1 / 45

Itay Hen

Adiabatic quantum computation – a tutorial for computer scientists. Itay Hen. Dept. of Physics, UCSC. Advanced Machine Learning class UCSC June 6 th 2012. Outline. introduction I : what is a quantum computer?

gay
Download Presentation

Itay Hen

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Adiabatic quantum computation –a tutorial for computer scientists Itay Hen Dept. of Physics, UCSC Advanced Machine Learning class UCSC June 6th 2012

  2. Outline • introduction I: what is a quantum computer? • introduction II: motivation for adiabatic quantum computing • adiabatic quantum computing • simulating adiabatic quantum computers • future of adiabatic computing • [AQC and machine learning(?)]

  3. What is a Quantum Computer?

  4. What is a quantum computer? 011 computer 010011 input state computation output state • both classical and quantum computers may be viewed as machines that perform computations on given inputs and produce outputs. Both inputs and outputs are strings of bits (0’s and 1’s). • classical computers are based on manipulations of bits. • at any given time the system is in a unique “classical” configuration (i.e., in a state that is a sequence of 0’s and 1’s).

  5. What is a quantum computer? 011 computer 010011 input state computation output state • a classical algorithm (circuit) looks like this: • at any given time the system is in a unique “classical” configuration (i.e., in a state that is a sequence of 0’s and 1’s). 0 1 0 1 1 1 1 1 0 1 0 1 0 1 0 1 0 0 1 0 input state computation output state

  6. What is a quantum computer? computer 011 010011 input state computation output state • quantum computers on the other hand manipulate objects called quantum bits or qubits for short.

  7. What is a quantum computer? computer 011 010011 input state computation output state • quantum computers on the other hand manipulate objects called quantum bits or qubits for short. what are qubits?

  8. What are qubits? • a qubit is a generalization on the concept of bit. • motivation / idea behind it: small particles, say electrons, obey different laws than ‘big’ objects (such as, say, billiard balls). • small particles obey the laws of Quantum Physics. big objects are classical entities – they obey the laws of Classical Physics. (however, big objects are collections of small particles!) • most notably: a quantum particle can be in a superposition of several classical configurations. classical world quantum world or and I’m here I’m there I’m here I’m there

  9. What are qubits? • while a bit can be either in the a state or a state (this is Dirac’s notation). • a qubit can be in a superposition of these two states, e.g., • imagine and a as being two orthogonal basis vectors. while a classical bit is either of the two, a qubit can be an arbitrary (yet normalized) linear combination of the two. • in order to access the information stored in a qubit one must perform a measurement that “collapses” the qubit. extremely non- intuitive. important: only 1 bit of information is stored in a qubit.

  10. What are qubits? • take for example a system of 3 bits. Classically, we have possible “classical configurations”: • the corresponding quantum state of a 3-qubit system would be: • this is a (normalized) 8-component vector over the complex numbers (8 complex coefficients). enumerates the 8 classical configurationslisted above.

  11. Encoding optimization problems • suppose that we are given a function and are asked to find its minimum and the corresponding minimizing configuration. • this is the same as having the diagonal matrix: • and asking which is the smallest eigenvalue (minimum of f ) and corresponding eigenvector (minimizing configuration). • here, the basis vectors correspond to and the elements on the diagonal are the evaluation of f on them.

  12. Encoding optimization problems • in matrix-form we can generalize classical optimization problems to quantum optimization problems by adding off-diagonal elements to the matrix. • we still look for the smallest eigen-pair, i.e., • but problem now is much harder. (this is Dirac’s notation). • in quantum mechanics, the minimal (eigen)vector is called the “ground state” of the system and the corresponding minimal (eigen)value is called the “ground state energy”. • also, the matrix F is called the Hamiltonian (usually denoted by H). • quantum mechanics is just linear algebra in disguise.

  13. What is a quantum computer? computer 011 010011 input state computation output state • quantum computers manipulate quantum bits or qubits for short. a quantum algorithm (circuit) may look like this: 1 1 0 1 0 +1 1 0 0 1 +1 1 1 0 1 0 1 0 1 0 +0 1 0 0 1 0 1 0 1 1 1 0 0 1 0 input state computation output state

  14. What is a quantum computer? computer 011 010011 input state computation output state • space of possible quantum states is huge. • range of operations on qubits is huge. • capabilities are greater (name dropping: superposition, entanglement, tunneling, interference…). can solve problems faster. • only problem with quantum computers: they do not exist! theory is very advanced but many technological unresolved challenges.

  15. Motivation for adiabaticquantum computing

  16. Motivation clearly, a quantum computer is a generalization of a classical computer in what ways quantum computers are more efficient than classical computers? what problems could be solved more efficiently on a quantum computer? these are two of the most basic and important questions in the field of quantum information/computation.

  17. Motivation in what ways quantum computers are more efficient than classical computers? what problems could be solved more efficiently on a quantum computer? • best-known examples are: • Shor’s algorithm for integer factorization. Solves the problem in polynomial time (exponential speedup).current quantum computers can factor all integers up to 21. • Grover's algorithm is a quantum algorithm for searching an unsorted database with entries in time (quadratic speedup). • importance is huge (cryptanalysis, etc.).

  18. Motivation what other problems can quantum computers solve? • people are considering “hard” satisfiability (SAT) and other optimization problems which are at least NP-complete. • these are hard to solve classically; time needed is exponential in the input (exponential complexity). • could a quantum computer solve these problems in an efficient manner? perhaps even in polynomial time? Adiabatic Quantum Computingis a general approach to solve a broad range of hard optimization problems using a quantum computer [Farhiet al.,2001]

  19. Adiabatic quantum computation

  20. The nature of physical systems • adiabatic quantum computation is different that circuit-based computation (there are still other models of computation). • adiabatic quantum computation is analog in nature (as opposed to 0/1 “digital” circuits). • it is based on the fact that the state of a system will tend to reach a minimum configuration. this is the principle of least action.

  21. Analog classical optimization • we are given a landscape f(x) for which we need to find the minimum configuration, i.e., the stable state of the ball. • if energy landscape is “simple”: • the ball will eventually end up at the bottom of the hill. f x

  22. Analog classical optimization • we are given a landscape f(x) for which we need to find the minimum configuration, i.e., the stable state of the ball. • if energy landscape is “jagged”: • ball will eventually end up in a minimum but is likely to end up in a local minimum. this is no good. f x

  23. Analog classical optimization • we are given a landscape f(x) for which we need to find the minimum configuration, i.e., the stable state of the ball. • if energy landscape is “jagged”: • ball will eventually end up in a minimum but is likely to end up in a local minimum. this is no good. • probability of jumping over the barrier is exponentially small in the height of the barrier. • ball is unlikely to end up in the true minimum. f x

  24. Analog quantum optimization • we are given a landscape f(x) for which we need to find the minimum configuration, i.e., the stable state of the ball. • if energy landscape is “jagged”: • ball will eventually end up in a minimum but is likely to end up in a local minimum. this is no good. • quantum mechanically, the ball can “tunnel through” thin but high barriers! • sometimes more likely to end up in the true minimum. • tunneling is a key feature of quantum mechanics. f x

  25. The adiabatic theorem of QM • the adiabatic theorem of QM tells us that a physical system remains in its instantaneous eigenstate if a given perturbation is acting on it slowly enough and if there is a gap between the eigenvalue and the rest of the Hamiltonian's spectrum.

  26. The adiabatic theorem of QM • the adiabatic theorem of QM tells us that a physical system remains in its instantaneous eigenstate if a given perturbation is acting on it slowly enough and if there is a gap between the eigenvalue and the rest of the Hamiltonian's spectrum. ??

  27. The adiabatic theorem of QM • the adiabatic theorem of QM tells us that a physical system remains in its instantaneous eigenstate if a given perturbation is acting on it slowly enough and if there is a gap between the eigenvalue and the rest of the Hamiltonian's spectrum. • rephrasing: • if the ball is at the minimum (ground state)of some function (Hamiltonian), and • the function is changes slowly enough, • ball will stay close to the instantaneousglobal minimum throughout the evolution. • here, initial function is easy to find the minimum of. final function is much harder. • take advantage of adiabatic evolution to solve the problem!

  28. The adiabatic theorem of QM • the adiabatic theorem of QM tells us that a physical system remains in its instantaneous eigenstate if a given perturbation is acting on it slowly enough and if there is a gap between the eigenvalue and the rest of the Hamiltonian's spectrum. • real QM example: change the strength of a harmonic potential of a system in the ground state: harmonicpotential

  29. The adiabatic theorem of QM • the adiabatic theorem of QM tells us that a physical system remains in its instantaneous eigenstate if a given perturbation is acting on it slowly enough and if there is a gap between the eigenvalue and the rest of the Hamiltonian's spectrum. • real QM example: change the strength of a harmonic potential of a system in the ground state: • an abrupt change (a diabaticprocess): harmonicpotential

  30. The adiabatic theorem of QM • the adiabatic theorem of QM tells us that a physical system remains in its instantaneous eigenstate if a given perturbation is acting on it slowly enough and if there is a gap between the eigenvalue and the rest of the Hamiltonian's spectrum. • real QM example: change the strength of a harmonic potential of a system in the ground state: • a gradual slow change (an adiabatic process): wave function can “keep up” with the change. harmonicpotential

  31. The quantum adiabatic algorithm (QAA) • the mechanism proposed by Farhiet al., the QAA: 1. take a difficult (classical) optimization problem. 2. encode its solution in the ground state of a quantum “problem” Hamiltonian 3. prepare the system in the ground state of another easily solvable “driver” Hamiltonian” . 4. vary the Hamiltonian slowly and smoothly from to until ground state of is reached.

  32. The quantum adiabatic algorithm (QAA) • the interpolating Hamiltonian is this: is the problem Hamiltonian whose ground state encodes the solution of the optimization problem is an easily solvable driver Hamiltonian, which does not commute with • the parameter obeys , with and . also: and . • here, stands for time and is the running time, or complexity, of the algorithm.

  33. The quantum adiabatic algorithm (QAA) • the interpolating Hamiltonian is this: • the adiabatic theorem ensures that if the change in is made slowly enough, the system will stay close to the ground state of the instantaneous Hamiltonian throughout the evolution. • one finally obtains a state close to the ground state of . • measuring the state will give the solution of the original problem with high probability. how fast can the process be?

  34. Quantum phase transition • bottleneck is likely to be something called a • happens when the energy differenece between the ground state and the first excited state (i.e., the gap) is small. • there, the probability to “get off track” is maximal. Quantum Phase Transition a schematic picture of the gap to the first excited state as a function of the adiabatic parameter . gap QPT

  35. Quantum phase transition • Landau-Zener theory tells us that to stay in the ground state the running time needed is: • exponentially closing gap (as a function of problem size N)  exponentially long running time  exponential complexity.

  36. Quantum phase transition • Landau-Zener theory tells us that to stay in the ground state the running time needed is: • exponentially closing gap (as a function of problem size N)  exponentially long running time  exponential complexity. a two-level avoided crossing system remains in its instantaneous ground state

  37. The quantum adiabatic algorithm • most interesting unknown about QAA to date: could the QAA solve in polynomial time “hard” (NP-complete) problems? or for which hard problems is polynomial in ?

  38. Simulating Adiabatic Quantum Computers

  39. Exact diagonalization • quantum computers are currently unavailable. how does one study quantum computers? • quantum systems are huge matrices. sizes of matrices for nqubits: . • one method: diagonalization of the matrices. however, takes a lot of time and resources to do so. can’t go beyond ~25 qubits. • another method: “Quantum Monte Carlo”.

  40. Quantum Monte Carlo methods • quantum Monte Carlo (QMC) is a generic name for classical algorithms designed to study quantum systems (in equilibrium) on a classical computer by simulating them. • in quantum Monte Carlo, one only samples the exponential number of configurations, where configurations with less energy are given more weight and are sampled more often. • this is usually a stochastic Markov process (a Markovian chain). • there are statistical errors. • not all quantum physical systems can be simulated (sign problem). • quantum systems have an additional “extra” dimension (called “imaginary time” and is periodic). for example a 3D classical system is similar to 2D quantum systems (with notable exceptions).

  41. Method • main goal: • study the dependence of the typical minimum gap • on the size (number of bits) of the problem. • this is because: • polynomial dependence  polynomial complexity! determine the complexity of the QAA for the various optimization problems

  42. Future of adiabatic Quantum Computing

  43. Future of adiabatic QC • so far, there is no clear-cut example for a problem that is solved efficiently using adiabatic quantum computation (still looking though). • the first quantum adiabatic computer (quantum annealer) has been built. ~128 qubits. built by D-Wave (Vancouver). • one piece has been sold to USC / Lockheed-Martin (~1M$). • a strong candidate for future quantum computers. • there are however a lot of technological challenges. • people are looking for other possible uses for it.

  44. Adiabatic QC and Machine Learning • one of many possible avenues of research is Machine Learning. people are starting to look into it. • supervised and unsupervised machine learning. could work if problem can be cast in the form of an optimization problem. • HartmutNeven’s blog: http://googleresearch.blogspot.com/2009/12/machine-learning-with-quantum.html#!/2009/12/machine-learning-with-quantum.html

  45. Adiabatic quantum computation –a tutorial for computer scientists Itay Hen Dept. of Physics, UCSC Advanced Machine Learning class UCSC June 6th 2012

More Related