Bob Lucas Federico Spedalieri Information Sciences Institute Viterbi School of Engineering USC

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Bob Lucas Federico Spedalieri

Information Sciences InstituteViterbi School of Engineering

USC

Adiabatic Quantum Computing with the D-Wave One

The End of Dennard Scaling

Need More Capability?

Application Specific SystemsD.E. Shaw Research Anton

Massive Scaling – ORNL Cray XK7

Exploit a New PhenomenonAdiabatic 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))H0 + (t)H1

• Prepare the computer in the ground state of H0

• Slowly vary (t) from 0 to 1

• Read out the final state: the ground state of H1

• Runtime associated with (gmin)-2

E

gmin

E0

E1

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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 H1

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 One128 (well, 108) qubit Rainier chip

20mK operating temperature1 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)

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Program API

Overview

• Adiabatic quantum computation

• Brief description of D-Wave One

• The three main thrusts of research:

1. Quantumness

2. Benchmarking

3. Applications

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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

+1

-1

-1

17-fold degenerate ground space:

+/- 1

+/- 1

+/- 1

1

+/- 1

1

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-1

-1

-1

-1-1

-1-1

-1

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Classical Thermalization

Several SA schedules

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Quantum annealing

Quantum Annealing

We want to find the ground state of an Ising Hamiltonian:

Instead of “temperature” fluctuations, we use quantum fluctuationsa 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 4th order in

Small gap ->small coupling!!!

QA closed system

QA open system

QA vs. SA

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Experiments

Embedding

Chip Connectivity Our Quantum Signature problemas it looks in the chip

DW1 Experiments

144 embeddings

Quantum Signature: this state is suppressed

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Entanglement

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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

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Separability criteria with partial information

• Some properties of this approach:

1. If the test fails the state may still be entangled

2. We can use a dual approach that checks if a state satisfying the linear constraints is separable (also a SDP)

3. If both tests fail, we need to go to higher k

4. All entangled states will be detected for some k

5. Going beyond k=2 may be tricky (the size of the SDP gets too big)

6. 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

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Benchmarking

Benchmarking hard problems10 – 108 qubits

Benchmarking hard problems108 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

Problem Application

Traveling salesman Logistics, vehicle routing

Minimum Steiner tree Circuit layout, network design

Graph coloring Scheduling, register allocation

MAX-CLIQUE Social networks, bioinformatics

QUBO Machine learning (H. Neven, Google)

Integer Linear Programming Natural language processing

Sub-graph isomorphism Cheminformatics, drug discovery

Job shop scheduling Manufacturing

Motion planning Robotics

MAX-2SAT Artificial 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

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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

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Counterexample-Guided Abstraction-Refinement (CEGAR)

Check Counterexample

Obtain Refinement Cue

Model CheckBuild New Abstract Model

M’M

No Bug

Pass

Fail

BugReal CESpurious CE

SATILPMachine 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)