ADIABATIC PREPARATION OF ENCODED QUANTUM STATES

Transcription

1 ADIABATIC PREPARATION OF ENCODED QUANTUM STATES Fernando Pastawski & Beni Yoshida USC Los Angeles, June 11, 2014

2 QUANTUM NOISE Storing single observable not enough. Decoherence deteriorates quantum observables. Can we store quantum information?

3 KITAEV S IDEA Locally defined quantum error correcting code (QECC). Hamiltonian with degenerate ground space (GS). Constructed examples of topologically protected GS. Kitaev, A. Y. (2003). Fault-tolerant quantum computation by anyons. Annals of Physics, 303(1), 2 30.

4 ROBUST ENERGY DEGENERACY Quantum information storage. Energy Decoherence = uncontrolled phase = uncontrolled energy or time Use robust zero-modes! I 2 I 1 I 0 J Code space = ground space Bravyi, S., Hastings, M. B., & Michalakis, S. (2010). Topological quantum order: Stability under local perturbations. Journal of Mathematical Physics, 51(9), doi: /

5 ACTUAL VS. IDEAS GROUND SPACE 2 Energy I 2 P = UP 0 U P 0 = U PU 1 I 1 Degeneracy is robust. code space is not! 0 Ideal I 0 J Actual H = UH 0 U = H 0 +(U 1)H 0 U + H 0 (U 1) U = Y s U j = Y j exp("t j ) tr[p 0 P ] / tr[p0 ]exp( "N) Ideal code space has large contribution of perturbed excitations. Excitations introduce errors in t < O(log(L)) for the toric code. Pastawski, F., Kay, A., Schuch, N., & Cirac, I. (2010). Limitations of Passive Protection of Quantum Information. Quantum Information and Computation, 10(7&8),

6 ADIABATIC PREPARATION OF ENCODED QUANTUM STATES Prepares actual GS by design.

7 CAN IT BE DONE? (The gap closes)

8 IS IT ROBUST?

9 CAN WE ENCODE INFORMATION?

10 OUTLINE Toric code Hamiltonian Adiabatic interpolation. Joint symmetries. Phase transitions & boundaries. Supercoherent qubit. Color code (Steane 7 qubit)accessible states. Conclusions

11 TORIC CODE & HAMILTONIAN Plaquete and vertex anyons 0 P p + O e of p Z e 1 A /2 P v = + O! X e /2 e at v Parameters: [[2L 2, 2,L]] Ground space projector Y P 0 = p H TC = X v P p Y v P v P v X p P p Kitaev, A. Y. (2003). Fault-tolerant quantum computation by anyons. Annals of Physics, 303(1), 2 30.

12 ADIABATIC PREPARATION OF TOPOLOGICAL ORDER Hamma, A., & Lidar, D. A. (2008) H(s) =H U +(1 s)h + sh g U g, H U = U X p X P p Z e H g = g v H = X e P v No plaquete anyons: HU is a joint symmetry of the system. Lattice gauge duality mapping gives gap: (L) O(L 1 ) Sectors are topologically protected from perturbations.

13 ADIABATIC INTERPOLATION From uniform field to toric code H(s) =(1 s)h P + sh TC Magnon quasiparticles Anyon quasiparticles H P = ~ h X ~ e H TC = X v S v X S p e p Unique GS. Constant gap 4-fold degenerate GS Topological gap ~ = {X, Y, Z}

14 JOINT SYMMETRIES & CONSERVED QUANTITIES No plaquete anyons ~ h ~ = ±Z ) 8p, s :[Sp,H(s)] = 0 8s, q 2 {1, 2} :[ Z q,h(s)] = 0 No vertex anyons ~ h ~ = ±X ) 8v, s :[Sv,H(s)] = 0 8s, q 2 {1, 2} :[ X q,h(s)] = 0 Less conserved quantities in Y direction. ~ h ~ = ±Y ) 8s, q 2 {1, 2} :[ Ȳ q,h(s)] = 0 Hamma, A., & Lidar, D. A. (2008). Adiabatic Preparation of Topological Order. Physical Review Letters, 100(3),

15 OTHER JOINT SYMMETRIES Translational invariance: 8s, ˆ :[Tˆ,H(s)] = 0 D4 symmetry on square: 8s, R 2 D4 :[R, H(s)] = 0 Unique initial GS shares symmetries (+1 eig). D2 is a logical swap between logical qubits 1 and 2. Restricted to symmetric subspace of qubits 1 and 2.

16 FINAL ADIABATIC STATE?

17 THE PHASE DIAGRAM Field orientationfull determines phase transition. phase diagram (J = 1/2)? S hx hz S hy Multicritical line with continuously varying critical exponents! Different anyon species condense.? S. Dusuel, M. Kamfor, R. Orus, K. P. Schmidt, and J. Vidal, Phys. Rev. Lett. 106, (2011) ipeps + PCUT numerical study. 15 Dusuel, S., Kamfor, M., Orús, R., Schmidt, K. P., & Vidal, J. (2011). Robustness of a Perturbed Topological Phase. Physical Review Letters, 106(10), doi: /physrevlett

18 NUMERICAL PHASE DIAGRAM Gap Z Field 0 Full phase diagram (J = 1/2)? hx S hz S hy Multicritical line with continuously varying critical exponents X Field 0.5? Qualitative for L=3 S. Dusuel, M. Kamfor, R. Orus, K. P. Schmidt, and J. Vidal, Phys. Rev. Lett. 106, (2011) 15

19 NUMERICAL PHASE DIAGRAM Topological gap E5 E Z Field X Field S hx Full phase diagram (J = 1/2)? hz hy 0.5 S Multicritical line with continuously varying critical exponen Qualitative for L=3? S. Dusuel, M. Kamfor, R. Orus, K. P. Schmidt, and J. Vidal, Phys. Rev. Lett. 106, (2011) 15

20 PROBABILITY OF REACHING GROUND STATE SPACE Full phase diagram (J = 1/2)? -Z S hx hz S hy Multicritical line with continuously varying critical ex? S. Dusuel, M. Kamfor, R. Orus, K. P. Schmidt, and J. Vidal, Phys. Rev. Lett. 106, (2011) -X -Y L=3, T=30

21 LOGICAL X POLARIZATION Full phase diagram (J = 1/2)? S Z hx hz S hy Multicritical line with continuously varying critical exp? -X Phys. Rev. Lett. 106, (2011) -Y L=3, T=30 S. Dusuel, M. Kamfor, R. Orus, K. P. Schmidt, and J. Vidal,

22 TWO REGIMES 1) Usual adiabatic theorem applicable for the magnon phase. 2) Quasi-adiabatic continuation within the topological phase. Unique: parameter space has no obstructions. Exponentially small splitting x polynomial time = exponentially small non-geometric phase Interesting stuff happens at the phase transition.

23 INFINITE TIME LIMIT Infinite time limit adiabatic theorem. Ground state gap closes at the end. Splitting determined by lowest order perturbation theory. For stabilizer codes, order is set by code distance. (H F + H I ) ni = E n ni E n = 1X j=0 j E (j) n ni = 1X j=0 j n (j) i Rigolin, G., & Ortiz, G. (2010). Adiabatic Perturbation Theory and Geometric Phases for Degenerate Systems. Physical Review Letters, 104(17),

24 DEGENERATE PERTURBATION THEORY Perturbative expansion around exact topological point. (H F + H I ) ni = E n ni 1X 1X E n = j E (j) n ni = j n (j) i j=0 j=0 Degeneracy preserved up to order L L H L X L X + Z L Z

25 OBSERVATIONS Preparation of encoded stabilizer states is stable - To increase in preparation time. - To Hamiltonian perturbations. Fast prep. time for 2 nd order phase transitions ( T=poly(L) ). These are associated to lowest weight logical operators. How about other codes?

26 CAN WE ALWAYS ADIABATICALY PREPARE CODE STATES?

27 SUPERCOHERENT QUBIT Decoherence-free subspace for uniform fields and d=2 code. H SS = 4X j6=k=1 ~ j ~ k H P H SS No avoided crossing [H SS,H P ]=0 Failure of naïve approach. E 0 s 1 j Bacon, D., Brown, K. R., & Whaley, K. B. (2001). Coherence-Preserving Quantum Bits. Physical Review Letters, 87(24),

28 CAN OTHER JOINT SYMMETRIES INDUCE OTHER STABLE STATES? MAGIC STATES?.

29 STEANE S 7 QUBIT CODE ( COLOR CODES ) [[7, 1, 3]] P A Transverse logical operators. A p = H CC = X 7 X Y 7 Ȳ Z 7 Z H 7 H R 7 R R = SH May commute with initial Hamiltonian. 4-body terms (stabilizers) O q2p X p2 face X j 3 A p O q2p 1 7 Z j 5 O q2p Y j New stable fixpoints? 2 6 4

30 GROUND SPACE OVERLAP Z X -Y Steane (N=7), T=4

31 LOGICAL X POLARIZATION Z X -Y Steane code (N=7), T=4

32 LOGICAL XYZ POLARIZATION Z X + Ȳ + Z p 3 X Y Steane code (N=7), T=4

33 LOGICAL X POLARIZATION Z P GS X -Y Steane code (N=7), T=256

34 LOGICAL XYZ POLARIZATION Z P GS X + Ȳ + Z p X -Y Steane code (N=7), T=256

35 CONCLUSIONS & OUTLOOK Efficient preparation of actual code space. Resource: encoded ancillas. Robust adiabatic preparation of stabilizer states. Joint symmetries are not enough to guarantee stability. Can we enhance joint symmetries to prepare magic states in a robust adiabatic manner?

36 THANK YOU!