1 / 65

Encoded Universality and Decoherence-Free Subspaces

Control Seminar – Quantum Information Science and Technology UC Berkeley, Feb. 9, 2004. Encoded Universality and Decoherence-Free Subspaces. Julia Kempe CS Division and Dept. of Chemistry, University of California, Berkeley LRI, Universite de Paris-Sud, Paris, France.

kennan
Télécharger la présentation

Encoded Universality and Decoherence-Free Subspaces

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Control Seminar – Quantum Information Science and Technology UC Berkeley, Feb. 9, 2004 Encoded Universality and Decoherence-Free Subspaces Julia Kempe CS Division and Dept. of Chemistry, University of California, Berkeley LRI, Universite de Paris-Sud, Paris, France www.lri.fr/~kempe

  2. Towards nanotechnology Gordon Moore 1965 Size of the components Number of components Speed Theoretical limitations reached in 2020 !!! Apparition of quantum phenomena prevent or use quantum effects ?

  3. Information is physical! Use the laws of quantum mechanics for the basic components of an information processing machine! • Quantum computing • Quantum cryptography • Quantum information • …

  4. Main applications • Cryptography • Protocol of unconditionally secure secret key distribution [Bennett, Brassard 84] Implementation : ~ 100 km • Quantum information • Teleportation [B, B, Crépeau, Jozsa, Peres, Wooters 93] Implementation[Bouwmeester, Pan, Mattle, Eibl, Weinfurter, Zeilinger 97] • Algorithms • Factoring, discrete logarithm, ... [Shor 94] • Database search [Grover 96] Num. of qubits ?1995 : 2, 1998 : 3, 2002 : 8 [Chuang (IBM)] - 10 [Los Alamos]

  5. Overview • Basic notions of quantum computing • Standard solutions: • Universal gate set • Quantum Error Correcting Codes (QECC) • Encoded Universality • Decoherence-free Subsystems

  6. The qubit Classical bit:b{0,1} Probabilistic bit:probability distribution dR+{0,1}such that||d||1 =1. d=(p,1-p)with p [0,1] Quantum bit:|C{0,1}such that|| |||2=1. |=  |0 +  |1 with | |2+ | |2=1 (Dirac notation)

  7. Qubit evolution • Measure:reads and modifies |0 | |2  |0 +  |1 Measure |1 | |2  Superposition  Probability distribution • Unitary transformation: U C22such thatUU†=Id U |’ = U | | unitary  reversible: U† | U|

  8. Example Superposition: Measure: |0 1/3 Measure | |1 2/3

  9. Example Superposition: Measure: Unitary transformations: • NOT:|0 |1 • Hadamard: |0 1/3 Measure | |1 2/3 U |’ = U | | H

  10. Quantum computer: n qubits • nqubits  tensor product| C{0,1}nsuch that|| |||2=1.  |=x{0,1}n x|xwith x|x|2 =1 • Measure • Partial Measure |x|2 x{0,1}n x|x |x Measure Second bit = 0 (||2 + | |2 )  |00+  |01+ |10+ |11 Measure

  11. Quantum computer: n qubits • nqubits  tensor product|C{0,1}nsuch that|| |||2=1.  |=x{0,1}nx|xwith x|x|2 =1 • Measure • Partial Measure • Unitary transformation |U|withU U(2n) ex: XOR= |x|2 x{0,1}n x|x |x Measure Second bit = 0 (||2 + | |2 )  |00+  |01+ |10+ |11 Measure |00 |01 |10 |11 |i |i |XOR(i,j) |j +

  12. Quantum computing a function Let f:{0,1}n {0,1}m x  f(x) Reversible: Rf :{0,1}n+m {0,1}n+m (x,y)  (x,yf(x)) Quantum: UfU(2n+m): Cn+m Cn+m |x|y |x|yf(x) 

  13. Simplest Quantum Algorithm:Deutsch’s Problem Input: function f:{0,1}{0,1} (in black box) Question:fconstant (f(0)=f(1)) or balanced (f(0)f(1)) ? Quantum black box (reversible): Algorithm: one query only!!! |x |x f |y |yf(x) |0 H H Measure f |1 H |0 -constant |1 -balanced

  14. Simplest Quantum Algorithm:Deutsch’s Problem Input: function f:{0,1}{0,1} (in black box) Question:fconstant (f(0)=f(1)) or balanced (f(0)f(1)) ? Quantum black box (reversible): Algorithm: one query only!!! |x |x f |y |yf(x) |0 H H Measure f |1 H |0 -constant |1 -balanced =0 if f constant =0 if f balanced

  15. Where are we in practice? Several physical architectures have been proposed: • NMR, solid state, ion traps, superconducting qubits, optical cavities, photons • Qubit = nuclear spin of atoms in a molecule (NMR), nuclear spin of (doped) atoms in a silicon donor (solid state), vibrational degrees of freedom of ions (ion trap), flux-degree of freedom (superconductors), polarisation (photons)

  16. Five requirements for the implementation of quantum computation* • A scalable physical system with well characterized qubits • The ability to initialise the qubits to a simple state, such as |00…0 • A “universal” set of quantum gates • Long relevant decoherence times, much longer than the gate operation time • A qubit-specific measurement capability * D. DiVincenzo, 97

  17. Overview • Basic notions of quantum computing • Standard solutions: • Universal gate set • Quantum Error Correcting Codes (QECC) • Encoded Universality • Decoherence-free Subsystems

  18. Five requirements for the implementation of quantum computation* • A scalable physical system with well characterized qubits • The ability to initialise the qubits to a simple state, such as |00…0 • A “universal” set of quantum gates • Long relevant decoherence times, much longer than the gate operation time • A qubit-specific measurement capability * D. DiVincenzo, 97

  19. Universal computation Classical circuit model: Quantum circuit model: • evaluates booleanfunctions • can be constructed fromuniversal local gates(ex.: NAND, COPY)  0 0 0 1 0 … 1  bits  • unitary transformations U |0 |0 |1 |1 |0 |0 U qubits Measure

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

  21. Five requirements for the implementation of quantum computation* • A scalable physical system with well characterized qubits • The ability to initialise the qubits to a simple state, such as |00…0 • A “universal” set of quantum gates • Long relevant decoherence times, much longer than the gate operation time • A qubit-specific measurement capability * D. DiVincenzo, 97

  22. Main obstacle – decoherence! • Quantum information is very fragile • Any interaction with the environmentdisturbs the stored information • Solutions: quantum error correcting codes, decoherence-free subspaces, fault-tolerant computation …

  23. Active noise protection: quantum error correction Quantum information is very fragile -how maintain quantum coherence? Quantum Error-Correcting Codes (QECCS)! Easiest classical code: repetition code: 0  000 1  111 Can correct one bitflip error (majority) 100,010,001 000 011,101,110 111

  24. Active noise protection: quantum error correction Quantum information is very fragile -how maintain quantum coherence? Quantum Error-Correcting Codes (QECCS)! Easiest classical code: repetition code: 0  000 1  111 Can correct one bitflip error (majority) 100,010,001 000 011,101,110 111 Quantum: many possible errors (bit flip, phase error, measurements…) “Quantum” repetition code? Impossible – no cloning principle.

  25. Active noise protection: quantum error correction No cloning principle: It is impossible to copy a quantum state. Proof: if then and By linearity:

  26. Active noise protection: quantum error correction 3-qubit QECC: protects against bitflip errors: 3-qubit QECC: protects against phase errors: 9-qubit QECC: protection against all one qubit errors (Shor-code):

  27. Overview • Basic notions of quantum computing • Standard solutions: • Universal gate set • Quantum Error Correcting Codes (QECC) • Encoded Universality • Decoherence-free Subsystems

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

  29. Hamiltonians=Interactions Unitaries are generated by Hamiltonians: + = U + Single-qubit gates and CNOT generate every unitary transformation!

  30. Hamiltonians=Interactions Unitaries are generated by Hamiltonians: Hamiltonians describe the interactions (of qubits) in physical systems. Quantum engineers tweak the Hamiltonians to produce single qubit gates and CNOT. + = U + Single-qubit gates and CNOT generate every unitary transformation!

  31. Universality: 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?

  32. 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 frequent interactions. Nature is not generic! qubit i qubit j qubit i qubit j Hij Hji

  33. Change of paradigm Traditionally: manipulate the physical system* to produce H1,H2,... + + * Independent of system’s natural talents (fast, robust interactions) often difficult, certain gates can only be implemented with noise; high decoherence ...

  34. Change of paradigm Traditionally: manipulate the physical system to produce H1,H2,... + + Universal encoded computation: interactions given by the physical system find a way to make them universal H H H Encoding?

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

  36. Language of Hamiltonians U(t) = exp(iHt) Which interactions are universal? Given =H1, H2,…, Hn can one generate any unitary transformation (exactly or approximatively)?

  37. Language of Hamiltonians U(t) = exp(iHt) Which interactions are universal? Given =H1, H2,…, Hn can one generate any unitary transformation (exactly or approximatively)? Possible compositions: 1) scalar multiple 2) linear combination 3) Lie bracket H has to generate Lie algebra su(N) of the unitary group SU(N)!

  38. Lie Algebra of H Lie(H) closed under: 1) scalar multiplication 2) linear combination 3) Lie-bracket H has to generate Lie algebra su(N) of the unitary group SU(N)!

  39. EX: Heisenberg interaction (Pauli matrices) • omnipresent in solid state physics (« Easy ») • is not universal: had to be supplemented with single qubit gates On three qubits: E12 E23 E13

  40. The algebra L3(E) of E (3 qubits) Lie algebra of E : the algebra L3(E) splits into irreducible components as: L3(E)  L3(E)  S1I4 S2I2 su(2)S2 Encoded qubit ? su(2) 2 su(2) 2

  41. The algebra L3(E) of E (3 qubits) Lie algebra of E : the algebra L3(E) splits into irreducible components 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 !

  42. The algebra Ln(E) of E (n qubits) ... the algebra Ln(E) splits into irreducible components as: Ln(E)  ... ...

  43. The algebra Ln(E) of E (n qubits) ... the algebra Ln(E) splits into irreducible components 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).

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

  45. Useful theorem Let L be a †-closed algebra closed under multiplication and linear combination. Then the underlying space H is isomorphic to and L and its commutant L’ 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:

  46. Problem of “Encoded Universality” Given an ensemble of generators H with Lie algebra Lie(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 ... ...

  47. EX: The algebra Ln(E) of E (n qubits) ... the algebra Ln(E) splits into irreducible components as: Ln(E)  ... ... Results: E is universal with encoding* introduce tensor structure, ex. blocks with 3 qubits** 19 operations for CNOT, 4 operations for 1-qubit *J.K., D.Bacon, D.A.Lidar, K.B.Whaley, Phys. Rev. A 63:042307 (2001) **D.DiVincenzo, D.Bacon, J.K., K.B.Whaley, NATURE 408 (2000)

  48. Ex: Heisenberg interaction (Pauli matrices) E12 E23 On three qubits: E13 su(2) 2 su(2) 2 Proof idea: su(2)=Lie(x,y,z) [x,y]= iz [y,z]= ix [z,x]= iy [i,j]= i ijkk su(2)

  49. Overview • Basic notions of quantum computing • Standard solutions: • Universal gate set • Quantum Error Correcting Codes (QECC) • Encoded Universality • Decoherence-free Subsystems

  50. Decoherence-free subsystems (DFS)(“Dual” approach to encoded universality) • interaction described by system-environment Hamiltonian: System (quantum computer) Environment interaction causes noise system environmt. Decoherence

More Related