Background

1. Generalized Deutsch Algorithms IPQI 5 Jan 2010

2. Background • Basic aim : Efficient determination of properties of functions. • Efficiency: No complete evaluation of the function itself. • Tools: Quantum Circuits. • Principles: Superposition, Entanglement

3. Example: Deutsch Algorithm f Let f : {0,1} → {0,1} There are four possibilities: Goal: Distinguish Constant from Non-constant Any classical method requires two queries

4. Deutsch contd….. Is There a Quantum Method? Answer: determine f(0)  f(1)

5. Quantum Oracle • Function black box or Oracle:- Unitary operation implementing unknown function • After applying a series of Gates and the Oracle, we “measure” the final state of the qubit in a suitable basis. • Membership to sets of functions with orthogonal final states determinable by measurement.

6. Deutsch Algorithm |0>+|1> • The final state is |0> for constant and |1> for non-constant • Operate |0><0|-|1><1| : one measurement distinguishes between constant and non constant functions. |0> H H |1> H |0>-|1>

7. Note: Vectors vis-à-vis Rays (0 –1) (0 +1)

8. Experimental Implementation: IISc group

9. Generalization: Deutsch Jozsa algorithm • F:{0,..,2n-1} → {0,1} Strong Restriction • Further Restrictions: Either balanced or constant • Recall: balanced functions send half the domain points to 0 and the other half to 1 • Question Posed : constant or balanced ?

10. The Circuit • Single Measurement/query does the job H

11. Deutsch Jozsa algorithm contd • Constant : outcome is |0> with probability 1 • Balanced : outcome is non |0> with probability 1 • Classical algorithm requires minimum of 2 and maximum of 2n-1+1 measurements H

12. AIM :Generalization to more general functions

13. Approach to generalization • Question posed must be ‘non-trivial’ • Functions must have symmetries that can be exploited. • Generalize Deutsch-Josza : Include a larger range and hence a larger class of functions.

14. Approach contd… • Constructive approach : Circuit is designed. • Allows the study of the relationship between quantum circuits and properties of functions,

15. Larger Aim • Theory of Quantum Circiits

16. 256 such functions 4 functions in each category obtained by uniform translation Illustration: The 2-qubit case CONSTANT PLATEAU/BASIN SAWTOOTH EVEN STEP

17. CONSTANT PLATEAU/BASIN SAWTOOTH EVEN STEP

18. Circuit: Ansatz • Category |0>,|1>,|2> or |3>. • Deutsch Jozsa is a subset • = identity, for the particular classification being attempted

19. 3 Circuit Continued… Can be written as a product of single qubit gates

20. Circuit Explanation:… Applying the oracle gives :

21. Circuit Continued… • The final state is :

22. Important Point Invariants within each category are in the parentheses.

23. Basic requirement : functions from different categories produce orthogonal final states • i.e. : if f belongs to the i-th category, |i> is obtained with probability 1 on measurement. • Eg : constant functions is the 0-th category CONSTANT PLATEAU/BASIN SAWTOOTH EVEN STEP |00> |11> |01> |10>

24. CONSTANT PLATEAU/BASIN SAWTOOTH EVEN STEP

25. Characteristics of categories: • Functions in a given category give the same ray as the output. • Such functions cannot be distinguished by the circuit. These indistinguishable functions form a category

26. Questions • Is a further generalization Possible? • Note: We still have 240 functions untouched • Is it possible to understand why the above circuit works?

27. The Underlying Group Structure • A unique Unitary transformation U(f) : action of the oracle on |x>|y>, corresponding to a function f. • {U(fi)} corresponding to the 256 functions form an abelian group, with composition as the group operation. .

28. Explicit form of U • For a given f which has f(0)=F0, f(1)=F1, f(2)=F2, f(3)=F3, U(f) is given by a 16 x 16 matrix Note: A A = A a b a  b

29. Basic idea behind the generalization • Cosets of the subgroup consisting of U(fi’) • Left and right cosets equivalent since the group is abelian • 16 cosets of order 16 each

30. Cosets • Each coset is “labelled” by the set {ki} • The 16 cosets {ki} required to exhaust all 256 group elements are as shown {0,0,0,0} {1,0,0,0} {2,0,0,0} {3,0,0,0} {0,1,1,0} {0,1,0,0} {1,0,3,0} {0,3,0,0} {0,0,1,1} {0,0,1,0} {1,3,0,0} {0,0,3,0} {0,1,0,1} {0,0,0,1} {1,0,0,3} {0,0,0,3} Note that within each coset, we can again distinguish between categories consisting of 4 functions each, as we shall now see.

31. Claim • within each coset, we can again distinguish four categories consisting of 4 functions each.

32. An Example: Coset generation and its labeling: {0,1,1,0} CONSTANT PLATEAU/BASIN SAWTOOTH EVEN STEP {0,1,1,0}

33. Modification of the circuit: Introduce

34. The structure of the Gate : =

35. Example 0000→0110 0220 → 0330 0202 → 0312 0022 → 0132 1111 → 1221 1331 → 1001 1313 → 1023 1133 → 1203 2222 → 2332 2002 → 2112 2020 → 2130 2200 → 2310 3333 → 3003 3113 → 3223 3131 → 3201 3311 → 3021 |00> |11> |01> |10>

36. Cost of the new gate Depends on the “entanglement” in the state!

37. Entanglement ? • Preparation of an entangled state is not required at any step in the Deutsch/Deutsch-Jozsa. • It is also not required if we were to consider only those 16 functions corresponding to the subgroup • However, the state shown, which is necessary for categorization of a coset, may be entangled for certain cosets.

38. is a measure of entanglement for : • Entanglement of the initial state {0,0,0,0} {1,0,0,0}{2,0,0,0}{3,0,0,0} {0,1,1,0}{0,1,0,0}{1,0,30}{0,3,0,0} {0,0,1,1} {0,0,1,0}{1,3,0,0 }{0,0,3,0} {0,1,0,1} {0,0,0,1}{1,0,0,3}{0,0,0,3} (k0 k1 k2 k3 for each coset) Partially entangled (E = 0.707) Entangled (E = 1) Not entangled (E = 0)

39. Entanglement of initial state • Identify a global property of the functions that is invariant within each coset • Measure of entanglement explicitly for the initial state. sin(π(k1 +k2 – k3 – k0)/4).

40. Entanglement of initial state • In the subgroup f(1)+f(2)-f(0)-f(3) = 0, • f(1)+f(2)-f(0)-f(3) = 0, we do not need an entangled state to categorize this coset. • f(1)+f(2)-f(0)-f(3) = 1 or 3, we need a partially entangled state • If f(1)+f(2)-f(0)-f(3) = 2, then we need a maximally entangled state

41. Points to note • The measure of entanglement which we have used has no non-trivial generalization to multipartite systems • In such cases, entropy of reduced density matrices is an indicator of entanglement • (In this particular case (bipartite), Entropy is a monotonic function of the measure used)

42. Generalization to n Qubits = =

43. Generalization to n Qubits • There are NN functions. ( N = 2n ) • Subgroup – N2 elements • NN-2 cosets = =

44. Thank You Collaborators : Vipul Ambasht Pronoy Sircar Sunil Yeshwanth