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Quantum Computers Quantum Computers ERIK LIND. PROFESSOR IN NANOELECTRONICS Erik Lind EIT, Lund University 1 Quantum Computers Classical Digital Computers Why quantum computers? Basics of QM • Spin-Qubits • Optical qubits • Superconducting Qubits • Topological Qubits Erik Lind EIT, Lund University 2 Classical Computing • We build digital electronics using CMOS • Boolean Logic • Classical Bit – 0/1 (Defined as a voltage level 0/+Vdd) +vDD +vDD +vDD Inverter NAND Logic is built by connecting gates Erik Lind EIT, Lund University 3 Classical Computing • We build digital electronics using CMOS • Boolean Logic • “Classical Bit” – 0/1 (0V or VDD) • Very rubust! • Inverter (and CMOS) has a large noise margin. • Modern CPU – billions of transistors (logic and memory) • We can keep a logical state for years (if attached to power) • We can easily copy a bit However – some problems are hard to solve on a classical computer Erik Lind EIT, Lund University 4 Computing Complexity Searching an unsorted list Grovers algorithm. BQP • P – easy to solve and check on a 푛 classical computer ~O(N) • NP – easy to check – hard to solve on a classical computer ~O(2N) P P • BQP – easy to solve on a quantum computer. ~O(N) NP • BQP is larger then P Factoring large • Some problems in P can be more numbers – efficiently solved by a quantum Shor’s Algorithm computer Simulating quantum systems Note – it is still not known if 푃 = 푁푃. It is very hard to build a quantum computer… Erik Lind EIT, Lund University 5 Application of QC:s Modeling of ‘quantum systems’ • Penicillin – 41 atoms in ground state • 1086 bits in a classical computer • 290 quantum bits Room temperature superconductivity (?) Erik Lind EIT, Lund University 6 Application of QC:s Erik Lind EIT, Lund University 7 Your Quantum Mechanics background? 푑 ℏ2 d2 푖ℏ Ψ 푥, 푡 = − + 푉 푥 Ψ 푥, 푡 Schrödinger equation in real space 푑푡 2푚 dx2 representation 푑 푝Ƹ2 푖ℏ |Ψ 푥, 푡 ۧ = + 푉 푥ො |Ψ 푥, 푡 ۧ Schrödinger equation in abstract vector 푑푡 2푚 space (Dirac) notation Erik Lind EIT, Lund University 8 Super-basics of linear algebra A vector V in 3D space 푉 = 훼푒푥 + 훽푒푦 + 훾푒푧 푒푦 V 푒푥, 푒푦, 푒푧 are the basis vectors (Normalized, orthogonal to each other) We can represent any vector with three numbers 푒푥 푒 푧 We can chose a different set of basis vectors ′ ′ ′ 푒푥, 푒푦, 푒푧 are the basis vectors The same vector can then be written with three different numbers 훼′, 훽′, 훾′ ′ ′ ′ ′ ′ ′ 푉 = 훼 푒푥 + 훽 푒푦 + 훾 푒푧 Erik Lind EIT, Lund University 9 Basics of Quantum Mechanics 푥 • Every physical system is a state. 3 푥4 • A state of a system is a vector (ray) in a abstract complex vector (Hilbert) Space. 휙 • Denoted 휙 using Dirac notation. (You can sort of think of this as a vector in nD space.) • Observable – something we can measure about the particle. • Energy 푥 푥 1 푥 2 • Position 푛 • Momentum • Spin direction 휙 = 훼 푥1 + 훽 푥2 + 훾 푥3 + ⋯ 푉 = 훼푒푥 + 훽푒푦 + 훾푒푧 • Orthonormal basis vectors – these are the observables • Each set of observables is represented by a set of basis vectors • These are ‘rotated’ with respect to each other • A state can be in a superposition of the basis vectors. The coefficients are complex numbers 휙 = 훼 푥 + 훽 푥 + 훾 푥 Coefficients 훼, 훽, 훾 are complex numbers 1 2 3 State is normalized: 훼 2 + 훽 2+ 훾 2 + ⋯ = 1 Erik Lind EIT, Lund University 10 Basics of Quantum Mechanics Examples • The position of a electron in a molecule 휙 = 훼 푥1 + 훽 푥2 + 훾 푥3 + ⋯ • Energy of a particle in a quantum well 휙 = 훼 퐸1 + 훽 퐸2 + 훾 퐸3 + ⋯ • Energy of an electron in an atom Example – the position of 0 1 an electron on a molecule • Energy of an electron a harmonic oscillator 0 • The spin on an electron or a neutron. 2 1 5 3 2 4 4 5 3 This would be a 6 dimensional space! Erik Lind EIT, Lund University 11 What about the wavefunction 횿(퐱)? Ψ 푥 x ℏ2 d2 • We are used of solving the S.E equation − + 푉 푥 Ψ 푥 = 퐸 Ψ 푥 2푚 dx2 푛 • Energies and wavefunction Ψۧ = ෍ 푐푛|xnۧ| 푛 • Expansion coefficients cn is the wave function! • Probability of finding electron at x=xn! • Position space – infinite (and uncountable) dimensions Erik Lind EIT, Lund University 12 Uncertainty Relations Depending on the choice of basis – the state of an particle can be described using different basis vectors! 푏푛 푏1 푎푛 Example – the position of a electron is known to be x1 푎4 휓 = 푥1 The electron is then in a superposition 휓 = 푎 푝1 + 푏 푝2 + 푐 푝3 + ⋯ 푏4 of different momentums! 푎1 Ex: Difference between position and momentum eigenstates 푏2 푎3 푏3 Uncertainty relation Δ푥Δ푝 ≥ 푖ℏ 푎2 Erik Lind EIT, Lund University 13 Measurements in Quantum Mechanics Measurement – apparatus which determines an 휓 = 푎1 푥1 + 푎2 푥2 + 푎3 푥3 + ⋯ observable (Energy, position, Spin direction…) 푥푛 푥4 푥1 푥3 A measurement can be though of running a state 푥2 through a machine (apparatus) • A pointer shows the measured value of the During a measurement of an observable a observable State collapses to an eigenstate with probability • The state after the measurement is an eigenstate 푃 = 푎 2 푛 푛 • The outcome is non-deterministic • Not deterministic and NOT time reversible! • Collapse of wavefunction Erik Lind EIT, Lund University 14 Quantum Bits ۧ↓ For quantum computation – effective 2D dimensional Hilbert space This is called a quantum bit (qubit, qbit) ۧ↑| Example - Spin ½ particle - |↑ۧ ↓ۧ : 2 dimensional Hilbert space • 휓푠 = 훼 0 + 훽 1 = 훼 ↑ + 훽 ↓ ห푥2ൿ ห퐸2ൿ Example : Two energy levels of an atom 휓퐸 = 훼 0 + 훽 1 = 훼 퐸1 + 훽 퐸2 ห푥1ൿ ห퐸1ൿ Example : Position of electron in two quantum dots 휓퐸 = 훼 0 + 훽 1 = 훼 푥1 + 훽 푥2 훼 휓 = 훼 0 + 훽 1 훽 a anb b – two complex numbers State is normalized: 훼2 + 훽2 = 1 Vector representation of a state Erik Lind EIT, Lund University 15 Quantum Bit – Bloch sphere 1 0 Normalized Complex coeff. 휓 = 훼 0 + 훽 1 휃 sin 1 푖 휃 휃 훼 2 푌 = 0 + 1 휓 = sin 0 + cos 푒푖휙 1 = 2 2 훽 휃 2 2 e푖휙cos 2 1 1 2 푖 0 1 1 1 푋 = 0 + 1 2 2 1 1 For one spin ½ particle – the Bloch sphere 2 1 sort-of represents the spatial orientation of the spin! Erik Lind EIT, Lund University 16 Quantum System Dynamics A quantum system evolves in time set by the 휓(푡) = 훼(푡) 푎1 + 훽(푡) 푎2 + 훾(푡) 푎3 + ⋯ Schrödinger Equation, where H is the Hamilton operator. 푎푛 푑 푖ℏ 휓(푡) = 퐻෡(푡) 휓(푡) 푎4 푑푡 푝Ƹ 2 퐻෡ = + 푉(푥ො) Electron in potential 2푚 푎1 퐻෡ = −훾퐵푧휎푧 Spin in magnetic field 푎3 푎 • This give a set of coupled differential equations for 2 the coefficients a b g… • By applying an Hamiltonian to a quantum state – move the state in the Hilbert space over time. We can in this way control a quantum state. • This is deterministic and time reversible! Erik Lind EIT, Lund University 17 Quantum Mechanical bit Operator - 푛 × 푛 matrix Time evolution U of a state – unitary matrix Preserves norm of quantum state Visualized as rotation of point on Bloch Sphere 푖 퐻푡 휓(푡) = 푒ℏ 휓(0) If the Hamiltonian H is independent of time By chosing H and t – we can induce any rotation on the Bloch Sphere 훼1 푎 푏 훼0 = All 1-bit gates can be represented by a unitary2x2 matrix 훽1 푐 푑 훽0 Erik Lind EIT, Lund University 18 Single quantum Gate – Bloch Sphere rotation t Quantum gates are 휓 휓′ reversible! U Let H(t) act on 휓 for some time t0. These can in general be build by applying time varying B/E field to a spin/position qbit in x,y and z 휃 휃 direction for a fixed time. 휓 = sin 0 + cos 푒푖휙 1 2 2 NOT gate p rotation around x axis 0 → 1 훼 휓 = 훼 0 + 훽 1 훽 1 Hadamard gate: 0 → 0 + 1 2 Matrix representation of a state These also work on superpositions: 휓 = 훼 0 + 훽 1 Erik Lind EIT, Lund University 19 Many qubits A Qubit A Tensor product 퐻퐴 휓 = 0 퐴 ⊗ 1 퐵 = 01 퐻퐵 B Qubit B Example: Two non-interacting qubits – product state 휓 퐴 = 훼퐴 0 퐴 + 훽퐴 1 퐴 휓 = 휓 퐴 ⊗ 휓 퐵 = 훼퐴훼퐵 00 + 훽퐴훼퐵 10 + 훼퐴훽퐵 01 + 훽퐴훽퐵 11 휓 퐵 = 훼퐵 0 퐵 + 훽퐵 1 퐵 Example: Two correlated qubits – entangled state 1 휓 = ( 10 + 01 ) This can NOT be written 2 as a product state! ۧ N Coupled qbits requires 2푁 complex Example: General 2 qubit state ↓↓ numbers to describe 휓 = 훼 00 + 훽 10 +훾 01 +훿 11 ۧ↑↑| ۧ↓↑ N=100 : 2100 ≈ 1030 complex numbers ۧ↑↓| Erik Lind EIT, Lund University 20 Equal superposition of qubit states 1 휓 = 0 + 1 1 qubit: 2 states 2 1 휓 = 00 + 10 + 01 + 11 2 qubit: 4 states 4 1 휓 = 000 + 001 + 010 + 100 + 011 + 110 + 101 + 111 8 states 8 휓 = ⋯ . 100 spin ½ : 2100 = 1030 dimensional space • There are 1080 atoms in the observable 휓 = ⋯ . 300 spin ½ : 2300 = 1090 dimensional space universe • Hilbert spaces are big! A quantum state can represent a huge number of 0’s and 1’ simultaneously! Erik Lind EIT, Lund University 21 Measurement of multiple qubit states 푁 = 100 ↔ 1030 푠푡푎푡푒푠 1 휓 = 000 … + 001 … + 010 … + 100 … + 011 … + 110 … + ⋯ 2푁 Will randomly select ONE of the states with equal probability! Gives randomly 100 bits of information or out 1030 Direct application of quantum parallelism is not an easy task! 1 휓 = 000 … + 001 … + 010 … + 휓 = 010 … 2푁 0 Picks ONE of states of |휓ۧ Measure A Erik Lind EIT, Lund University 22 Entanglement – ”spooky action” at a distance 1 휓 = ( 10 + 01 ) 2 A B A B Qubit A Qubit B 1) Create entangled qbit state 2) Separate qubits 3) Measurement on qbit A 1 directly sets state of qbit B.
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