Encoded Universality – an Overview

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

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 Can we avoid “hard” interactions? *quantum dots, donor-atom nuclear spins, electron spins 1 Ralph, Munroe and Milburn, 2001

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! qubit i qubit j qubit i qubit j Hij Hji

7. Change of paradigm H1,H2,... Traditionally: manipulate the physical system* to produce + + * 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 H1,H2,... Traditionally: manipulate the physical system* to produce + + Universal encoded computation: interactions given by the physical system find a way to make them universal H H H Encoding? * Independent of system’s natural talents (fast, robust interactions) often difficult, certain gates can only be implemented with noise; high decoherence ...

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

10. Language of Hamiltonians U(t) = exp(iHt)Which interactions are universal? Given =H1, H2,…, Hn 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)! 1) scalar multiple 2) linear combination 3) Lie bracket

11. Heisenberg interaction (Pauli matrices) • omnipresent in solid state physics (« Easy ») • is not universal: preserves the total spin of the qubits Lie algebra of E : On three qubits: E12 E23 E13 su(2)

12. The algebra L(E) of E (3 qubits) the algebra L3(E) splits as: L3(E)  L3(E)  S1I4 S2I2 su(2)S2 Encoded qubit ? su(2) 2 su(2) 2 Simulation of all operations of one qubit (su(2)) with L3(E) on the encoded qubit !

13. The algebra Ln(E) of E (n qubits) ... the algebra Ln(E) splits as: Ln(E)  ... ... Commutant L’ of Ln(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(Cd) (M(Cn)) is the algebra of all matrices on Cd (Cn). ... ... 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(Cd) (M(Cn)) is the algebra of all matrices on Cd (Cn). NO! The multiplicative algebra is not at our disposition! However the Lie algebra splits into irreducible components in the same basis:

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(nj )? Encode the quantum information into the corresponding sub-space. dimension: nj ... 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** • 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 *Kempe, Bacon, Lidar, Whaley, Phys. Rev. A 63:042307 (2001) **DiVincenzo, Bacon, Kempe,Whaley, NATURE 408 (2000)

18. Exchange-only CNOT Nearest neighbor exchange coupling exchange gate i j DiVincenzo, Bacon, Kempe, Burkard, Whaley, Nature408, 339 (2000) Tradeoffs factor of 3 in space (encoding) factor of ~ 10 in time

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. Anisotropic Exchange* encode into qutrit: conjoin qutrits: “XY”-interaction i.e., 1-qutrit operations • HXY 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

21. Gate sequences:7 operations for single qubit operations (serial) • 5 operations for Sqrt (-ZZ) (equiv. to controlled phase) • “P3”-gate: The Gates* /4 -/4 /2 P3() = /2 -/2 Truncated qubit: Single qubit operations: Two-qubit operation: P3(-/2) =  = P3(-) = (Euler angles) *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

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

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

24. Results: Asymmetric Anisotropic Exchange* h23 h23 h13 H13 h13 H13 h12 H12 h12 H12 H23 H23 H4 H4 1 4 1 4 |000> |011> |111> |100> |110> |101> |001> |010> 2 3 2 3 The total exchange Hamiltonian consists of two components: 1) symmetric, which couples the physical qubit states |01> and |10> Hij = J ( sx,isx,j + sy,isy,j ) + K ( sx,isy,j - sy,isx,j ) 2) and antisymmetric , coupling the states |00> and |11> hij = j ( sx,isx,j - sy,isy,j ) + k ( sx,isy,j + sy,isx,j ) which both simultaneously transform pairs of code words in two code-subspaces. code space II code space I This allows to apply similar techniques as in the symmetric XY-case! *Vala and Whaley, in preparation 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 Poster No. 38 by Mike Hsieh 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:

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 codesBacon et al., PRL85, 1758 (2000) • Bacon et al., quant-ph/0102140 • find entangling operations • Lie algebraic analysisKempe et al., PRA63, 042307 (2001) • Kempe et al., JQIC (2001) • efficient implementation • numerical search e.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, Nature408, 339 (2000)

31. Results encode into qutrit: conjoin qutrits: • 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: i.e., 1-qutrit operations HXY generates su(9) on this subspace (Kempe, Bacon, DiVincenzo, Whaley IQC’01)

32. Single-Qubit and Two-Qubit Gates 1 2 6 LOGICAL 1 2 LOGICAL 3 4 5 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. [H13,H23] = i (J2 - j2) sy,12and[H12 ,sy,12 ] = i 2 J sz,12 2) entangling two-qubit operation C(Z) results from application of the encoded sz 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