1. DIGITAL SWITCHING IN THE QUANTUM DOMAIN Riccardo RICCI, Francesco VITULO A.A. 2002/03 Corso di Nanotecnologie 1

2. R. Ricci, F. Vitulo2 QUANTUM STATE Each particle has its own quantum state. It can be represented as a linear combination of two eigenstates: |0  and |1 . |0  and |1  can be used to simulate the classical binary logic, so the particle is called qubit. A particle usually is in condition of superposition, and has a part |0  and a part |1 . When a particle is measured, it is projected to one of its states, |0  or |1 .

3. R. Ricci, F. Vitulo3 QUANTUM STATE The quantum state can be written in various ways. |  = c 0 |0  + c 1 |1  as a linear combination as a column matrix | c 0 | 2 + | c 1 | 2 = 1; | c 0 | 2, | c 1 | 2 : probability of obtaining the state |0  or |1 , respectively.

4. R. Ricci, F. Vitulo4 QUANTUM STATE Two or more qubits form a quantum system. |  AB = c 0 |00  AB + c 1 |01  AB + c 2 |10  AB + c 3 |11  AB | c 0 | 2 + | c 1 | 2 + | c 2 | 2 + | c 3 | 2 = 1 In this way we can generalize to a n-qubits system.

5. R. Ricci, F. Vitulo5 QUANTUM GATE A quantum gate manipulates a quantum system. It can be represented in form of a matrix operation. Example 1: NOT gate It changes the state from |0  to |1  and vice-versa.

6. R. Ricci, F. Vitulo6 QUANTUM GATE Example 2: Control-NOT (CN) gate. It has one control qubit and one target qubit. Target qubit changes his state if control qubit state = |1 .

7. R. Ricci, F. Vitulo7 QUANTUM GATE Symbols of quantum gates. (a): NOT gate (b): CN gate The horizontal line connecting input and output represents a qubit under time evolution.

8. R. Ricci, F. Vitulo8 QUBIT PERMUTATION We write a permutation in this way: The permutation P makes the following changes: a  db  ec  c d  ae  ff  b Any quantum boolean logic can be represented using a permutation.

9. R. Ricci, F. Vitulo9 QUBIT PERMUTATION A cycle is defined as: C = (e 1, e 2, …, e n-1, e n ) It changes: e 1  e 2… e n-1  e n e n  e 1 Special cases: c 1 = (e 1 )trivial cycle c 2 = (e 1, e 2 )transposition A trivial cycle can be ignored as it does not change anything.

10. R. Ricci, F. Vitulo10 QUBIT PERMUTATION A permutation can be expressed as disjoint cycles: P is equivalent to: P = (a, d) (c) (b, e, f) = (a, d) (b, e, f) The implementation consists of executing cycles of various lenghts in parallel.

11. R. Ricci, F. Vitulo11 QUBIT PERMUTATION The transposition of two qubits can be done using three CN gates, as shown in the picture below: The proof is in [1].

12. R. Ricci, F. Vitulo12 IMPLEMENTATION OF CYCLES A n-qubit cycle C can be done by six layers of CN gates. C = (q 0, q 1, …, q n-1 ) Case 1: if n is even (n = 2m), we define: X = (q m-1, q m+1 ) … (q 2, q n-2 ) (q 1, q n-1 ) Y = (q m, q m+1 ) … (q 2, q n-1 ) (q 1, q 0 ) The cycle is implemented as: U = YX

13. R. Ricci, F. Vitulo13 IMPLEMENTATION OF CYCLES Case 2: if n is odd (n = 2m + 1), we define: X = (q m, q m+1 ) … (q 2, q n-2 ) (q 1, q n-1 ) Y = (q m, q m+2 ) … (q 2, q n-1 ) (q 1, q 0 ) The cycle is implemented as: U = YX Case 1 Case 2

14. R. Ricci, F. Vitulo14 SWITCHING NETWORKS Example of circuit switching

15. R. Ricci, F. Vitulo15 CONNECTION DIGRAPH Given a n  n switch, a Connection Digraph is defined as: G t = {V, E t } 1) v i  V is a I/O port, i = 1, …, n – 1. 2) v m v n  E t if and only if there is a connection from input port v m to output port v n at time t.

16. R. Ricci, F. Vitulo16 CONNECTION DIGRAPH A null point has not neither input nor output. A loopback is a trvial cycle in which input traffic goes to the same port for output. 1) Null Points (N) & Loopbacks (L):

17. R. Ricci, F. Vitulo17 CONNECTION DIGRAPH A queue has an head and a tail node. Each node except head and tail has exactly one input and one output. A null point is a special queue with only one node. 2) Queue (Q):

18. R. Ricci, F. Vitulo18 CONNECTION DIGRAPH Each node of the cycle has exactly one input and one output. A cycle is obtained by a queue connecting the tail with the head. A loopback is a special cycle with only one node. 3) Cycle (C):

19. R. Ricci, F. Vitulo19 CONNECTION DIGRAPH A tree has one root node with no input and a collection of leaves with no output. Each node except these has only one input and at least one output. A queue is a special case of tree. 4) Tree (T):

20. R. Ricci, F. Vitulo20 CONNECTION DIGRAPH A forest has only one cycle and a collection of disjointed null points, queues and/or trees. It can be obtained from a tree connecting a leaf with the root. A cycle is a special case of forest. 5) Forest (F):

21. R. Ricci, F. Vitulo21 DIGITAL QUANTUM SWITCHING

22. R. Ricci, F. Vitulo22 DIGITAL QUANTUM SWITCHING The I/O port can be either quantum or classical oriented. Switching can be done efficiently using CN gates. It can also switch classical information using C/Q converters in input and Q/C converters in output.

23. R. Ricci, F. Vitulo23 DIGITAL QUANTUM SWITCHING Unicasting connection digraph is a collection of disjointed null points, loopbacks, queues and/or cycles as subdigraphs. Multicasting connection digraph contains trees and forests as subdigraphs. All these topologies are inter-related each other.

24. R. Ricci, F. Vitulo24 DIGITAL QUANTUM SWITCHING Sx means “is a special case of”. Ex means “can be extended to”. Tx represents the operations of “cycle extraction” and “link recovery”.

25. R. Ricci, F. Vitulo25 1) Cycle extraction: DIGITAL QUANTUM SWITCHING

26. R. Ricci, F. Vitulo26 1) Cycle extraction: It transforms a forest into one cycle and a collection of null points, queues and/or trees. If there are still any trees, they can be transformed in a forest and can be applied again the process of cycle extraction. In order to implement a connection digraph we need to transform every subdigraph into cycles or loopbacks. DIGITAL QUANTUM SWITCHING

27. R. Ricci, F. Vitulo27 DIGITAL QUANTUM SWITCHING 2) Link recovery:

28. R. Ricci, F. Vitulo28 DIGITAL QUANTUM SWITCHING 2) Link recovery: It recovers the links that have been cut. All the elementary topologies can be reduced to a collection of loopbacks and cycles: this allows an efficient implementation of the switching process.

29. R. Ricci, F. Vitulo29 UNICAST QUANTUM SWITCHING A typical unicast connection is the following: We need to implement the subdigraphs: G C = (q 3, q 4, q 6, q 7, q 5 ) G Q = [q 0, q 1, q 2 ]

30. R. Ricci, F. Vitulo30 UNICAST QUANTUM SWITCHING First, we extend G Q to G C’ = (q 0, q 1, q 2 ). The subdigraph G C can be done applying: X = (q 6, q 7 ) (q 4, q 5 ) Y = (q 6, q 5 ) (q 4, q 5 ) Then, we implement G C and G C’ using six layers of CN gates (see picture on the next slide).

31. R. Ricci, F. Vitulo31 UNICAST QUANTUM SWITCHING

32. R. Ricci, F. Vitulo32 MULTICAST QUANTUM SWITCHING It can be achieved reading a data packet once and writing it to multiple destinations. A typical configuration is the following: We need to implement the subdigraphs: G T = [q 0, q 1 ] [q 1, q 4 ] [q 1, q 3 ] [q 3, q 5, q 2 ] [q 3, q 6, q 7 ]

33. R. Ricci, F. Vitulo33 MULTICAST QUANTUM SWITCHING They are realized in the following way: For the details, see [1]

34. R. Ricci, F. Vitulo34 MULTICAST QUANTUM SWITCHING The switching circuit is the following: The total number of layers is 6 +  log 2 (r + 1)  where r is the number of connections that are to be recovered.

35. R. Ricci, F. Vitulo35 ADVANTAGES OF QUANTUM SWITCHING Quantum switching is strict-sense non- blocking: the network can always connect each idle inlet to an arbitrary idle outlet independent of the current network permutation. In fact, quantum switching is a unitary transformation, which is always possible.

36. R. Ricci, F. Vitulo36 ADVANTAGES OF QUANTUM SWITCHING Unicast quantum switching has time complexity O(1) as a space switch, because the circuit can be implemented with only six layers of CN gates, and has space complexity O(n), where n is the number of input qubits. Multicast quantum switching has time complexity O(log 2 n) and space complexity O(n). These values cannot be achieved in the same time with a classical switch.

37. R. Ricci, F. Vitulo37 ISSUES OF QUANTUM SWITCHING: DECOHERENCE Decoherence is a coupling between two initially isolated quantum systems (qubit and environment) that randomizes the relative phases of the states. It is the probability that quantum information spread out the computer, compromising the computation results. To avoid it, engineers should produce sub-micro systems in which qubits influence each other, but are completely insulated from the external environment.

38. R. Ricci, F. Vitulo38 ISSUES OF QUANTUM SWITCHING: DECOHERENCE In this case, we need to maximize: S max = t 0 / t d where t d is decoherence time and t 0 is the time of a single operation. If 6  t 0  t d, the speed of the switch can be: 1 / (6  t 0 ) bit/sec For most details about this issue, see [2].

39. R. Ricci, F. Vitulo39 ISSUES OF QUANTUM SWITCHING: ERRORS To reduce the probability of errors, there are a lot of error correction schemes. A bit of information can be encoded using m qubits. However, if operation time 6  t 0 is short compared with decoherence time, errors tend to be very small.

40. R. Ricci, F. Vitulo40 ISSUES OF QUANTUM SWITCHING: C/Q AND Q/C If the architecture is used to switch classical information, we need an interface formed by C/Q and Q/C converters. We assume that classical data are in optical form: C/Q converter must excite the state |0  (|1  ) if the incoming value is “0” (“1”). On the other hand, Q/C converter must convert the quantum state |0  or |1  back to the optical form, performing a measurement on the qubit.

41. R. Ricci, F. Vitulo41 ISSUES OF QUANTUM SWITCHING: QUBIT COPY It is not clear how a CN gate can make the copy of a qubit. It works only in two cases: (|0 , |0  ) and (|1 , |0  ). Control INTarget INControl OUT`Target OUT |0  |1  |0  |1  |0  |1  |0 

42. R. Ricci, F. Vitulo42 PHYSICAL REALIZATIONS OF QUANTUM GATES We have found two possible experimental realizations of CN gates: 1. Ramsey atomic interferometry. 2. Selective driving of optical resonances of two qubits undergoing a dipole-dipole interaction. We do not deal with the first (see [3] for more details).

43. R. Ricci, F. Vitulo43 REALIZATION WITH QUANTUM DOTS The qubits can be: 1. Magnetic dipoles, such as nuclear spins in external magnetic fields. 2. Electric dipoles, such as single-electron quantum dots in static electric fields. Mathematically these two cases are isomorphic, so we describe only the second.

44. R. Ricci, F. Vitulo44 REALIZATION WITH QUANTUM DOTS There are two quantum dots separated by a distance R, embedded in a semiconductor. Each dot represents a qubit. Control qubit has resonant frequency  1. Target qubit has resonant frequency  2. The ground state corresponds to state |0 , while the first excited state corresponds to state |1 .

45. R. Ricci, F. Vitulo45 REALIZATION WITH QUANTUM DOTS There is the quantum-confined Stark effect. In presence of an external static electric field, the charge distribution in the ground state (first excited state) is shifted in the direction of the field (in the opposite direction).

46. R. Ricci, F. Vitulo46 REALIZATION WITH QUANTUM DOTS The coordinates of the system are chosen such that dipole moments in states |0  and |1  are ±d i, where i = 1, 2 refers to control or target qubit. Approximation: the electric field from the electron in the first quantum dot may shift energy levels in the second one (and vice- versa), but it does not cause transitions.

47. R. Ricci, F. Vitulo47 REALIZATION WITH QUANTUM DOTS The previous approximation is valid because the total Hamiltonian: Ĥ = Ĥ 1 + Ĥ 2 + Û 12 is dominated by the dipole-dipole interaction term Û 12. Let’s define:

48. R. Ricci, F. Vitulo48 REALIZATION WITH QUANTUM DOTS Due to these interactions, the resonant frequency for transitions depends on the neighboring dot’s state. First (second) dot’s resonant frequency becomes    (   ±  ) if second (first) dot is in state |0  or |1 , respectively (see picture on next slide).

49. R. Ricci, F. Vitulo49 REALIZATION WITH QUANTUM DOTS

50. R. Ricci, F. Vitulo50 REALIZATION WITH QUANTUM DOTS Thus, a light  -pulse at frequency   +  causes the transitions |0   |1  in the second dot if and only if the first is in state |1 . In this way, a two quantum dots system can simulate the behavior of a control-NOT quantum gate.

51. R. Ricci, F. Vitulo51 ISSUES OF THIS APPROACH Decoherence time must be greater than time scale of the optical interaction. It is estimated as about 10 -6, but impurities and thermal vibration can reduce it to about 10 -9 or worse. Optical interaction time scale is about 10 -9.

52. R. Ricci, F. Vitulo52 ISSUES OF THIS APPROACH Length of  -pulse must be greater than the inverse of dipole-dipole interaction coupling constant. In this case:   10 12 Hz. This model is more difficult to implement than the one based on Ramsey atomic interferometry.

53. R. Ricci, F. Vitulo53 ADVANTAGES OF THIS APPROACH Effects of decoherence can be reduced with a more precise fabrication technology and by cooling the crystal. This approach allows an easy integration of quantum dots into complex quantum circuits, as required for quantum information processing.

54. R. Ricci, F. Vitulo54 CONCLUSION: QUANTUM COMPUTING Quantum computing is a good challenge for physicist and engineers for the coming years. In fact, quantum computers can solve exponentially complex problems in polynomial time. So, they give an answer in few seconds to problems that today require lots of years of computation.

55. R. Ricci, F. Vitulo55 CONCLUSION: QUANTUM COMPUTING The works we presented here are not about the realization of a quantum computer, but they can be considered a big part of it. Recently, IBM researchers built a 7-qubit quantum computer which factorized the number 15 in its prime factors 3 and 5. Despite its simplicity, it is the most complex quantum calculation ever carried out.

56. R. Ricci, F. Vitulo56 REFERENCES [1] I. M. Tsai and S. Y. Kuo, “Digital Switcing in the Quantum Domain”, IEEE Trans. on Nanotechnology, vol. 1, no. 3, pp. 154-164, Sep. 2002 [2] C. P. Williams and S. H. Clearwater, “Explorations in Quantum Computing”, Springer-Verlag

57. R. Ricci, F. Vitulo57 REFERENCES [3] A. Barenco, D. Deutsch, A. Ekert and R. Jozsa, “Conditional Quantum Dynamics and Logic Gates”, Physical Review Letters, vol. 74, pp. 4083-4086, May 1995