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    Adiabatic Quantum Optimization Processor Performance

    Richard Harris

    D-Wave Systems Inc.Burnaby, BC Canada

    September 2009

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    Outline

    1 C1 Ising Spin Processor

    2 AQC Processor Performance

    3 Problem Dissection

    4 AQO Processor Performance

    5 Is This A Quantum Processor?

    6 Conclusions and Whats Next

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    C1 Ising Spin Processor

    Outline

    1 C1 Ising Spin Processor

    2 AQC Processor Performance

    3 Problem Dissection

    4 AQO Processor Performance

    5 Is This A Quantum Processor?

    6 Conclusions and Whats Next

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    C1 Ising Spin Processor

    Optimization Problem Specification

    Let there be an optimization problem of the form

    Ec(s) = i

    hisi +i,j>i

    Ksisj (1)

    where

    1

    hi

    +1 and

    1

    K

    +1, for which there exists an

    optimal solution s that minimizes the energy Ec(s), where the individualelements si = 1. This problem can be mapped onto a dimensionless Isingspin Hamiltonian of the form

    H(t) =

    i

    hi(i)z +

    i,j

    Ki,j(i)z

    (j)z (2)

    Here, the groundstate configuration of the spins s represents the lowestenergy solution to the optimization problem. The objective is toimplement an algorithm in hardware that reliably finds this groundstate.

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    C1 I i S i P

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    C1 Ising Spin Processor

    Adiabatic Quantum Optimization (AQO)

    Introduce quantum mechanical degrees of freedom to create a quantumIsing spin Hamiltonian of the form

    H(t) =

    i

    hi(i)z +

    i,j

    Ki,j(i)z

    (j)z

    i

    f(t)(i)x (3)

    Use physical system to find the groundstate by evolving f(t) such that

    f(0) hi,Kf(t ) hi,Ki,j

    See Farhi et al, Science 292, 472 (2001).

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    C1 I i S i P

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    C1 Ising Spin Processor

    Implementation

    CJJ rf-SQUID Flux Qubits

    Low energy rf-SQUID Hamiltonian can be approximated as a qubit:

    Hq = 12

    [z + qx] (4)

    where = 2I

    pq

    xq is the bias energy and q is the tunneling energy. Assuch,

    Ipq and q are the defining properties of a flux qubit.

    We use compound-compound Josephson junction (CCJJ) rf-SQUID flux

    qubits. Here, an external flux bias

    x

    ccjj controls the tunnel barrier height,thus allowing us to initialize qubits with q kBT and then annealingthe qubit by slowly raising the tunnel barrier such that q 0. A fulldisclosure of the CCJJ rf-SQUID flux qubit has been posted asarXiv:0909.4321.

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    C1 Ising Spin Processor

    Implementation

    Example Ipq and

    0.67 0.665 0.66 0.655 0.65 0.6450.4

    0.6

    0.8

    1

    1.2

    1.4

    xccjj/0

    |Ip q|(A)

    a) DataTheory

    0.67 0.665 0.66 0.655 0.65 0.64510

    5

    106

    107

    108

    109

    1010

    xccjj /0

    q

    /h(Hz)

    b)kbT/h

    MRT Data

    LZ Data

    g| Ipq |g Data

    Theory

    Theory from CCJJ rf-SQUID Hamiltonian with independentlycalibratedparameters. MRT (incoherent regime), see Harris et al, PRL 101, 117003(2008). LZ (incoherent regime), see Johannson et al, PRB 80, 012507

    (2009). g| Ipq |g (coherent regime), see Harris et al, arXiv:0909.4321.

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    C1 Ising Spin Processor

    Implementation

    Flux Qubits are Used in the Flux Basis in AQO

    Qubits are used in the flux basis in AQO, not the energy eigenbasis as inmany (but not all) implementations of gate model quantum computation.

    5 2.5 0 2.5 5

    0

    5

    10

    15

    20

    25

    (q 0/2)/Lq (A)

    Energy/h(GH

    z)

    ||

    |g

    |e

    State Energy Basis Flux Basis

    |0 |g | |1 |e |

    Groundstate Eigenstate, |0 (|0 + |1) /2Excited State Eigenstate, |1 (|0 |1) /2

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    C1 Ising Spin Processor

    Implementation

    A Network of Inductively Coupled rf-SQUID Flux Qubits

    Use rf-SQUID flux qubits as quantum Ising spins. Provide tunable mutualinductances between qubits to implement pairwise interactions. Lowenergy Hamiltonian for general bias conditions can be written as

    H(t) = 1

    2i

    x

    i (t)2|Ip

    i

    x

    ccjj(t) |

    (i)

    z + q

    x

    ccjj(t)

    (i)

    x

    +i,j

    Mi,j

    cpli,j

    Ipq

    xccjj(t)2 (i)z (j)z (5)

    x

    i ,x

    ccjj qubit external flux biasesMi,j(

    cpli,j ) tunable mutual inductance between qubitsq qubit tunnel energy|Ipq| qubit persistent current

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    C1 Ising Spin Processor

    Implementation

    Mapping Problems onto Hardware

    Assume that all qubits are identical: |Ipi (t)| |Ipq(t)| and i(t) (t).Map problem parameters hi and K in the dimensionless problemHamiltonian onto the hardware Hamiltonian. Scale problem energies tothe antiferromagnetic (AFM) coupling energy.

    H(t) = JAFM(t)

    i

    hi(i)z +

    i,j

    Ki,j(i)z

    (j)z

    1

    2

    i

    (t)(i)x (6)

    hi JAFM(t) = |Ipq(t)|xi (t)K JAFM(t) = M|Ipq(t)|2

    JAFM(t) = MAFM|Ipq(t)|2

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    g p

    Implementation

    Removing Time-Dependence In Problem Mapping

    The problem parameters hi and K are supposed to be time-independent.However, the mapping onto physical hardware yielded

    hi =|Ipq(t)|qi (t)

    MAFM|Ipq(t)

    |2

    =xi (t)

    MAFM|Ipq(t)

    |(7)

    K =M|Ipq(t)|2

    MAFM|Ipq(t)|2 =M

    MAFM(8)

    Clearly K are naturally time-independent. However, in order to render hi

    time-independent one must apply time-dependent qubit flux biases

    xi (t) = hi MAFM|Ipq(t)| (9)

    See Harris et al, arXiv:0903.3906.

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    g p

    Implementation

    Running the AQO Algorithm

    To run AQO on this hardware, use given mappings for hi and K and sweepqubit tuning parameters from regime where JAFM to JAFM.

    H(t) = JAFM(t)

    i

    hi(i)z +

    i,j

    Ki,j(i)z

    (j)z

    12

    i

    (t)(i)x (10)

    0.63 0.64 0.65 0.66 0.670

    0.05

    0.1

    0.15

    0.2

    xcjj/0

    Temperature(K)

    T

    /2kBJAFM/kB

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    g

    C1 Processor Architecture

    Processor Unit Cell

    Hardware has been detailed in several publications. Key references:

    Programmable Control Circuitry: Johnson et al, arXiv:0907.3757

    High Fidelity Readout: Berkley et al, arXiv:0905.0891

    Inter-qubit Coupler: Harris et al, PRB 80, 052506 (2009)

    CCJJ Qubit: Harris et al, arXiv:0909.4321

    A new publication describing how this hardware is operated is currently

    being drafted and will be posted on arXiv soon.

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    C1 Processor Architecture

    Processor Unit Cell

    8 flux qubits (shaded) connected by 16 inter-qubit couplers (dark).

    q0 q1 q2 q3

    q4

    q5

    q6

    q7RO

    CCJJLT

    CO

    IPC

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    Embedding Problems on C1 Processor

    Optimization Problems As Graphs

    Problems posed as connected graphs. One can assign 1 hi +1 toany vertex and 1 K +1 to any edge.

    0

    12

    3

    4

    5 6

    7

    +1

    1

    0

    Qubit

    Coupler

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    Embedding Problems on C1 Processor

    Optimization Problems on Hardware

    Assign vertices and edges of graphs (problems) to qubits and couplers,respectively. Qubits are extended objects in the hardware and couplers arelocalized at intersections of qubits.

    0 1 2 3

    4

    5

    6

    7

    +1

    1

    0

    Qubit

    Coupler

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    Embedding Problems on C1 Processor

    Symmetry Leads to Multiple Embeddings

    There are many ways to embed the same graph. Some examples . . .

    0 1 2 3

    4

    5

    6

    7

    Reference

    0 1 2 3

    4

    5

    6

    7

    /2 Rotation

    0 1 2 3

    4

    5

    6

    7

    Rotation

    0 1 2 3

    4

    5

    6

    7

    3/2 Rotation

    0 1 2 3

    4

    5

    6

    7

    Reflect Across x=0

    0 1 2 3

    4

    5

    6

    7

    Reflect Across y=0

    0 1 2 3

    4

    5

    6

    7

    Reflect Across y=-x

    0 1 2 3

    4

    5

    6

    7

    Reflect Across y=x

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    AQC Processor Performance

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    Outline

    1 C1 Ising Spin Processor

    2 AQC Processor Performance

    3 Problem Dissection

    4 AQO Processor Performance

    5

    Is This A Quantum Processor?

    6 Conclusions and Whats Next

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    AQC Processor Performance

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    Benchmarking The Hardware

    Definition of Benchmark Problem Set

    A set of 8 permutations of 800 unique graphs (total 6400 problems) weregenerated. The graphs were created by choosing sets of 8 values of hi and16 values of K from a uniform random distribution of values within the set

    hi,K [ 1 6/7 5/7 . . . +5/7 +6/7 +1 ].

    We denote such a problem as being specified to 4 bits of precision (4BOP),where hi,K are sampled from 2

    4 1 evenly spaced values between 1.

    Connections To Complexity and Noise

    The precision to which one wants to specify values of hi and K is relatedto the complexity or difficulty of the optimization problem that is to besolved. The precision to which one can specify values of hi and K inpractice is intimately linked to the physics behind dephasing via flux noisein superconducting gate model quantum computing hardware.

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    Control Signals

    Only 2 time-dependent control signals during annealing: xccjj(t) and asingle persistent current compensation signal Ig(t) that simultaneously

    drives all qubit flux biases xi (t).

    0

    0

    0

    -0/2

    -0/2

    ~0/3

    iro(t)

    ccjj(t)x

    ro(t)x

    xlatch(t)

    t=0 trep

    QFPs

    }d

    c-SQUID

    -0

    -0

    Ig(t)

    100 s

    }}

    Qubits

    ReadoutAnneal

    ccjjm

    ccjjb

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    AQC Processor Performance

    Adiabatic quantum computation (AQC) implies system should be in

    groundstate at all times with P = 1. Run each of the 6400 problems 1024times and bin the probability of finding the groundstate. Data showstructure around P 0.6 and a tail that is rapidly suppressed at lower P.

    0 0.2 0.4 0.6 0.8 10

    200

    400

    600

    800

    1000

    Probability

    Frequency

    8Q4BOP, Probability of Correct SolutionW444.15.R3C13

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    AQC Processor Performance

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    Failures of the AQC Processor

    Track the probability of finding the system in the most prominent incorrect

    state. If the errors were simply random, P should be very low for all cases.Data indicate otherwise.

    0 0.2 0.4 0.6 0.8 10

    200

    400

    600

    800

    1000

    1200

    1400

    Probability

    Frequency

    8Q4BOP, Probability of Most Probable Incorrect AnswerW444.15.R3C13

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    AQC Processor Performance

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    Summing It Up Thusfar ...

    The Key Experimental Observations

    The hardware seemingly only yields a groundstate with P = 1 for only aportion of the benchmarking problems. However, the failures do not

    appear to be random.

    Should one even expect this system to return a groundstate with P = 1?

    The remainder of this lecture will address the question posed above. It willbe demonstrated that acknowledging that the AQO processor is operated

    at finite temperature is essential to understanding its performance.

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    Problem Dissection

    O li

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    Outline

    1 C1 Ising Spin Processor

    2 AQC Processor Performance

    3 Problem Dissection

    4 AQO Processor Performance

    5 Is This A Quantum Processor?

    6 Conclusions and Whats Next

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    Problem Dissection

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    A Typical Problem

    Consider Problem 793 (permutation 0 pictured below). Graph contains a

    seemingly typical mix of hi and K, nothing special in its structure. Thiswas a problem that returned the groundstate with P 0.6 0.1 for allpermutations.

    0

    12

    3

    4

    5 6

    7

    +1

    1

    0

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    Problem Dissection

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    States Above the Single Count Level

    Permutation State (

    i

    si+1

    2 28i

    ) State (s) Probability0 200 (1 1 -1 -1 1 -1 -1 -1) 563/1024 *0 201 (1 1 -1 -1 1 -1 -1 1) 91/10240 232 (1 1 1 -1 1 -1 -1 -1) 362/1024

    . . . .

    . . . .

    . . . .

    7 140 (1 -1 -1 -1 1 1 -1 -1) 656/1024 *7 142 (1 -1 -1 -1 1 1 1 -1) 212/10247 156 (1 -1 -1 1 1 1 -1 -1) 152/1024

    Key Observation

    Only 3 states showed up for all 8 embeddings of the same problem. Theerror states are always related to the groundstate (*) by a single bit flip.

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    Problem Dissection

    A P tt I th E St t s

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    A Pattern In the Error States

    Permutation Bits That Fail Mapping Transformed Bits

    0 2,7 Reference 2,71 0,6 Rotated +pi/2 2,72 1,4 Rotated pi 2,73 3,5 Rotated -pi/2 2,7

    4 1,7 Reflect across V 2,75 2,4 Reflect across H 2,76 0,5 Reflect across x=-y 2,77 3,6 Reflect across x=+y 2,7

    Key Conclusion

    The failures are independent of permutation. The loss of probability in thegroundstate is due to something intrinsic to the problem structure.

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    Problem Dissection

    The Failure States are Low Lying States

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    The Failure States are Low Lying States

    Choose permutation 0 and calculate the energy for all classical states s:

    Ec(s) = JAFM

    7i=0

    hisi +7

    i,j>i

    Ksisj

    (11)

    0 50 100 150 200 2500

    5

    10

    15

    20

    State (dec)

    (EE

    g)/JAFM

    8Q4BOP, Problem 793 Classical Energy Versus State

    State 200 (Groundstate) State 201

    State 232

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    Problem Dissection

    Temperature Matters

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    Temperature Matters

    Thermalization of the Processor State

    If our hardware really does implement an internally adiabatic evolutioncoupled to a bath at temperature T, then one really ought to observe athermal distribution of processor states si:

    Pi =

    eEc(si)/kBT

    Z (12)Z=

    i

    eEc(si)/kBT

    For the C1 chip, we know T

    50mK and MAFM = 1.563 pH. Since JAFM

    grows during annealing, one can speculate that the device will successfullythermalize until it enters the incoherent regime where the dynamics will befrozen: the dividing line is roughly located at xcjj 0.66 0, at whichpoint

    Ipq

    1.1A. This yields JAFM/kB = MAFM

    Ipq

    2/kB 150 mK.

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    Problem Dissection

    Temperature Matters

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    Temperature Matters

    A Quick Sanity Check

    Compare observed (averaged over 8 permutations) and predictedprobabilities for problem 793:

    State P Observed P Predicted

    200 0.55 0.58232 0.35 0.29201 0.09 0.12

    Others 0.01 0.01The thermalization picture does pick out the right states and gives adecent estimate of their probabilities.

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    Problem Dissection

    Annealing Trajectories and Thermalization

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    Annealing Trajectories and Thermalization

    Consider lowest 16 levels from a simulation of the system running Problem

    793: System localizes into classical configurations when x

    ccjj 0.65 0(where q 2JAFM). We see evidence of the device thermalizing up toxccjj 0.66 0, which is well beyond the phase transition.

    0.635 0.64 0.645 0.65 0.655 0.66 0.665 0.670

    0.1

    0.2

    0.3

    0.4

    0.5

    0.6

    0.7

    xccjj/0

    (E

    E0)/kB

    (K)

    Coherent Incoherent

    |20 0

    |23 2

    |20 1

    |0

    |1,m

    |2,m,n

    3T

    Second Excited Stateof rfSQUID

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    Problem Dissection

    Is This a 1-Qubit Fluke?

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    Is This a 1-Qubit Fluke?

    The low lying states for problem 793 were both 1 bit flip from the

    groundstate. Is there evidence of low lying states that are multiple bit flipsfrom the groundstate being thermally occupied? Consider Problem 698:

    0

    12

    3

    4

    5 6

    7

    +1

    1

    0

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    Problem Dissection

    Experimental (Theoretical) Occupation of Ground State

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    Experimental (Theoretical) Occupation of Ground State

    0 50 100 150 200 2500

    2

    4

    6

    8

    10

    12

    14

    State (dec)

    (EE0

    )/JAFM

    8Q4BOP, Problem 698 Classical Energy Versus State

    Groundstate: 0.36 (0.37)

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    Problem Dissection

    Experimental (Theoretical) Occupation of Excited States

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    Experimental (Theoretical) Occupation of Excited States

    0 50 100 150 200 2500

    2

    4

    6

    8

    10

    12

    14

    State (dec)

    (EE0

    )/JAFM

    8Q4BOP, Problem 698 Classical Energy Versus State

    1st Excited: 0.16 (0.17) 1st Excited: 0.20 (0.17)

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    Problem Dissection

    Experimental (Theoretical) Occupation of Excited States

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    Experimental (Theoretical) Occupation of Excited States

    0 50 100 150 200 2500

    2

    4

    6

    8

    10

    12

    14

    State (dec)

    (EE0

    )/JA

    FM

    8Q4BOP, Problem 698 Classical Energy Versus State

    2nd Excited: 0.04 (0.08) 2nd Excited: 0.06 (0.08)

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    Problem Dissection

    Experimental (Theoretical) Occupation of Excited States

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    Experimental (Theoretical) Occupation of Excited States

    0 50 100 150 200 2500

    2

    4

    6

    8

    10

    12

    14

    State (dec)

    (EE0

    )/JA

    FM

    8Q4BOP, Problem 698 Classical Energy Versus State

    3rd Excited: 0.02 (0.04)3rd Excited: 0.07 (0.04)

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    Problem Dissection

    Experimental (Theoretical) Occupation of Excited States

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    p ( ) O p S

    0 50 100 150 200 2500

    2

    4

    6

    8

    10

    12

    14

    State (dec)

    (EE0

    )/JA

    FM

    8Q4BOP, Problem 698 Classical Energy Versus State

    4th Excited: 0.02 (0.02) 4th Excited: 0.02 (0.02)

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    Problem Dissection

    Experimental (Theoretical) Occupation of Excited States

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    p ( ) p

    0 50 100 150 200 2500

    2

    4

    6

    8

    10

    12

    14

    State (dec)

    (EE0

    )/JA

    FM

    8Q4BOP, Problem 698 Classical Energy Versus State

    5th Excited: 0.01 (0.01)

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    Problem Dissection

    Observation of Multiple Bit Flips

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    p p

    State Hamming Distance P Observed P Predicted

    97 - 0.36 0.37101 1 0.20 0.1754 5 0.16 0.1784 4 0.06 0.0820 5 0.04 0.08

    117 2 0.07 0.04190 7 0.02 0.0455 4 0.02 0.0239 4 0.02 0.02

    227 2 0.01 0.01

    Others - 0.01 0.01

    Multi-bit differences are observed with appreciable probability. Thediscrepancies between experiment and theory may be due to the target

    Hamiltonian being slightly miscalibrated.Copyright, D-Wave Systems (2009) AQO Processor Performance September 2009 39 / 59

    Problem Dissection

    Simulated Processor Performance

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    Nature Wont Allow a Perfect AQC ...

    The humped structure appears to be due to a conspiracy betweenthermalization and problem set structure - it will never go away at finite T.

    0 0.5 10

    200

    400

    600

    800

    1000

    Probability

    Frequency

    8Q4BOP, Probability of Correct SolutionW444.15.R3C13

    0 0.5 10

    200

    400

    600

    800

    1000

    Probability

    Frequency

    8Q4BOP, Probability of Correct SolutionTHERMAL SIMULATION

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    Problem Dissection

    Simulated Processor Performance

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    Nature Wont Allow a Perfect AQC ...

    Adding a small amount of Gaussian distributed random noise on hi to thesimulation reproduces the salient features.

    0 0.5 10

    200

    400

    600

    800

    1000

    Probability

    Frequency

    8Q4BOP, Probability of Correct SolutionW444.15.R3C13

    0 0.5 10

    200

    400

    600

    800

    1000

    Probability

    Frequency

    8Q4BOP, Probability of Correct SolutionSIMULATION, Gaussian Randomization

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    AQO Processor Performance

    Outline

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    1 C1 Ising Spin Processor

    2 AQC Processor Performance

    3 Problem Dissection

    4 AQO Processor Performance

    5 Is This A Quantum Processor?

    6 Conclusions and Whats Next

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    AQO Processor Performance

    Choosing a New Performance Metric

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    AQO = AQC when T > 0Simply counting the probability of achieving the groundstate alone candistort ones perception of processor performance. When characterizingAQO, one really ought to specify the probability of obtaining a low energystate within a thermal energy window above the groundstate.

    So what is a fair width E for this window? Consider a system that hastwo low lying levels that are well separated from all others. Let us definethe success metric as being a probability of 0.95 of being within thegroundstate. Solving for the excited state probability then yields:

    0.05 =eE/kBT

    1 + eE/kBT

    E 3kBT

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    AQO Processor Performance

    A Dividing Line Between Low and High Energy States

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    Revisit the classical energy plot for problem 793, permutation 0: The

    suggested definition of E = 3kBT makes sense.

    0 50 100 150 200 2500

    5

    10

    15

    20

    State (dec)

    (EEg

    )/JAFM

    8Q4BOP, Problem 793 Classical Energy Versus State

    EState 200 (Groundstate) State 201

    State 232

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    AQO Processor Performance

    AQO Performance

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    Low Energy Solutions

    The hardware returns low energy solutions with very high probability. Thisis the expected behaviour of a AQO processor operated at finite T.

    0 0.5 10

    500

    1000

    1500

    2000

    2500

    3000

    3500

    Probability

    Frequency

    8Q4BOP, Solution Within Thermal BandW444.15.R3C13

    0 0.5 10

    500

    1000

    1500

    2000

    2500

    3000

    3500

    Probability

    Frequency

    8Q4BOP, Solution Within Thermal BandTHERMAL SIMULATION

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    AQO Processor Performance

    Number of Low Lying States

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    How many states lie within the thermal window?

    The number is 1, which seems to be a feature ofthe choice of test problems. This means that finding the true groundstateout of relatively few occupied states is easy.

    0 5 10 15 20 250

    200

    400

    600

    800

    1000

    1200

    1400

    1600

    Number of States

    F

    requency

    8Q4BOP, Number of States Within Thermal BandW444.15.R3C13

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    AQO Processor Performance

    A Scalar Success Metric

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    Is the groundstate the most probable state?Thermodynamics dictates that the groundstate ought to be the mostprobable state. We ran each problem 1024 times, identified the state sthat showed up with the highest probability and then asked whether thisstate was a groundstate: This was confirmed to be the case for 6073 out

    of 6400 problems (success rate 95.89%).

    Even for those cases where a groundstate was not the most probable state,a groundstate was always observed with finite probability. Since only a

    small number of states ( 28

    ) showed up with finite probability for allproblems in the benchmarking set, then it was very easy to identify anexact solution to a given optimization problem.

    Copyright, D-Wave Systems (2009) AQO Processor Performance September 2009 47 / 59

    AQO Processor Performance

    Implications for Hardware Operation

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    Differing Modes of Hardware OperationSuggested modes for operation of the AQO processor:

    Run problem a statistically large number of times, calculate

    Ec(s) =

    i

    hisi +

    i,j

    Ki,jsisj

    for all states s that appear above the readout noise level. Take statethat yields the lowest Ec as the solution.

    Run hardware once, calculate Ec using measured state s. If Ec is less

    than some threshold that is dictated by the user, then answer can bedeemed an acceptable solution. If Ec > threshold, iterate.

    Copyright, D-Wave Systems (2009) AQO Processor Performance September 2009 48 / 59

    AQO Processor Performance

    Number of Repetitions

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    How many times must one run the hardware to find an exact solution?

    Assume for some particular problem that one obtains the groundstate withprobability 0.05, so 95% of the time one gets another state. In order toobserve the groundstate at least once in M measurements with probabilityPgs,

    1

    Pgs = (0.95)

    M (13)

    Pgs M

    0.95 590.99 90

    0.999 135. . . . . .

    Even in the worst case scenario for this processor, one could find an exactsolution to an optimization problem in a small number of iterations.

    Copyright, D-Wave Systems (2009) AQO Processor Performance September 2009 49 / 59

    Is This A Quantum Processor?

    Outline

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    1 C1 Ising Spin Processor

    2 AQC Processor Performance

    3

    Problem Dissection

    4 AQO Processor Performance

    5 Is This A Quantum Processor?

    6 Conclusions and Whats Next

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    Is This A Quantum Processor?

    Is This a Quantum Processor?

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    Point 1:

    The data shown herein have simply demonstrated that we have builthardware that solves optimization problems in a manner that is consistent

    with thermodynamics. As such, they cannot be used to argue whether thisis a classical or quantum mechanical system as all physical systems mustobey thermodynamics. Rather, the details must be in how the systemthermalizes.

    Copyright, D-Wave Systems (2009) AQO Processor Performance September 2009 51 / 59

    Is This A Quantum Processor?

    Is This a Quantum Processor?

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    Point 2:

    The fact that measured single qubit parametersIpq and agree with the

    results from an independently calibrated quantum mechanical rf-SQUID

    Hamiltonian indicates that the constituent elements of this processor arequantum mechanical. However, this does not immediately imply quantummechanical behavior of the collective system. We need to perform specific,targeted experiments to characterize the dynamics of the collective systemand compare the results to quantitative models, be they quantum or

    classical, of the system evolution.

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    Is This A Quantum Processor?

    ... But, System Thermalizes in the Presence of Large Tunnel Barriers

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    The system thermalizes when the single qubit tunnel barrier heightU kBT, but only in the presence of finite tunneling energy . We see

    signatures of thermalization up to x

    cjj 0.66 0, where U/kBT 10but /h 50 MHz. Thermal activation over such a barrier seems unlikely,but tunneling is still relatively fast.

    0.63 0.64 0.65 0.66 0.670

    0.1

    0.2

    0.3

    0.4

    0.5

    xcjj/0

    T

    emperature(K)

    T

    /2kB

    JAFM/kB

    U/kB

    Copyright, D-Wave Systems (2009) AQO Processor Performance September 2009 53 / 59

    Conclusions and Whats Next

    Outline

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    1 C1 Ising Spin Processor

    2 AQC Processor Performance

    3 Problem Dissection

    4 AQO Processor Performance

    5 Is This A Quantum Processor?

    6 Conclusions and Whats Next

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    Conclusions and Whats Next

    Conclusions

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    It is critical to include the effects of finite T when discussing theperformance of a AQO processor.

    The dominant failure mode for the AQO processor can be describedby thermalization processes. These are arguably soft failures to lowenergy solutions. The width of the band of probable low lying statesis given roughly by 3kBT.

    Copyright, D-Wave Systems (2009) AQO Processor Performance September 2009 55 / 59

    Conclusions and Whats Next

    Whats Next - The Key Issues

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    Two Key Issues That Need to Be Addressed:

    Scaling: 8-qubit system is too small to state anything meaningfulregarding the scaling of AQO. We need to experimentally characterizelarger processors at a size that at least pushes the performance of

    classical optimization systems (hardware and software).Mechanism: We need to perform carefully crafted experiments toconfirm or refute that quantum mechanics is essential to theprocessor operation. This boils down to determining whether theoff-diagonal elements of the single qubit Hamiltonians (q) affect thedynamics of the collective processor.

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    Conclusions and Whats Next

    Whats Next - Scaling

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    We currently have three 128-qubit chips cold and being calibrated at themoment. Our intention is to benchmark these devices against large sets ofrandom Ising spin glass problems.

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    Conclusions and Whats Next

    Whats Next - Mechanism

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    We are looking into using antiferromagentic chains of varying length toprobe the dynamics of collective systems. Such chains have 2 low lyingstates that are well separated from all others, thus simplifying modelling.

    Meff

    1Q:F2

    x

    F1x

    F2x

    F2x

    F3x=0

    F3x=0 F4

    x=0

    MeffMeffMeff

    MeffMeff

    F1x

    2Q:

    3Q:

    A

    F1x

    0

    0

    -0/2

    -0/2

    ~0/3

    i t

    ccjj(t)x

    ro(t)x

    xlatch(t)

    QFPs

    dc-S

    QUID

    -0

    -0

    Ig(t)

    }}

    Qubits

    ccjjm

    ccjjb

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    Conclusions and Whats Next

    Whats Next - Mechanism

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    By carefully choosing static flux biases applied to the antiferromagnetic

    chain one can steer the magnitude of the gap at the anticrossing0 < g < q. This provides a means for slowing down the dynamics of thecollective system such that it can be probed with our low bandwidth biaslines.

    0 0.005 0.01 0.015 0.02 0.025 0.031

    0.5

    0

    0.5

    1

    xcjj refcjj

    (E

    E)/h(GHz)

    |

    | |

    |

    C180 MHz

    0.015 0.02 0.025 0.03 0.035 0.04 0.0451

    0.5

    0

    0.5

    1

    xcjj refcjj

    (E

    E)/h(GHz)

    |

    | |

    |

    D

    180 MHz

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    http://find/