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## Quantum Information Processing

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**A. Hamed Majedi**Institute for Quantum Computing (IQC) and RF/Microwave & Photonics Group ECE Dept., University of Waterloo Quantum Information Processing**Outline**• Limits of Classical Computers • Quantum Mechanics Classical vs. Quantum Experiments Postulates of quantum Mechanics • Qubit • Quantum Gates • Universal Quantum Computation • Physical realization of Quantum Computers • Perspective of Quantum Computers**Moore’s Law**The # of transistors per square inch had doubled every year since the invention of ICs.**Limits of Classical Computation**• Reaching the SIZE & Operational time limits: 1- Quantum Physics has to be considered for device operation. 2- Technologies based on Quantum Physics could improve the clock-speed of microprocessors, decrease power dissipation & miniaturize more! (e.g. Superconducting processors based on RSFQ, HTMT Technology) Is it possible to do much more? Is there any new kind of information processing based on Quantum Physics?**Quantum Computation & Information**• Study of information processing tasks can be accomplished using Quantum Mechanical systems. Quantum Mechanics Computer Science Information Theory Cryptography**Quantum Mechanics History**• Classical Physics fail to explain: 1- Heat Radiation Spectrum 2- Photoelectric Effect 3- Stability of Atom • Quantum Physics solve the problems Golden age of Physics from 1900-1930 has been formed by Planck, Einstein, Bohr, Schrodinger, Heisenberg, Dirac, Born, …**Classical vs Quantum Experiments**• Classical Experiments • Experiment with bullets • Experiment with waves • Quantum Experiments • Two slits Experiment with electrons • Stern-Gerlach Experiment**detector**P1(x) Gun wall H1 H2 wall (a) Exp. With Bullet (1)**detector**Gun P2(x) wall H1 H2 wall (a) Exp. With Bullet (2)**P1(x)**Gun P2(x) (c) H1 H2 wall (a) Exp. With Bullet (3) (b) (c)**detector**I1(x) I2(x) (b) wall Exp. with Waves (1) wave source H1 H1 H2**detector**I1(x) I2(x) (b) (c) wall Exp. with Waves (2) wave source H1 H2**Results intuitively expected**P1(x) H1 detector H2 P2(x) source of electrons wall (c) (a) (b) wall Two Slit Experiment (1) (c)**Results observed**P1(x) H1 detector H2 P2(x) source of electrons wall (c) (a) (b) wall Two Slit Experiment (2)**light source**P1(x) detector P2(x) source of electrons (c) (b) wall Two Slit Exp. With Observer Interference disappeared! H1 H2 ⇨ “Decoherence”**Results from Experiments**• Two distinct modes of behavior (Wave-Particle Duality): 1- Wave like 2- Particle-like • Effect of Observations can not be ignored. • Indeterminacy (Heisenberg Uncertainty Principle) • Evolution and Measurement must be distinguished**S**N Stern-Gerlach Experiment**QM Physical Concepts**• Wave Function • Quantum Dynamics (Schrodinger Eq.) • Statistical Interpretation (Born Postulate)**V(t)**1 t V(t) 0 t Bit & Quantum Bits (1)**Qubit (1)**• A qubit has two possible states: • Unlike Bits, qubits can be in superposition state • A qubit is a unit vector in 2D Vector Space (2D Hilbert Space) • are orthonormal computational basis • We can assume that & & &**Qubit (2)**• A measurement yields 0 with probability & 1 with probability • Quantum state can not be recovered from qubit measurement. • A qubit can be entangled with other qubits. • There is an exponentially growing hidden quantum information.**Math of Qubits**• Qubits can be represented in Bloch Sphere.**Quantum Gates**• A Quantum Gate is any transformation in Bloch sphere allowed by laws of QM, that is a Unitary transformation. • The time evolution of the state of a closed system is described by Schrodinger Eq.**P**X • Z gate: Z H Example of Quantum Gates • NOT gate: • Hadamard gate: • Phase gate:**Universal Computation**• Classical Computing Theorem : Any functions on bits can be computed from the composition of NAND gates alone, known as Universal gate. •Quantum Computing Theorem: Any transformation on qubits can be done from composition of any two quantum gates. e.g. 3 phase gates & 2 Hadamard gates, the universal computation is achieved. • No cloning Theorem: Impossible to make a copy from unknown qubit.**Measurement**• A measurement can be done by a projection of each in the basis states, namely and . • Measurement can be done in any orthonormal and linear combination of states & . • Measurement changes the state of the system & can not provide a snapshot of the entire system. Probabilistic Classical Bit M Probabilistic Classical Bit**Multiple Qubits**• The state space of nqubits can be represented by Tensor Product in Hilbert space with orthonormal base vectors. E.g. states produced by Tensor Product is separable & measurement of one will not affect the other. • Entangledstate can not be represented by Tensor Product E.g.**Multiple Qubit Gates**C-NOT Gate Any Multiple qubit logic gate may be composed from C-NOT and single qubit gate. C-NOT Gate is Invertible gates. There is not an irretrievable loss of information under the action of C-NOT.**Physical Realization of QC**• Storage:Store qubits for long time • Isolation:Qubits must be isolated fromenvironment to decrease Decoherence • Readout:Measuring qubits efficiently & reliably. • Gates: Manipulate individual qubits & induce controlled interactions among them, to do quantum networking. • Precision: Quantum networking & measurement should be implemented with high precision.**DiVinZenco Checklist**• A scalable physical system with well characterized qubits. • The ability to initialize the state of the qubits. • Long decoherence time with respect to gate operation time • Universal set of quantum gates. • A qubit-specific measurement capability.**Quantum Computers**• Ion Trap • Cavity QED (Quantum ElectroDynamics) • NMR (Nuclear Magnetic Resonance) • Spintronics • Quantum Dots • Superconducting Circuits (RF-SQUID, Cooper-Pair Box) • Quantum Photonic • Molecular Quantum Computer • …**Spintronics**Cavity QED Atom Chip Cooper Pair Box RF-SQUID**Perspective of Quantum Computation & Information**• Quantum Parallelism • Quantum Algorithms solve some of the complex problems efficiently (Schor’s algorithm, Grover search algorithm) • QC can simulate quantum systems efficiently! • Quantum Cryptography: A secure way of exchanging keys such that eavesdropping can always be detected. • Quantum Teleportation: Transfer of information using quantum entanglement.