1 Bob Lucas Federico Spedalieri Information Sciences Institute Viterbi School of Engineering USC Adiabatic Quantum Computing with the D-Wave One

The End of Dennard Scaling

Need More Capability? Application Specific Systems D.E. Shaw Research Anton Massive Scaling – ORNL Cray XK7 Exploit a New Phenomenon Adiabatic Quantum Processor D-Wave One

Overview Adiabatic quantum computation Brief description of D-Wave One The three main thrusts of research: 1.Quantumness 2.Benchmarking 3.Applications

Quantum computer Hamiltonian: H(t) = (1  (t))H 0 + (t)H 1 Prepare the computer in the ground state of H 0 Slowly vary (t) from 0 to 1 Read out the final state: the ground state of H 1 Runtime associated with (g min ) -2 E g min E0E0 E1E Adiabatic Quantum Computation

Adiabatic QC is universal (can compute any function, just like circuit model) But universality may be too much to ask for. Consider only “classical” final Hamiltonians, i.e.: Diagonal (in computational basis) Off-diagonal (in computational basis) The final state is a classical state that minimizes the energy of H 1 Adiabatic Quantum Optimization

Solving Ising models with AQC Ising problem: Find Adiabatic quantum optimization:

Overview Adiabatic quantum computation Brief description of D-Wave One The three main thrusts of research : 1.Quantumness 2.Benchmarking 3.Applications

USC/ISI’s D-Wave One 128 (well, 108) qubit Rainier chip 20mK operating temperature 1 nanoTesla in 3D across processor

Qubits and Unit Cell One qubit SC loop; qubit = flux generated by Josephson current Unit cell compound-compound Josephson junction (CCJJ) rf SQUIDs  flux qubit

Eight Qubit Unit Cell

Tiling of Eight-Qubit Unit Cells

Adiabatic Quantum Optimization Problem: find the ground state of Use adiabatic interpolation from transverse field (Farhi et al., 2000) Graph Embedding implemented on DW-1 via Chimera graph retains NP-hardness V. Choi (2010) 13

Program API

Overview Adiabatic quantum computation Brief description of D-Wave One The three main thrusts of research: 1.Quantumness 2.Benchmarking 3.Applications

16 Experimental Quantum Signature (S. Boixo, T. Albash, F. S., N. Chancellor, D. Lidar)

Classical Simulated Annealing Minimizing a complex cost function we can get trapped in local minima. Add temperature to go “uphill”. Temperature decreases with time.

Quantum resources: tunneling

Degenerate Ising Hamiltonian fold degenerate ground space: +/

20 Classical Thermalization

Several SA schedules

22 Quantum annealing

Quantum Annealing We want to find the ground state of an Ising Hamiltonian: Instead of “temperature” fluctuations, we use quantum fluctuations a transverse field Slowly remove the transverse field to stay on the ground state:

DW1 Gap Gap 1.5 GHz (Temp: 0.35 GHz) Transitions to 4 th order in Small gap -> small coupling!!!

QA closed system

QA open system

QA vs. SA

28 Experiments

Embedding Chip ConnectivityOur Quantum Signature problem as it looks in the chip

DW1 Experiments 144 embeddings Quantum Signature: this state is suppressed

31 Entanglement

32 Entanglement: a definition Separable states Entangled states It is a classical mixture of product states It can be constructed locally

Entanglement in DW2 Even numerically, determining if a state is entangled is NP-hard How can we experimentally show entanglement? 1.Measure the complete density matrix of the system Quantum State Tomography 2.Measure an observable that distinguishes entangled states Entanglement Witnesses Requires a large number of measurements (exponential in the number of qubits) The reconstructed density matrix may not be physical (not PSD) For DW2, these measurements are not even possible For every entangled state there is an entanglement witness Measuring the expectation of Z can prove entanglement But to find Z we need to know the state (or be very lucky) The measurements required will likely not be available in DW2 Separable States

Magnetic susceptibilities in DW2 Use a weakly coupled probe to measure Compute the magnetic susceptibilities as Use perturbation theory (and some assumptions) to write

Separability criteria Apply PPTSE separability criteria to this general state All separable states have a PPTSE for any k Search for PPTSE can be cast as a semidefinite program Produces a hierarchy of separability tests If state is entangled, dual SDP computes an entanglement witness

36 Separability criteria with partial information Some properties of this approach: 1.If the test fails the state may still be entangled 1.We can use a dual approach that checks if a state satisfying the linear constraints is separable (also a SDP) 2.If both tests fail, we need to go to higher k 3.All entangled states will be detected for some k 1.Going beyond k=2 may be tricky (the size of the SDP gets too big) 2.In theory, this is the best you can do with partial information: if you could run the tests for all k, this approach will eventually prove that all states satisfying the linear constraints are entangled or that there is one such state that is separable

Overview Adiabatic quantum computation Brief description of D-Wave One The three main thrusts of research: 1.Quantumness 2.Benchmarking 3.Applications

38 Benchmarking

Benchmarking hard problems 10 – 108 qubits

Benchmarking hard problems 108 qubits, 5us – 20ms

Classical repetition cost r

Some benchmarking Exponential run time for the best exact classical solver Faster exponential run time for the D-Wave (Vesuvius)

Some benchmarking (3BOP h and J)

Overview Adiabatic quantum computation Brief description of D-Wave One The three main thrusts of research: 1.Quantumness 2.Benchmarking 3.Applications

Some NP-complete problems and their applications ProblemApplication Traveling salesmanLogistics, vehicle routing Minimum Steiner treeCircuit layout, network design Graph coloringScheduling, register allocation MAX-CLIQUESocial networks, bioinformatics QUBOMachine learning (H. Neven, Google) Integer Linear ProgrammingNatural language processing Sub-graph isomorphismCheminformatics, drug discovery Job shop schedulingManufacturing Motion planningRobotics MAX-2SATArtificial intelligence The problem addressed by quantum annealing is NP-Complete

Given a: – Finite transition system M – A temporal property p The model checking problem: – Does M satisfy p ? It typically requires analyzing every possible path the system can take Workarounds: – Binary Decision Diagrams (BDDs) – Abstractions The Model Checking Problem Complexity is exponential on the number of states State space explosion problem 46

Abstraction Group states together – Eg., localization: neglect some state variables (make them invisible) Eliminate details irrelevant to the property Obtain smaller models sufficient to verify the property using traditional model checking tools Disadvantage: — Loss of Precision — False positives/negatives Spurious counterexamples 47

Counterexample-Guided Abstraction-Refinement (CEGAR) Check Counterexample Obtain Refinement Cue Model Check Build New Abstract Model M’M No Bug Pass Fail Bug Real CE Spurious CE SATILP Machine learning AQC 48

Summary DW1 is a programmable superconducting quantum adiabatic processor It solves a particular type of combinatorial optimization problem We have investigated the quantum nature of the device We chose a problem for which classical thermalization and quantum annealing predict different statistics Experiments agree with quantum annealing prediction suggesting quantum annealing is surprisingly robust against noise Working on an experimental test for entanglement Benchmarks show promising scaling when compared with classical solvers Currently working to bridge the gap between the device and real applications (non-trivial issues to be addressed)