1. Quantum Computing Reversibility & Quantum Computing

2. Mandate “Why do all these Quantum Computing guys use reversible logic?”

3. Material  Logical reversibility of computation Bennett ’73  Elementary gates for quantum computation Berenco et al ’95  […] quantum computation using teleportation Gottesman, Chuang ’99

4. Material  Logical reversibility of computation Bennett ’73  Quantum computing needs logical reversibility  Elementary gates for quantum computation Berenco et al ’95  Gates can be thermodynamically irreversible  […] quantum computation using teleportation Gottesman, Chuang ’99

5. Material  Logical reversibility of computation Bennett ’73  Quantum computing needs logical reversibility  Elementary gates for quantum computation Berenco et al ’95  Gates can be thermodynamically irreversible  […] quantum computation using teleportation Gottesman, Chuang ’99

6. Heat Generation in Computing  Landauer’s Principle –Want to erase a random bit? It will cost you –Storing unwanted bits just delays the inevitable  Bennett’s Loophole –Computed bits are not random –Can uncompute them if we’re careful

7. Example Input (11) Work bits

8. Example Input (11) Work bits

9. Example Input (11) Work bits

10. Example Input (11) Work bits Output (1)

11. Example Input (11) Work bits Output

12. Example ComputeUncompute Copy Result

13. Thermodynamic Reversibility a b c c?b:a c?a:b c

14. Material  Logical reversibility of computation Bennett ’73  Quantum computing needs logical reversibility  Elementary gates for quantum computation Berenco et al ’95  Gates can be thermodynamically irreversible  […] quantum computation using teleportation Gottesman, Chuang ’99

15. Quantum State

16.  Two Distinguishable States

17.  ab+ Continuous State Space

18. Two Spin-½ Particles

19.         Four Distinguishable States

20. c+d+b+a         Continuous State Space

23. State Evolution  H is Hermitian, U is Unitary  Linear, deterministic, reversible  (Continuous form) (Discrete form)

24. Measurement  Outcome m occurs with probability p(m)  Operators M m non-unitary  Probabilistic, irreversible

25. Deriving Measurement “It can be done up to a point… But it becomes embarrassing to the spectators even before it becomes uncomfortable for the snake” – Bell “Like a snake trying to swallow itself by the tail”

26. A Simple Measurement  ab+   Outcome with probability

27. A Simulated Measurement  ab+ 

28.  ab+ 

29.  ab+ 

30.   Terms remain orthogonal – evolve independently, no interference or

31. Density Operator Representation

32. Mixed States

33. Partial Trace +     A B

34. Discarding a Qubit

35. Material  Logical reversibility of computation Bennett ’73  Quantum computing needs logical reversibility  Elementary gates for quantum computation Berenco et al ’95  Gates can be thermodynamically irreversible  […] quantum computation using teleportation Gottesman, Chuang ’99

36. Toffoli Gate a b c  ab a b c

37. Deutsch’s Controlled-U Gate a b c’c’ a b c U for Toffoli gate

38. = VV†V† VU Equivalent Gate Array for Toffoli gate

39. U = CBA P Equivalent Gate Array

40. Almost Any Gate is Universal

41. Material  Logical reversibility of computation Bennett ’73  Quantum computing needs logical reversibility  Elementary gates for quantum computation Berenco et al ’95  Gates can be thermodynamically irreversible  […] quantum computation using teleportation Gottesman, Chuang ’99

42. Protecting against a Bit-Flip (X) EvenOdd InputOutput Syndrome  third qubit flipped (reveals nothing about state)

43. Protecting against a Phase-Flip (Z) Phase flip (Z)

44. General Errors  Pauli matrices form basis for 1-qubit operators:  I is identity, X is bit-flip, Z is phase-flip  Y is bit-flip and phase-flip combined (Y = iXZ)

45. 9-Qubit Shor Code  Protects against all one-qubit errors  Error measurements must be erased  Implies heat generation

46. Material  Logical reversibility of computation Bennett ’73  Quantum computing needs logical reversibility  Elementary gates for quantum computation Berenco et al ’95  Gates can be thermodynamically irreversible  […] quantum computation using teleportation Gottesman, Chuang ’99

47. Fault Tolerant Gates H S

48. H H H H H H H encoded control qubit (Steane code) encoded target qubit (Steane code) encoded input qubit (Steane code) ZS encoded input qubit (Steane code)

49. Clifford Group  Encoded operators are tricky to design  Manageable for operators in Clifford group using stabilizer codes, Heisenberg representation  Map Pauli operators to Pauli operators  Not universal

50. H ZM1ZM1 M2M2 M1M1 Teleportation Circuit XM2XM2

51. Simplified Circuit

52. HX Equivalent Circuit

53. HXT Implementing a Gate

54. HSXT Implementing a Gate

55. HSXT Implementing a Gate

56. Conclusions  Quantum computing requires logical reversibility –Entangled qubits cannot be erased by dispersion  Does not require thermodynamic reversibility –Ancilla preparation, error measurement = refrigerator