Adiabatic quantum computation a tutorial for computer scientists

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1 Adiabatic quantum computation a tutorial for computer scientists Itay Hen Dept. of Physics, UCSC Advanced Machine Learning class UCSC June 6 th 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? computer 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? computer input state computation a classical algorithm (circuit) looks like this: output state input state computation output state 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).

6 What is a quantum computer? computer 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 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 I m here or I m there quantum world I m here and I m there

9 What are qubits? while a bit can be either in the a 0 state or a 1 state (this is Dirac s ket notation). a qubit can be in a superposition of these two states, e.g., φ = imagine 0 and a 1 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 nonintuitive. 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 2 3 = 8 possible classical configurations : 000, 001, 010, 011, 100, 101, 110, 111, the corresponding quantum state of a 3-qubit system would be: 7 φ = i=0 this is a (normalized) 8-component vector over the complex numbers (8 complex coefficients). i enumerates the 8 classical configurations listed above. c i i

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: f: {0,1} 3 R F = and asking which is the smallest eigenvalue (minimum of f ) and corresponding eigenvector (minimizing configuration). here, the basis vectors e i correspond to i 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., min φ φ F φ = min φ φ t Fφ but problem now is much harder. (this is Dirac s bra 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 input state computation output state quantum computers manipulate quantum bits or qubits for short. a quantum algorithm (circuit) may look like this: input state computation output state

14 What is a quantum computer? computer input state computation space of possible quantum states is huge. output state 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 adiabatic quantum 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 N entries in O N 1/2 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 Computing is a general approach to solve a broad range of hard optimization problems using a quantum computer [Farhi et 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. f probability of jumping over the barrier is exponentially small in the height of the barrier. x ball is unlikely to end up in the true minimum.

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. f quantum mechanically, the ball can tunnel through thin but high barriers! x sometimes more likely to end up in the true minimum. tunneling is a key feature of quantum mechanics.

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 instantaneous global 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: harmonic potential ψ(x, t) 2

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 diabatic process): harmonic potential ψ(x, t) 2

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. harmonic potential ψ(x, t) 2

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

32 The quantum adiabatic algorithm (QAA) the interpolating Hamiltonian is this: H(t) = s(t)h p + 1 s(t) H d H p is the problem Hamiltonian whose ground state encodes the solution of the optimization problem H d is an easily solvable driver Hamiltonian, which does not commute with H p the parameter s obeys 0 s t 1, with s 0 = 0 and s T = 1. also: H 0 = H d and H T = H p. here, t stands for time and T is the running time, or complexity, of the algorithm.

33 The quantum adiabatic algorithm (QAA) the interpolating Hamiltonian is this: H(t) = s(t)h p + 1 s(t) H d the adiabatic theorem ensures that if the change in s t 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 H p. 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 Quantum Phase Transition 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. gap E 1 a schematic picture of the gap to the first excited state as a function of the adiabatic parameter s. 0 QPT 1 s H = H d H = H p

35 Quantum phase transition Landau-Zener theory tells us that to stay in the ground state the running time needed is: T 1 2 ΔE min 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: T 1 2 ΔE min 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 T polynomial in N?

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 n qubits: N = 2 n. 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: determine the complexity of the QAA for the various optimization problems study the dependence of the typical minimum gap ΔE min = min ΔE s 0,1 on the size (number of bits) of the problem. this is because: T 1 2 ΔE min polynomial dependence polynomial complexity!

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. Hartmut Neven s blog:

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

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