non-trivial phase Abelian anyons

7 Topological Oder Toric Code is the Standard Model for topological order Topological Phases: e.g. in fractional quantum Hall systems and spin liquids states locally indistinguishable no local order parameters for phase transitions ground states have higher symmetry than Hamiltonian no Ginzburg-Landau theory of symmetry breaking Interesting, novel class of order intriguing correlation and entanglement properties excitations with exotic statistics anyons quantum simulator for topological order desirable here: Toric Code in superconducting circuits Kitaev, Ann. Phys. (2003), Stormer et al., Rev. Mod. Phys. 71, 298 (1999), Balents, Nature 464, 199 (2010)

8 Superconducting Circuits picture: Wallraff group, ETH electromagnetic excitations non-equilibrium physics aim at keeping circuit simple transmon qubits connected by tunable couplers Google/UCSB, Delft, ETH, make use of possibilities for driving system ex = dc + ac (t) - single site addressing - can build periodic boundary conditions crucial for testing topological order picture: dicarlo group, Delft

9 Implementation: Spins ˆ' ex B p A s H =4E C ˆn 2 E J ( ex / 0 ) cos ˆ' nonlinear oscillator: H = ~! q b b ~! q = p 8E C E J E C 2 b b bb tuneable E C all interactions between qubits H q = ~! q +

10 Implementation: Interactions B p A s

11 Coupling Circuit ' 2 auxiliary nodes eliminate ' b L ' a 2C J L ' 3 ' 1 ex = dc ex + ac ex(t) L E J ' j L ' 4 C q;j E Jq;j

12 Coupling Circuit: Hamiltonian ' b ' + = ' a + ' b, ' = ' a ' b ' a 2C J E J L H L = = H S =4 e2 2C J 2 very high frequency ignore 2 0 4L (' ' j) L ('2 + ' 2 j 2' j ' ) ~ 2 2E J cos 'ext 2 cos(' ) SQUID: drive: H S =4 e2 2C J 2 ~ L '2 H D = E J ' ac (t) cos(' ) coupling: ' ext = + ' ac (t) H I = 2 0 2L ' j'

13 Concept rotating frame at! j! 1 6=! 2 6=! 3 6=! 4 qubits SQUID! 1 a 1 e i! 1t cos(! d t)! S 6=! d! S 6=! j! 4 a 4 e i! 4t! d =! 1 +! 2 +! 3 +! 4 ) a 1 a 2 a 3 a 4! d =! 1 +! 2! 3! 4 ) a 1 a 2 a 3a 4! S! 3 a 3 e i! 3t! 2 a 2 e i! 2t multi-tone X drive F (t) = c cos(! t) generates stabilizers

14 Adiabatic Elimination of SQUIDs coupling between qubit and SQUID H I = 2 0 2L ' j' Schrieffer-Wolff transformation: H! e S He S eliminates H I and generates S = i 2~ X 'j E J 8 ' ac(t) cos(' )(' 1 ' 2 ' 3 + ' 4 ) 4 = h0 cos ' 0i ' 1 ' 2 ' 3 ' 4 + X i,j b ij' i ' j + X (ijkl) c ijkl' i ' j ' k ' l two indices identical

15 Rotating Frame include time-dependent pre-factors V = A(t)' 1 ' 2 ' 3 ' 4 + X i,j B ij (t)' i ' j + X (ijkl) C ijkl (t)' i ' j ' k ' l ' j = ' j (a j + a j )! ' j( j + + j ) in frame where each qubit rotates at its transition frequency A(t)' 1 ' 2 ' 3 ' 4! negligible in rotating wave appr. ' j! ' j j e i! jt + + j ei! jt! 4Y j=1 ' j X a,b,c,d2{+, } A(t) e i(a! 1+b! 2 +c! 3 +d! 4 )t a 1 b 2 c 3 d 4 / ac ex(t) choose time dependence to enable desired terms

16 Multi Tone Driving 8 < F s (t) =F Y (t)+f X (t) A(t) / ' ac : F p (t) =F Y (t) F X (t) contain 4 drive frequencies each different transition frequencies for all 4 qubits in one unit-cell A s F s (t)! J x x x x s ! A s B p F p (t)! J p y 1 y 2 y 3 y 4! B p star and plaquette differ by phase π for half the drive frequencies

17 Perturbations transition frequencies of all 4 qubits different V =A(t)' 1 ' 2 ' 3 ' 4 + X i,j B ij (t)' i ' j + desired 4-body interactions + fast rotating terms X (ijkl) fast rotating 2-body interactions always two indices identical leading contributions can be compensated by capacitances remaining terms lead to frequency shifts compensate by modified drive frequencies C i,j,k,l B j,l terms negligible C ijkl (t)' i ' j ' k ' l

18 Toric Code Hamiltonian in frame that rotates with H 0 = X j ~! j + j j B p A s H = J s X s A s J p X p B p A s = Y x j B p = Y y j j2star(s) j2plaq(p) J s = J p = E J ' ac 16 4Y i=1 2 E C;i E Jq;i 1 4 = h0 cos(' a ' b ) 0i

19 Rotating Frame and Dissipation superconducting qubits face dissipation and dephasing Toric Code Hamiltonian exists in rotating frame topologically ordered states are not ground states transformation into rotating frame is local unitary } topological order entanglement of topologically ordered states untouched

20 Minimal Size Toric Code experimental challenge for any 2d lattice: complexity crossing conductors, air bridges, stray fields What is the smallest lattice to study Toric Code physics? : qubit, periodic repetition : star interaction : plaquette interaction 8 qubits 8 coupling SQUIDs

21 Towards Experiments 8-qubit Toric Code: qubit transition frequencies:! 1 = 11.4 GHz! 3 =4.8GHz! 4 =4.2GHz! 5 = 11.9 GHz! 2 =! MHz! 6 =! MHz! 7 =! MHz! 8 =! MHz plasma frequency of SQUIDs:! S =8.1 GHz

22 Parameters qubits: 25 ff apple C q apple 90 ff 9.5 GHz apple E Jq apple 29 GHz L 2C J E J SQUIDs: 2.5fFapple C J apple 3.5fF 8.0 GHz apple E J apple 9.5 GHz inductors: 50 nh apple L apple 70 nh 4-body interactions: build as superinductors Josephson junction arrays J s = J p = 2 MHz 2-body interactions: 0.2 MHz Bell et al., PRL 2012 Masluk et al., PRL 2012 T 1 = T 2 = 50 µs! =0.02 MHz

23 Preparation of Correlated States without driving the SQUIDs: all physical qubits in ground state 0 i = g, g,..., gi 1. initially detune qubits 2. turn down detuning and ramp up drive H = X j ~ j (t) + j j J s (t) X s A s J p (t) X p B p j(t) : j! 0 J s (t) : 0! J s J p (t) : 0! J p adiabatic sweep into correlated states Hamma and Lidar, PRL (2008)

24 Adiabatic Sweep H = X j ~ X + J j j s s A s J s = J p = J p X p B p = (1 ) 5 MHz 2 MHz MHz g, g,..., gi topological order 1i, 2i, 3i, 4i

25 Preparation via Adiabatic Sweep start from 0 i = g, g,..., gi and ramp up J s = J p F = X h (t) i P =Tr[ (t) 2 ] F P T S [µ s] 2 [MHz] T S [µ s] [MHz] J s = J p = h 2 MHz T 1 = T 2 = 50 µs topological order can be prepared via adiabatic sweep Hamma and Lidar, PRL (2008)

26 Measuring Topological Order on a torus: 4 topologically correlated states locally indistinguishable: reduced density matrices of individual qubit identical for all 4 topologically correlated states can be distinguished by correlations DY i2loop E x j along irreducible loops transitions between topologically correlated states Y apply i2loop y j along irreducible loop

27 Measurements: Minimal Model locally indistinguishability as before irreducible loop: 2-qubit correlation e.g. h 1 x 2 x i 1. prepare correlated state 2. measure individual qubits and y 1 y 2 h x 1 3. apply different correlated state 4. again measure individual qubits and results for individual qubits are the same results for correlations are different x 2 i h x 1 x 2 i

28 The Team Edmund Owen Dan Browne postdoc Mahdi Sameti PhD Oliver Brown PhD quantumdynamics.eps.hw.ac.uk Anton Potočnik Andreas Wallraff

29 Thanks for listening. H = J s X A s J p X B p s p B p A s 1. arxiv: [pdf, other] A Superconducting Quantum Simulator for Topological Order and the Toric Code Mahdi Sameti, Anton Potocnik, Dan E. Browne, Andreas Wallraff, Michael J. Hartmann