Logic gates. Quantum logic gates. α β 0 1 X = 1 0. Quantum NOT gate (X gate) Classical NOT gate NOT A. Matrix form representation

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1 Quantum logic gates Logic gates Classical NOT gate Quantum NOT gate (X gate) A NOT A α 0 + β 1 X α 1 + β 0 A N O T A Matrix form representation 0 1 X = 1 0 The only non-trivial single bit gate α β X = β α 1

2 More single qubit gates Any unitary matrix U will produce a quantum gate! Z 1 0 = 0 1 α 0 + β 1 α 0 β 1 Z Hadamard gate: H = α 0 + β 1 H α + β 2 2 Single qubit gates,two-qubit gates, three-qubit gates How many gates do we need to make? Do we need three-qubit and four-qubit gates? Where do we find such physical interactions? Coming up with one suitable controlled interaction for physical system is already a problem! 2

3 Universality: classical computation Only one classical gate (NAND) is needed to compute any function on bits! A B A A B A AND B NOT A A NAND B A B A AND B A NAND B Universality: quantum computation How many quantum gates do we need to build any quantum gate? Any n-qubit gate can be made from 2-qubit gates. (Since any unitary n x n matrix can be decomposed to product of two-level matrices.) Only one two-qubit gate is needed! Example: CNOT gate 3

4 Quantum CNOT gate Quantum algorithms Unique features of quantum computation Superposition: n qubits can represent 2 n integers. Problem: if we read the outcome we lose the superposition and we can t know with certainty which one of the values we will obtain. Entanglement: measurements of states of different qubits may be highly correlated. 4

5 Quantum algorithms Strategy: Use superposition to calculate 2 n values of function simultaneously and do not read out the result until a useful result is expected with reasonably high probability. Use entanglement Quantum algorithms Shor's quantum Fourier transform provides exponential speedup over known classical algorithms. Applications: solving discrete logarithm and factoring problems which enables a quantum computer to break public key cryptosystems such as RSA. Quantum searching (Grover's algorithm) allows quadratic speedup over classical computers. Simulations of quantum systems. 5

6 Quantum cryptography Classical cryptography Scytale - the first known mechanical device to implement permutation of characters for cryptographic purposes 6

7 Classical cryptography Private key cryptography How to securely transmit a private key? Scientific American 314, (2016) 7

8 Key distribution A central problem in cryptography: the key distribution problem. 1) Mathematics solution: public key cryptography. 2) Physics solution: quantum cryptography. Public-key cryptography relies on the computational difficulty of certain hard mathematical problems (computational security) Quantum cryptography relies on the laws of quantum mechanics (information-theoretical security). Public key distribution RSA cryptosystems Basic idea of public key cryptosystems (much like a mailbox) Alice sets a mailbox. Public key is available to the public Public Private Anyone can send mail Alice has secret key Only Alice can get the mail out of the mailbox Result: anyone in the world can communicate with Alice privately. Note:there are two distinct keys; a public key and a private key (which only Alice has). 8

9 How does it work? Suppose Bob wishes to send private message to Alice. (1) Alice generates two keys, a public key (P) and a secret (private) key (S). (2) Bob gets a copy of a public key (P). (3) Bob encrypts the message using P. Encryption stage is very difficult to reverse! Like a trap door for the mail: if you put in your mail you can not get it out. Bob sends the encrypted message. (4) Alice uses a secret key to decrypt the message. Problem: There is no known scheme which is proven to be secure. It is just widely believed that it is! Scientific American 314, (2016) 9

10 Why public key encryption works? Because some mathematical operations are easy to do but very hard to undo: Multiply and 22307: Easy Now try to factor back into these two numbers Very hard 2014 factoring record: 1199-bit number (360 decimal digits) 7500 CPU-years on 2.2 GHz Opterons The Mathematical Guts of RSA Encryption ( How to factor numbers? Modular arithmetic working only with remainders For any positive integers k is a non-negative integer and. Modular arithmetic = ordinary arithmetic in which we pay attention to remainders only. Notation (mod n) is used to indicate that we are working in modular arithmetic. Class exercise: Prove that 2=5=8=11 (mod 3) 10

11 Class exercise: Prove that 2=5=8=11 (mod 3) Class exercise: calculate Class exercise: calculate 11

12 How to factor numbers? Classical factoring algorithm: How to factor 15? (1) Pick a number less than 15 (for example 7). (2) Calculate How to factor numbers? The point of calculating for classical computers for large n. was to find period R. This is the step that is hard 3) Calculate greatest common divisor 12

13 Quantum factoring Large-scale quantum computer will be able to break publickey encryption. Key distribution A central problem in cryptography: the key distribution problem. 1) Mathematics solution: public key cryptography. 2) Physics solution: quantum cryptography. Public-key cryptography relies on the computational difficulty of certain hard mathematical problems (computational security) Quantum cryptography relies on the laws of quantum mechanics (information-theoretical security). 13