1

Quantum logic gates

Logic gates

Classical NOT gateQuantum NOT gate

(X gate)

A N O T A

0 1

1 0

A NOT A

The only non-trivial

single bit gate

X10α β+ 01α β+

Matrix form representation

0 1

1 0X

=

Xβα

β α

=

2

More single qubit gates

Any unitary matrix U will produce a quantum gate!

Z0 1α β++++ 0 1α β−−−−1 0

0 1Z

= −

1 11

1 12

H

= −

H0 1α β+0 1 0 1

2 2α β

+ −+

Hadamard gate:

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!

3

Universality:classical computation

Only one classical gate (NAND) is needed to

compute any function on bits!

A

BA AND B

A NOT A

A

BA NAND B

A B A AND B A NAND B

0 0 0 1

0 1 0 1

1 0 0 1

1 1 1 0

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

4

Quantum Quantum Quantum Quantum CNOTCNOTCNOT

CNOT gategategategate

Quantum algorithms

Superposition: n qubits can represent 2n 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.

Unique features of quantum computation

5

Quantum algorithms

Strategy:

Use superposition to calculate 2n valuesof function simultaneously and

do not read out the result until a usefulresult 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.

6

Quantum cryptography

Classical cryptography

Scytale - the first known mechanical device to implement

permutation of characters for cryptographic purposes

7

Classical cryptography

Private key cryptography

How to securely transmit a private key?

Scientific American 314, 48-55 (2016)

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).

Basic idea of public key cryptosystems (much like a mailbox)

Alice sets a mailbox.Public key is available to the public

Public

Alice has secret key

Anyone can send mail

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).

Private

Public key distribution

RSA cryptosystems

9

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!

How does it work?

Scientific American 314, 48-55 (2016)

10

Why public key encryption works?

Because some mathematical operations are easy

to do but very hard to undo:

The Mathematical Guts of RSA Encryption(http://fringe.davesource.com/Fringe/Crypt/RSA/Algorithm.html)

Multiply 18313 and 22307: 408 508 091 - Easy

Easy Easy

Easy

Now try to factor 408 508 091 back into these two numbers – Very hard

Very hardVery hard

Very hard

2014 factoring record: 1199

2014 factoring record: 11992014 factoring record: 1199

2014 factoring record: 1199-

--

-bit number (360 decimal digits)

bit number (360 decimal digits) bit number (360 decimal digits)

bit number (360 decimal digits)

7500 CPU-years on 2.2 GHz Opterons

For any positive integers

k is a non-negative integer and .

Modular arithmetic = ordinary arithmetic in which we pay attention to remaindersonly. Notation (mod n) is used to indicate that we are working in modular arithmetic.

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

How to factor numbers?

Modular arithmetic – working only with remainders

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

Class exercise: calculate

Class exercise: calculate

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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 was to find period R. This is the step that is hard for classical computers for large n.

3) Calculate greatest common divisor

How to factor numbers?

13

Quantum factoring

Large-scale quantum computer will be able to break public-

key 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).