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 Let f : {0,1} → {0,1} f There are four possibilities: xf1(x)f1(x) 0 1 0 1 0 xf2(x)f2(x) 0 1 0 1 1 xf3(x)f3(x) 0 1 0 1 0 1 xf4(x)f4(x) 0 1 0 1 0 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  The final state is |0> for constant and |1> for non-constant  Operate |0> <1| : one measurement distinguishes between constant and non constant functions. |0> |1> H H |0>-|1> |0>+|1> H

7. Note: Vectors vis-à-vis Rays xf1(x)f1(x) 0 1 0 1 0 xf2(x)f2(x) 0 1 0 1 1 xf3(x)f3(x) 0 1 0 1 0 1 xf4(x)f4(x) 0 1 0 1 0 (0 + 1)(0 + 1) (0 – 1)(0 – 1)

8. Experimental Implementation: IISc group

9. Generalization: Deutsch Jozsa algorithm F:{0,..,2 n -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 2 n-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. Illustration: The 2-qubit case 256 such functions 4 functions in each category obtained by uniform translation 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. Circuit Continued… 3 Can be written as a product of single qubit gates

20. Applying the oracle gives : Circuit Explanation:…

21. The final state is : Circuit Continued…

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

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

29. Basic idea behind the generalization Cosets of the subgroup consisting of U(f i ’ ) 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 {k i } The 16 cosets {k i } 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 |00> |11> |01> |10> 0000→01100220 → 03300202 → 03120022 → 0132 1111 → 12211331 → 10011313 → 10231133 → 1203 2222 → 23322002 → 21122020 → 21302200 → 2310 3333 → 30033113 → 32233131 → 32013311 → 3021

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} (k 0 k 1 k 2 k 3 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(k 1 + k 2 – k 3 – k 0 )/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. There are N N functions. ( N = 2 n ) Subgroup – N 2 elements N N-2 cosets = =

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