1. Holonomic quantum computation in decoherence-free subspaces Lian-Ao Wu Center for Quantum Information and Quantum Control In collaboration with Polao Zanardi and Daniel Lidar

2. Background:  Decoherence-free subspace (DFS): symmetry- aided protection of quantum information, against such as collective noise  Holonomic (adiabatic) quantum computation (HQC): all-geometrical Quantum Information Processing strategy, robust against operational errors Question is:  Can we combine the advantages of the two? Bringing together the best of two worlds....!! Bringing together the best of two worlds....!!

3. First recall Univerisal Quantum Computation (requires) Have 2-level product space (qubits), prepare initial state Have universal set of gates (gates can take one state to arbitrary state): e.g. 1-qubit X, Z gates for each qubit plus CPHASE gates. Gates usually are evolution operators given by Measure the final state

4. Introduce DFS Decoherence-Free Subspace (DFS) I Decoherence? a quantum information processor (system) cannot be isolated from its environment (Bath), due to the interaction of the system with its bath, i.e.

5. Introduce DFS Decoherence-Free Subspace (DFS) II Invariant subspaces if there is symmetry in H e.g. collective dephasing For example, subspace spanned by |01> and |10> will be invariant. (|0001>,|0010>,|0100> & |1000>) B-S interaction not harmful for the system

6. Control time-dependent periodical Hamiltonian through M parameters where Introduce HQC Holonomic Quantum Computation (HQC) I T- period, evolution operator: HQC is based on the adiabatic theorem, which shows if H has non-degenerate eigenvalues If start with an eigenstate of H(t), the system will stick on it but 2 phases

7. One is interested in the case at time T, if start with The Berry phase is all geometrical, independent of speed of parameters Introduce HQC Holonomic Quantum Computation (HQC) II Example, The Berry phase is the solid angle swept out by the vector Allow operational error, as long as solid angle same, the Berry phase same. Geometrical Phase Gate: if

8. Dark Eigen State: Using dark state to generate all-geometrical phase gate: Introduce HQC Holonomic Quantum Computation (HQC) III We need 4 states |0>, |1>,|2> & |3> for 1 particle, a controllable Hamiltonian in terms of parameters  (t) and  (t): The dark state: The Hamiltonian does nothing on |0> and add a Berry phase on state |1> after evolution from 0 to T if  (0)=0 If we define our qubit by |0> & |1>

9. Using dark state to generate all-geometrical X gate: e i  X/2 Introduce HQC Holonomic Quantum Computation (HQC) V In the 4-state space by |0>, |1>,|2> & |3> for one particle, we need a controllable Hamiltonian The dark state: The Hamiltonian will do nothing on |+> and add a Berry phase for state |-> after evolution from 0 to T.

10. Using dark state to generate all-geometrical 2-qubit gate Introduce HQC Holonomic Quantum Computation (HQC) V We have 16 states for 2 particles, |00>, |01>, |02>,…. Chose a controllable Hamiltonian, The dark state: The Hamiltonian will do nothing on |00>,|01>& |10> and add a Berry phase for state |11> after evolution from 0 to T.

11. Use dark states to generate all-geometrical universal set of gates, 1 qubit Z, X gate & 2-qubit CPHASE gate (by controlling H z, H x and H 4 ) A brief Sum-Up of HQC Holonomic Quantum Computation (HQC) VI For a dark state, the wave function at T Using this relation to perform phase gates. In above cases,  is half of solid angle swept out by the vector (  ) We have to use ancillas |2> and |3> for each qubit when make gates. We pay more price. We need to Have 4-dimensional working space to support 1 qubit.

12. Come to our work A Decoherence-Free Subspace as working space If interaction between system and bath is 4 qubit DFS: C=span { |1000>, |0100>, |0010>, |0001>} C is a DFS against collective dephasing, will be our working space to support encoded logical qubit

13. Time-dependent controllable Hamiltonian Set Every eigenspace of Z is invariant under action of Assume that the system dynamics is generated by the Hamiltonian where parameters are dynamically controllable.

14. Dark-states in the DFS In the basis {|100>,|010>,|001>} for qubits l, m and n, the above Hamiltonian has a dark state Satisfying Turn on the parameters in such a way to get

15. One-qubit geometrical gates Turn on the parameters in such a way to get Acting only on 2, 3 and 4 qubit states|1000> 4321, |0100> 4321 and |0010> 4321 nothing on |0001> 4321. Define the logic qubit supported by state |0> L =|0001> and |1> L =|0010>. Dark state in qubits 2,3 and 4 Adiabatically changing parameters a Berry Phase for |1> L

16.  x  gate  x  gate Turn on the parameters in such a way to get where Adiabatically changing parameters a Berry Phase for |-> L Easy to prove Dark state is

17. Two-qubit geometrical gates Suppose that one can engineer the four-body interaction Dark state is Adiabatically changing parameters a Berry Phase for |11> L

18. Compare General HQC with HQC in DFS General HQCHQC in DFS 4D working space: |0>, |1>, |2>, |3> 4D working space: |0001>, |0010>,|0100>,|1000> Qubit by |0> and |1> Logic qubit by |0> L =|0001> and |1> L =|0010> Controllable Hamiltonian H x, H z and H 4 Experimental implementation: depend on the system H x, H z and H 4 have same matrix representations but acting different spaces Interesting to note X, Z need only 2-body interaction Robust against operational error Robust: operational error and collective dephasing

19. Implementations Spin-based quantum dot proposals One qubit Hamiltonians achievable: confining potential, pulse shaping (Stepanenko etal al 2003,2004) Ion Traps Sorensen-Moelmer scheme (two lasers control) (Kielpinski et al 2002) Realizable as well! SM over two pairs of trapped ions geometrical gates already realized (Leibfried et al 2003)

20. Summary Summary We have discussed how to merge together universal HQC & DFS by using dark-states in decoherence-free subspace against collective dephasing. The scheme can be extended to the cases against general collective noise. We have discussed how to merge together universal HQC & DFS by using dark-states in decoherence-free subspace against collective dephasing. The scheme can be extended to the cases against general collective noise. LA Wu, PZ, DA Lidar, PRL 95, 130501 (2005) Thank you for the attention !

21. geometrical phase factor is precisely the holonomy in a Hermitian line bundle since the adiabatic theorem naturally defines a connection in such a bundle.