* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Talk - Quantum Device Lab
Quantum decoherence wikipedia , lookup
Measurement in quantum mechanics wikipedia , lookup
Copenhagen interpretation wikipedia , lookup
Quantum field theory wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Scalar field theory wikipedia , lookup
Density matrix wikipedia , lookup
Particle in a box wikipedia , lookup
Path integral formulation wikipedia , lookup
Quantum dot wikipedia , lookup
Bell's theorem wikipedia , lookup
Hydrogen atom wikipedia , lookup
Probability amplitude wikipedia , lookup
Quantum entanglement wikipedia , lookup
Molecular Hamiltonian wikipedia , lookup
Coherent states wikipedia , lookup
Many-worlds interpretation wikipedia , lookup
Quantum electrodynamics wikipedia , lookup
Quantum fiction wikipedia , lookup
Orchestrated objective reduction wikipedia , lookup
History of quantum field theory wikipedia , lookup
EPR paradox wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
Interpretations of quantum mechanics wikipedia , lookup
Quantum group wikipedia , lookup
Quantum teleportation wikipedia , lookup
Quantum computing wikipedia , lookup
Hidden variable theory wikipedia , lookup
Quantum key distribution wikipedia , lookup
Quantum state wikipedia , lookup
Ants Remm, Roland Matt Quantum Adiabatic Computation Overview The problems solved The methods Simulated annealing Quantum adiabatic computation Simulated quantum annealing Results Comparing SA,QA, SQA Is there a quantum speedup? D-PHYS first part by Ants second part by Roland Ants Remm, Roland Matt | 29.04.2016 | 2 The spin glass system 𝐽𝑖𝑗 𝜎𝑖𝑧 𝜎𝑗𝑧 + 𝐻= 𝑖<𝑗 ℎ𝑖 𝜎𝑖𝑧 𝑖 A set of coupled qubits Couplings 𝐽𝑖𝑗 between neighbours or all qubits, could be random Couplings between more that two qubits also possible: 𝐽𝑖𝑗𝑘 𝜎𝑖𝑧 𝜎𝑗𝑧 𝜎𝑘𝑧 Local fields ℎ𝑖 Not ergodic – many local minima, frustration D-PHYS Ants Remm, Roland Matt | 29.04.2016 | 3 Problem equivalence Many different optimization problems can be reduced to finding the ground state of the spin glass system: D-PHYS The knapsack problem Finding Hamiltonian cycles Graph colouring Image restoration, etc. Ants Remm, Roland Matt | 29.04.2016 | 4 Problem equivalence: Image restoration D-PHYS Ants Remm, Roland Matt | 29.04.2016 | 5 Problem equivalence: Fair partitioning Given a set of numbers 𝑛𝑖 we want to divide the numbers into two groups of equal sum For example 40, 25, 5, 11, 15, 11, 5, 26, 14 : 40 + 5 + 11 + 15 + 5 = 25 + 11 + 26 + 14 = 76 Equivalent Hamiltonian: 2 𝑛𝑖 𝜎𝑖𝑧 𝐻= 𝑖 𝐽𝑖𝑗 = 2𝑛𝑖 𝑛𝑗 D-PHYS Ants Remm, Roland Matt | 29.04.2016 | 6 Problem equivalence: The 𝒌-satisfiability problem Given a set of Boolean clauses 𝐶 on 𝑁 bits, find a configuration of the bits that satisfies all (the most) clauses Each clause acts on 𝑘 bits and specifies for each of them whether they should be 0 or 1. A Hamiltonian describing a clause for bits 𝑖1 , 𝑖2 , … , 𝑖𝑘 can be written as 1 ± 𝜎𝑖𝑧𝑘 1 ± 𝜎𝑖𝑧1 1 ± 𝜎𝑖𝑧2 𝐻𝐶 = … 2 2 2 Entire system Hamiltonian to be minimized: 𝐻 = − 𝐻𝐶 D-PHYS Ants Remm, Roland Matt | 29.04.2016 | 7 Simulated annealing An ensemble of classical systems, stochastically exploring the state space Probability for a system to cross a potential barrier of Δ𝐸 − 𝑘𝑇 height Δ𝐸 is proportional to 𝑒 Slowly lower the temperature so that Boltzmann − 𝐸 𝑘𝑇 distribution 𝑝(𝐸) ∝ 𝑒 is maintained At the end of the simulation only ground state is occupied D-PHYS Ants Remm, Roland Matt | 29.04.2016 | 8 Simulated annealing: Why it works Probability of crossing the barrier left to right (right to left) 𝑝1(2) ~𝑒 − Δ𝐸1(2) 𝑘𝑇 For 𝑘𝑇 > Δ𝐸1 , Δ𝐸2 the barrier is irrelevant: easily crossed both ways For Δ𝐸1 < 𝑘𝑇 < Δ𝐸2 crossings from right to left are dominant For 𝑘𝑇 < Δ𝐸1 , Δ𝐸2 potential wells are isolated D-PHYS Δ𝐸1 Δ𝐸2 Ants Remm, Roland Matt | 29.04.2016 | 9 Simulated annealing: How it works: The Metropolis algorithm Prepare a random ensemble of system configurations Sequentially select a configuration from the ensemble and take a random step with it (a bit flip in our example) If the energy change Δ𝐸 is negative: accept the step positive: accept the step with probability exp − Δ𝐸 𝑘𝑇 . Repeated application of this step gives us the Boltzmann distribution for the configurations in the ensemble D-PHYS Ants Remm, Roland Matt | 29.04.2016 | 10 Simulated annealing: The shortcomings D-PHYS Thermal jump Energy The spin glass system has typically many local minima High probability of having relatively high and narrow potential barriers For simulated annealing, only the height of the barrier matters Idea: what if we could tunnel through the barrier Quantum tunnelling Configuration Ants Remm, Roland Matt | 29.04.2016 | 11 Quantum adiabatic computation A single quantum system at zero temperature (totally coherent) Start with the system in the ground state of a simple Hamiltonian 𝐻0 , e.g. 𝜎𝑖𝑥 𝐻0 = − 𝑖 Slowly change the Hamiltonian to the problem Hamiltonian 𝐻𝑃 : 𝑡 𝑡 𝐻 𝑡 = 1 − 𝐻0 + 𝐻𝑃 𝑇 𝑇 D-PHYS Ants Remm, Roland Matt | 29.04.2016 | 12 Quantum adiabatic computation: The adiabatic theorem If the Hamiltonian is changing slow enough, the system stays near its instantaneous ground state. How slow is slow enough: 𝜀 𝑇≫ 𝑔min 2 with 𝜀 = max 𝐸1 (𝑡) 𝐻𝑃 − 𝐻0 𝐸0 (𝑡) 0≤𝑡≤𝑇 𝐸0 (𝑡) and 𝐸1 𝑡 the two lowest energy instantaneous eigenvalues (-states) and 𝑔min the minimal gap between 𝐸0 (𝑡) and 𝐸1 𝑡 . D-PHYS Ants Remm, Roland Matt | 29.04.2016 | 13 Quantum adiabatic computation: Motivation for the initial Hamiltonian 1 1 𝐻0 = − 2 2𝑁 D-PHYS 𝜎𝑖𝑥 𝑖 Simple to implement Describes a transverse interaction field Does not commute with the problem Hamiltonian Avoids energy level crossings (degeneracy) Ground state is the equal superposition state, spanning all of the solution space Ants Remm, Roland Matt | 29.04.2016 | 14 Quantum adiabatic computation: Single qubit example Let’s consider the simplest example: finding the ground state of a single qubit 1 𝐻𝑃 = 1 − 𝜎 𝑧 2 1 𝑠 𝐻 𝑡 = 2 −𝑠 𝐸0,1 D-PHYS 1 𝐻0 = 1 − 𝜎 𝑥 2 −𝑠 2−𝑠 𝑡 𝑠= 𝑇 1 = 1 ∓ 1 − 2𝑠 + 2𝑠 2 2 Ants Remm, Roland Matt | 29.04.2016 | 15 Quantum adiabatic computation: Single qubit example D-PHYS Ants Remm, Roland Matt | 29.04.2016 | 16 Quantum adiabatic computation: Single qubit example Commuting Hamiltonians means that energy levels cross 1 𝐻𝑃 = 1 − 𝜎 𝑧 2 1 𝐻0 = 1 + 𝜎 𝑧 2 D-PHYS Ants Remm, Roland Matt | 29.04.2016 | 17 Quantum adiabatic computation: NP-hard example D-PHYS Ants Remm, Roland Matt | 29.04.2016 | 18 Quantum adiabatic computation: NP-hard example Random instances of a specific type of 3-SAT problem, the “Exact cover” Each clause requires that exactly one of the three bits involved must be 1 Remember: 𝐻𝐶 𝑧1 𝑧2 … 𝑧𝑁 = 1 ∙ 𝑧1 𝑧2 … 𝑧𝑁 if clause 𝐶 is satisfied 𝐻𝐶 𝑧1 𝑧2 … 𝑧𝑁 = 0 ∙ 𝑧1 𝑧2 … 𝑧𝑁 otherwise Problem Hamiltonian: 𝐻𝑃 = − 𝐻𝐶 D-PHYS Ants Remm, Roland Matt | 29.04.2016 | 19 Quantum adiabatic computation: NP-hard example D-PHYS Ants Remm, Roland Matt | 29.04.2016 | 20 Quantum adiabatic computation: NP-hard example From exponential to quadratic complexity But: Median does not indicate worst case scaling They looked at a specific subset of “Exact Cover” problems Don’t know the behaviour for large 𝑁 Requires coherent quantum evolution! D-PHYS Ants Remm, Roland Matt | 29.04.2016 | 21 Simulated quantum annealing Is a quantum Monte Carlo algorithm Same annealing schedule as the quantum annealer Start with a strong initial transverse field which goes to zero during simulation Start ramping up the problem Hamiltonian from zero Monte Carlo dynamics instead of unitary evolution Could use discrete or continuous time path integrals D-PHYS Ants Remm, Roland Matt | 29.04.2016 | 22 Comparing SA, QA and SQA D-PHYS Ants Remm, Roland Matt | 29.04.2016 | 23 One needs to note the difference D-PHYS Ants Remm, Roland Matt | 29.04.2016 | 24 How the problem is set up D-PHYS Ants Remm, Roland Matt | 29.04.2016 | 25 Similarities between D-Wave and SQA D-PHYS Ants Remm, Roland Matt | 29.04.2016 | 26 Correlation plots Another way of comparing algorithms D-PHYS Ants Remm, Roland Matt | 29.04.2016 | 27 D-PHYS Ants Remm, Roland Matt | 29.04.2016 | 28 Pitfalls of detecting quantum speed-up D-PHYS Ants Remm, Roland Matt | 29.04.2016 | 29 Speedup of DW2 compared to SA • Scaling of the highest quantile is the most informative • High quantiles are hard for DW2 • Range 7 is harder than Range 1 • Overall positive slope shows that DW is better D-PHYS Ants Remm, Roland Matt | 29.04.2016 | 30 We have a quantum device, but is there an universal quantum speedup? D-PHYS Ants Remm, Roland Matt | 29.04.2016 | 31 References http://people.fas.harvard.edu/~lucas/zurichCStalk.pdf Cohen, Eliahu, and Boaz Tamir. "D-Wave and predecessors: From simulated to quantum annealing." International Journal of Quantum Information 12.03 (2014): 1430002. Farhi, Edward, et al. "Quantum computation by adiabatic evolution." arXiv preprint quant-ph/0001106 (2000). Farhi, Edward, et al. "A quantum adiabatic evolution algorithm applied to random instances of an NP-complete problem." Science 292.5516 (2001): 472-475. Rønnow, Troels F., et al. "Defining and detecting quantum speedup."Science 345.6195 (2014): 420-424. Boixo, Sergio, et al. "Evidence for quantum annealing with more than one hundred qubits." Nature Physics 10.3 (2014): 218-224 D-PHYS Ants Remm, Roland Matt | 29.04.2016 | 32 Thank you! D-PHYS Ants Remm | 29.04.2016 | 33