1

Benchmarking D-Wave One

Sergio Boixo USC Viterbi School of Engineering

Information Sciences Institute

Harvard Associate

People at USC / ISI

Federico Spedalieri Greg Ver Steeg Kristen Pudenz

Stephan Haas Paolo Zanardi Todd Brun Ben Reichardt Lorenzo Campos Tony Levi Edmun Jonckheere Massoud Pedram Susumu Takahashi

Bob Lucas John Damoulakis

Daniel Lidar

Tameem Albash Milad Marvian

Coupled CJJ

maximum antiferromagnetic !AFM" coupling, Mij!!co,ijx "

=MAFM. Define this energy scale as JAFM!t"#MAFM$Iqp!t"$2.

Rearranging Hamiltonian !5" yields

H0!t"JAFM!t"

= ! %i

hi"z!i" + %

i,j#iKij"z

!i""z!j" ! $!t"%

i"x

!i",

!6a"

hi =$Iq

p!t"$!!ix!t" ! !i

0"MAFM$Iq

p!t"$2=

!ix!t" ! !i

0

MAFM$Iqp!t"$

, !6b"

Kij =Mij!t"$Iq

p!t"$2

MAFM$Iqp!t"$2

=Mij!t"MAFM

, !6c"

$!t" =%q!t"

2JAFM!t". !6d"

In order to solve a particular optimization problem, hi andKij must be time independent. According to Eq. !6c", onemust hold all Mij constant during operation. This is conve-nient as it obviates the application of individually tailoredtime-dependent flux bias signals to each interqubit coupler.Rather, one need only apply a static control signal !co,ij

x toeach coupler, as depicted in Fig. 1, that can be provided byPMM. On the other hand, according to Eq. !6b", one mustapply time-dependent qubit-flux biases of the form

!ix!t" = !i

0 + hi & MAFM$Iqp!t"$ !7"

to render hi time independent. Thus, it is necessary to pro-vide a custom-tailored time-dependent control signal plus astatic offset to every qubit. The static component !i

0 can beprovided by PMM. As for the time-dependent component,providing these signals with one external bias per qubitwould not constitute a scalable approach for building a mul-

tiqubit processor. Rather, one can take advantage of the factthat, according to Eq. !7", all qubits must receive a controlsignal with the same time-dependent shape but with custom-tailored time-independent scale factors 'hi. A scalable archi-tecture for providing these signals is depicted in Fig. 1 andhas been further expounded upon in Ref. 31. Here, a singleglobal current bias Ig!t"=($Iq

p!t"$, where ( is a convenientscale factor, is coupled to multiple qubits via in situ tunablemutual inductances of magnitude Mi#hiMAFM /(. The verysame type of device that is used to provide in situ tunableinterqubit coupling29 can be retooled to provide coupling be-tween flux qubits and a global bias line. Moreover, each Mican be controlled with a static flux bias !Ip,i

x provided byPMM. We will refer to this architecture as persistent current!$Iq

p$" compensation, as it is a means of compensating forchanging $Iq

p!t"$ such that the ISG problem specified by hiand Kij remains on target throughout annealing.

To summarize up to this point, a prescription for imple-menting AQO to solve ISG problems using a network ofinductively coupled CCJJ rf-SQUID flux qubits has beenpresented. A problem specified by a set of hi and Kij can beembedded in the hardware using time-independent interqubitcouplings controlled by PMM and time-dependent qubit-fluxand CCJJ biases. The qubit-flux bias signals can be suppliedusing a combination of static flux offsets provided by PMMand a single global signal Ig!t" that is applied to each qubitthrough in situ tunable couplers that are also controlled byPMM. The CCJJ bias can, in principle, also be provided toall qubits simultaneously using a single global control signal.

III. DEVICE ARCHITECTURE AND CALIBRATION

With the mapping of the AQO algorithm onto hardwarecompleted, we turn to a high-level description of a supercon-ducting chip whose architecture embodies that algorithm. Allof the principal components of the processor, namely, thequbits,28 couplers,29 readout,30 and PMM !Ref. 31" have beendescribed in detail in other publications. As such, we willonly provide brief summaries of the important points as per-taining to the functioning of the collective system herein.

As stated previously, we have incorporated CCJJ rf-SQUID flux qubits in our design.28 A schematic of an iso-lated flux qubit with the two external bias controls relevantfor this study is shown in Fig. 2!a". This particular qubit isrobust against fabrication variations in the Josephson-junction critical currents and facilitates the homogenizationof the net critical current among a population of such qubits.This device also contains an inductance !L" tuner that can beused to compensate for variations in qubit inductance due tofabrication and from tuning the interqubit couplers.29 To eachqubit we have added an $Iq

p$ compensator, as introduced inSec. II. We have provided PMM to flux bias the two minorlobes of each CCJJ,28 the L tuner, the $Iq

p$ compensator, andthe qubit body !!i

0" for all qubits on the chip.31

The chip that was used for this study was composed of 16eight-qubit unit cells that were tiled on a 4&4 square grid.To limit the scope of this paper, we focus upon a single unitcell near the center of the chip. A discussion of the completeprocessor, with multiple unit cells acting in concert, will be

Ig(t)

!1 !1

Iccjj(t)

"Ip,ix x

"Ip,j{ {{

Mij

Mi Mj

"ccjj(t)"ccjj(t)xx

"co,ijx

"i0 "j

0

Qubit

i

Qubit

j

CouplerC ijr

|IqII | Comp. ipII |IqII | Comp. jpII

"ccjj(t)"ccjj(t)xx

Kij

hi hj

FIG. 1. !Color online" Mapping of ISG problems, specified by aset of values denoted as hi and Kij, onto superconducting hardware.Two qubits, two $Iq

p$ compensators, and one interqubit coupler areshown. A global current bias Iccjj!t" provides the fluxes !ccjj

x !t" thatdrive the annealing process to multiple CCJJ rf-SQUID flux qubits.Interqubit coupling is mediated by tunable mutual inductancesMij 'Kij that are controlled by static fluxes !co,ij

x . Qubit bodies aresubjected to the sum of static flux biases !i

0 and time-dependentflux biases driven by a global current bias Ig!t". The latter signalsare mediated to each qubit via tunable mutual inductances Mi'hithat are controlled by static fluxes !Ip,i

x .

EXPERIMENTAL INVESTIGATION OF AN EIGHT-QUBIT… PHYSICAL REVIEW B 82, 024511 !2010"

024511-3

R. Harris et. al.

Adiabatic Interpolation

Energies (GHz) A(t) B(t)

H(t) = A(t)X

j

�x

+B(t)HIsing

Time / Total time

Temperature

Scalable design… but no microwave control!!!

Scalable

Supercond. Sci. Technol. 23 (2010) 065004 M W Johnson et al

Figure 3. Top: schematic showing two eight-qubit unit cells tiledtogether. The qubits are schematically shown as the extended blackloops, similar to the way these devices are physically implemented.The couplers (shown as blue and red squares) are local to theintersections of qubits. Bottom: photograph of a standaloneeight-qubit unit cell occupying a 700 µm ! 700 µm square on a3 mm ! 7 mm chip.

Algorithm 2: The annealing algorithm.Input : A run-time t f ;

a set of qubits with HamiltonianH (s) = A(s)HI + B(s)HF ,where s = t/t f , 0 ! s ! 1,HI = !N

j=1 !x, j ,

HF = !Nj=1 h j!z, j + !

i, j"E Ki j!z,i !z, j ,where !x, j and !z, j are Pauli matrices forqubit j ,and A(s) and B(s) are envelope functions withunits of energysuch that A(0)/B(0) # 1 andA(1)/B(1) $ 1.

Output: Evolution of H (s) from H (0) to H (1).

set s = 0;wait 1 ms for hardware to reach ground state of H (0);Ramp currents on global analog lines to drive evolutions % 1

the problem to be solved, and the algorithm used to solveit, are implemented by manipulating the system Hamiltonian.Primarily motivated by this observation, the approach wehave taken to design hardware is a top-down one. For thecircuits considered here, the requirements are driven by whatis required to run the AQO algorithm. To provide context forthe material in this section we first provide an overview of thealgorithm itself.

Table 1. Parts count versus number of unit cells.

Unit cells Qubits Couplers DACS JJs

1 8 16 56 1 5004 32 72 232 6 000

16 128 328 968 24 00064 512 1416 3 976 96 000

256 2048 5896 16 136 384 000

Consider the following discrete optimization problem.Given a vector &h and upper diagonal matrix K̂ , where theelements of both are real numbers, find the set {s'

i } thatminimizes the objective function

E(s1, . . . , sN ) =N"

j=1

h j s j +"

i, j"E

Ki j si s j (1)

where si = {(1,+1}, and E is an set of (i, j) pairs whereKi j is allowed to be non-zero. We call E the allowed edge set.The necessity for explicitly defining the set E arises becauseultimately we will connect this term in the objective functionto physical couplings between pairs of qubits, and for a varietyof reasons the number of elements in E will generally be muchless than the total number of possible pairs N(N ( 1)/2. Adesign constraint on processor architecture is that qubits mustbe connected in such a way so that finding the minimumof equation (1) is NP-hard. Even with this constraint, it isstraightforward to find realizable sets E for which this holds,and we will focus exclusively on these cases.

An AQO algorithm exists for solving this problem. Theapproach is outlined in algorithms 1 and 2. The controlsystem reported on here enters into these in the load step ofalgorithm 1.

2.1. Processor interconnect architecture

There are many possibilities for how one might try to builda hardware system capable of running algorithms 1 and 2.Here we focus on a unit cell consisting of eight qubits and24 couplers. See figure 3 for a schematic and photographshowing the interconnect pattern. Copies of this unit cell canbe connected together, as indicated in the top of figure 3, andthis is how we design larger systems. This choice of unit cellfixes the allowed edge set E , and satisfies the constraint thatminimizing equation (1) be NP-hard.

2.2. Number of DACs

The total number of DACs required for circuits of increasingcomplexity is shown in table 1. Here we provide a briefoverview of how these numbers arise, and refer the readerto [14, 15] for further details.

2.2.1. One DAC per coupler. A tunable compound Josephsonjunction rf-SQUID coupler inductively coupled to qubits i andj is used to set each desired value of Ki j [14]. One suchphysical device is required per element of the allowed edge setE . Couplers are controlled using a static dc flux bias applied totheir compound-junction—no time dependence in this signal is

3

Johnson et. al.

T2 and T1 times worse than other superconducting qubits.

Ising Spin Glasses

•  Ground state is an NP-Complete problem (Barahona 1982)

•  Complex graphs can be embedded into simpler graphs using strong ferromagnetic couplings (Kaminsky and Lloyd, 2002)

•  The strength of the ferromagnetic couplings grows with the degree of

the embedded graph (Choi 2008)

•  In principle, an N-complete graph can be embedded in the geometry implemented by Dwave using N2 vertices (Choi 2010)

G can be solved in an adiabatic quantum computer that implements the spin-1/2Ising Hamiltonian, by reduction through minor-embedding of G in the quantumhardware graph U . By reduction through minor-embedding, we mean that onecan reduce the original Ising Hamiltonian on the input graph G to the embed-ded Ising Hamiltonian Hemb on its minor-embedding Gemb, i.e., the solutionto the embedded Ising Hamiltonian gives rise to the solution to the originalIsing Hamiltonian. We proved the correctness of the minor-embedding reduc-tion. There are two components to the reduction: embedding and parametersetting. The embedding problem is to find a minor-embedding Gemb of a graphG in U . The parameter setting problem is to set the corresponding parameters,qubit bias and coupler strengths, of the embedded Ising Hamiltonian. In [6],we solved the parameter setting problem. The embedding problem, though, isdependent on the hardware graph design problem discussed in the followingsections.

4 TRIAD: Optimal Hardware Graph for Embedding CompleteGraph Kn

In this section, we describe a Kn

-minor hardware graph, where Kn

is a completegraph of n vertices. A triangular layout of a K

n

-minor graph [18], called TRIAD,is shown in Figure 3.

1

2

3 4

5

6

78

1

1

1

1

1

1

1

2 2

2

2

2

2

2

3

3

3

3

3

3

3

4

4

4

4

4

4

4

5

5

5

5

5

5

5

6

6

6

6

6

6

67

7

7

7

7

7

7 8

8

8

8

8

8

8

Fig. 3: Left, K8. Right, a triangular layout of a K8-minor. Each vertex of K8 is mapped to a chainof 7 “virtual” vertices (with the same color).

4.1 Construction of TRIAD

The idea behind the construction of TRIAD is to map each vertex of Kn

to achain of n � 1 “virtual” vertices. The inductive construction is illustrated inFigure 4.

Exact “chimera” solver (Belief propagation)

M. Drew-Brook

Gaps of spin glasses

Karimi et al. 2010

108100908070605040302010

00.1

0.20.3

0.40.5

0.60.7

0.80.9

1

0

50

100

150

200

SpinsProbability

N. o

f ins

tanc

es (o

ver 1

000)

Spin Glasses Prob. vs. spins (5us)

Spin Glasses (108 spins) Prob. vs. time

520

40100

5001000

1000020000

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

50

100

150

Adiabatic Times (us)

Probability

N. o

f ins

tanc

es (o

ver 1

000)

Spin Glasses Median times vs. spins

0 20 40 60 80 100 1200

50

100

150

200

250

Number of spins

Estim

ated

tim

e (9

9%) i

n us

Estimated time (99%) in us

Spin Glasses 90th percentile

0 20 40 60 80 100 1200

1000

2000

3000

4000

5000

6000

Number of spins

Estim

ated t

ime (

99%)

in us

(90 p

ercen

tile)

Estimated time (99%) in us (90 percentile)

13

Questions?