Quantum Speedups

1. Quantum Speedups DoRon Motter August 14, 2001

2. Introduction • Two main approaches are known which produce fast Quantum Algorithms • The first, and main approach is the Quantum Fourier Transform • This is used in number factoring, discrete logarithm, and other algorithms • The second approach is Grover’s search

3. Notation • Let 0.j1j2j3…= j1/2+j2/4+j3/8… • Hn denote taking the tensor product n times

4. Discrete Fourier Transform • Defined as • where 0  k  N – 1 • Fast implementation takes O(N lg N)

5. Quantum Fourier Transform • The same transformation • On an orthonormal basis defined as

6. Quantum Fourier Transform • Alternately, on an n qubit computer, use the basis • N=2n

7. Quantum Fourier Transform • That’s great, but how do you build it? • Rewriting QFT expression gives rise to a circuit which implements it

8. Quantum Fourier Transform

9. Quantum Fourier Transform

10. QFT: Gates • To build the circuit for the QFT, we will need an additional gate: • We’ll also need the Hadamard gate

11. QFT: Implementation • Using R and H we can produce an efficient circuit for the QFT • The circuit comes naturally since the formula for QFT has been decomposed into the product representation

12. QFT: Implementation

13. QFT: Implementation • What is the effect of ? • Recall: • Symbolically

14. QFT: Implementation • What is the effect of ? • Notice:

15. QFT: Implementation • Using these effects we can verify the circuit

16. QFT: Implementation • Begin in initial state • The first gate is the Hadamard gate on bit 1 • Next, apply R2

17. QFT: Implementation

18. QFT: Implementation • Continue to apply R3…Rn, giving • This produces the desired ‘factor’ in the product representation

19. QFT: Implementation

20. QFT: Implementation • The next level of the circuit acts similarly: • Applying the Hadamard gate gives

21. QFT: Implementation • Applying the gates R2…Rn-1 gives After all gates are applied, the state will be

22. QFT: Summary • The QFT uses n(n+1)/2 gates not counting swaps • QFT is unitary, since each gate is unitary • Using the QFT is subtle • There is no way of directly accessing the result • There is no way (in general) of preparing the initial state efficiently

23. Search Algorithms • A simple example of search: • Everyone’s second C++ program: for(int x = 2; x <= sqrt(n); ++x) { if( “n is divisible by x” ) return Composite; } return Prime;

24. Search Algorithms • Oracle Search • Oracle(x) takes the value 1 iff x is a solution to the search problem • Grover’s search uses an oracle • In general, a unitary operator • The specific oracle depends on the search desired

25. Grover Iteration • Grover’s Search is the repeated application of a single operation • This operation is called the Grover operator, G • Understanding G is key to understanding Grover’s search

26. Grover Iteration • G consists of four ‘steps’ • Apply the Oracle operator O • Apply the Hadamard transform Hn • Give every basis state except a phase shift of –1 • Apply the Hadamard transform Hn

27. Grover Iteration • Give every basis state except a phase shift of –1 • This can be written

28. Grover Iteration • Consider the last 3 steps • Apply the Hadamard transform Hn • Apply the Hadamard transform Hn Together these give:

29. Grover Iteration • G consists of four ‘steps’ • Apply the Oracle operator O • Apply the Hadamard transform Hn • Apply the Hadamard transform Hn Together these give:

30. Grover’s Search • Takes: A black box oracle O which performs • n+1 qubits in the state

31. Grover’s Search • Runtime: • Procedure: • Initialize states: • Apply Hn to the first n quibits, and HX to the last qubit • Apply the Grover iteration • Measure first n qubits

32. Grover’s Search • Procedure: • Initialize states: • Apply Hn to the first n quibits, and HX to the last qubit • Apply the Grover iteration • Measure first n qubits

33. Conclusion