For computer scientists Quantum computation For computer scientists Unit vector Prof. Dorit Aharonov School of computer science and engineering Hebrew university, Jerusalem, Israel

What is a computation La Segrada Familia (Barcelona) Architect: Gaudi Very different than the turing machine… La Segrada Familia (Barcelona) Architect: Gaudi

Computation (Algorithm) What is a computation? Computation (Algorithm) Input 0111001 Output 011000 Universal computation models: A Q B C ≈ Very different than the turing machine… Turing machine, 1936 Uniform Circuits ≈ ≈

Game of life Rules: A living site: stays alive if it has 2 or 3 live neighbors Otherwise dies A dead site: comes to life if it has exctly 3 living neighbors Very different than the turing machine…

≈ ≈ The Extended Church-Turing thesis (ECTT) A corner stone thesis in computer science: The Extended Church Turing thesis: “Any physically realizable computational model can be simulated efficiently by a randomized Turing machine” ≈ ≈ Quantum computation is the only computational model which credibly challenges the Extended Church Turing thesis

Bird’s view on Quantum computation Inherently different from standard “classical” computers. We believe that it will be exponentially more powerful for certain tasks. Polynomial time Quantum algorithm for factoring Deutsch Josza [‘92] Bernstein Vazirani[‘93] Simon [‘94] Shor[’94] 6 6 Physics of many particles (non universal computations) Philosophy of Science Algorithms Cryptography technology

About this school Goal: Intro to quantum computation & complexity Some important notions, results, open questions Note: We will not cover many important things… (A partial list will be provided & updated ) Two remarks: 1) The lectures are intertwined, not independent! 2) The TA sessions are mainly exercises. Do them! We rely on them in the next lecture. ---------------------------------------

Intro Lecture: Qubits Part 1: The principles of quantum Physics Part2: The qubit Part 3: Measurements Part 4: Dynamics Part 5: Two qubits ---------------------------------------

The principles of Quantum Physics Part I: The principles of Quantum Physics ---------------------------------------

The two slit experiment Part A: bullets Unit vector

Two slit experiments Part B: Water waves Unit vector

Two slit experiment Part C: electrons Unit vector Interference pattern for particles!

Explanation: superpositions and measurements The particle passes through both paths simultaneously! If measured, it collapses to one of the options Unit vector 1) The superposition principle 2) Measurement gives one option & changes the state

Part II: The Qubit Very different than the turing machine…

1st quantum principle: Superposition + b A quantum particle can be in a Superposition of all its possible “classical” states + a b + a b Unit vector

The elementary quantum information unit: Qubit = Quantum bit The qubit can be in either one of the States:0,1 As well as in any linear combination!  a vector in a 2 dim Hilbert space |0 Very different than the turing machine… On the board: Dirac notation Vector notation Transpose Inner products density matrix We can also talk about qudits, Of higher dim. + a b 1 |1

Part III: Measurements Very different than the turing machine…

The quantum measurement + a b 1 |a|2 |b|2 1 When a quantum particle is measured the answer is Probabilistic The Superposition collapses to one of its possible classical states Unit vector 2nd principle measurement Those (weird!) aspects have been tested in thousands of experiments

Projective measurements A projective measurement is described by a Hermitian matrix M. M has eigenvectors (eigenspaces) with associated real eigenvalues The classical outcome is eigenvalue 𝛌 with probability = norm squared of projection on the corresponsing eigenspace & the state collapses to this projection and renormlized On the board: Measure with respect to Z Prob=inner product squared Measure X: The +/- basis Probablity for 𝛱 a projection Expected value of measurement: Direct expression and as Tr(Mρ) Uncertainty principle Very different than the turing machine… + a b 1 1

Part IV: Dynamics Very different than the turing machine…

Dynamics Schrodinger’s equation: The Hamiltonian   Schrodinger’s equation: The Hamiltonian (A Hermitian operator) Discrete time evolution:     Very different than the turing machine… On the board: From the differential equation to unitary evolution (eigenvalues which are primitive roots of unity) Unitary as preserving inner product

Quantum Gates On the board:               Very different than the turing machine… On the board: Applying X,Z on computational basis states of a qubit Linearity Applying Hadamard on basis states and measuring (“coin flip”)

Interference & path integrals     |0 |0 |0 |1 |1 Nucleos spin – in a large constant magnetic field, apply a perturbation of an oscilating magnetic field in resonant frequency. Generates rotations of the spin. H H On the board: compute weights, repeat with measurement in the middle

Part V: Two qubits From time to time I will be using more technical terms, as parts of remarks. Ignore than if you don’t understand them- they are meant for references for those who do understand, but are not necessary For the rest of the talk. The few technical terms I will need will be explained in detail.

The superposition principle for more qubits one two three The state of n quantum bits is a superposition of all 2n possible configurations, each with its own weight!

The space of two qubits The EPR state: The computational Basis     The computational Basis for the two qubits space     The EPR state:

1st ex. of entanglement: The CHSH game They win if: > 0.85! Pr(success) with EPR