Review on quantum search algorithms

2. GROVER’S ALGORITHM FINDS THE TARGET ITEM
IN A DATABASE

3. ORACLE
• THE ORACLE IS DESIGNED SUCH THAT IT WILL FLIP THE PROBABILITY AMPLITUDE OF THE TARGET ITEMS
• GROVER’S ALGORITHM ONLY NEEDS 𝑂( 𝑁) QUERIES ( AGAINST 𝑂(𝑁) IN CLASSICAL COUNTERPART)
• THE ACTION OF ORACLE MAY BE WRITTEN AS: 𝑥 → (−1)𝑓 𝑥
𝑥

4. DIFFUSION OPERATOR
• IT INVERTS THE AMPLITUDE OF QUANTUM STATES ABOUT AVERAGE AMPLITUDE
• AN N-QUBIT DIFFUSION OPERATOR CAN BE EXPRESSED AS: 2|Ψ⟩⟨Ψ| − 𝐼⊗𝑛 WHERE |𝜓⟩ IS THE
NORMALIZED SUM OF EQUALLY DISTRIBUTED SUPERPOSITION STATE

5. LET’S SEE SOME RESULTS

6. Board please

8. SCHEMATIC

9. DIFFERENCE
• INSTEAD OF FINDING THE TARGET ITEM, THE PARTIAL SEARCH ALGORITHMFINDS THE TARGET BLOCK
• 𝑁 ITEMS IS DIVIDED INTO 𝐾 BLOCKS, EACH BLOCK CONSISTS OF 𝑏 ITEMS 𝑁 = 𝑘𝑏
• ASSUME 𝑁 = 2𝑛 & 𝑏 = 2𝑚 & 𝑘 = 2𝑛−𝑚, SO :
• THE PARTIAL SEARCH ALGORITHM FINDS THE ITEMS WITH THE LAST 𝑛 − 𝑚 QUBITS MATCH TO THE
RIGHT STATE |𝑡𝑛−𝑚⟩
• GUARANTEE THE CORRECT SOLUTION BUT REQUIRES LESS QUERY.(O(√𝑁 − √𝑏))

10. RESULTS

12. SCHEMATIC

13. DIFFERENCE
• NO ORACLE OPTIMIZATION(SIMILAR TO GROVER )
• FOCUS ON OPTIMIZING THE DIFFUSION OPERATOR
• N-QUBIT ADDRESS OF THE TARGET STATE IS DIVIDED INTO |𝑡𝑛⟩ = |𝑡𝑚⟩ |𝑡𝑛−𝑚⟩ . PERFORMING THE LOCAL GROVER
SEARCH OPERATOR FOR THE FIRST 𝑚 QUBITS WILL AMPLIFY THE PROBABILITIES OF THE ITEMS WITH THE LAST 𝑛 −
𝑚 QUBITS IN THE CORRECT STATE. PERFORMING THE LOCAL GROVER SEARCH OPERATOR WILL AMPLIFY THE
PROBABILITIES OF THE ITEMS WITH FIRST 𝑚 QUBITS IN THE CORRECT STATE. WITH THESE TWO OPERATORS
WORKING COLLECTIVELY, WE CAN AMPLIFY THE PROBABILITY OF THE TARGET ITEM

14. More explanation please

15. RESULTS

16. INTERPRETATION OF IDEA
GROVER PARTIAL EFFICIENT

17. IBMQ-BOGOTA TEST
GROVER PARTIAL EFFICIENT