Grover. Part 2

Components of Grover Loop The Oracle -- O The Hadamard Transforms -- H The Zero State Phase Shift -- Z O is an Oracle H is Hadamards H is Hadamards Z is Zero State Phase Shift Grover Iterate

Inputs oracle We need to initialize in a superposed state This is action of quantum oracle

This is a typical way how oracle operates This is a typical way how oracle operation is described Encodes input combination with changed sign in a superposition of all

Role of Oracle We want to encode input combination with changed sign in a superposition of all states. This is done by Oracle together with Hadamards. We need a circuit to distinguish somehow globally good and bad states.

Vector of Hadamards

Notation Reminder a Control with value a=1 a Control with value a=0 a equivalent

This is value of oracle bit Flips the data phase All information of oracle is in the phase but how to read it? This is just an example of a single minterm, but can be any function Zero State Phase Shift Circuit

Flips the oracle bit when all bits are zero Rewriting matrix Z to Dirac notation, you can change phase globally This is state of all zeros

= With accuracy to phase

In each G This is a global view of Grover. Repeatitions of G Here you have all components of Grover’s loop

Generality Observe that a problem is described only by Oracle. So by changing the Oracle you can have your own quantum algorithm. You can still improve the Grover loop for particular special cases

proof Here we explain in detail what happens inside G. This can be generalized to G- like circuits Grover iterate has two tasks: (1) invert the solution states and (2) invert all states about the mean

Here we prove that |  > <  | used inside HZH calculates the mean a Vector of mean values Will be explained in next slide Explanation of the first part of Grover iterate formula

This proof is easy and it only uses formalisms that we already know. () ( ) From previous slide What does it mean invert all states about the mean?

For every bit Amplitudes of bits after Hadamard Positive or negative amplitudes in other explanations All possible states

Amplitudes of bits after one stage of G This value based on previous slide

This slides explains the basic mechanism of the Grover-like algorithms

You can verify it also in simulation Additional Exercise This is a lot calculations, requires matrix multiplication

Here we calculate analytically when to stop The equations taken from the previous slides “Grover Iterate” For marked state For unmarked state

We found k from these equations recursion We want to find how many times to iterate

But you can do better if you have knowledge, for instance the upper bound of chromatic number in graph coloring

Grover search example. Here is an example of Grover search for n = 3 qubits, where N = 2 n =8. –We omit reference to qubit n+1, which is in state 1 /√2 (|0>−|1>i) and does not change. The dimension of the unitary operators for this example is thus 2 n = 8 also.)

(Remember that numbering starts with 0 and ends with 7, so that the -1 here is in the slot for |5>.) This matrix reverses the sign on state |5>, and leaves the other states unchanged. Suppose the unknown number is |a> = |5>. The matrix or black box oracle Ufa is oracle

The Walsh matrix W 8 is Now we use normalization

The matrix −U f 0 is

This matrix changes the sign on all states except |0>. Finally, we have the repeated step R s R a in the Grover algorithm: oracle shift hadamards

After second rotation we get

Summary and our work When you know anything about the problem (symmetry, observation, bounds, function within some classification class) you can design a better Grover like algorithm but for your data only. This is enough in real life like CAD or Image Processing, since data are always specific, not the worst case data as in Mathematic proofs

Problem for students Build the Grover algorithm for ternary quantum logic. First you need to generalize Hadamard transform to Chrestenson transform. Next you need to have some kind of ternary reversible gates to build oracle. The same gates will be used for Zero State Phase Shift circuit.