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Quantum algorithms Daniel J. Bernstein Quantum algorithm means an - PDF document

Apr 24, 2023 •169 likes •1.53k views

1 Quantum algorithms Daniel J. Bernstein Quantum algorithm means an algorithm that a quantum computer can run. i.e. a sequence of instructions, where each instruction is in a quantum computers supported instruction set. How do we

  1. 9 e.g.: Say 3 qubits have state (1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1). Measurement produces 000 = 0 with probability 1 = 8; 001 = 1 with probability 1 = 8; 010 = 2 with probability 1 = 8; 011 = 3 with probability 1 = 8; 100 = 4 with probability 1 = 8; 101 = 5 with probability 1 = 8; 110 = 6 with probability 1 = 8; 111 = 7 with probability 1 = 8. “Quantum RNG.”

  2. 9 e.g.: Say 3 qubits have state (1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1). Measurement produces 000 = 0 with probability 1 = 8; 001 = 1 with probability 1 = 8; 010 = 2 with probability 1 = 8; 011 = 3 with probability 1 = 8; 100 = 4 with probability 1 = 8; 101 = 5 with probability 1 = 8; 110 = 6 with probability 1 = 8; 111 = 7 with probability 1 = 8. “Quantum RNG.” Warning: Quantum RNGs sold today are measurably biased.

  3. 10 e.g.: Say 3 qubits have state (3 ; 1 ; 4 ; 1 ; 5 ; 9 ; 2 ; 6).

  4. 10 e.g.: Say 3 qubits have state (3 ; 1 ; 4 ; 1 ; 5 ; 9 ; 2 ; 6). Measurement produces 000 = 0 with probability 9 = 173; 001 = 1 with probability 1 = 173; 010 = 2 with probability 16 = 173; 011 = 3 with probability 1 = 173; 100 = 4 with probability 25 = 173; 101 = 5 with probability 81 = 173; 110 = 6 with probability 4 = 173; 111 = 7 with probability 36 = 173.

  5. 10 e.g.: Say 3 qubits have state (3 ; 1 ; 4 ; 1 ; 5 ; 9 ; 2 ; 6). Measurement produces 000 = 0 with probability 9 = 173; 001 = 1 with probability 1 = 173; 010 = 2 with probability 16 = 173; 011 = 3 with probability 1 = 173; 100 = 4 with probability 25 = 173; 101 = 5 with probability 81 = 173; 110 = 6 with probability 4 = 173; 111 = 7 with probability 36 = 173. 5 is most likely outcome.

  6. 11 e.g.: Say 3 qubits have state (0 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0).

  7. 11 e.g.: Say 3 qubits have state (0 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0). Measurement produces 000 = 0 with probability 0; 001 = 1 with probability 0; 010 = 2 with probability 0; 011 = 3 with probability 0; 100 = 4 with probability 0; 101 = 5 with probability 1; 110 = 6 with probability 0; 111 = 7 with probability 0.

  8. 11 e.g.: Say 3 qubits have state (0 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0). Measurement produces 000 = 0 with probability 0; 001 = 1 with probability 0; 010 = 2 with probability 0; 011 = 3 with probability 0; 100 = 4 with probability 0; 101 = 5 with probability 1; 110 = 6 with probability 0; 111 = 7 with probability 0. 5 is guaranteed outcome.

  9. 12 NOT gates NOT 0 gate on 3 qubits: (3 ; 1 ; 4 ; 1 ; 5 ; 9 ; 2 ; 6) �→ (1 ; 3 ; 1 ; 4 ; 9 ; 5 ; 6 ; 2).

  10. 12 NOT gates NOT 0 gate on 3 qubits: (3 ; 1 ; 4 ; 1 ; 5 ; 9 ; 2 ; 6) �→ (1 ; 3 ; 1 ; 4 ; 9 ; 5 ; 6 ; 2). NOT 0 gate on 4 qubits: (3 ; 1 ; 4 ; 1 ; 5 ; 9 ; 2 ; 6 ; 5 ; 3 ; 5 ; 8 ; 9 ; 7 ; 9 ; 3) �→ (1 ; 3 ; 1 ; 4 ; 9 ; 5 ; 6 ; 2 ; 3 ; 5 ; 8 ; 5 ; 7 ; 9 ; 3 ; 9).

  11. 12 NOT gates NOT 0 gate on 3 qubits: (3 ; 1 ; 4 ; 1 ; 5 ; 9 ; 2 ; 6) �→ (1 ; 3 ; 1 ; 4 ; 9 ; 5 ; 6 ; 2). NOT 0 gate on 4 qubits: (3 ; 1 ; 4 ; 1 ; 5 ; 9 ; 2 ; 6 ; 5 ; 3 ; 5 ; 8 ; 9 ; 7 ; 9 ; 3) �→ (1 ; 3 ; 1 ; 4 ; 9 ; 5 ; 6 ; 2 ; 3 ; 5 ; 8 ; 5 ; 7 ; 9 ; 3 ; 9). NOT 1 gate on 3 qubits: (3 ; 1 ; 4 ; 1 ; 5 ; 9 ; 2 ; 6) �→ (4 ; 1 ; 3 ; 1 ; 2 ; 6 ; 5 ; 9).

  12. 12 NOT gates NOT 0 gate on 3 qubits: (3 ; 1 ; 4 ; 1 ; 5 ; 9 ; 2 ; 6) �→ (1 ; 3 ; 1 ; 4 ; 9 ; 5 ; 6 ; 2). NOT 0 gate on 4 qubits: (3 ; 1 ; 4 ; 1 ; 5 ; 9 ; 2 ; 6 ; 5 ; 3 ; 5 ; 8 ; 9 ; 7 ; 9 ; 3) �→ (1 ; 3 ; 1 ; 4 ; 9 ; 5 ; 6 ; 2 ; 3 ; 5 ; 8 ; 5 ; 7 ; 9 ; 3 ; 9). NOT 1 gate on 3 qubits: (3 ; 1 ; 4 ; 1 ; 5 ; 9 ; 2 ; 6) �→ (4 ; 1 ; 3 ; 1 ; 2 ; 6 ; 5 ; 9). NOT 2 gate on 3 qubits: (3 ; 1 ; 4 ; 1 ; 5 ; 9 ; 2 ; 6) �→ (5 ; 9 ; 2 ; 6 ; 3 ; 1 ; 4 ; 1).

  13. � � � � 13 state measurement (1 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0) 000 � (0 ; 1 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0) 001 (0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 0 ; 0) 010 � (0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 0) 011 (0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0) 100 � (0 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0) 101 (0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0) 110 � (0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 1) 111 Operation on quantum state: NOT 0 , swapping pairs. Operation after measurement: flipping bit 0 of result. Flip: output is not input.

  14. 14 Controlled-NOT (CNOT) gates e.g. C 1 NOT 0 : (3 ; 1 ; 4 ; 1 ; 5 ; 9 ; 2 ; 6) �→ (3 ; 1 ; 1 ; 4 ; 5 ; 9 ; 6 ; 2).

  15. 14 Controlled-NOT (CNOT) gates e.g. C 1 NOT 0 : (3 ; 1 ; 4 ; 1 ; 5 ; 9 ; 2 ; 6) �→ (3 ; 1 ; 1 ; 4 ; 5 ; 9 ; 6 ; 2). Operation after measurement: flipping bit 0 if bit 1 is set; i.e., ( q 2 ; q 1 ; q 0 ) �→ ( q 2 ; q 1 ; q 0 ⊕ q 1 ).

  16. 14 Controlled-NOT (CNOT) gates e.g. C 1 NOT 0 : (3 ; 1 ; 4 ; 1 ; 5 ; 9 ; 2 ; 6) �→ (3 ; 1 ; 1 ; 4 ; 5 ; 9 ; 6 ; 2). Operation after measurement: flipping bit 0 if bit 1 is set; i.e., ( q 2 ; q 1 ; q 0 ) �→ ( q 2 ; q 1 ; q 0 ⊕ q 1 ). e.g. C 2 NOT 0 : (3 ; 1 ; 4 ; 1 ; 5 ; 9 ; 2 ; 6) �→ (3 ; 1 ; 4 ; 1 ; 9 ; 5 ; 6 ; 2).

  17. 14 Controlled-NOT (CNOT) gates e.g. C 1 NOT 0 : (3 ; 1 ; 4 ; 1 ; 5 ; 9 ; 2 ; 6) �→ (3 ; 1 ; 1 ; 4 ; 5 ; 9 ; 6 ; 2). Operation after measurement: flipping bit 0 if bit 1 is set; i.e., ( q 2 ; q 1 ; q 0 ) �→ ( q 2 ; q 1 ; q 0 ⊕ q 1 ). e.g. C 2 NOT 0 : (3 ; 1 ; 4 ; 1 ; 5 ; 9 ; 2 ; 6) �→ (3 ; 1 ; 4 ; 1 ; 9 ; 5 ; 6 ; 2). e.g. C 0 NOT 2 : (3 ; 1 ; 4 ; 1 ; 5 ; 9 ; 2 ; 6) �→ (3 ; 9 ; 4 ; 6 ; 5 ; 1 ; 2 ; 1).

  18. 15 Toffoli gates Also known as CCNOT gates: controlled-controlled-NOT gates. e.g. C 2 C 1 NOT 0 : (3 ; 1 ; 4 ; 1 ; 5 ; 9 ; 2 ; 6) �→ (3 ; 1 ; 4 ; 1 ; 5 ; 9 ; 6 ; 2).

  19. 15 Toffoli gates Also known as CCNOT gates: controlled-controlled-NOT gates. e.g. C 2 C 1 NOT 0 : (3 ; 1 ; 4 ; 1 ; 5 ; 9 ; 2 ; 6) �→ (3 ; 1 ; 4 ; 1 ; 5 ; 9 ; 6 ; 2). Operation after measurement: ( q 2 ; q 1 ; q 0 ) �→ ( q 2 ; q 1 ; q 0 ⊕ q 1 q 2 ).

  20. 15 Toffoli gates Also known as CCNOT gates: controlled-controlled-NOT gates. e.g. C 2 C 1 NOT 0 : (3 ; 1 ; 4 ; 1 ; 5 ; 9 ; 2 ; 6) �→ (3 ; 1 ; 4 ; 1 ; 5 ; 9 ; 6 ; 2). Operation after measurement: ( q 2 ; q 1 ; q 0 ) �→ ( q 2 ; q 1 ; q 0 ⊕ q 1 q 2 ). e.g. C 0 C 1 NOT 2 : (3 ; 1 ; 4 ; 1 ; 5 ; 9 ; 2 ; 6) �→ (3 ; 1 ; 4 ; 6 ; 5 ; 9 ; 2 ; 1).

  21. 16 More shuffling Combine NOT, CNOT, Toffoli to build other permutations.

  22. 16 More shuffling Combine NOT, CNOT, Toffoli to build other permutations. e.g. series of gates to rotate 8 positions by distance 1: 3 1 4 1 ▲ 5 9 2 6 ▲ sssssssss ▲ ▲ ▲ ▲ C 0 C 1 NOT 2 ▲ ▲ ▲ 3 1 ✾ 4 6 5 9 ✾ 2 1 ✾ ✾ ✆ ✆ ✾ ✾ ✆ ✆ ✾ ✾ ✆ ✆ C 0 NOT 1 ✾ ✾ ✆ ✆ ✆ ✆ 3 ✱ 6 4 ✱ 1 5 ✱ 1 2 ✱ 9 ✱ ✱ ✱ ✱ ✒✒✒✒ ✒✒✒✒ ✒✒✒✒ ✒✒✒✒ ✱ ✱ ✱ ✱ NOT 0 ✱ ✱ ✱ ✱ 6 3 1 4 1 5 9 2

  23. 17 Hadamard gates Hadamard 0 : ( a; b ) �→ ( a + b; a − b ). 3 1 4 1 5 9 2 6 ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✡✡✡✡ ✡✡✡✡ ✡✡✡✡ ✡✡✡✡ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ 14 − 4 − 4 4 2 5 3 8

  24. 17 Hadamard gates Hadamard 0 : ( a; b ) �→ ( a + b; a − b ). 3 1 4 1 5 9 2 6 ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✡✡✡✡ ✡✡✡✡ ✡✡✡✡ ✡✡✡✡ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ 14 − 4 − 4 4 2 5 3 8 Hadamard 1 : ( a; b; c; d ) �→ ( a + c; b + d; a − c; b − d ). 3 ❋ 1 ❋ 4 1 5 ❋ 9 ❋ 2 6 ❋ ❋ ❋ ❋ ①①①①①①① ①①①①①①① ①①①①①①① ①①①①①①① ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ − 1 7 2 0 7 15 3 3

  25. 18 Some uses of Hadamard gates Hadamard 0 , NOT 0 , Hadamard 0 : 3 1 4 1 5 9 2 6 ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✡✡✡✡ ✡✡✡✡ ✡✡✡✡ ✡✡✡✡ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ 4 2 5 3 14 ✹ − 4 8 ✹ − 4 ✹ ✹ ✹ ✹ ✹ ✹ ✡✡✡✡ ✡✡✡✡ ✡✡✡✡ ✡✡✡✡ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ 2 4 3 5 − 4 ✹ 14 − 4 8 ✹ ✹ ✹ ✹ ✹ ✡✡✡✡ ✡✡✡✡ ✡✡✡✡ ✡✡✡✡ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ 6 − 2 8 − 2 10 − 18 4 − 12

  26. 18 Some uses of Hadamard gates Hadamard 0 , NOT 0 , Hadamard 0 : 3 1 4 1 5 9 2 6 ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✡✡✡✡ ✡✡✡✡ ✡✡✡✡ ✡✡✡✡ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ 4 2 5 3 14 ✹ − 4 8 ✹ − 4 ✹ ✹ ✹ ✹ ✹ ✹ ✡✡✡✡ ✡✡✡✡ ✡✡✡✡ ✡✡✡✡ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ 2 4 3 5 − 4 ✹ 14 − 4 8 ✹ ✹ ✹ ✹ ✹ ✡✡✡✡ ✡✡✡✡ ✡✡✡✡ ✡✡✡✡ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ 6 − 2 8 − 2 10 − 18 4 − 12 “Multiply each amplitude by 2.” This is not physically observable.

  27. 18 Some uses of Hadamard gates Hadamard 0 , NOT 0 , Hadamard 0 : 3 1 4 1 5 9 2 6 ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✡✡✡✡ ✡✡✡✡ ✡✡✡✡ ✡✡✡✡ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ 4 2 5 3 14 ✹ − 4 8 ✹ − 4 ✹ ✹ ✹ ✹ ✹ ✹ ✡✡✡✡ ✡✡✡✡ ✡✡✡✡ ✡✡✡✡ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ 2 4 3 5 − 4 ✹ 14 − 4 8 ✹ ✹ ✹ ✹ ✹ ✡✡✡✡ ✡✡✡✡ ✡✡✡✡ ✡✡✡✡ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ 6 − 2 8 − 2 10 − 18 4 − 12 “Multiply each amplitude by 2.” This is not physically observable. “Negate amplitude if q 0 is set.” No effect on measuring now .

  28. 19 Fancier example: “Negate amplitude if q 0 q 1 is set.” Assumes q 2 = 0: “ancilla” qubit. 3 1 4 1 ▲ 0 0 0 0 ▲ sssssssss ▲ ▲ ▲ ▲ C 0 C 1 NOT 2 ▲ ▲ ▲ 3 ▲ 1 ▲ 4 ▲ 0 ▲ 0 0 0 1 ▲ ▲ ▲ ▲ sssssssss sssssssss sssssssss sssssssss ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ Hadamard 2 ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ 3 ▲ 1 ▲ 4 ▲ 1 ▲ 3 1 4 − 1 ▲ ▲ ▲ ▲ sssssssss sssssssss sssssssss ▲ ▲ ▲ ▲ sssssss ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ NOT 2 ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ 3 ▲ 1 ▲ 4 − 1 3 1 4 1 ▲ ▲ ▲ ▲ ▲ sssssssss sssssssss sssssssss sssssssss ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ Hadamard 2 ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ 6 2 8 0 ▲ 0 0 0 − 2 ▲ ▲ sssssss ▲ ▲ ▲ C 0 C 1 NOT 2 ▲ ▲ ▲ 6 2 8 − 2 0 0 0 0

  29. 20 Affects measurements: “Negate amplitude around its average.” (3 ; 1 ; 4 ; 1) �→ (1 : 5 ; 3 : 5 ; 0 : 5 ; 3 : 5).

  30. 20 Affects measurements: “Negate amplitude around its average.” (3 ; 1 ; 4 ; 1) �→ (1 : 5 ; 3 : 5 ; 0 : 5 ; 3 : 5). 3 1 4 1 0 0 0 0 ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ☞☞☞☞ ☞☞☞☞ ☞☞☞☞ ☞☞☞☞ ✷ ✷ ✷ ✷ H 0 ✷ ✷ ✷ ✷ 4 ❉ 2 ❉ 5 3 0 ❉ 0 ❉ 0 0 ❉ ❉ ❉ ❉ ③ ③ ③ ③ ❉ ❉ ❉ ❉ ③ ③ ③ ③ ❉ ❉ ❉ ❉ ③ ③ ③ ③ ❉ ❉ ❉ ❉ ③ ③ ③ ③ H 1 ❉ ❉ ❉ ❉ ③ ③ ③ ③ ❉ ❉ ③ ③ ③ ③ ③ ③ ③ ③ 9 5 − 1 − 1 0 0 0 0 : : : − 9 ✷ 5 − 1 ✷ − 1 0 0 0 0 ✷ ✷ ✷ ✷ ☞☞☞☞ ✷ ✷ ☞☞☞☞ ☞☞☞☞ ☞☞☞☞ ✷ ✷ ✷ ✷ H 0 ✷ ✷ ✷ ✷ − 4 − 14 − 2 0 0 ❉ 0 ❉ 0 0 ❉ ❉ ❉ ❉ ③ ③ ③ ❉ ❉ ③③③③③ ❉ ❉ ③ ③ ③ ❉ ❉ ❉ ❉ ③ ③ ③ ❉ ❉ ❉ ❉ ③ ③ ③ H 1 ❉ ❉ ❉ ❉ ③ ③ ③ ❉ ❉ ③ ③ ③ ③ ③ − 6 − 14 − 2 − 14 0 0 0 0

  31. 21 Simon’s algorithm Assumptions: • Given any u ∈ { 0 ; 1 } n , can efficiently compute f ( u ). • Nonzero s ∈ { 0 ; 1 } n . • f ( u ) = f ( u ⊕ s ) for all u . • f has no other collisions. Goal: Figure out s .

  32. 21 Simon’s algorithm Assumptions: • Given any u ∈ { 0 ; 1 } n , can efficiently compute f ( u ). • Nonzero s ∈ { 0 ; 1 } n . • f ( u ) = f ( u ⊕ s ) for all u . • f has no other collisions. Goal: Figure out s . Traditional algorithm to find s : compute f for many inputs, hope to find collision.

  33. 21 Simon’s algorithm Assumptions: • Given any u ∈ { 0 ; 1 } n , can efficiently compute f ( u ). • Nonzero s ∈ { 0 ; 1 } n . • f ( u ) = f ( u ⊕ s ) for all u . • f has no other collisions. Goal: Figure out s . Traditional algorithm to find s : compute f for many inputs, hope to find collision. Simon’s algorithm finds s with ≈ n quantum computations of f .

  34. 22 Example of Simon’s algorithm Step 1. Set up pure zero state: 1 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 :

  35. 22 Example of Simon’s algorithm Step 2. Hadamard 0 : 1 ; 1 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 :

  36. 22 Example of Simon’s algorithm Step 3. Hadamard 1 : 1 ; 1 ; 1 ; 1 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 :

  37. 22 Example of Simon’s algorithm Step 4. Hadamard 2 : 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 : Each column is a parallel universe.

  38. 22 Example of Simon’s algorithm Step 5. C 0 NOT 3 : 1 ; 0 ; 1 ; 0 ; 1 ; 0 ; 1 ; 0 ; 0 ; 1 ; 0 ; 1 ; 0 ; 1 ; 0 ; 1 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 : Each column is a parallel universe performing its own computations.

  39. 22 Example of Simon’s algorithm Step 5b. More shuffling: 1 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 : Each column is a parallel universe performing its own computations.

  40. 22 Example of Simon’s algorithm Step 5c. More shuffling: 1 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 1 : Each column is a parallel universe performing its own computations.

  41. 22 Example of Simon’s algorithm Step 5d. More shuffling: 1 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 0 : Each column is a parallel universe performing its own computations.

  42. 22 Example of Simon’s algorithm Step 5e. More shuffling: 1 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 : Each column is a parallel universe performing its own computations.

  43. 22 Example of Simon’s algorithm Step 5f. More shuffling: 0 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 1 ; 0 : Each column is a parallel universe performing its own computations.

  44. 22 Example of Simon’s algorithm Step 5g. More shuffling: 0 ; 1 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 0 ; 1 : Each column is a parallel universe performing its own computations.

  45. 22 Example of Simon’s algorithm Step 5h. More shuffling: 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 1 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 : Each column is a parallel universe performing its own computations.

  46. 22 Example of Simon’s algorithm Step 5i. More shuffling: 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 : Each column is a parallel universe performing its own computations.

  47. 22 Example of Simon’s algorithm Step 5j. Final shuffling: 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 1 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 : Each column is a parallel universe performing its own computations.

  48. 22 Example of Simon’s algorithm Step 5j. Final shuffling: 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 1 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 : Each column is a parallel universe performing its own computations. Surprise: u and u ⊕ 101 match.

  49. 22 Example of Simon’s algorithm Step 6. Hadamard 0 : 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 1 ; 1 ; 0 ; 0 ; 1 ; 1 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 1 ; 1 ; 0 ; 0 ; 1 ; 1 ; 1 ; 1 ; 0 ; 0 ; 1 ; 1 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 1 ; 1 ; 0 ; 0 ; 1 ; 1 ; 0 ; 0 : Notation: 1 means − 1.

  50. 22 Example of Simon’s algorithm Step 7. Hadamard 1 : 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 :

  51. 22 Example of Simon’s algorithm Step 8. Hadamard 2 : 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 2 ; 0 ; 2 ; 0 ; 0 ; 2 ; 0 ; 2 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 2 ; 0 ; 2 ; 0 ; 0 ; 2 ; 0 ; 2 ; 2 ; 0 ; 2 ; 0 ; 0 ; 2 ; 0 ; 2 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 2 ; 0 ; 2 ; 0 ; 0 ; 2 ; 0 ; 2 :

  52. 22 Example of Simon’s algorithm Step 8. Hadamard 2 : 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 2 ; 0 ; 2 ; 0 ; 0 ; 2 ; 0 ; 2 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 2 ; 0 ; 2 ; 0 ; 0 ; 2 ; 0 ; 2 ; 2 ; 0 ; 2 ; 0 ; 0 ; 2 ; 0 ; 2 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 2 ; 0 ; 2 ; 0 ; 0 ; 2 ; 0 ; 2 : Step 9: Measure. Obtain some information about the surprise: a random vector orthogonal to 101.

  53. 23 Repeat to figure out 101.

  54. 23 Repeat to figure out 101. Generalize Step 5 to any function u �→ f ( u ) with f ( u ) = f ( u ⊕ s ). “Usually” algorithm figures out s .

  55. 23 Repeat to figure out 101. Generalize Step 5 to any function u �→ f ( u ) with f ( u ) = f ( u ⊕ s ). “Usually” algorithm figures out s . Shor’s algorithm replaces ⊕ with more general + operation. Many spectacular applications.

  56. 23 Repeat to figure out 101. Generalize Step 5 to any function u �→ f ( u ) with f ( u ) = f ( u ⊕ s ). “Usually” algorithm figures out s . Shor’s algorithm replaces ⊕ with more general + operation. Many spectacular applications. e.g. Shor finds “random” s with 2 u mod N = 2 u + s mod N . Easy to factor N using this.

  57. 23 Repeat to figure out 101. Generalize Step 5 to any function u �→ f ( u ) with f ( u ) = f ( u ⊕ s ). “Usually” algorithm figures out s . Shor’s algorithm replaces ⊕ with more general + operation. Many spectacular applications. e.g. Shor finds “random” s with 2 u mod N = 2 u + s mod N . Easy to factor N using this. e.g. Shor finds “random” s; t with 4 u 9 v mod p = 4 u + s 9 v + t mod p . Easy to compute discrete logs.

  58. 24 Grover’s algorithm Assume: unique s ∈ { 0 ; 1 } n has f ( s ) = 0. Traditional algorithm to find s : compute f for many inputs, hope to find output 0. Success probability is very low until #inputs approaches 2 n .

  59. 24 Grover’s algorithm Assume: unique s ∈ { 0 ; 1 } n has f ( s ) = 0. Traditional algorithm to find s : compute f for many inputs, hope to find output 0. Success probability is very low until #inputs approaches 2 n . Grover’s algorithm takes only 2 n= 2 reversible computations of f . Typically: reversibility overhead is small enough that this easily beats traditional algorithm.

  60. 25 Start from uniform superposition over all n -bit strings u .

  61. 25 Start from uniform superposition over all n -bit strings u . Step 1: Set a ← b where b u = − a u if f ( u ) = 0, b u = a u otherwise. This is fast.

  62. 25 Start from uniform superposition over all n -bit strings u . Step 1: Set a ← b where b u = − a u if f ( u ) = 0, b u = a u otherwise. This is fast. Step 2: “Grover diffusion”. Negate a around its average. This is also fast.

  63. 25 Start from uniform superposition over all n -bit strings u . Step 1: Set a ← b where b u = − a u if f ( u ) = 0, b u = a u otherwise. This is fast. Step 2: “Grover diffusion”. Negate a around its average. This is also fast. Repeat Step 1 + Step 2 about 0 : 58 · 2 0 : 5 n times.

  64. 25 Start from uniform superposition over all n -bit strings u . Step 1: Set a ← b where b u = − a u if f ( u ) = 0, b u = a u otherwise. This is fast. Step 2: “Grover diffusion”. Negate a around its average. This is also fast. Repeat Step 1 + Step 2 about 0 : 58 · 2 0 : 5 n times. Measure the n qubits. With high probability this finds s .

  65. 26 Normalized graph of u �→ a u for an example with n = 12 after 0 steps: 1.0 0.5 0.0 −0.5 −1.0

  66. 26 Normalized graph of u �→ a u for an example with n = 12 after Step 1: 1.0 0.5 0.0 −0.5 −1.0

  67. 26 Normalized graph of u �→ a u for an example with n = 12 after Step 1 + Step 2: 1.0 0.5 0.0 −0.5 −1.0

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