Quantum Information Processing A. Hamed Majedi Institute for Quantum Computing (IQC) and RF/Microwave & Photonics Group ECE Dept., University of Waterloo

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

Exp. With Bullet (1) detector wall P1(x) wall H1 H2 (a) Gun

Exp. With Bullet (2) detector wall P2(x) wall H1 H2 (a) Gun

Exp. With Bullet (3) P2(x) P1(x) (c) wall H1 H2 (a) Gun (b) (c)

Exp. with Waves (1) detector wall wave source H1 I1(x) I2(x) (b) H1 H2

Exp. with Waves (2) detector wall wave source I1(x) I2(x) (b) (c) H1

Two Slit Experiment (1) detector wall P2(x) P1(x) (b) (c) Results intuitively expected wall H1 H2 (a) source of electrons (c)

Two Slit Experiment (2) detector wall P2(x) P1(x) (b) (c) Results observed wall H1 H2 (a) source of electrons

Two Slit Exp. With Observer light source detector wall P2(x) P1(x) (b) (c) Interference disappeared! H1 source of electrons 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

Stern-Gerlach Experiment

QM Physical Concepts Wave Function Quantum Dynamics (Schrodinger Eq.) Statistical Interpretation (Born Postulate)

Bit & Quantum Bits (1) V(t) t 1

More Quantum Bits

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.

Example of Quantum Gates NOT gate: X • Z gate: Z • Hadamard gate: H • Phase gate: P

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 M 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 n qubits 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. • Entangled state 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.

Physics & Math Connections in QIP Postulate 1 Postulate 2 Postulate 3 Postulate 4 Isolated physical system Evolution of a physical system Measurements of a physical system Composite physical system Hilbert Space Unitary transformation Measurement operators Tensor product of components

Physical Realization of QC Storage: Store qubits for long time Isolation: Qubits must be isolated from environment 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 RF-SQUID Cooper Pair Box

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.