1. Encoded Universality – an Overview Julia Kempe University of California, Berkeley Department of Chemistry & Computer Science Division Sponsors:

2. Encoded Universality Whaley group people: Dave Bacon (now Caltech) Mike Hsieh (undergraduate) Julia Kempe (postdoc) Simon Myrgren (graduate student) Prof. Birgitta Whaley Jiri Vala (postdoc) Jerry Vinokurov (undergraduate) Sponsors:

3. Overview Universal quantum computation - a bit of history Change of paradigm Example : Heisenberg interaction Lie algebra formalism for encoded universal computation Results: Heisenberg interaction, symmetric “XY” interaction, asymmetric “XY” with crossterms

4. Quantum circuits + + =U Barenco et al. ’95: Single-qubit gates and CNOT generate every unitary transformation! Mantra

5. The problem “Easy” and “hard” interactions (system-dependent) “Easy”: intrinsic interactions “natural” to the system, easy to tune, rapid “Hard”: slower, require higher device complexity, high decoherence System“Easy”“Hard” Photon-qubits Single qubit operations (linear optics) Two qubit operations (non-linearity, non-deterministic qc) Solid state* Two qubit operations (ex. J-gates) Single qubit operations (ex. local focused B-fields) “Coherent- state” qubits 1 Two qubit operations (beam-splitter) Single qubit operations (non-linearity, non-det.) *quantum dots, donor-atom nuclear spins, electron spins 1 Ralph, Munroe and Milburn, 2001 Can we avoid “hard” interactions?

6. Almost every interaction is universal! Deutsch et al.(’95), Lloyd (‘95) : Almost any interaction on two qubits is universal. In the generic sense. Does not include the most frequentes interactions. Nature is not generic! H ij H ji qubit iqubit jqubit iqubit j

7. Change of paradigm Traditionally: manipulate the physical system* to produce + + H 1,H 2,... * Independent of system’s natural talents (fast, robust interactions) often difficult, certain gates can only be implemented with noise; high decoherence...

8. Change of paradigm Traditionally: manipulate the physical system* to produce + + H 1,H 2,... Universal encoded computation Universal encoded computation: interactions given by the physical system find a way to make them universal * Independent of system’s natural talents (fast, robust interactions) often difficult, certain gates can only be implemented with noise; high decoherence... HHH Encoding?

9. Classical « Analogy » Two coins can only flip the two coins together « encode » « 0 »- « 1 »- 11 00 flip 100111 00 11 00 00 11 01 10 0 0 Encoded « coin »

10. Language of Hamiltonians U(t) = exp(iHt) Which interactions are universal? Given  =  H 1, H 2,…, H n  can one generate any unitary transformation (exactly or approximatively)? H has to generate the Lie algebra of su(N) of the unitary group SU(N)! scalar multiple 1) scalar multiple 2) linear combination 3) Lie bracket Lie bracket

11. Heisenberg interaction omnipresent in solid state physics (« Easy ») is not universal: preserves the total spin of the qubits Lie algebra of E : On three qubits: su(2) E 12 E 13 E 23 (Pauli matrices)

12. The algebra L(E) of E (3 qubits) the algebra L 3 (E) splits as: L 3 (E)  L 3 (E)  S 1  I 4  S 2  I 2 su(2)  S 2 Encoded qubit ?  su(2) 2 2 Simulation of all operations of one qubit (su(2)) with L 3 (E) on the encoded qubit !

13. The algebra L n (E) of E (n qubits) the algebra L n (E) splits as: L n (E) ... Commutant L’ of L n (E) : L’ is generated by (« spin » algebra su(2)) As a Lie algebra L’ splits into irreducible representations of su(2).

14. Useful theorem Let S be a †-closed algebra closed under multiplication and linear combination. Then the underlying space H is isomorphic to such that S and its commutant S’ split as: where M(C d ) (M(C n )) is the algebra of all matrices on C d (C n ).... Universal computation “for free”?

15. Useful theorem Let S be a †-closed algebra closed under multiplication and linear combination. Then the underlying space H is isomorphic to such that S and its commutant S’ split as: where M(C d ) (M(C n )) is the algebra of all matrices on C d (C n ). The multiplicative algebra is not at our disposition! However the Lie algebra splits into irreducible components in the same basis: NO!

16. Problem of “Encoded Universality” Given an ensemble of generators H with Lie algebra L(H) which splits as can one find a component s.t. contains su(n j )? Encode the quantum information into the corresponding sub-space. dimension: n j... Yes D. Bacon, J. Kempe, D.P. DiVincenzo, D.A. Lidar, K.B. Whaley, “ Encoded Universality in Physical Implementations of a Quantum Computer”, Proceedings of IQC ’01, Australia

17. Previous Results - Heisenberg interaction E is universal with encoding* introduce tensor structure, ex. blocks with 3 qubits** *Kempe, Bacon, Lidar, Whaley, Phys. Rev. A 63:042307 (2001) **DiVincenzo, Bacon, Kempe,Whaley, NATURE 408 (2000) efficient implementation of encoded gates: numerical search** serial coupling - 19 operations for CNOT, 4 operations for 1-qubit parallel coupling - 7 operations for CNOT, 3 operations for 1-qubit

18. i j exchange gate DiVincenzo, Bacon, Kempe, Burkard, Whaley, Nature 408, 339 (2000) Nearest neighbor exchange coupling Tradeoffs factor of 3 in space (encoding) factor of ~ 10 in time Exchange-only CNOT

19. Conjoining – a new tensor structure Introduce a cutoff that defines a single “qudit”. In principle:... For larger n one could find larger component with better encoding ratio ? Need to guarantee uniformity of quantum circuits! (“Form” of the circuit should not depend on size of problem.) Introduce cutoff -> tensor product structure. Conjoining subsystems:

20. encode into qutrit: i.e., 1-qutrit operations conjoin qutrits: H XY generates su(9) on this subspace “Truncated qubit”: use and only effectively with an ancillary qubit for gate-applications *J. Kempe, D. Bacon, D.P. DiVincenzo, K.B. Whaley, “ Encoded universality from a single physical interaction”, in «Quantum Information and Computation»; Special Issue, Vol. 1, 2001 Anisotropic Exchange* “XY”-interaction

21. The Gates* Gate sequences: 7 operations for single qubit operations (serial) 5 operations for Sqrt (-ZZ) (equiv. to controlled phase) “P3”-gate: P3(  ) = /4/4 /2/2 -/4-/4 -/2-/2 /2/2 Truncated qubit: P3(-  ) = =  (Euler angles) Two-qubit operation: *J. Kempe and K.B.Whaley, “Exact gate-sequences for universal quantum computation using the XY- interaction alone ”, quant-ph/0112014, to appear in Phys. Rev. A Single qubit operations: P3(-  / 2 ) =

22. Layout – Anisotropic Exchange a) triangular array (qutrit) b) “truncated qubit” or

23. Results: Asymmetric Anisotropic Exchange* *J. Vala and K.B. Whaley, “Encoded Universality with Generalized Anisotropic Exchange Interactions”, in preparation 2002 Poster No. 21 by Jiri Vala Asymmetric exchange: Asymmetric exchange with crossterm: Universal Encodings and Gate-Sequences

24. Results: Asymmetric Anisotropic Exchange* |000>|011> |101>|110> |111>|100> |010>|001> 14 23 14 23 H4H4 H4H4 h 12 H 12 h 12 H 12 h 23 H 23 h 23 H 23 h 13 H 13 h 13 H 13 The total exchange Hamiltonian consists of two components: 1) symmetric, which couples the physical qubit states |01> and |10> H ij = J (  x,i  x,j +  y,i  y,j ) + K (  x,i  y,j -  y,i  x,j ) 2) and antisymmetric, coupling the states |00> and |11> h ij = j (  x,i  x,j -  y,i  y,j ) + k (  x,i  y,j +  y,i  x,j ) which both simultaneously transform pairs of code words in two code-subspaces. code space I code space II *Vala and Whaley, in preparation This allows to apply similar techniques as in the symmetric XY-case! Poster No. 21 by Jiri Vala

25. Is encoded universality always possible? NO! non-interacting fermions (Valiant, Terhal&DiVincenzo, Knill ’01) nearest-neighbor XY-interaction linear optics quantum computation Criterion: If a set of Hamiltonians (over n qubits) allows for (encoded) universal computation then the Lie algebra L(H) contains exponentially many linearly independent elements.... Some component has to contain where is a polynomial function of n. ex: is not universal with any encoding.

26. Continuing Work How find the gate sequences that implement the encoded one- and two-qubit gates? Numeric search – genetic algorithms (Hsieh, Kempe) Developed numeric tools: Preliminary results in 4-qubit encoding for Heisenberg interaction: Poster No. 38 by Mike Hsieh

27. Summary Lie-algebra methods and redefinition of the tensor structure of Hilbert space allow for universality! Use of encoding - the implementation of one-qubit gates is obsolete! (change of paradigm) Heisenberg interaction is omnipresent (e.g. in solid state physics) and easy to implement, whereas one-qubit gates are extremely hard to obtain  encoded universality gives attractive qc proposals Evaluation of the trade-offs in space and time Open Questions: Find better than ad hoc ways to generate the gate sequences General easy criteria to determine encoded power of interaction (e.g. amount of symmetry…) a general theory allowing for measurements and prior entanglement (to incorporate Briegel/Rauschendorff and Nielson schemes into analysis)

28. References J. Kempe and K.B.Whaley, “Exact gate-sequences for universal quantum computation using the XY-interaction alone ”, quant-ph/0112014, to appear in Phys. Rev. A J. Kempe, D. Bacon, D.P. DiVincenzo, K.B. Whaley, “ Encoded universality from a single physical interaction”, in «Quantum Information and Computation»; Special Issue, Vol. 1, 2001, quant-ph/0112013 D. Bacon, J. Kempe, D.P. DiVincenzo, D.A. Lidar, K.B. Whaley, “ Encoded Universality in Physical Implementations of a Quantum Computer”, Proceedings of IQC ’01, Australia, quant-ph/0102140 D.P. DiVincenzo, D. Bacon, J. Kempe, K.B. Whaley, “ Universal Quantum Computation with the Exchange Interaction”, NATURE 408, 339 (2000), quant- ph/0005116 J. Vala and K.B. Whaley, “Encoded Universality with Generalized Anisotropic Exchange Interactions”, in preparation 2002 Earlier related work on DFSs: J. Kempe, D. Bacon, D. Lidar,K.B. Whaley, Phys. Rev. A 63:042307 (2001) D. Bacon, J. Kempe, D. Lidar, K.B. Whaley, Phys. Rev. Lett. (2000)

29. Conclusions/Open Questions Encoding into sub-spaces allows to make certain interactions universal Representation theory of Lie groups - powerful tool E and XY alone are universal - important simplification of physical implementations Which other interactions to investigate? General theory when interactions allow for encoded universality? How find the gate sequences that implement the encoded one- and two-qubit gates?

30. tensor product of encoded qubits conjoined codes Bacon et al., PRL 85, 1758 (2000) Bacon et al., quant-ph/0102140 find entangling operations Lie algebraic analysis Kempe et al., PRA 63, 042307 (2001) Kempe et al., JQIC (2001) efficient implementation numerical searche.g. Heisenberg exchange serial coupling - 19 operations for CNOT, 4 operations for 1-qubit parallel coupling - 7 operations for CNOT, 3 operations for 1-qubit DiVincenzo, Bacon, Kempe, Burkard, Whaley, Nature 408, 339 (2000)

31. encode into qutrit: i.e., 1-qutrit operations conjoin qutrits: H XY generates su(9) on this subspace (Kempe, Bacon, DiVincenzo, Whaley IQC’01)Results tensor product of encoded qubits: conjoined codes (Bacon, Kempe, DiVincenzo, Lidar, Whaley ICQ’01) universality: Lie algebraic analysis (Kempe, Bacon, DiVincenzo, Whaley IQC’01) Anisotropic Exchange Interaction:

32. Single-Qubit and Two-Qubit Gates 1) the full su(2) algebra over a single logical qubit is generated via the commutation relations between exchange interactions over physical qubits: e.g. [H 13,H 23 ] = i (J 2 - j 2 )  y,12 and [H 12,  y,12 ] = i 2 J  z,12 2) entangling two-qubit operation C(Z) results from application of the encoded  z operation onto the physical qubits 2-3-4 in the triangular architecture and single-qubit operations 3) the commutation relations are applied via selective recoupling 4) a similar construction is valid for a general anisotropic interaction containing the cross-product terms 12 345 6 LOGICAL 1 2 LOGICAL