Quantum Computing. Quantum Computing. Sushain Cherivirala. Bits and Qubits

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1 Quantum Computing Bits and Qubits Quantum Computing Sushain Cherivirala Quantum Gates Measurement of Qubits More Quantum Gates Universal Computation Entangled States Superdense Coding Measurement Revisited Quantum Teleportation Surprise?

2 Quantum Computing Bits and Qubits Quantum Gates Measurement of Qubits More Quantum Gates Universal Computation Entangled States Superdense Coding Measurement Revisited Quantum Teleportation Surprise? Bits and Qubits The fundamental units of information

3 Classical Bits (light switch) 0 and 1 1 ψ Bits and Qubits Quantum Bits β Computational Basis States 0 and 1 (Dirac/bra- ket notation) 1 0 and 0 (vector in complex space notation) 1 General Quantum State α 0 α 0 + β 1 = α β = ψ with α and β as amplitudes α ) + β ) = 1 (normalization condition unit vector) Start with mathematical description then develop physical intuition for results

4 Quantum Computing Bits and Qubits Quantum Gates Measurement of Qubits More Quantum Gates Universal Computation Entangled States Superdense Coding Measurement Revisited Quantum Teleportation Surprise? Quantum Gates Manipulation of bits (computation) builds logical systems (computers)

5 Classical NOT gate Quantum NOT gate (Pauli- X gate) NOT Gate α 0 + β 1 α 1 + β 0 X = 0 1 (Pauli matrix) 1 0 XX = 1 0 = I (Identity matrix) 0 1 X X X Quantum NOT gate and Classical NOT gate are practically equivalent

6 Hadamard Gate Hadamard Gate / ) ) α 0 + β 1 α + β α β 2 1 H = (Hadamard matrix) ) 1 1 HH = = I H 8 = 2 ) , H8 0 1 (not normalized) Useful in taking shortcuts, similar to quantum tunneling, by expanding the states computer can assume H H H

7 Quantum Computing Bits and Qubits Quantum Gates Measurement of Qubits More Quantum Gates Universal Computation Entangled States Superdense Coding Measurement Revisited Quantum Teleportation Surprise? Measurement of Qubits Computation is pointless without measurement of results

8 Measurement in the computation basis Quantum state not directly observable from a qubit 0 with P = α ) α 0 + β 1 : 1 with P = β ) Measurement disturbs state, results in a computational basis state α ) + β ) = 1 (normalization constraint)

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10 Quantum Computing Bits and Qubits Quantum Gates Measurement of Qubits More Quantum Gates Universal Computation Entangled States Superdense Coding Measurement Revisited Quantum Teleportation Surprise? More Quantum Gates Gates, unitary transformations, serve as primitives for computation

11 General single- qubit Gates Pauli- X (NOT) Gate X = Hadamard Gate H = ) 1 1 General Single- qubitgate U = a b (Unitary matrix) c d U E U = I where U E = (U G ) Preservation of length U ψ = ψ X H Preservation of length is the unique property of unitary matrices

12 : 0 ekl 0 1 e KM 1 Phase shift Gates α 0 + β 1 αe KL 0 + βe KM 1 R L,M = ekl 0 0 e KM At first glance, αe KL 0 + βe KM 1 is not distinguishable from α 0 + β 1 but can be made so through the Hadamard gate R L = cos θ sinθ sin θ cos θ

13 Pauli X- Gate X = X Pauli Gates Pauli Y- Gate Y = 0 i i 0 Y Pauli Z- Gate Z = Z

14 Controlled- NOT Gate (C- NOT) Final gate that is essential, all else can be built upon it Computational basis: α 00 + β 01 + γ 10 + δ 11 = α ) + β ) + γ ) + δ ) = xy x y x xyz xy z y CNOT = Control qubit Target qubit α β γ δ

15 Quantum Computing Bits and Qubits Quantum Gates Measurement of Qubits More Quantum Gates Universal Computation Entangled States Superdense Coding Measurement Revisited Quantum Teleportation Surprise? Universal Computation Only a small set of primitives is necessary for building complex systems of grand proportions

16 Universal single gates are the NOR and NAND gates. Classical Universality NOR gate = NOT gate + OR gate NAND gate = NOT gate + AND gate Universal set of gates sufficient for all classical computation

17 Quantum Universality CNOT and single- qubit gates provide quantum universality for any unitary quantum operation on n qubits

18 Circuit Models Can be shown that all classical systems can be converted into equivalent quantum systems of roughly the same size Numerous equivalent models exist that describe quantum systems, quantum circuit model has an analogy in Classical computing Classical Computing Quantum Computing Sometimes, the quantum circuit can be significantly shorter than the classical circuit as in Shor s algorithm that can factor N in polynomial time O log N p whereas the fastest classical factoring algorithm, the general number field sieve works in sub- exponential time O e 2.r stuv w y x stustu v x, subsequently breaking public- key cryptography such as RSA and destroying the world as we know it

19 Tangent

20 Quantum Computing Bits and Qubits Quantum Gates Measurement of Qubits More Quantum Gates Universal Computation Entangled States Superdense Coding Measurement Revisited Quantum Teleportation Surprise? Entangled States Entangled states enable quantum computation to be more powerful than classical computation

21 H Entangled States: Intro CNOT H 00 CNOT ) ) ) α 0 + β 1 )(γ 0 + δ 1 ) = ψ φ (non- separable) Result is a non- classical state, an entangled state, useful for all sorts of things to come; essential difference between a quantum computer from a classical computer Provable that quantum algorithm without entangled states can be made equivalent to a similarly performant classical algorithm

22 Entangled States: Intuition Nobody knows precisely why entangled states are required for performant quantum algorithms ψ = ψ ψ amplitudes (information) hidden within n qubits

23 Quantum Computing Bits and Qubits Quantum Gates Measurement of Qubits More Quantum Gates Universal Computation Entangled States Superdense Coding Measurement Revisited Quantum Teleportation Surprise? SuperdenseCoding Optimal protocol for equating two classical bits with one qubit

24 Superdense Coding Eve 0 0 H Alice : I 01: X 10: Z 11:XZ Bob H The entangled state that Eve produces enables the protocol Discovered in 1992 by Charles Bennet and Steven Wiesner (Paper)

25 H Bell State More generally, for all normalized states ψ and φ, there exists a unitary U such that U ψ = φ. Symbolically, ψ, φ, U U ψ = φ where ψ = 1! φ = 1! U E U = I

26 Bell Basis Bell Basis ) ) ) ) = 2 ) = 2 ) = 2 ) = 2 ) Orthonomal quantum states and orthogonal normalized quantum states can be distinguished via a unitary function In each state, determining one bit determines the other

27 EPR Paradox God does not play dice with the Universe ~ Albert Einstein Quantum mechanics is an incomplete theory in Einstein s view Flipping a coin is not random; knowledge of position and momentum degrees of freedom makes it deterministic Speed of light limits determination of second bit in a bell state John Bell s Bell Experiment discredits Einstein s hidden local variables theory and supports quantum mechanical probabilities

28 Quantum Computing Bits and Qubits Quantum Gates Measurement of Qubits More Quantum Gates Universal Computation Entangled States Superdense Coding Measurement Revisited Quantum Teleportation Surprise? Measurement Revisited Generalization of measurement concepts

29 ψ = 2 ) 0 + Š L ) 1 Standard Orthonormal basis Measurement in a non- standard basis P Œ 0 = P Œ 1 = 2 ) Non- standard +, basis + 2 ( ) 0 ) ) 2 / ( 0 1 ) 1 2 ) ) ψ = 2 ) 0 + Š L ) 1 = 21Š ) 2 ( + + ) ( + ) Š ) 1 / P Œ + = cos ) L ) P Œ = sin ) L ) from e KL = cos θ + i sin θ (Euler relation)

30 ψ = α 0 + β 1 1 ψ Measurement in the general orthonormal basis General Orthonormal basis + ν = a 0 + b 1 ν = b 0 a 1 ν ν 8 = 0 P Œ ν = ν ψ ) = a α + b β ) ν 8 ν 0 P Œ ν 8 = ν 8 ψ ) = bα aβ )

31 Partial Measurements in the computational basis α 00 + β 01 + γ 10 + δ 11 Measured Qubit P 0 = P( 00 ) + P( 01 ) = α ) + β ) Posterior Qubit ψ = 0 α 0 + β (γ 0 + δ 1 ) ψ = y 1 y

32 Quantum Computing Bits and Qubits Quantum Gates Measurement of Qubits More Quantum Gates Universal Computation Entangled States Superdense Coding Measurement Revisited Quantum Teleportation Surprise? Quantum Teleportation Cloning quantum state with two classical bits

33 Quantum Teleportation: Problem P We have time for this = P I will ever understand any of this = 0

34 Google and NASA's Quantum Artificial Intelligence Lab October 11, 2013