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Quantum Computing MAS 725

Quantum Computing MAS 725. Hartmut Klauck NTU 13.2.2012. Organization. Lectures: Mo. 10:00, TR+9 Lecturer: Hartmut Klauck Office: SPMS-MAS-05-44 Website: http://www.ntu.edu.sg/home/hklauck/QC12.html. Grading. Homework (biweekly): 40% Final exam: 60%

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Quantum Computing MAS 725

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  1. Quantum ComputingMAS 725 HartmutKlauckNTU 13.2.2012

  2. Organization • Lectures: Mo. 10:00, TR+9 • Lecturer: HartmutKlauck • Office: SPMS-MAS-05-44 • Website: http://www.ntu.edu.sg/home/hklauck/QC12.html

  3. Grading • Homework (biweekly): 40% • Final exam: 60% • Homework must be written individually! • And handed in on time

  4. Required Background • Linear algebra • Some basic probability theory • No background in physics required

  5. Textbook • Nielsen/Chuang: Quantum Computation and Quantum Information(Cambridge)

  6. Recommended Reading • http://homepages.cwi.nl/~rdewolf/qcnotes.pdf • http://www.cs.berkeley.edu/~vazirani/s09quantum.html • http://www.cs.uwaterloo.ca/~watrous/lecture-notes.html

  7. Quantum Mechanics • Quantum mechanics is one of the basic theories of physics • Quantum mechanics is concerned with states, and how they evolve/change • Includes many “strange” effects that are different from “classical”, Newtonian mechanics: • Superposition • Entanglement • Such effects usually appear in very small systems

  8. Quantum Mechanics and Computing • Moore’s Law: The number of transistors that can be placed on a chip doubles every two years • I.e., the computational power doubles • This trend has been approximately true for more than 50 years • Main way to achieve this is by making smaller transistors! • Even today quantum mechanical effects are important to chip design

  9. Quantum Mechanics and Computing • Another problem: heat generation in integrated circuits • This heat is the result of erasing information • Quantum computations are (for the most part) reversible • Reversible computations (ideally) do not generate (much)heat

  10. Quantum Mechanics and Computing • Chip designers nowadays mostly “combat” quantum effects • Is it possible to make good use of quantum effects?

  11. Quantum Computing • First suggested by Feynman and Benioff in the 1980’s • Feynman’s observation: • Simulating quantum systems on classical computers takes exponential time in the ‘size’ of the quantum system • Conclusion: build universal quantum systems • Quantum systems that can simulate all other quantum systems (up to a size) • I.e., quantum computers

  12. Quantum Computing • Hence reasons for investigating quantum computing are: • Making good use of quantum effects instead of trying to force microscopic system to adhere to classical physics • There are quantum algorithms and protocols that achieve things that seem to be impossible for classical algorithms/protocols • If the world is quantum mechanical, the ultimate limits of computation are determined by quantum physics

  13. Quantum Computing Examples • Some example of tasks that quantum computers can do: • Efficiently factor natural numbers • Break public key cryptosystems like RSA • Search an unordered database in sublinear time • Provide cryptographic protocols that are secure without placing assumptions on the computational power of an eavesdropper

  14. Quantum Computing Models • There are several models of quantum computing • E.g. Deutsch (1985) defined Quantum Turing Machines as a universal model of quantum computation • Another (easier to handle)model are quantum circuits • But first we need to understand some basics about quantum mechanics

  15. Quantum Mechanics • The double slit experiment for light

  16. Quantum Mechanics • Perform the “same” experiment with electrons • We observe the same outcome of the experiment • Even when single electrons are emitted • The wave-like behavior is not just statistical

  17. Quantum Mechanics • The name “Quantum Mechanics” (coined by Planck) derives from the fact that certain quantities can change only by a discrete amount • E.g. The smallest unit of electromagnetic radiation is a photon (a quantum of light) • It is possible to emit and detect single photons

  18. Quantum Mechanics

  19. Quantum Mechanics

  20. Some History • Development of quantum mechanics:Planck 1900, Schrödinger, Heisenberg, Bohr, Einstein..... • 1930’s: von Neumann’s formalism • 1935: Einstein, Podolsky, Rosen describe “entanglement” in an attempt to show that quantum mechanics is not a “complete” theory of reality (German “Verschränkung”) • Today quantum mechanics is the best established theory in physics

  21. (Quantum) Computer Science • 1936: Turing defines a “universal” machine (Church Turing Thesis) • 1948: Shannon’s Information Theory • 1965: Moore’s Law • 1982: Feynman proposes quantum computers (to simulate quantum systems) • 1982 Wiesner: first proposal of quantum cryptography published (after more than 10 years)

  22. (Quantum) Computer Science • 1985: Deutschfinds the first quantum algorithm • 1993: quantum teleportation • 1994: Shor finds a quantum algorithm for factorization • 1996: Grover’s algorithm finds a marked element in a database with n elements in time • Since then the field is steadily growing…

  23. Quantum States • Quantum mechanics is an abstract theory of states and transformations on states • Can be derived from certain axioms • Quantum states are vectors in a Hilbert space • Hilbert Space: • A real or complex vector space with an inner product that maps vectors to their length • Must be complete • We will only consider finite dimensional spaces • Usually either Rn or Cn

  24. Bits • A bit is either the value 0 or the value 1, stored in a register • We will write the state of a bit as|0i, |1i • Examples: • A bit stored in the memory of a computer • The path that a ball took in a giant double slit experiment

  25. Bits and Qubits • We identify the states |0i, |1i with the basis vectors of a two dimensional space (say C2):(1,0) and (0,1) • The states of a quantum bit (qubit) are arbitrary unit vectors in C2 • Hence all the states of a qubit are:|0i + |1iwith ||2 +||2=1

  26. Qubits • |0i, |1i are two basis vectors in C2 • Qubits have states:  |0i+ |1iwith ||2 +||2=1 • ,are called amplitudes • Qubit states are unit vectors under the euclidean norm

  27. Comparison to Probability Theory • Suppose we have a random bit (say a coin flip) • Then we need to specify the probability of 1 and 0 (coin may not be fair) • For example0 has probability p, 1 has probability 1-p) probability distributions on bits are unit vectors under 1-Norm • Qubits: , are complex numbers,possibly negative! • The squares of the absolute values of the amplitudes form a probability distribution

  28. The Quantum Formalism • Quantum states are vectors in a Hilbert space • The Hilbert space corresponds to a register that can hold a quantum state • Hilbert space here: Ckwith the inner producth (vi) | (wi) i = i=1…k vi* wix*: complex conjugate

  29. Dirac Notation • h | “BRA” row vector • |  i “KET” column vector • h |  i inner product (product of a row and a column vector) • |ÁihÃ| outer (matrix valued) product

  30. Many Qubits • To hold k qubits we need a Hilbert space of dimension 2k • I.e. 2k basis vectors (corresponding to the 2k values of k bits) • First notation: |ii, i=1,...,2k. Unit vectors are of the formii |ii; i=1....2k with i |i |2 = 1 • Better notation: identify i=1...2k with x2{0,1}k • Basis states are |xi, x2 {0,1}k • Basis states correspond to classical values a register can hold • General quantum states are linear combinations of the 2k classical (basis) states • Also called “superpositions”

  31. Tensor Product • For Hilbert spacesH, K, with dimensions dHanddKtheir tensor product H­K is a Hilbert space of dimension dH¢dK • Tensor product of vectors: (a1,..., al) ­ (b1,...,br)= (a1b1,a1b2,...,a1br,a2 b1,......,albr) • Example: |0i = (10)T; |1i= (01)Tand |01i= |0i­|1i = (0100)T • A basis of H­K: all|xi­|yi=|xyiwhere |xi,|yi are basis vectors of H,K • Not all vectors in H­K are tensor products of vectors in H and K

  32. Example • Basis of C4: • |00i, |01i, |10i, |11i • Another basis: • (|00i + |11i)(|00i - |11i)(|01i + |10i)(|01i - |10i)(scaled by square root of 2) • None of these are tensor products of vectors in C2

  33. What can we do with one or more qubits? • Quantum systems evolve according to the Schrödinger equation • The result can be described as the application of a unitary transformation to the quantum state • Additionally quantum states can be measured • This leads to observable output • Need some background from linear algebra…

  34. Linear Algebra • Linear transformations: A(x+y)=Ax + Ay • x,y: vectors in Ck, A: k£k matrix (complex entries) • Over the reals a linear transformation O is orthogonal, ifOOT=I • Over the complex numbers a matrix U is unitary,ifUUy=IU*: take the complex conjugate of all entriesUy= (U*)T • Unitary transformation preserve the euclidean length of vectors • Transformations in QM: unitary(i.e., reversible and length preserving)

  35. Examples • On one qubit:classical transformations:identity, negation • Hadamard Transformation:

  36. Applying Hadamard

  37. Applying Hadamard

  38. Applying Hadamard

  39. Unitary Transformations • Define U |xifor allx2 {0,1}k) U is defined. The U|xi need to be unit vectors andU|xi?U|yi for all xy • Tensor product for matrices: • A ­ B=

  40. Unitary Transformations

  41. Hadamard Transformation • x,z2{0,1}n and x¢z=xizi • For any x we have H­n |xi=1/2n/2 (|0i +(-1)x(1) |1i)­­(|0i +(-1)x(n) |1i)

  42. Applying many unitary transformations • Later we will construct unitary transformations as the product of many “simple” unitary transformation • First applying a unitary U, and then a unitary V is the same as applying the product VU. • Note that the product of two unitary matrices is unitary • Careful: matrix multiplication is not commutative! • The exact sequence of multiplications matters

  43. Measurements • Quantum states (unit vectors in Ck) can be changes by applying a unitary transformation • Computations on quantum states consist of unitary transformations and measurements • Measurements allow us to access the result of a computation • What happens if we measure ii |ii ? • The result will be i with probability |i|2 • i |i2|=1 is very helpful now! • After measuring the value i the state “collapses” to |ii

  44. Example • Measuring the state • Will result in the outcome 0 or 1, each with probability ½ • If we measured 1, the resulting state after the measurement will be

  45. Overview • Quantum states: unit vectors in a Hilbert space, the log of the dimension corresponds to the number of qubits • States in a Hilbert space of dimension 2k correspond to superpositions of strings of length k and the space is a register of k qubits • Evolution: by applying unity transformations • Measurement: i |ii results in output iwith probability |i|2, the state collapses to |iiif i is the measurement result

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