WIXLIB

Quantum Information in A Nutshell吳致盛Jhih-Sheng Wu陽明交大光電Department of Photonics NYCU2021

OutlineQuantum Information Types of Quantum Technologies Qubits and Gates Entanglement AlgorithmsReference:Physics 160 Lecture Notes by Prof. Mikhail LukinJSWOutline2 / 20

Types of Quantum TechnologiesClassification by Prof. Lukin Quantum Metrology: superposition and entanglement to makemore precise measurements Quantum Communication: superposition and entanglement totransmit information in a secure way. No-cloning theorem! Quantum Computing: a superposition of input states, differentinputs interfere, provides “quantum parallelism”. Quantum FourierTransform.JSWTypes of Quantum Technologies3 / 20

Qubits and Bloch SphereA qubit is a two-state system: Q⟩ c0 0⟩ c1 1⟩. 0⟩ and 1⟩ are arbitrary two different states. For example, they can be H⟩and V⟩ of a plane wave.θθc0 cosc1 sin eiϕ22JSWQubits and Gates4 / 20

Single Qubit Operations (Gates)OperationsSingle qubit operations are rotations on the Blcoh sphere.X gate180 -Rotation about the x-axis(0 1X 1 0( ) ( )01X 10)Arbitrary-angle θ-rotationRX ( θ) e iJSW θX2Qubits and Gates5 / 20

Single Qubit Operations (Gates)OperationsSingle qubit operations are rotations on the Blcoh sphere.Y gate180 -Rotation about the y-axis(Y 12 12)(0 iY i 0)( 1 ) i 2 12Arbitrary-angle θ-rotationRY ( θ) e iJSW θY2Qubits and Gates6 / 20

Single Qubit Operations (Gates)OperationsSingle qubit operations are rotations on the Blcoh sphere.Z gate180 -Rotation about the z-axis(Z 12 12)(1 0Z 0 1) ( 1 ) 2 1 2Arbitrary-angle θ-rotationRz ( θ) e iJSW θZ2Qubits and Gates7 / 20

Single Qubit Operations (Gates)Hadamard gate180 -Rotation about the(x̂ ẑ 2)-axis1H 2()1 11 1QuestionH X, ⟩ ?JSWQubits and Gates8 / 20

Single Qubit Operations (Gates)JSWQubits and Gates9 / 20

Two-qubit Operations (Gates)Controlled NOT gate (Controlled X gate)If the first qubit is 0⟩, do nothing on the second qubit.If the first qubit is 1⟩, apply X on the second qubit.Controlled Z gateIf the first qubit is 0⟩, do nothing on the second qubit.If the first qubit is 1⟩, apply Z on the second qubit.JSWQubits and Gates10 / 20

More GatesJSWQubits and Gates11 / 20

Universal Set of Quantum GatesUniversality TheoremAny n-dimensional unitary operator can be decomposed as a product oftwo-dimensional operators.Specific Sets of Quantum Gates1. H, CNOT, and T2. Toffoli and HJSWQubits and Gates12 / 20

Quantum ProgramingIBM QiskitJSWQubits and Gates13 / 20

Quantum ProgramingQuantum circuitJSWQubits and Gates14 / 20

Quantum ProgramingYou can run the quantum code with python and real qubits!JSWQubits and Gates15 / 20

Bell StatesDefinitionA state is entangled if ψ⟩ ̸ ψ⟩A ψ⟩B .Bell StatesTwo-level systems A and B. Bell states are 00⟩ 11⟩ 2 01⟩ 10⟩ 2 Φ ⟩AB Ψ ⟩ABMeasurement by Alice affects the reduced density matrix of Bob!JSWEntanglement16 / 20

Einstein–Podolsky–Rosen ParadoxEPR ParadoxEinstein did not believe that measurement by Alice can affect the spatiallyseparate state of Bob. His thought was local reality. In stead of 100% in 01⟩ 10⟩ ,2he thought that 50% in 10⟩and 50% in 01⟩He thought the state was not superposition! There was a reality beforemeasurement.JSWEntanglement17 / 20

Bell InequalityTo determine whether50% in100% in 10⟩ 01⟩ 10⟩ ,250% in 01⟩John Bell in 1964, if local reality, then Ch (a, b) Ch (a, c) 1 Ch (b, c)The inequality basically describes the correlations. But, this inequalitycan be violated if it is the left case.Experiments showed that the Bell’s inequality is violated.Superposition and entanglement are true!!JSWEntanglement18 / 20

Application of Entanglement Quantum key distribution Superdense coding Quantum teleportationJSWEntanglement19 / 20

Quantum Algorithms Deutsch’s AlgorithmDetermine if a function f(x) is constant for x 0, 1 and f(x) 0, 1. Deutsch-Jozsa algorithmDetermine if a function f(xn ) is constant for all xn 0, 1 andf(xn ) 0, 1. Grover’s AlgorithmFind xi such f(xi ) 1 when all other f(xn ) 0. If the number of xn isN, the classical algorithm takes O(N) steps. Grover’s algorithm takesO( N) steps. Shor’s AlgorithmDecompose a product of two large prime number N pq.Acceleration from exponential time to polynomial time.JSWQuantum Algorithms20 / 20

Quantum Algorithms Deutsch's Algorithm Determine if a function f(x) is constant for x 0,1 and f(x) 0,1. Deutsch-Jozsa algorithm Determine if a function f(xn) is constant for all xn 0,1 and f(xn) 0,1. Grover's Algorithm Find xi such f(xi) 1 when all other f(xn) 0.If the number of xn is N, the classical algorithm takes O(N) steps.