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.