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Quantum Information and Computation for Dummies

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Quantum computers are devices capable of performing computations using quantum bits, or qubits.

The first thing we need to understand is what a qubit actually is. A “classical computer,” like the one you’re reading this on (either desktop, laptop, tablet, or phone), is also referred to as a “binary computer” because all of the functions it performs are based on either ones or zeros.

On a binary computer the processor uses transistors to perform calculations. Each transistor can be on or off, which indicates the one or zero used to compute the next step in a program or algorithm.

There’s more to it than that, but the important thing to know about binary computers is the ones and zeros they use as the basis for computations are called “bits.”

Quantum computers don’t use bits; they use qubits. Qubits, aside from sounding way cooler, have extra functions that bits don’t. Instead of only being represented as a one or zero, qubits can actually be both at the same time. Often qubits, when unobserved, are considered to be “spinning.” Instead of referring to these types of “spin qubits” using ones or zeros, they’re measured in states of “up,” “down,” and “both.”

Superposition
Qubits can be more than one thing at a time because of a strange phenomenon called superposition. Quantum superposition in qubits can be explained by flipping a coin. We know that the coin will land in one of two states: heads or tails. This is how binary computers think. While the coin is still spinning in the air, assuming your eye isn’t quick enough to ‘observe’ the actual state it’s in, the coin is actually in both states at the same time. Essentially until the coin lands it has to be considered both heads and tails simultaneously.

A clever scientist by the name of Schrodinger explained this phenomenon using a cat which he demonstrated as being both alive and dead at the same time.

Observation theory
Qubits work on the same principle. An area of related study called “observation theory” dictates that when a quantum particle is being watched it can act like a wave. Basically the universe acts one way when we’re looking, another way when we aren’t. This means quantum computers, using their qubits, can simulate the subatomic particles of the universe in a way that’s actually natural: they speak the same language as an electron or proton, basically.

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Quantum Information and Computation for Dummies

  1. 1. Quantum Information and Computation for Dummies Peter Samuelsson, Mathematical Physics
  2. 2. Outline What is quantum information?1 What is a quantum computer?2 Why build a quantum computer?3 How to build a quantum computer?4
  3. 3. Outline What is quantum information?1 What is a quantum computer?2 Why build a quantum computer?3 How to build a quantum computer?4
  4. 4. Basic unit of information Classical bit 0 1 Quantum bit - qubit “Quantum information is physical information that is held in the state of a quantum system” - wikipedia Definition Two-state quantum system V 1 0 Two-state classical system
  5. 5. More qubit ”hardware” examples Electrons spin Photons polarization position atomic level n n+1
  6. 6. Bloch sphere representation Vector on the unit sphere State parametrization More qubits - classical 00,01,10,11 2 N - ”analog information”
  7. 7. Quantum vs classical Teleportation No cloning theoremCopy information Quantum teleportation
  8. 8. Entanglement Qubit A and B Resource for quantum information processing Hotly debated non-local properties Bob Alice Alice measures Bob must measure No information is transferred
  9. 9. Teleportation Photon experiment source of entangled pairs Alice initial state Bob final state classical communication two-qubit measurement single qubit rotation
  10. 10. Superdense Coding Cryptography - sending 2 classical bits in 1 qubit 1 Alice Bob 2 00, 01, 10, 11 3 00 01 10 11 Share en- tangled pair A rotates - codes B measures - decodes - quantum key distribution Eve classical quantum
  11. 11. What is a quantum computer? Outline What is quantum information?1 2 Why build a quantum computer?3 How to build a quantum computer?4
  12. 12. “A quantum computer make use of quantum-mechanical phenomena, such as superposition and entanglement, to perform operations on data.” - wikipedia Definition Fundamentals Quantum computers can be programmed to carry out the same operations as classical computers – Deutch Any operation on a reversible Turing machine can be simulated quantum mechanically – Benioff A reversible Turing machine can perform the same operations as a standard Turing machine – Bennett 1 2 3 4 Any algorithm can be implemented on a (classical) Turing machine – Church,Turing
  13. 13. One bit gates Classical one-bit gate = NOT aa a c 0 1 1 0 NOT c Single qubit gates X Rθ H gate unitary transformation operator X Quantum mechanics
  14. 14. Two qubit gates a b ANDc a bAND 0 0 0 0 1 0 1 0 0 1 1 1 a b c = Classical two-bit gate …. also AND, OR, XOR, NAND, Two qubit gate Controlled NOT ::
  15. 15. Circuits Universal computation Entangler H … and multi qubit circuits With (for example) Rπ/4H single qubit gates two qubit gate any multi qubit circuit (quantum computer) can be constructed.
  16. 16. Quantum parallelism f Superposition of computations f f f f Classical circuit Quantum circuit All function evaluations in a single run
  17. 17. Why build a quantum computer? Outline What is quantum information?1 What is a quantum computer?2 3 How to build a quantum computer?4
  18. 18. Motivation 1 Computationally hard problems 2 Quantum simulations 3 Energy saving Quantum computers can solve certain problems much faster than any classical computer Grover Shor Quantum computers can simulate many-particle systems much faster than any classical computer Feynman Quantum computers are reversible and can consume much less energy than (standard) classical computers Landauer
  19. 19. Shor’s algorithm Prime factorization problem: Given an integer N, find its prime factors 15 = 3 x 5 9999999942014077477 = 3162277633 x 3162277669 Time to solve: classical quantum Public key cryptography email transfers identity … internet security Product of two unknown primes Exponential speed-up
  20. 20. Grovers algorithm Unsorted data base search: Find a given element in a data base of size N Time to solve: classical quantum Quadratic speed-up
  21. 21. Quantum simulations Classical computers require exponentially long time to simulate many-particle quantum systems Feynmans observation Quantum computers require polynomial time Exponential speed-up Many-body fermion system Lloyd v  Initial state preparation v  Time evolution v  Charge density v  Correlation functions v  …. Simulate Bosonic systems, chemical reactions, …
  22. 22. Outline What is quantum information?1 What is a quantum computer?2 Why build a quantum computer?3 How to build a quantum computer?4
  23. 23. Di Vincenzo criteria Requirement on physical system (hardware) Scalable with well characterized qubits1 2 3 4 5 Initial state preparation A universal set of quantum gates Long (relevant) decoherence times Reading out the result Quantum error correction
  24. 24. Linear quantum optics Trapped ions Nuclear magnetic resonance Superconducting circuits
  25. 25. Electrons in solid state Spins in quantum dots Dopants in silicon Nitrogen vacancies in diamond Charge states in quantum dots
  26. 26. Status and outlook Achieved to date Controlling & coupling individual qubits Needed to beat classical computers Full quantum computer : 104 - 105 qubits Limited tasks: 102 - 103 qubits Running algorithms with <10 qubits Error correction with <10 qubits Coherent transfer of individual qubits … Still plenty of work to do….

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