Statistics and Quantum Computing

Transcription

1 Statistics and Quantum Computing Yazhen Wang Department of Statistics University of Wisconsin-Madison yzwang Workshop on Quantum Computing and Its Application George Washington University, March 16, 2017 Yazhen (at UW-Madison) 1 / 31

2 Outline Quantum computation Quantum physics and quantum probability Quantum annealing and Monte Carlo methods Statistical analysis of quantum computing data Yazhen (at UW-Madison) 2 / 31

3 Quantum Yazhen (at UW-Madison) information 4 / 31 Computing Paradigm Classical (today s) computers based on electronic devices Obey laws of classical physics Computer chip technology Size limitation: Quantum effects at atomic scale New computing paradigm Quantum computing uses quantum devices for computing: Quantum Information Quantum Computation Moore s Law is Dead

4 Complexity Classic vs quantum Classical physical system with b components: Dimension d b Quantum system with b components: Dimension d = 2 b Classical computer simulation 2 b bits of memory to simulate quantum system of size b Quantum computing Quantum device: Handle exponential complexity of data Quantum information: Atoms and molecules for information Yazhen (at UW-Madison) 5 / 31

5 Quantum Computing Quantum Bits A quantum computer taps directly into the fundamental fabric of reality the strange and counter-intuitive world of quantum mechanics to speed computation. Rather than store information as 0s or 1s as conventional computers do, a quantum computer uses qubits which can be 1 or 0 or both at the same time. This quantum superposition enables quantum computers to consider and manipulate all combinations of bits simultaneously, making Classic bit vs quantum bit quantum computation powerful and fast. Classic bit: 0 1 D-Wave systems use quantum annealing to solve problems. Quantum Quantum annealing tunes qubits bit from (qubit): their superposition 0 state to and a classical 1 state to return a set of answers scored to show the best solution. The D-Wave Two System Qubit superposition In order for the quantum effects to take place, a quantum processor must operate in an extreme environment - temperatures close to absolute zero, shielded from magnetism, and isolated from vibration and external signals of A superposition of qubits 0 & 1 any form. The environmental enclosure shields the processor to 50,000 less than the Earth s magnetic field. ψ = α 0 0 +α 1 1, α α 1 2 = 1 Inside the box the closed cycle dilution refrigerator lowers the temperature at each level until it reaches almost absolute zero (0.02 Kelvin), 150x colder than interstellar space. Qubit measurement The quantum processor contains a lattice of tiny superconducting circuits (qubits) made from the metal niobium, which exhibits quantum behaviors at low temperatures. Qubits are the basic elements that the system uses to solve optimization problems. Measure ψ : Obtain either result 0 with probability α 0 2 or The quantum processor is surrounded by electronics used to program the processor and read out the results. Quantum processor Qubits in red result 1 with probability α 1 2. Qubits can exist in a state of 1 or 0 simultaneously ψ : both 0 & 1 Dilution refrigerator 0 = 1, 1 = 0, ψ = α Bloch sphere α 1 Yazhen (at UW-Madison) 6 / 31

6 Quantum System Quantum state Quantum system may be in a pure state ψ ( which is a unit vector ) Quantum evolution Time evolution of quantum system: the Schrödinger equation ψ(t) 1 = H ψ(t) ψ(t) = e 1H ψ(0) t H: Hamiltonian Yazhen (at UW-Madison) 9 / 31

7 Quantum + Statistical Learning Quantum State Tomography Recover a quantum state (a density matrix) Statistical Learning Handle sparse or low rank matrices Quantum Tomography via Statistical Learning Adopt statistical learning methods and algorithms to quantum state tomography Key Insight Quantum tomography E(N k ) = tr(ρm k ) Matrix completion tr(x k ρ) Gross et al. (2010, Phys. Rev. Lett.), Wang (2013, Ann. Statist.) Optimal density matrix estimation as n, p Minimax theory: Binomial distribution vs normal distribution under spectral norm (Cai, Kim, Wang, Yuan & Zhou, Ann Statist. 2016) Yazhen (at UW-Madison) 13 / 31

8 Annealing 7000 year old technology to slowly cool metal and glass for property improvement Ising Model lattice sites: Hamiltonian (or energy) H Ising (s) = J ij s i s j i j j J ij : coupling, h j : local field Optimization Problem h j s j s = {s j = ±1, 1 j N} ARTICLES a c Ferrimagnet (FR) b d Antiferromagnet (AF) Given an instance {J ij & h j }, among total 2 N configurations find N phase θ defined by Φ = Φ e iπ/2 and m we call AF, illustra forms first when th The pattern of using the η-expans state has the lowest to the state found f Putting togethe h > 0, h < 0 and phase diagram con Fig. 3a. On general either be direct and of coexistence cor Isosceles triangl Now we adapt a complicated case w Neel 1 (N1) Neel 2 (N2) distortion of the t s = {sj, j = 1, 2,, N} to minimize energy H Ising(s). in the real materi Yazhen (at UW-Madison) Figure 2 Magnetic ordering patterns of FR, AF, N1 and N2 phases. The arrows represent the projection of the spin on the local easy axis When h < 0, associated 17 / with 31 the

9 Simulated Annealing (SA) Rugged Energy Landscape NP-Hard Many interesting hard problems (e.g. traveling salesman problem, protein folding, portfolio optimization, integer factoring) can be mapped onto the optimization problem. Simulated Annealing (Kirkpatrick et. al 1983, Science) 30 year old technique to slowly cool a model in Monte Carlo simulations for solving an optimization problem. Yazhen (at UW-Madison) 18 / 31

10 Markov Chain Monte Carlo (MCMC) Metropolis-Hastings algorithm Sample from Boltzmann distribution P(s) = exp[ H Ising(s)/T ], Z T = Z T s Thermal Fluctuation [ exp H ] Ising(s), T =temperature T Slowly lower the temperature to reduce the escape probability of trapping in local minima, Annealing schedule : T i 1 i + 1 or 1 log(i + 1) Many Tries for Solution The minimum energy of one run can be the global minimum with some probability. Try many runs to explore the search space, and find the minimum energy of each run to search for the solution of the minimization problem. Yazhen (at UW-Madison) 19 / 31

11 Quantum Annealing (QA): Basic Idea Classical optimization: Min{H Ising (s) : s { 1, 1} N } Find a target quantum system with Hamiltonian H(1) whose energies match H Ising (s): H(1) = diag{h Ising (s 1, ), H Ising (s 2 N )}. Create an initial quantum system with Hamiltonian H(0) whose lowest energy state is known and easy to prepare. QA: Engineer H(0) in its lowest energy state and gradually move H(0) H(1) Energy State H(0) Ground State H(1) Yazhen (at UW-Madison) 21 /

12 Quantum Model Quantum Ising Hamiltonian H Ising = i j D-Wave Qubits J ij σ z i σz j j h j σ z j R x x cjj Initial Hamiltonian H X = j σx j : a transverse magnetic field Ground state of H X : spins all in x direction= H X H Ising via weight shifting Qubit M eff H(t) = xa(t)h X + B(t)H Ising { Quantum annealing Coupler schedule A(t): transverse magnetic M co,i field M H X Φx q-qfp B(t): Ising spin qfp glass H Ising x QFP Φ co latch Quantum dc SQUID Model: Spin glass in a transverse magnetic field Φx ro i ro Φ q x x Φ LT M qfp-ro H X : induce quantum fluctuations & turn classical H Ising to quantum Harris et al., Experimental demonstration of a robust and scalable flux qubit, Yazhen (at UW-Madison) 24 / 31

13 Classical Thermal vs Quantum Annealing flips and thermal escape rule quantum tunneling out of local minima Thermal Jump Energy Energy Quantum Tunneling Configuration Configuration Yazhen (at UW-Madison) 25 / 31

14 Experiments on D-Wave One machine (128 qubits) Random Instance: Assign ±1 to couplings J ij at random Solve the problem of min H Ising on (i) D-Wave machine and by (ii) simulated classical annealing & (iii) simulated quantum annealing Run each annealing algorithm 1000 times and compute success probability = frequency of finding the lowest energy over 1000 repetitions Repeat the whole procedure 1000 times to obtain success probabilities for 1000 randomly selected instances Supplementary material for Quantum a Minimize H Ising = J ij σ z i σz j Yazhen (at UW-Madison) 51 /

15 Histogram of Ground State Success Probability Data DW: Work SA (a) DW (b) SA Number of Instance Number of Instance Success Probability Success Probability Yazhen (at UW-Madison) 52 / 31

16 Simulated Quantum Annealing (SQA) Spin glass in transverse field H = A(t)H X + B(t)H Ising, two parts non-commuting Path integral representation via Suzuki-Trotter expansion H H 2+1 = classical (2+1)-dimensional anisotropic Ising system (2 + 1)-dimensional system Two directions: along the original 2-dimensional direction spins have Chimera graph couplings, and along the extra (imaginary-time) direction spins have uniform couplings Quantum Monte Carlo H 2+1 : a collection of 2-dimensional classical Ising systems, that can be simulated by MCMC with moves in both directions A B C Yazhen (at UW-Madison) 53 / 31

17 Histogram of Ground State Success Probability Data DW: Work SA (a) DW (b) SA Number of Instance Number of Instance Success Probability Success Probability SQA (c) SQA Number of Instance Success Probability Yazhen (at UW-Madison) 55 / 31

18 What to do? Multiple hypothesis tests: Instance by instance check for ground state probabilities among annealing methods Goodness-of-fit test: Check ground state successful probabilities among annealing methods over 1000 instances Test for distribution shapes of ground state successful probabilities among annealing methods (Wang et al. 2016, Statistical Science) Investigae factors lead to the distribution shapes Study sampling properties of quantum annealing & D-Wave machines Find types of problems (machine learning, classification, portfolio allocation, protein folding, statistical computing, etc) that are good for D-Wave quantum computer to solve Yazhen (at UW-Madison) 59 / 31

19 Concluding Remarks Monte Carlo methods are important for quantum computing Statistics has an important role to play in quantum science High-dimensional statistics and compressed sensing quantum for tomography and quantum information. Statistical inference & Monte Carlo methods for quantum computing. Current & Future Research How to characterize D-Wave s sampling performance? Not sample from Boltzmann distribution, more on ground and high energy states. What kinds of computing problems can be benefited from D-Wave sampling and are good for D-Wave quantum computer? Machine learning? Statistical computing? Computational finance?... Yazhen (at UW-Madison) 60 / 31

20 References Wang, Y., Wu, S. and Zou, J. (2016). Quantum annealing with MCMC and D-Wave quantum computers. Statistical Science 31, Cai, Kim, Wang, Yuan & Zhou (2016). Optimal large-scale quantum state tomography with Pauli measurements. Ann. Statist. 44, Wang, Y. and Xu, C. (2015). Density matrix estimation in quantum homodyne tomography. Statistica Sinica 25, Wang, Y. (2013). Asymptotic equivalence of quantum state tomography and noisy matrix completion. Ann. Statist. 41, Wang, Y. (2012). Quantum computation and quantum information. Statistical Science 27, Wang, Y. (2011). Quantum Monte Carlo simulation. Ann. Appl. Statist. 5, Nielsen, M. and Chuang, I. (2010). Quantum Computation and Quantum Information. Cambridge: Cambridge University Press. Yazhen (at UW-Madison) 61 / 31