Holonomic quantum computation in decoherence-free subspaces

1. Holonomic quantum computation in decoherence-free subspaces Center for Quantum Information and Quantum Control Lian-Ao Wu 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....!!

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

7. Introduce HQC Holonomic Quantum Computation (HQC) II One is interested in the case at time T, if start with The Berry phase is all geometrical, independent of speed of parameters 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. Introduce HQC Holonomic Quantum Computation (HQC) III Dark Eigen State: Using dark state to generate all-geometrical phase gate: We need 4 states |0>, |1>,|2> & |3> for 1 particle, a controllable Hamiltonian in terms of parameters q(t) and f(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 q(0)=0 If we define our qubit by |0> & |1>

9. Introduce HQC Holonomic Quantum Computation (HQC) V Using dark state to generate all-geometrical X gate:eigX/2 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. Introduce HQC Holonomic Quantum Computation (HQC) V Using dark state to generate all-geometrical 2-qubit gate 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. A brief Sum-Up of HQC Holonomic Quantum Computation (HQC) VI Use dark states to generate all-geometricaluniversal set of gates, 1 qubit Z, X gate & 2-qubit CPHASE gate (by controlling Hz, Hx and H4) For a dark state, the wave function at T Using this relation to perform phase gates. In above cases,gis half of solid angle swept out by the vector (q,f) 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 4 qubit DFS: C=span{ |1000>, |0100>, |0010>, |0001>} If interaction between system and bath is C is a DFS against collective dephasing, will be our working space to support encoded logical qubit

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

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

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

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 HQC HQC in DFS 4D working space: |0>, |1>, |2>, |3> 4D working space: |0001>, |0010>,|0100>,|1000> Logic qubit by |0>L=|0001> and |1>L =|0010> Qubit by |0> and |1> Hx, Hz and H4 have same matrix representations but acting different spaces Controllable Hamiltonian Hx, Hz and H4 Robust against operational error Robust: operational error and collective dephasing Experimental implementation: depend on the system Interesting to note X, Z need only 2-body interaction

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 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. Thank you for the attention! LA Wu, PZ, DA Lidar, PRL 95, 130501 (2005)

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.