Itay Hen

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