Quantum talk
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This document discusses quantum computing. It explains that qubits, the basic unit of quantum computers, can exist in superposition and be entangled. Common qubit operations like CNOT gates are used to entangle qubits and generate Bell pairs. Quantum algorithms like Grover's, Shor's, and HHL are described, though current quantum computers have only demonstrated small problems compared to what the algorithms enable. Quantum computers are not expected to replace classical computers but will be useful for problems involving combinatorics.
Quantum talk
- 2. © Len Bass 2019 2 In the beginning of the computer age • Programmers worked in machine language • Required a good knowledge of the architecture of the underlying hardware • Limited in terms of size and complexity of programs.
- 3. © Len Bass 2019 3 In the beginning of the quantum computer age • Programmers work in quantum machine language • Requires a good knowledge of the architecture of the underlying hardware • Limited in terms of size and complexity of programs.
- 4. © Len Bass 2019 4 This talk • Will focus on the logic of the quantum hardware • Qubits • And how you program for this hardware • Gates • Operators • Will sketch some common algorithms
- 5. © Len Bass 2019 5 What is a bit in a classical computer? • A bit has a value of either 0 or 1. • There is no ambiguity • Reading a classical bit does not affect its value.
- 6. © Len Bass 2019 6 What is a bit (qubit) in a quantum computer? • A qubit has the value 0 with some probability • It has the value 1 with 1-probability(0) • i.e you can think of a qubit has having both values simultaneously
- 7. © Len Bass 2019 7 A qubit is fundamentally different from a classical bit • Measuring (reading) a qubit has the following effects • It returns a 0 or a 1. The value returned is based on probabilities of the values. • It replaces the original values in the qubit with the returned value (0 or 1)
- 8. © Len Bass 2019 8 Copying qubits not possible • A copy would result in two instances of the same value • A copy is a read followed by a store • A read destroys the value and results in a 0 or a 1 • Therefore – cannot copy qubits • It is possible to move the value in one qubit to another qubit at the cost of destroying the original value
- 9. © Len Bass 2019 9 Probabilities • A measurement of a qubit will return 0 with a probability of α and 1 with a probability of b = 1- α • I.e. if α = 40% then 4 out of 10 measurements will return 0 and 6 will return 1 (roughly).
- 10. © Len Bass 2019 10 Superposition • The ability of a qubit to hold values with different probabilities is called “superposition”
- 11. © Len Bass 2019 11 Third value for qubit specification • A qubit also has a “phase”. • A phase is a value between 0 and 2π • The phase provides an additional handle for manipulating qubits but does not enter into probabilities for reading 0 or 1.
- 12. © Len Bass 2019 12 Notations • Two notations: • Ket (Dirac) notation |0> for 0 and |1> for 1 • Matrix notation |0> = 1 0 |1> = 0 1
- 13. © Len Bass 2019 13 Arbitrary qubit • An arbitrary qubit g is • In ket notation |g> = α|0> + b |1> • In matrix notation g = α b • The phase is captured by making α and b be complex numbers. We won’t go into phases.
- 14. © Len Bass 2019 14 How do you manipulate qubit? • A qubit is manipulated using an operator O • g → O → d • An operator is realized as a gate.
- 15. © Len Bass 2019 15 Single qubit operators • Single qubit operators include • NOT • Hanamard • Rotate
- 16. © Len Bass 2019 16 Representing qubit operators • A qubit operator can be represented as • a gate in a circuit for ket notation or • a matrix for matrix notation
- 17. © Len Bass 2019 17 NOT • NOT reverses α and b Matrix form Circuit form 0 1 1 0 0 1 1 0 = |0> |1>NOT
- 18. © Len Bass 2019 18 Hanamard • Hanamard places the qubit into superposition with probabilities of 50% for both α and b 1 2 1 1 1 −1 1 0 = 1 2 1 1 Matrix form Circuit form |0> H |0>+|1> 2
- 19. © Len Bass 2019 19 Rotate • Rotate changes the phase by π. It does not affect the probabilities. Matrix form Circuit form = |0> |0>Z1 0 0 −1 1 0 1 0
- 20. © Len Bass 2019 20 Gates can be cascaded • What is the output of this? |0> H NOT
- 21. © Len Bass 2019 21 Two qubit notation • |00> = • |01> = 1 0 0 0 0 1 0 0 • |10> = • |11> = 0 0 1 0 0 0 0 1
- 22. © Len Bass 2019 22 Measuring two qubits |gh> = a|00> + b|01> + c|10> + d|11> • Measuring the two qubits will yield two classical bits (0 0, 0 1, 1 0, 1 1) with probabilities a, b, c, d • It will also collapse the two qubits. Suppose, b is chosen as the value, then the measured values is 0 1 and g is collapsed to 0 and h is collapsed to 1.
- 23. © Len Bass 2019 23 Entanglement • Suppose |gh> = a|00> + d|11> • i.e. b and c are 0. (The probabilities for the middle two terms of the definition.) • Now measure gh It will be 0 0 or 1 1 with probabilities a and d. • Both qubits will have the same value after the measurement. • The two qubits are “entangled”
- 24. © Len Bass 2019 24 Weirdness of entanglement • A subsequent measurement of h can be at a different time from a subsequent measurement of g. • h can be in a different location than g. • They will always have the same value. • Physicists have verified this phenomenon over kilometers.
- 25. © Len Bass 2019 25 Exploiting entanglement • We will use entanglement to communicate information over distances • First we show how to generate entangled qubits • This requires the Controlled Not operation
- 26. © Len Bass 2019 26 Operation on two qubits • Controlled Not – CNOT • The first qubit acts as a control. If it is 0 then there is no change to the second qubit. If it is 1 then the second qubit is flipped.
- 27. © Len Bass 2019 27 CNOT |0> |0> CNOT |0> |0> Circuit form 1 0 0 0 1 0 0 0 = 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 Matrix form
- 28. © Len Bass 2019 28 Bell Pair • Entangling two qubits can be done by creating a Bell Pair |0> H |00>+|11> 2=== = |0> CNOT
- 29. © Len Bass 2019 29 Quantum teleportation Three qubits involved: A, B, • A and are in one location. B in another. • Is teleported to B and is destroyed in the process. • Although a qubit cannot be copied, its contents can be moved at the cost of destroying the original qubit.
- 30. © Len Bass 2019 30 Steps in teleportation - 1 1. Entangle qubits A and B. 2. Prepare the payload. The payload qubit will have the state to be teleported.
- 31. © Len Bass 2019 31 Steps in teleportation - 2 3. Propagate the payload. The propagation involves two classical bits that are transferred to the location of B. The propagation also involves measuring A and This will destroy the state of both of these qubits. 4. Recreate the state of in B.
- 32. © Len Bass 2019 32 Why is this interesting? • It is not possible to copy qubits but we can transfer the state of a qubit to a different location at the cost of destroying the state of the original qubit. • This will be th the basis of quantum based communication protocols. • NIST is currently considering creation of a httpq protocol to ultimately replace https
- 33. © Len Bass 2019 33 Other quantum algorithms • Other algorithms exist that are not currently realizable • Notably: • Grover’s – breaks password hash • Shor’s – breaks RSA encryption • HHL – matrix inversion – used in machine learning • …
- 34. © Len Bass 2019 34 Grover’s Algorithm • Computes inverse of a function – in particular a hash function • Uses superposition to identify inverse of a value • Problem is the identified value has same probability of being measured as all of the other values • Phase manipulation and amplitude amplification are used to increase probability of measuring inverse value.
- 35. © Len Bass 2019 35 Shor’s algorithm • Uses number theory results to break RSA • Quantum used to find period of an exponential factor. • Results are not directly readable but must be inferred
- 36. © Len Bass 2019 36 HHL • Inverts large matrices. • Used in machine learning • Uses Amplitude amplification as in Grover’s
- 37. © Len Bass 2019 37 Current state of quantum computers • Google has announced they have achieved “quantum supremacy” • This means they have demonstrated a problem that can be solved exponentially faster on a quantum computer than on a classical computer. (the problem is not interesting)
- 38. © Len Bass 2019 38 Future state • Suppose Moore’s Law holds (exponential growth in computing power over time) • Then quantum computers will become real in the 5-10 year time frame. • Quantum computers will NOT replace classical computers. • Quantum computers will be used for problems involving combinatorics.
- 39. © Len Bass 2019 39 Summary • Qubits are the basic computation unit of a quantum computer • Qubits can be in superposition • Qubits can be entangled. • Algorithms exist for some problems intractable on classic computers but, as yet, none of them are realizable.