DIGITAL SWITCHING IN THE QUANTUM DOMAIN

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

2. 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 calledqubit. • A particle usually is in condition of superposition, and has a part|0anda part |1. • When a particle is measured, it is projected to one of its states, |0or |1. R. Ricci, F. Vitulo

3. QUANTUM STATE • The quantum state can be written in various ways. • |y = c0|0+c1|1 as a linear combination • as a column matrix • | c0 |2 + | c1 |2 = 1; • | c0 |2, | c1 |2: probability of obtaining the state |0or |1, respectively. R. Ricci, F. Vitulo

4. QUANTUM STATE • Two or more qubits form a quantum system. • |yAB = c0|00AB+c1|01AB+ c2|10AB+c3|11AB • | c0 |2 + | c1 |2 +| c2 |2 + | c3 |2 = 1 • In this way we can generalize to a n-qubits system. R. Ricci, F. Vitulo

5. 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. R. Ricci, F. Vitulo

6. 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. R. Ricci, F. Vitulo

7. 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. R. Ricci, F. Vitulo

8. QUBIT PERMUTATION • We write a permutation in this way: • The permutation P makes the following changes: a  d b  e c  c d  a e  f f  b • Any quantum boolean logic can be represented using a permutation. R. Ricci, F. Vitulo

9. QUBIT PERMUTATION • A cycle is defined as: C = (e1, e2, …, en-1, en) • It changes: e1 e2 … en-1 en en  e1 • Special cases: c1 = (e1) trivial cycle c2 = (e1, e2) transposition • A trivial cycle can be ignored as it does not change anything. R. Ricci, F. Vitulo

10. 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. R. Ricci, F. Vitulo

11. 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]. R. Ricci, F. Vitulo

12. IMPLEMENTATION OF CYCLES • A n-qubit cycle C can be done by six layers of CN gates. C = (q0, q1, …, qn-1) • Case 1: if n is even (n = 2m), we define: X = (qm-1, qm+1) … (q2, qn-2) (q1, qn-1) Y = (qm, qm+1) … (q2, qn-1) (q1, q0) • The cycle is implemented as: U = YX R. Ricci, F. Vitulo

13. IMPLEMENTATION OF CYCLES • Case 2: if n is odd (n = 2m + 1), we define: X = (qm, qm+1) … (q2, qn-2) (q1, qn-1) Y = (qm, qm+2) … (q2, qn-1) (q1, q0) • The cycle is implemented as: U = YX Case 1 Case 2 R. Ricci, F. Vitulo

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

15. CONNECTION DIGRAPH • Given a nn switch, a Connection Digraph is defined as: Gt = {V, Et} • vi V is a I/O port, i = 1, …, n – 1. • vmvn  Et if and only if there is a connection from input port vm to output port vn at time t. R. Ricci, F. Vitulo

16. CONNECTION DIGRAPH 1)Null Points (N) & Loopbacks (L): • 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. R. Ricci, F. Vitulo

17. CONNECTION DIGRAPH 2)Queue (Q): • 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. R. Ricci, F. Vitulo

18. CONNECTION DIGRAPH 3)Cycle (C): • 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. R. Ricci, F. Vitulo

19. CONNECTION DIGRAPH 4)Tree (T): • 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. R. Ricci, F. Vitulo

20. CONNECTION DIGRAPH 5)Forest (F): • 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. R. Ricci, F. Vitulo

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

22. 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. R. Ricci, F. Vitulo

23. 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. R. Ricci, F. Vitulo

24. 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”. R. Ricci, F. Vitulo

25. DIGITAL QUANTUM SWITCHING • Cycle extraction: R. Ricci, F. Vitulo

26. DIGITAL QUANTUM SWITCHING • 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. R. Ricci, F. Vitulo

27. DIGITAL QUANTUM SWITCHING • Link recovery: R. Ricci, F. Vitulo

28. DIGITAL QUANTUM SWITCHING • 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. R. Ricci, F. Vitulo

29. UNICAST QUANTUM SWITCHING • A typical unicast connection is the following: • We need to implement the subdigraphs: GC = (q3, q4, q6, q7, q5) GQ = [q0, q1, q2] R. Ricci, F. Vitulo

30. UNICAST QUANTUM SWITCHING • First, we extend GQ to GC’ = (q0, q1, q2). • The subdigraph GC can be done applying: X = (q6, q7) (q4, q5) Y = (q6, q5) (q4, q5) • Then, we implement GC and GC’ using six layers of CN gates (see picture on the next slide). R. Ricci, F. Vitulo

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

32. 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: GT = [q0, q1] [q1, q4] [q1, q3] [q3, q5, q2] [q3, q6, q7] R. Ricci, F. Vitulo

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

34. MULTICAST QUANTUM SWITCHING • The switching circuit is the following: • The total number of layers is 6 + log2 (r + 1) where r is the number of connections that are to be recovered. R. Ricci, F. Vitulo

35. 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. R. Ricci, F. Vitulo

36. 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(log2 n) and space complexity O(n). • These values cannot be achieved in the same time with a classical switch. R. Ricci, F. Vitulo

37. 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. R. Ricci, F. Vitulo

38. ISSUES OF QUANTUM SWITCHING: DECOHERENCE • In this case, we need to maximize: Smax = t0 / td where td is decoherence time and t0 is the time of a single operation. • If 6  t0 td, the speed of the switch can be: 1 / (6  t0) bit/sec • For most details about this issue, see [2]. R. Ricci, F. Vitulo

39. 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  t0 is short compared with decoherence time, errors tend to be very small. R. Ricci, F. Vitulo

40. 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. R. Ricci, F. Vitulo

41. 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). R. Ricci, F. Vitulo

42. PHYSICAL REALIZATIONS OF QUANTUM GATES • We have found two possible experimental realizations of CN gates: • Ramsey atomic interferometry. • 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). R. Ricci, F. Vitulo

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

44. 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 w1. • Target qubit has resonant frequency w2. • The ground state corresponds to state |0, while the first excited state corresponds to state |1. R. Ricci, F. Vitulo

45. 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). R. Ricci, F. Vitulo

46. REALIZATION WITH QUANTUM DOTS • The coordinates of the system are chosen such that dipole moments in states |0 and |1 are ±di, 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. R. Ricci, F. Vitulo

47. REALIZATION WITH QUANTUM DOTS • The previous approximation is valid because the total Hamiltonian: Ĥ = Ĥ1 + Ĥ2 + Û12 is dominated by the dipole-dipole interaction term Û12. • Let’s define: R. Ricci, F. Vitulo

48. 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 w1 w(w2± w) if second (first) dot is in state |0 or |1, respectively (see picture on next slide). R. Ricci, F. Vitulo

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

50. REALIZATION WITH QUANTUM DOTS • Thus, a light p-pulse at frequency w2+ w 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. R. Ricci, F. Vitulo