D-Wave Quantum Annealer

Graham Norris

Graham Norris 2016-04-29 1 A Different Quantum Device

Physical quantum annealer • (not a GMQC) Solves optimization • problems (no algorithms) Commercial product • Mature technology with large • number of qubits (>1000)

Graham Norris 2016-04-29 2 Outline

Introduction Architecture Details Brief Theoretical Background Connectivity, Tuning, Ising Spin Glass Problem Readout Quantum Annealing Physical Implementation RF–SQUID Results Flux Qubit Is It Quantum? Potential Future Directions Advanced RF–SQUIDs Conclusions

Graham Norris 2016-04-29 3 Ising Spin Glass Problem

Ising spin glass Hamiltonian

X z z X z H = − Jij σi σj − hi σi i

J : couplings between spins • ij h : local ﬁelds • i σα: Pauli matrix σα applied to spin i • i

Find the ground state of this Hamiltonian (NP Hard)

Graham Norris 2016-04-29 4 Quantum Annealing LETTER RESEARCH coupled together using programmable coupling elements29 which pro- a vide a spin–spin coupling energy that is continuously tunable between Classical ferromagnetic and antiferromagnetic coupling. This allows spins to thermal favour alignment or anti-alignment, respectively. annealing The behaviour of this system can be described with an Ising model Hamiltonian Quantum N N z z z annealing P~ hisi z Jijsi sj 1 H i~1 i,j~1 ð Þ X X where for spin i sz is the Pauli spin matrix with eigenvectors { æ, æ} b i j" j# 1 and 2hi is the energy bias; and 2Jij is the coupling energy between the spins i and j. Our implementation allows each J and h Startto be pro- in coherent superposition ij i Γ Λ grammed independently within the constraints of the• connectivity of our devices. Add time–dependent0.5 quantum ﬂuctuations to Ising Hamiltonian The quantum mechanical properties of the individual• devices have been well characterized13, but we are interested in what happens when X x 0 H(t) = −A(t) σ + B(t)H several of them are coupled together. It is reasonable to ask whether 0 0.2 0.4 0.6 i 0.8Ising 1.0 t/t this manufactured, macroscopic (,1 mm) system of artificial spins fnal i behaves quantum mechanically. We report here on an experiment that c demonstrates a signature of quantum annealing in a coupled set of (A(t) decreasesh with increasing time, B(t) increases) eight artificial Ising spins. t Whereas thermal annealing uses progressively weaker thermalChange fluc- A and B slowly; should remain in ground state tuations to allow a system to explore its energy landscape and arrive at h

• Early a low-energy configuration, quantum annealing uses progressivelyDo not have to protect excited stateLate from environment weaker quantum fluctuations, mediated by tunnelling. In• both thermal and quantum annealing, a system starts with a mixture of all possible 0 states: a classical mixed state in the former and a coherent superposi- 0 0.2 0.4 0.6 0.8 1.0 Graham Norris 2016-04-29 5 t/tfnal tion in the latter. d Quantum annealing can be performed by slowly changing the system Hamiltonian Early N x (t)~C(t) Disi zL(t) P H i~1 H X where C decreases from one to zero and L increases from zero to one Late monotonically with time, and Di parameterizes quantum mechanical tunnelling between æ and æ. j" j# At the beginning of the annealing, C 5 1, L 5 0 and the system is Figure 2 Quantum annealing. a, Annealing is performed by gradually N x | fully characterized by the transverse terms, i~1 Disi . The ground raising the energy barrier between states. In thermal annealing, when the state of this is a superposition of all states in the sz basis. It is straight- barrier becomes much larger than kBT thermal excitation over the barrier P TA forward to initialize the system in this state. During quantum anneal- eventually ceases, at some time tfreeze. In quantum annealing, tunnelling QA ing, the transverse term is gradually turned off (C R 0) and the weight between states also will eventually cease, at a time tfreeze. b, The value of the parameters C and L during annealing are not independent of each other in the of the Ising Hamiltonian, P, is increased (L R 1) (Fig. 2b). If this annealing is done slowly enough,H the system should remain in the flux qubit. The annealing ends at tfinal 5 148 ms. c, Changing the value of h(t) (see d) at various points during annealing can be used to probe the freeze-out ground state at all times, thus ending up in the ground state of P H time, tfreeze. d, Double-well potential during annealing. If h is turned on early (ref. 4). enough (blue line), the system follows the ground state through annealing and The above description of quantum annealing is in the language of an reaches the final ground state of equation (1) with high probability. If h(t)is ideal Ising spin system. Let us look more closely at what this means for turned on too late (red line), the state probabilities are determined by the earlier an individual flux qubit. During annealing, the energy barrier, dU(t), Hamiltonian, for which h 5 0. between the two wells is gradually raised (Fig. 2a). If thermal fluctuations are dominant, then the qubit dynamics may be viewed as thermal activation or quantum tunnelling is the dominant effect governing activation over the barrier with a rate that is proportional to e{dU=kBT qubit dynamics. at a temperature T (kB, Boltzmann’s constant). This suggests that the Here we modify this annealing procedure to perform a specialized dynamics stops when dU?kBT. Because dU is increasing with time, experiment that permits us to distinguish between these two cases, by TA TA this freezing out happens at t

Quantum annealing with manufactured spins

M. W. Johnson1, M. H. S. Amin1, S. Gildert1, T. Lanting1, F. Hamze1, N. Dickson1,R.Harris1, A. J. Berkley1, J. Johansson2, P. Bunyk1, E. M. Chapple1, C. Enderud1, J. P. Hilton1, K. Karimi1, E. Ladizinsky1, N. Ladizinsky1,T.Oh1, I. Perminov1, C. Rich1, M. C. Thom1, E. Tolkacheva1, C. J. S. Truncik3, S. Uchaikin1, J. Wang1, B. Wilson1 & G. Rose1

Many interesting but practically intractable problems can be reduced implemented such a spin system, interconnected as a bipartite graph, to that of finding the ground state of a system of interacting spins; using an in situ reconfigurable array of coupled superconducting flux however, finding such a ground state remains computationally qubits14. The device fabrication is discussed in Methods and in Sup- difficult1. It is believed that the ground state of some naturally occur- plementary Information. The simplified schematic in Fig. 1a shows ring spin systems can be effectively attained through a process called two superconducting loops in the qubit, each subject to an external flux 2,3 quantum annealing . If it could be harnessed, quantum annealing bias W1x or W2x, respectively. The device dynamics can be modelled as a might improve on known methods for solving certain types of quantum mechanical double-well potential with respect to the flux, W1, 4,5 problem . However, physical investigation of quantum annealing in loop 1 (Fig. 1b). The barrier height, dU, is controlled by W2x. The has been largely confined to microscopic spins in condensed-matter energy difference between the two minima, 2h, is controlled by W1x. systems6–12. Here we use quantum annealing to find the ground The two lowest energy states of the system, corresponding to clockwise state of an artificial Ising spin system comprising an array of eight or anticlockwise circulating current in loop 1, are labelled æ and æ, superconducting flux quantum bits with programmable spin–spin with flux localized in the left- or the right-hand well (Fig. 1b),j# respec-j" couplings. We observe a clear signature of quantum annealing, tively. If we consider only these two states (a valid restriction at low Realizingdistinguishable from classical a Quantum thermal annealing through Annealer the tem- temperature), the qubit dynamics is equivalent to those of an Ising perature dependence of the time at which the system dynamics spin, and we treat the qubits as such in what follows. Qubits (spins) are freezes. Our implementation can be configured in situ to realize a wide variety of different spin networks, each of which can be ‘Spin 13,14 a -up’ c monitored as it moves towards a low-energy configuration . ircula ting cu This programmable artificial spin network bridges the gap between rre Does not map onto anything we nt the theoretical study of ideal isolated spin networks and the experi- Φ1x mental investigation of bulk magnetic samples. Moreover, with an haveincreased seen number so of spins,far such a system may provide a practical physical means to implement a quantum algorithm, possibly allow- ing more-effective approaches to solving certain classes of hard com- Φ binatorial optimization problems. 1 nt Physically interesting in their own right, systems of interacting spins urre ing c Wealso want have practical something importance for like quantum the computation ﬁgure15. One ‘Spin-down’ circulat widely studied example is the Ising spin model, where spins may take Φ2x on one of two possible values: up or down along a preferred axis. Many Φ2 seeminglySpin unrelated states: yet important Two hard rotating problems, in fields ranging •from artificial intelligence16 to zoology17, can be reformulated as the Josephson problemcurrents of finding the lowest energy configuration, or ground state, of junction an Ising spin system. b Quantum annealing has been proposed as an effective way for find- U ingh suchi : a groundtunable state2–5. To well implement offset a processor (Φ that1 usesx ) quantum •annealing to help solve difficult problems, we would need a programmable quantum spin system in which we could control individual spinsAnnealing: and their couplings, tunable perform quantum barrier annealing and then ωp δU •determine the state of each spin. Until recently, physical investigation ωp of quantumheight annealing (Φ has2x been) confined to configurations achievable in condensed-matter systems, such as molecular nanomagnets6–10 or 2h bulk solids with quantum critical behaviour11,12. Unfortunately, these Φ1 systems cannot be controlled or measured at the level of individual spins, and are typically investigated through the measurement of bulk Figure 1 | Superconducting flux qubit. a, Simplified schematic of a properties. They are not programmable. Nuclear magnetic resonance superconducting flux qubit acting as a quantum mechanical spin. Circulating techniques have been used to demonstrate a quantum annealing algo- current in the qubit loop gives rise to a flux inside, encoding two distinct spin rithm on three quantum spins18. Recently, three trapped ions were states that can exist in a superposition. b, Double-well potential energy diagram and the lowest quantum energy levelsGraham corresponding Norris to the 2016-04-29 qubit. States | æ 6 used to perform a quantum simulation of a small, frustrated Ising spin and | æ are the lowest two levels, respectively. The intra-well energy spacing" is system19. # vp. The measurement detects magnetization, and does not distinguish between, One possible implementation of an artificial Ising spin system say, | æ and excited states within the right-hand well. In practice, these involves superconducting flux quantum bits20–28 (qubits). We have excitations" are exceedingly improbable at the time the state is measured.

1D-Wave Systems Inc., 100-4401 Still Creek Drive, Burnaby, British Columbia V5C 6G9, Canada. 2Department of Natural Sciences, University of Agder, Post Box 422, NO-4604 Kristiansand, Norway. 3Department of Physics, Simon Fraser University, Burnaby, British Columbia V5A 1S6, Canada.

Cs EJ CJ EJ CJ

Φext

Transmon Qubit Flux Qubit Capacitively shunted Inductively shunted • • Josephson junction Josephson junction Relevant parameter: number Relevant parameter: • • of Cooper pairs on the island magnetic ﬂux (or phase) Persistent rotating currents •

Graham Norris 2016-04-29 7 Hamiltonians Charge Qubit

x 2 HC = 4EC(nˆ − n ) − EJ cos (ϕ)

Flux Qubit 1 H = 4E n2 − E cos (ϕ ˆ) + E (ϕ ˆ − ϕx )2 F C J 2 L

E : Josephson energy (E = I Φ /2π) • J J c 0 E : Coulomb energy (E = e2/2C ) • C C Σ 2 E : Inductor energy (E = Φ0 1 ) • L L 2π L nx , ϕx : external voltage or ﬂux bias •

Graham Norris 2016-04-29 8 Shifted Cosine Potential Fixed offset 1 H = 4E n2 − E cos (ϕ ˆ) + E (ϕ ˆ − ϕx )2 F C J 2 L

Interesting physics

180 Devoret and Martinis ϕx = π 3

E/EJ 1

-1-0.5 0 0.5 1

Φ/Φ0

Fig. 8. Schematic potential energy landcape for the RF-SQUID.

Graham Norris 2016-04-29 9 a superconducting transformer rather than a gate capacitor to adjust the hamiltonian. The two sides of the junction with capacitance CJ are connected by a superconducting loop with inductance L. An external ﬂux !ext is imposed through the loop by an auxiliary coil. Using the methods of Appendix 1, we obtain the hamiltonian(8)

q2 φ2 2e H E cos (φ !ext) . (9) = 2C + 2L − J − J ! ! " We are taking here as degrees of freedom the integral φ of the voltage across the inductance L, i.e., the flux through the superconducting loop, and its conjugate variable, the charge q on the capacitance CJ; they obey [φ,q] i!. Note that in this representation, the phase θ, corresponding to the branch= flux across the Josephson element, has been eliminated. Note also that the flux φ, in contrast to the phase θ, takes its values on a line and not on a circle. Likewise, its conjugate variable q, the charge on the capacitance, has continuous eigenvalues and not integer ones like N. Note 2 that we now have three adjustable energy scales: EJ, ECJ (2e) /2CJ and 2 = EL !0/2L. =The potential in the flux representation is schematically shown in Fig. 8 together with the first few levels, which have been seen experimentally for the first time by the SUNY group.(24) Here, no analytical expressions exist for the eigenvalues and the eigenfunctions of the problem, which has two aspect ratios: EJ/ECJ and λ LJ/L 1. Whereas in the Cooper box the potential is= cosine-shaped− and has only one well since the variable θ is 2π-periodic, we have now in general a parabolic potential with a cosine corrugation. The idea here for cur- ing the detrimental effect of the offset charge fluctuations is very different 170 Circuit QED: superconducting qubits coupled to microwave photons Tunable Potential Minima π ϕg = π ϕ = ϕg = 0 g 2

~10 GHz

Potential energy ϕ 2π 1 1 2 (a) (b) (c)

Changing ϕx (≡ ϕ ) shifts cosine relative to quadratic; • (d) g U changes the shape of theEJ potential Flux qubit can change well offset (h in Ising Hamiltonian) • i through biasing external ﬂux 6 Changing EJ could turn cosine on and off (tunable barrier) • 4 Graham Norris 2016-04-29 10 2

ϕ–6 π –10 –5 5 10

2 Fig. 3.21 (a) Extended cosine potential U = EJ cos(ϕ + ϕg)+ELϕ , with dimensionless − offset flux ϕg = π.Thisisusedinthefluxoniumqubittoproducea“Λ”levelconfiguration; (b) Same as (a) but with ϕg = π/2.(c)Sameas(a)butwithϕg =0to produce a “V” level configuration. (d) Same as (a) but displaced a distance ϕg =6π to illustrate the current (flux) biased phase qubit. EL =0.01EJ . the Josephson cosine term allows us to create a number of different potential energy well shapes and thus generate different interesting qubit spectra. Before delving into this, we need to recognize that there is one more “control knob” at our disposal, namely externally applied flux, which we can view as the inductive analog of the offset charge studied previously. If our inductor is part of a transformer with dc current applied in the other winding, then there is a flux offset and the Hamiltonian becomes

1 H =4E (ˆn n )2 E cosϕ ˆ + E (ˆϕ ϕ )2. (3.253) C − g − J 2 L − g It is convenient to translateϕ ˆ andn ˆ using the unitary transformation

iϕ nˆ in ϕˆ U = e g e− g , (3.254) which yields

UϕˆU † =ˆϕ + ϕg, (3.255)

UnUˆ † =ˆn + ng. (3.256) RF–SQUID

HARRIS et al. PHYSICAL REVIEW B 81, 134510 ͑2010͒ includeSimple a tunable inductanceflux qubit in the (a) flux also qubit body known to ac- • x countas for both RF–SQUID differences in inductance between qubits in a jq multiqubit device and to compensate for changes in qubit j1 Lbody jq I1 inductance during operation. Thereafter, the focus ofx the paper shiftsShifts toward well an experimental offset through demonstrationϕ ofq the rf- (a) SQUID• flux qubit. The architecture of the experimental device and its operation are discussed in Sec. III and then a Need many identical qubits for L /2 L /2 series of experiments to characterize the rf SQUID and to cjj cjj jx • q highlight its control are presented in Sec. IV. Section V con- j1 x j Lbody j quantum computer I jcjj 2 q tains measurements of properties that indicate that this more 1 I2 complex rf SQUID is indeed a flux qubit. Flux and critical (b) currentHard noise measurements to make and uniform a formula for converting the measured• flux noise spectral density into a free-induction L /2 L /2 ccjj ccjj x ͑RamseyJosephson͒ decay time are presented junctions in Sec. VI. A summary of jq x key conclusions is provided in Sec. VII. Detailed calcula- jccjj jl j j x x 1 L j2 j3 jR j j Lbody jq tions of(exponential multiple Josephson-junction dependence rf-SQUID Hamiltonians on I I 4 r 1 I2 3 I4 have been placed in Appendixes A and B. oxide thickness) (c) II. RF-SQUID FLUX QUBIT DESIGN FIG. 1. ͑Color online͒͑a͒ A single-junction rf-SQUID qubit. ͑b͒ TheWant behavior to of change most superconductingEJ in situ devices is gov- CJJ rf-SQUID qubit. ͑c͒ CCJJ rf-SQUID qubit. Junction critical erned• by three types of macroscopic parameters: the critical currents Ii and junction phases ␸i ͑1ՅiՅ4͒ as noted. Net device phases are denoted as ␸ , where ␣ ,r,q . External fluxes, ⌽x, currents of any Josephson junctions, the net capacitance ␣ ෈͑ᐉ ͒ n are represented as phases ␸x 2␲⌽x ⌽ , where n across the junctions, and the inductance of the superconduct- n ϵ n / 0 ෈͑L,R,cjj,ccjj,q͒. Inductances of the rf-SQUID body, CJJ loop, ing wiring. The Hamiltonian for many of these devices can Graham Norris 2016-04-29 11 and CCJJ loop are denoted as L , L , and L , respectively. generically be written as body cjj ccjj

Q2 ␸ − ␸x 2 x 2 i ͑ n n͒ ͑␸q − ␸q͒ = − EJi cos͑␸i͒ + Un , ͑1͒ V ␸ = U − ␤ cos ␸ , 2b H ͚ ͫ 2C ͬ ͚ 2 ͑ q͒ q ͑ q͒ ͑ ͒ i i n ͭ 2 ͮ where Ci, EJi=Ii⌽0 /2␲, and Ii denote the capacitance, Jo- c 2␲LqI sephson energy, and critical current of Josephson junction i, ␤ = q , ͑2c͒ respectively. The terms in the first sum are readily recog- ⌽0 nized as being the Hamiltonians of the individual junctions with the qubit inductance Lq ϵLbody, qubit capacitance Cq for which the quantum-mechanical phase across the junction c ϵC1, and qubit critical current Iq ϵI1 in this particular case. ␸i and the charge collected on the junction Qi obey the com- If this device has been designed such that ␤Ͼ1 and is flux mutation relation ⌽ ␸ 2␲,Q =iប␦ . The index n in the x ͓ 0 i / j͔ ij biased such that ␸ Ϸ␲, then the potential energy V͑␸ ͒ will second summation is over closed inductive loops. External q q x be bistable. With increasing ␤ an appreciable potential- fluxes threading each closed loop, ⌽n, have been represented x x energy barrier forms between the two local minima of V͑␸q͒, as phases ␸n ϵ2␲⌽n /⌽0. The quantum-mechanical phase through which the two lowest-lying states of the rf SQUID drop experienced by the superconducting condensate circu- may couple via quantum tunneling. It is these two lowest- lating around any closed loop is denoted as ␸n. The overall lying states, which are separated from all other rf-SQUID potential-energy scale factor for each closed loop is given by states by an energy of order of the rf-SQUID plasma energy U ⌽ 2␲ 2 L . Here, L can be either a geometric induc- n ϵ͑ 0 / ͒ / n n ប␻ ϵប/ͱL C , that form the basis of a qubit. One can write tance from wiring or Josephson inductance from large p q q an effective low-energy version of Hamiltonian ͑2a͒ as9 junctions.8 Hamiltonian ͑1͒ will be used as the progenitor for all device Hamiltonians that follow. 1 = − ͓⑀␴ + ⌬ ␴ ͔, ͑3͒ Hq 2 z q x A. Compound-compound Josephson-junction structure p x p where ⑀=2͉Iq͉͑⌽q −⌽0 /2͒, ͉Iq͉ is the magnitude of the per- A sequence of rf-SQUID architectures is depicted in Fig. sistent current that flows about the inductive q loop when the 1. The most primitive version of such a device is depicted in device is biased hard ͓⑀ӷ⌬q͔ to one side, and ⌬q represents Fig. 1͑a͒ and more complex variants in Figs. 1͑b͒ and 1͑c͒. the tunneling energy between the otherwise degenerate coun- x For the single-junction rf SQUID ͓Fig. 1͑a͔͒, the phase tercirculating persistent current states at ⌽q =⌽0 /2. across the junction can be equated to the phase drop across A depiction of the one-dimensional potential energy and the body of the rf SQUID: ␸1 =␸q. The Hamiltonian for this the two lowest-energy states of an rf SQUID at degeneracy x device can then be written as ͑⌽q =⌽0 /2͒ for nominal device parameters is shown in Fig. 2 2. In this diagram, the ground and first-excited states are Qq denoted by ͉g͘ and ͉e͘, respectively. These two energy levels = + V͑␸q͒, ͑2a͒ H 2Cq constitute the energy eigenbasis of a flux qubit. An alternate

134510-2 Compound Josephson Junction

HARRIS et al. PHYSICAL REVIEW B 81, 134510 ͑2010͒ CJJ (b) replaces JJ with SQUID include• a tunable inductance in the flux qubit body to ac- x countTwo for both Josephson differences in inductance junctions between qubits that in a jq multiqubit device and to compensate for changes in qubit j1 Lbody jq • I inductancebehave during operation. as single Thereafter, tunable the focus of the pa- 1 per shifts toward an experimental demonstration of the rf- (a) SQUIDjunction flux qubit. The architecture of the experimental device and its operation are discussed in Sec. III and then a L /2 L /2 series of experiments to characterize the rf SQUID and to cjj cjj jx Tunes I (E ) through extra q highlight its controlc are presentedJ in Sec. IV. Section V con- j1 x j Lbody j I jcjj 2 q • 1 I2 tains measurementsexternal of flux properties term thatϕ indicatex that this more complex rf SQUID is indeed a flux qubit.cjj Flux and critical (b) current noise measurements and a formula for converting the measured flux noise spectral density into a free-induction If these junctions are L /2 L /2 ccjj ccjj x ͑Ramsey͒ decay time are presented in Sec. VI. A summary of jq • x key conclusions is provided in Sec. VII. Detailed calcula- jccjj asymmetric, we get complicated jl j j x x 1 L j2 j3 jR j j Lbody jq tions of multiple Josephson-junction rf-SQUID Hamiltonians I I 4 r 1 I2 3 I4 have been placed in Appendixes A and B. interdependence (c)

II. RF-SQUID FLUX QUBIT DESIGN FIG. 1. ͑Color online͒͑a͒ A single-junction rf-SQUID qubit. ͑b͒ x x x x CJJ rf-SQUID qubit. ͑c͒ CCJJ rf-SQUID qubit. Junction critical Thecos( behaviorϕq of) most7→ cos( superconductingϕq − ϕ devicesq0(ϕ iscjj gov-)) erned by three types of macroscopic parameters: the critical currents Ii and junction phases ␸i ͑1ՅiՅ4͒ as noted. Net device phases are denoted as ␸ , where ␣ ,r,q . External ﬂuxes, ⌽x, currents of any Josephson junctions, the net capacitance ␣ ෈͑ᐉ ͒ n are represented as phases ␸x 2␲⌽x ⌽ , where n across the junctions, and the inductance of the superconduct- n ϵ n / 0 ෈͑L,R,cjj,ccjj,q͒. InductancesGraham of Norris the rf-SQUID 2016-04-29 body, CJJ12 loop, ing wiring. The Hamiltonian for many of these devices can and CCJJ loop are denoted as L , L , and L , respectively. generically be written as body cjj ccjj

134510-2 Compound–Compound Josephson Junction

HARRIS et al. PHYSICAL REVIEW B 81, 134510 ͑2010͒ include a tunable inductance in the flux qubit body to ac- x count for both differences in inductance between qubits in a jq multiqubit device and to compensate for changes in qubit j1 Lbody jq I inductanceCCJJ during (c) operation. replaces Thereafter, each the focus JJ of in the pa- 1 per• shifts toward an experimental demonstration of the rf- (a) SQUIDCJJ flux qubit. with The a architecture SQUID of the experimental device and its operation are discussed in Sec. III and then a L /2 L /2 series of experiments to characterize the rf SQUIDx and to cjj cjj jx Tunes barrier height via ϕccjj , q highlight its control are presented in Sec. IV. Section V con- j1 x j Lbody j • I jcjj 2 q tains measurementsjust like CJJ of properties that indicate that this more 1 I2 complex rf SQUID is indeed a flux qubit. Flux and critical (b) current noise measurements and a formula for converting the measuredBalances flux noise spectral left and density right into a loops free-induction L /2 L /2 • ccjj ccjj x ͑Ramsey͒ decay time are presented in Sec. VI. A summary of jq x x key conclusionswith static is provided flux in offsets Sec. VII. Detailed (ϕ and calcula- jccjj L jl j j x x 1 L j2 j3 jR j4 jr Lbody jq tions of multiple Josephson-junction rf-SQUID Hamiltonians I1 I I x 2 3 I4 have beenϕ placed) in Appendixes A and B. R (c)

II. RF-SQUID FLUX QUBIT DESIGN FIG. 1. ͑Color online͒͑a͒ A single-junction rf-SQUID qubit. ͑b͒ The behavior of most superconducting devices is gov- CJJ rf-SQUID qubit. ͑c͒ CCJJ rf-SQUID qubit. Junction critical erned by three types of macroscopic parameters: the critical currents Ii and junction phases ␸i ͑1ՅiՅ4͒ as noted. Net device phases are denoted as ␸ , where ␣ ,r,q . External ﬂuxes, ⌽x, currents of any Josephson junctions, the net capacitance ␣ ෈͑ᐉ ͒ n are represented as phases ␸x 2␲⌽x ⌽ , where n across the junctions, and the inductance of the superconduct- n ϵ n / 0 ෈͑L,R,cjj,ccjj,q͒. Inductances of the rf-SQUID body, CJJ loop, ing wiring. The Hamiltonian for many of these devices can Graham Norris 2016-04-29 13 and CCJJ loop are denoted as L , L , and L , respectively. generically be written as body cjj ccjj

134510-2 From Qubit to Quantum Processor

Big difference between CCJJ • and processor Still need coupling, readout, • and tuning Utility of the ﬂux qubit • (tunable inductive coupling) Interesting aspects of • commercial engineering

Graham Norris 2016-04-29 14 So Far

We want a processor that can anneal an Ising spin glass. Currently, we have a qubit that can control the following:   X x X z z X z H(t) = − A(t) σi − B(t)  Jij σi σj + hi σi  i i

We need to add couplings Jij

Graham Norris 2016-04-29 15 HARRIS et al. PHYSICAL REVIEW B 81, 134510 ͑2010͒ include a tunable inductance in the flux qubit body to ac- x count for both differences in inductance between qubits in a jq multiqubit device and to compensate for changes in qubit j1 Lbody jq I inductance during operation. Thereafter, the focus of the pa- 1 per shifts toward an experimental demonstration of the rf- (a) SQUID flux qubit. The architecture of the experimental device and its operation are discussed in Sec. III and then a Lcjj/2 Lcjj/2 x series of experiments to characterize the rf SQUID and to jq highlight its control are presented in Sec. IV. Section V con- j1 x j Lbody j I jcjj 2 q tains measurementsCoupling of properties that and indicate Tuning that this more 1 I2 complex rf SQUID is indeed a flux qubit. Flux and critical (b) current noise measurements and a formula for converting the measured flux noise spectral density into a free-induction Top portion: Well offset L /2 L /2 ccjj ccjj x ͑Ramsey͒ decay• time are presented in Sec. VI. A summary of jq x x key conclusions iscontrol provided in (ϕ Sec.);VII single. Detailed control calcula- jccjj q jl j j x x 1 L j2 j3 jR j j Lbody jq tions of multiple Josephson-junction rf-SQUID Hamiltonians I I 4 r 1 I2 3 I4 have been placed inline Appendixes plus Aindividual and B. couplers EXPERIMENTAL(c) INVESTIGATION OF AN EIGHT-QUBIT… PHYSICAL REVIEW B 82, 024511 ͑2010͒

Middle: qubits and qubit

II. RF-SQUID• FLUX QUBIT DESIGN FIG. 1. ͑Color online͒͑a͒ A single-junction rf-SQUID qubit. ͑b͒ I (t) The behavior ofinductive most superconducting couplers devices (Jij ) is gov- CJJ rf-SQUIDg qubit. ͑c͒ CCJJ rf-SQUID qubit. Junction critical tiqubit processor. Rather, one can take advantage of the fact { currents I and junction phases ␸ ͑1ՅiՅ4͒ as noted. Net{ device erned by three types of macroscopic parameters: the critical i x i x that, according to Eq. 7 , all qubits must receive a control phases areΦ denoted as ␸ , where ␣ Φᐉ,r,q . External fluxes, ⌽x, ͑ ͒ currents of any JosephsonBottom: junctions, Barrier the height net capacitance Ip,i ␣ ෈͑ Ip,j ͒ n are represented as phases ␸x 2␲⌽x ⌽ , where n signal with the same time-dependent shape but with custom- across the junctions,• and the inductance of the superconduct- χ Mi hniϵ n / 0 χ Mj hj x ෈͑L,R,cjj,ccjj,p q͒. Inductances1 of thep rf-SQUID body,1 CJJ loop, ing wiring. The Hamiltoniancontrol for (ϕ manyccjj ) of these devices can |Iq | Comp. i |Iq | Comp. j tailored time-independent scale factors ϰhi. A scalable archi- and CCJJ loop are denoted as Lbody, Lcjj, and Lccjj, respectively. generically be written as x tecture for providing these signals is depicted in Fig. 1 and Not shown: L tuner, to Φco,ij Q•2 ␸ − ␸x 2 i x 2 j has been further expounded upon in Ref. 31. Here, a single i ͑ n n͒ ͑␸q − ␸q͒ = homogenize− EJi cos͑␸i͒ + inductanceUn , ͑1͒ V͑␸ ͒ = U 0 − ␤ cos͑␸ ͒ ,0 ͑2b͒ p H ͚ ͫ 2C ͬ ͚ 2 q qΦi CouplerC r ij q Φj global current bias Ig͑t͒=␣͉I ͑t͉͒, where ␣ is a convenient i i n 2 q Qubit ͭ ͮ Qubit scale factor, is coupled to multiple qubits via in situ tunable between qubits { where Ci, EJi=Ii⌽0 /2␲, and Ii denote the capacitance, Jo- Mc ij Kij 2␲LqIq mutual inductances of magnitude M h M ␣. The very sephson energy, and critical current of Josephson junction i, ␤ = , ͑2c͒ i ϵ i AFM/ All individual offsets stored in Iccjj(t) ⌽ respectively. The• terms in the first sum are readily recog- 0 same type of device that is used to provide in situ tunable nized as being the Hamiltonians of the individual junctions x x 29 Permanent Magnetic with theΦ qubitccjj(t) inductance Lq ϵLΦbodyccjj,( qubitt) capacitance Cq interqubit coupling can be retooled to provide coupling be- for which the quantum-mechanical phase across the junction C , and qubit critical current Ic I in this particular case. ϵ 1 Grahamq ϵ Norris1 2016-04-29 16 tween flux qubits and a global bias line. Moreover, each M ␸i and the charge collectedMemory on the junction Qi obey the com- i FIG.If this 1. deviceColor has online been designedMapping such of ISGthat ␤ problems,Ͼ1 and is fluxspecified by a x mutation relation ͓⌽0␸i /2␲,Qj͔=iប␦ij. The index n in the ͑ x ͒ can be controlled with a static flux bias ⌽ provided by biased such that ␸q Ϸ␲, then the potential energy V͑␸q͒ will Ip,i second summation is over closed inductive loops. Externalset ofbe values bistable. denoted With as increasinghi and K␤ij,an onto appreciable superconducting potential- hardware. fluxes threading each closed loop, ⌽x, have been represented p PMM. We will refer to this architecture as persistent current n Two qubits,energy barrier two ͉I forms͉ compensators, between the two and local one minima interqubit of V͑␸ ͒ coupler, are p as phases ␸x 2␲⌽x ⌽ . The quantum-mechanical phase q q ͉͑I ͉͒ compensation, as it is a means of compensating for n ϵ n / 0 shown.through A global which current the two bias lowest-lyingI t provides states of the the rf fluxes SQUID⌽x t that q drop experienced by the superconducting condensate circu- ccjj͑ ͒ ccjj͑ ͒ changing Ip t such that the ISG problem specified by h drive themay annealing couple via process quantum totunneling. multiple It isCCJJ these rf-SQUID two lowest- flux qubits. ͉ q͑ ͉͒ i lating around any closed loop is denoted as ␸n. The overall lying states, which are separated from all other rf-SQUID potential-energy scale factor for each closed loop is givenInterqubit by coupling is mediated by tunable mutual inductances and Kij remains on target throughout annealing. 2 states by an energy of order of the rf-SQUID plasma energy Un ϵ͑⌽0 /2␲͒ /Ln. Here, Ln can be either a geometric induc- x To summarize up to this point, a prescription for imple- MijϰKបij␻pthatϵប/ areͱLqC controlledq, that form bythe basisstatic of fluxes a qubit.⌽ Oneco,ij can. Qubit write bodies are tance from wiring or Josephson inductance from large 0 9 menting AQO to solve ISG problems using a network of 8 subjectedan effective to the low-energy sum of static version flux of Hamiltonian biases ⌽ ͑2aand͒ as time-dependent junctions. Hamiltonian ͑1͒ will be used as the progenitor for i inductively coupled CCJJ rf-SQUID flux qubits has been all device Hamiltonians that follow. flux biases driven by a global1 current bias Ig͑t͒. The latter signals q = − ͓⑀␴z + ⌬q␴x͔, ͑3͒ presented. A problem specified by a set of h and K can be are mediated to eachH qubit via2 tunable mutual inductances Mi ϰhi i ij x A. Compound-compound Josephson-junction structure that are controlledp byx static fluxesp ⌽ . embedded in the hardware using time-independent interqubit where ⑀=2͉Iq͉͑⌽q −⌽0 /2͒, ͉Iq͉ is theIp magnitude,i of the per- A sequence of rf-SQUID architectures is depicted in Fig. sistent current that flows about the inductive q loop when the couplings controlled by PMM and time-dependent qubit-flux 1. The most primitive version of such a device is depicted in device is biased hard ⑀ӷ⌬ to one side, and ⌬ represents ͓ q͔ q x and CCJJ biases. The qubit-flux bias signals can be supplied Fig. 1͑a͒ and more complex variants in Figs. 1͑b͒ and 1͑maximumc͒. the tunneling antiferromagnetic energy between the otherwise͑AFM͒ degeneratecoupling, coun-Mij͑⌽co,ij͒ x using a combination of static flux offsets provided by PMM For the single-junction rf SQUID ͓Fig. 1͑a͔͒, the phase tercirculating persistent current states at ⌽ =⌽ /2. p 2 =MAFM. Define this energy scale as qJAFM0 ͑t͒ϵMAFM͉I ͑t͉͒ . across the junction can be equated to the phase drop across A depiction of the one-dimensional potential energy and q and a single global signal I ͑t͒ that is applied to each qubit Rearranging Hamiltonian ͑5͒ yields g the body of the rf SQUID: ␸1 =␸q. The Hamiltonian for this the two lowest-energy states of an rf SQUID at degeneracy through in situ tunable couplers that are also controlled by x device can then be written as ͑⌽q =⌽0 /2͒ for nominal device parameters is shown in Fig. 0͑t͒ PMM. The CCJJ bias can, in principle, also be provided to 2 2. In this diagram, the͑ groundi͒ and first-excited͑i͒ ͑j͒ states are ͑i͒ Qq H = − hi␴ + Kij␴ ␴ − ⌫͑t͒ ␴ , = + V ␸ , 2a denoted by ͉g͘ and͚͉e͘, respectively.z ͚ Thesez twoz energy levels͚ x all qubits simultaneously using a single global control signal. ͑ q͒ ͑ ͒ JAFM͑t͒ i i,jϾi i H 2Cq constitute the energy eigenbasis of a flux qubit. An alternate ͑6a͒ 134510-2 III. DEVICE ARCHITECTURE AND CALIBRATION

p x 0 x 0 ͉Iq͑t͉͒͑⌽i ͑t͒ − ⌽i ͒ ⌽i ͑t͒ − ⌽i With the mapping of the AQO algorithm onto hardware hi = = , ͑6b͒ completed, we turn to a high-level description of a supercon- M ͉Ip͑t͉͒2 M ͉Ip͑t͉͒ AFM q AFM q ducting chip whose architecture embodies that algorithm. All

p 2 of the principal components of the processor, namely, the Mij͑t͉͒Iq͑t͉͒ Mij͑t͒ qubits,28 couplers,29 readout,30 and PMM ͑Ref. 31͒ have been Kij = p 2 = , ͑6c͒ MAFM͉Iq͑t͉͒ MAFM described in detail in other publications. As such, we will only provide brief summaries of the important points as per-

⌬q͑t͒ taining to the functioning of the collective system herein. ⌫͑t͒ = . ͑6d͒ 2J ͑t͒ As stated previously, we have incorporated CCJJ rf- AFM SQUID flux qubits in our design.28 A schematic of an iso- In order to solve a particular optimization problem, hi and lated flux qubit with the two external bias controls relevant Kij must be time independent. According to Eq. ͑6c͒, one for this study is shown in Fig. 2͑a͒. This particular qubit is must hold all Mij constant during operation. This is conve- robust against fabrication variations in the Josephson- nient as it obviates the application of individually tailored junction critical currents and facilitates the homogenization time-dependent flux bias signals to each interqubit coupler. of the net critical current among a population of such qubits. x Rather, one need only apply a static control signal ⌽co,ij to This device also contains an inductance ͑L͒ tuner that can be each coupler, as depicted in Fig. 1, that can be provided by used to compensate for variations in qubit inductance due to PMM. On the other hand, according to Eq. ͑6b͒, one must fabrication and from tuning the interqubit couplers.29 To each p apply time-dependent qubit-flux biases of the form qubit we have added an ͉Iq͉ compensator, as introduced in x 0 p Sec. II. We have provided PMM to flux bias the two minor ⌽i ͑t͒ = ⌽i + hi ϫ MAFM͉Iq͑t͉͒ ͑7͒ 28 p lobes of each CCJJ, the L tuner, the ͉Iq͉ compensator, and 0 31 to render hi time independent. Thus, it is necessary to pro- the qubit body ͑⌽i ͒ for all qubits on the chip. vide a custom-tailored time-dependent control signal plus a The chip that was used for this study was composed of 16 0 static offset to every qubit. The static component ⌽i can be eight-qubit unit cells that were tiled on a 4ϫ4 square grid. provided by PMM. As for the time-dependent component, To limit the scope of this paper, we focus upon a single unit providing these signals with one external bias per qubit cell near the center of the chip. A discussion of the complete would not constitute a scalable approach for building a mul- processor, with multiple unit cells acting in concert, will be

024511-3 Single Flux Quantum ReadoutEXPERIMENTAL DEMONSTRATION OF A ROBUST AND… PHYSICAL REVIEW B 81, 134510 ͑2010͒

Φx Qubit Quantum Flux Parametron { M Φx Since much of what follows depends on a clear under- R eff LT Φx x standing of our QFP-enabled readout mechanism, we present RF–SQUID L Φco Φx a brief review of its operation herein. The flux and readout Locks the final qubit state q • Coupler current wave form sequence involved in a single-shot read- based on external control M 47 Φx co,i Mq-qfp out is depicted in Fig. 5͑b͒. Much like the CJJ qubit, the ccjj Φx Insulates qubit from readout qfp QFP can be adiabatically annealed from a state with a • QFP x measurement monostable potential ͑⌽latch=−⌽0 /2͒ to a state with a M x DC–SQUID Φx qfp-ro bistable potential ͑⌽latch=−⌽0͒ that supports two countercir- latch dc SQUID culating persistent current states. The matter of which persis- Sensitive flux detector • Φx tent current state prevails at the end of an annealing cycle ro i x Reads state of QFP (a) ro depends on the sum of ⌽ and any signal from the qubit • qfp mediated via M . The state of the QFP is then determined Produces unwanted Φ q−qfp - 0/2 1 • x with high fidelity using a synchronized flux pulse and current microwave signals Φlatch(t) bias ramp applied to the dc SQUID. The readout process was -Φ0 Graham Norris 2016-04-29 170.75 x QFP typically completed within a repetition time t Ͻ50 ␮s. Φqfp(t) rep 0 } An example trace of the population of one of the QFP 0.5 x qfp persistent current states Pqfp versus ⌽qfp, obtained using the P Φ latching sequence depicted in Fig. 5͑b͒, is shown in Fig. 5͑c͒. ~ 0/3 x Φro(t) 0 0.25 This trace was obtained with the qubit potential held x monostable ͑⌽ccjj=−⌽0 /2͒ such that it presented minimal i (t) ro 0 flux to the QFP and would therefore not influence P . The 0 } dc SQUID 1 0.5 0 0.5 1 qfp Φ x Φ data have been fit to the phenomenological form (b) t=0 trep (c) qfp (m 0) x 0 FIG. 5. Color online a Schematic representation of a portion 1 ⌽qfp − ⌽qfp ͑ ͒͑ ͒ P = 1 − tanh , ͑8͒ of the circuit reported upon herein. Canonical representations of all qfp 2ͫ ͩ w ͪͬ x externally controlled flux biases ⌽␣, readout current bias iro, and key mutual inductances M are indicated. ͑b͒ Depiction of latching ␣ with width wϳ0.18 m⌽0 for the trace shown therein. When readout wave form sequence. ͑c͒ Example QFP state population biased with constant ⌽x =⌽0 , which we refer to as the ⌽x qfp qfp measurement as a function of the dc level qfp with no qubit signal. QFP degeneracy point, this transition in the population sta- Data have been fit to Eq. ͑8͒. tistics can be used as a highly nonlinear flux amplifier for qubit system will be the topic of a separate publication. To sensing the state of the qubit. Given that Mq−qfp clearly establish the lingua franca of our work, we have =6.28Ϯ0.01 pH for the devices reported upon herein and depicted a portion of the multiqubit circuit in Fig. 5͑a͒. Ca- that typical qubit persistent currents in the presence of neg- p nonical representations of the external flux biases needed to ligible tunneling ͉Iq͉տ1 ␮A, then the net flux presented by a p operate a qubit, a coupler, and a QFP-enabled readout are qubit was 2Mq−qfp͉Iq͉տ6m⌽0, which far exceeded w. By x x x x labeled on the diagram. The fluxes ⌽L, ⌽R, ⌽LT, and ⌽co this means, one can achieve the very high qubit state readout were only ever subjected to dc levels in our experiments that fidelity reported in Ref. 46. On the other hand, the QFP can x were controlled by PMM. The remaining fluxes and readout be used as a linearized flux sensor by engaging ⌽qfp in a current biases were driven by a custom-built 128-channel 0 feedback loop and actively tracking ⌽qfp. This latter mode of room-temperature current source. The mutual inductances operation has been used extensively in obtaining many of the between qubit and QFP ͑Mq−qfp͒, between QFP and dc results presented herein. SQUID ͑Mqfp−ro͒, qubit and coupler ͑Mco,i͒, and x ⌽co-dependent interqubit mutual inductance ͑Meff͒ have also been indicated. Further details concerning cryogenics, mag- IV. CCJJ RF-SQUID CHARACTERIZATION netic shielding, and signal filtering have been discussed in previous publications.44–47 We have calibrated the relevant The purpose of this section is to present measurements mutual inductances between devices in situ and have at- that characterize the CCJJ, L tuner, and capacitance of a tempted to measure a significant portion of the parasitic cross CCJJ rf SQUID. All measurements shown herein have been x couplings between bias controls ͑both analog lines and PMM made with a set of standard bias conditions given by ⌽L x x elements͒ into unintended devices. These cross couplings =98.4 m⌽0, ⌽R =−89.3 m⌽0, ⌽LT=0.344 ⌽0, and all in- were typically below our measurement threshold Շ0.1 fH terqubit couplers tuned to provide Meff=0, unless indicated while designed mutual inductances between bias controls otherwise. The logic behind this particular choice of bias and intended devices were typically ͑1 pH͒. Much of the 4 conditions will be explained in what follows. This section orders in magnitude separation betweenO intended versus un- will begin with a description of the experimental methods for c intended mutual inductances can be attributed to the exten- extracting Lq and Iq from persistent current measurements. sive use of ground planes and the encapsulation of trans- Thereafter, data that demonstrate the performance of the formers described above. Such careful shielding is critical CCJJ and L tuner will be presented. Finally, this section will for producing high-density and scalable superconducting de- conclude with the determination of Cq from macroscopic vice architectures. resonant tunneling data.

134510-7 HARRIS et al. PHYSICAL REVIEW B 82, 024511 ͑2010͒

1 m µ WIRC Ig IPC SiO2 WIRB SiO2 WIRA VIA LT TRI BASE RESI CCJJ RO Φx FIG. 3. Scanning electron micrograph of fabrication cross sec- Medium Connectivity (a) ccjj tion. Superconducting Nb layers identified as BASE, WIRA, WIRB, and WIRC. Resistive TiPt layer identified as RESI. Pla- XCO narized dielectric denoted as SiO2. An example Josephson-junction IPC q q q q trilayer and a via between WIRA and WIRB have been denoted as Eight qubits in a block 0 1 2 3 TRI and VIA, respectively. q4 • right, top, and bottom in order to generate larger graphs. ICO Each qubit attached to four q Each vertex is connected to a minimum ͑maximum͒ of four • 5 ͑six͒ other vertices, depending on the number and arrange- interblock qubits and two LT ment of unit cells used to form a larger graph. This hardware q extrablock qubits (red) 6 does not provide full connectivity within a population of N CCJJ qubits in which each physical qubit is connected to N−1 other physical qubits. This limitation can potentially be over- Bipartite complete q7 • come, at the cost of reducing the number of unique vertices ϽN, by using a ferromagnetically coupled chain of physical connectivity graph K4,4 RO XCO qubits to form a single logical qubit, as suggested in Ref. 22. Nonplanar (NP–Hard Ising) (b) The chip used in these experiments was fabricated on an • oxidized Si wafer with Nb/Al/Al2O3 /Nb trilayer junctions and four Nb wiring layers separated by planarized plasma- 0 1 enhanced chemical vapor deposited SiO2. A scanning elec- 2 tron micrograph of the fabrication cross section can be found 3 4 in Fig. 3. The Nb metal layers are referred to as BASE, 5 WIRA, WIRB, and WIRC, from bottom to top, respectively. Planar graph 6 Flux-qubit wiring was primarily located in WIRB and con- (c) 7 sisted of 2-␮m-wide leads arranged as an approximately 900-␮m-long differential microstrip located 200 nm above a FIG. 2. ͑Color online͒ DeviceGraham schematic. Norris͑ 2016-04-29a͒ A single CCJJ 18 rf ground plane in WIRA. Coupler wiring was primarily lo- SQUID with the two time-dependent biases relevant for this study. x cated in WIRC, stacked on top of the qubit wiring to provide The flux bias ⌽ccjj͑t͒ drives the annealing process. The global p inductive coupling. PMM flux storage loops were imple- ͉Iq͉-compensation bias is provided by Ig͑t͒. ͑b͒ Schematic layout of the eight-qubit unit cell. Qubits are shown in gray. Example read- mented as stacked spirals of 13–20 turns of 0.25-␮m-wide p wiring with 0.25 ␮m separation in BASE and WIRA out, CCJJ, L tuner ͑LT͒, ͉Iq͉ compensator ͑IPC͒, internal coupler ͑ICO͒, and portions of external couplers ͑XCO͒ have been noted. ͑WIRB͒. Stored flux was picked up by one-turn washers in ͑c͒ Graph representation of the hardware connectivity. Qubits are WIRB ͑WIRA͒ and fed into transformers for flux-biasing represented by vertices ͑solid dots͒ and couplings by edges ͑solid devices. External control lines were mostly located in BASE lines͒. Dark ͑shaded͒ edges correspond to ICO ͑XCO͒ couplings. and WIRA. Resistors that were used in the PMM demultiplexing circuit were made from a TiPt layer referred to as reserved for a future publication. A schematic layout of the RESI. All of these control elements resided below a ground unit cell is shown in Fig. 2͑b͒. Qubits are depicted as plane in WIRC. The ground planes under the qubits and over extended horizontal and vertical loops with 16 compound the PMM/external control lines were electrically connected Josephson junction ͑CJJ͒ couplers located at the intersections using extended vias in WIRB so as to form a nearly continu- of the qubits. Additional couplers at both extrema of the ous superconducting shield between the analog devices on qubit bodies facilitate connections to qubits in neighboring top and the bias circuitry below. Transformers for biasing unit cells to the left, right, top, and bottom. These latter cou- qubits, couplers, QFPs, and dc SQUIDs were enclosed in plers were set to zero coupling to isolate the single unit cell superconducting boxes with BASE and WIRC forming the for this study. Each qubit was connected to its own quantum top and bottom, respectively, and vertical walls formed by flux parametron ͑QFP͒ enabled readout.28,30 Not shown in extended vias in WIRA and WIRB. Minimally sized open- Fig. 2͑b͒ are the PMM elements, the demultiplexing tree for ings were placed in the vertical walls through which the bias addressing the PMM or the analog bias lines used to cali- and target device wiring passed at opposing ends of each brate and operate the unit cell. box. This design reduced most on-chip parasitic crosstalk to A simplified representation of the hardware connectivity a negligible level. is depicted in Fig. 2͑c͒. Here, qubits and couplers correspond An optical image of a unit cell completed through the to the vertices and edges of a graph.38 As with the unit-cell processing of WIRB is shown in Fig. 4͑a͒. One can discern schematic, this eight-vertex graph can be tiled to the left, the trenches in which the qubit wiring resides, where the

024511-4 SUPPLEMENTARY INFORMATION DOI: 10.1038/NPHYS2900

Supplementary material for “Evidence for quantum annealing with more than one SupplementaryEvidence material for forquantum “Evidencehundred annealing for qubits” quantum annealing with more with morethan than one hundred qubits” Sergio Boixo,1 Troelsone F. Rønnow, hundred2 Sergei V. qubits Isakov,2 Zhihui Wang,3 David 4 1 5 2 6 2 3 2 SergioWecker, Boixo,DanielTroels A. F. Lidar, Rønnow,JohnSergei M. Martinis, V. Isakov,andZhihui Matthias Wang, TroyerDavid∗ 1 4 5 6 2 InformationWecker, SciencesDaniel Institute, A. Lidar, UniversityJohn of Southern M. Martinis, California,and MatthiasLos Angeles, Troyer CA∗ 90089, USA 2 1 Theoretische Physik, ETH Zurich, 8093 Zurich, Switzerland Information3 Sciences Institute, University of Southern California, Los Angeles, CA 90089, USA Department2 ofTheoretische Chemistry andPhysik, Center ETH for Zurich, Quantum 8093 Information Zurich, Switzerland Science & Technology, 3DepartmentUniversity of Chemistry of Southern and CenterCalifornia, for Quantum Los Angeles, Information California Science 90089, & USA Technology, 4Quantum Architectures and Computation Group, Microsoft Research, Redmond, WA 98052, USA University5 of Southern California, Los Angeles, California 90089, USA 4Quantum ArchitecturesDepartments and Computation of Electrical Group, Engineering, Microsoft Chemistry Research, and Redmond, Physics, WA 98052, USA 5 andDepartments Center for of QuantumElectrical InformationEngineering,Larger Science Chemistry & Technology, and Connectivity Physics, University of Southern California, Los Angeles, California 90089, USA 6 and Center for Quantum Information Science & Technology, DepartmentUniversity of Physics, of Southern University California, of California, Los Angeles, Santa CaliforniaBarbara, CA 90089, 93106-9530, USA USA 6Department of Physics, University of California, SantaTile Barbara, the CA blocks 93106-9530, of USA qubits

I. OVERVIEW 0 4 8 12 16 20 24 28

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Here we provide additional details in support of the 21 65 109 1413 1817 2221 2625 3029 mainHere text. we Sectionprovide II additional shows details details of the in support chimera ofgraph the 32 76 1110 1514 1918 2322 2726 3130 used in our study and the choice of graphs for our simula- main text. Section II shows details of the chimera graph 3 7 11 15 19 23 27 31 usedtions. in Section our study III expandsand the choice upon the of graphs algorithms for our employed simula- 32 36 40 44 48 52 56 60 tions.in our study.Section Section III expands IV presents upon the additional algorithms success employed prob- 3332 3736 4140 4544 4948 5352 5756 6160 inability our study. histograms Section for IV diff presentserent numbers additional of qubits success and prob- for 3433 3837 4241 4645 5049 5453 5857 6261 instances with magnetic fields, explains the origin of easy ability histograms for different numbers of qubits and for 3534 3938 4342 4746 5150 5554 5958 6362 instancesand hard withinstances, magnetic and fields, explains explains how the the final origin state of easy can 35 39 43 47 51 55 59 63 andbe improved hard instances, via a simple and explains error reduction how the scheme. final state Section can 64 68 72 76 80 84 88 92 beV presentsimproved further via a simple correlation error reduction plots and scheme. provide Section more 6564 6968 7372 7776 8180 8584 8988 9392

Vdetails presents on gauge further averaging. correlation Section plots VI and gives provide details more on 6665 7069 7473 7877 8281 8685 9089 9493 how we determined the scaling plots and how quantum details on gauge averaging. Section VI gives details on 6766 7170 7574 7978 8382 8786 9190 9594 howspeedup we determined can be detected the scaling on future plots devices. and how Finally, quantum sec- 67 71 75 79 83 87 91 95 speeduption VII can explains be detected how the on spectral future devices.gaps were Finally, calculated sec- 96 100 104 108 112 116 120 124 tionby quantum VII explains Monte how Carlo the (QMC) spectral simulations. gaps were calculated 9796 101100 105104 109108 113112 117116 121120 125124 by quantum Monte Carlo (QMC) simulations. 9897 102101 106105 110109 114113 118117 122121 126125

9998 103102 107106 111110 115114 119118 123122 127126 II. THE CHIMERA GRAPH OF THE D-WAVE 99 103 107 111 115 119 123 127 II. THE CHIMERADEVICE. GRAPH OF THE D-WAVE FIG. 1: Qubits and couplers in the D-Wave device. DEVICE. FIG.The D-Wave 1: Qubits One“Chimera” and Rainer couplers chip consistsin graph the of D-Wave4 4 unit device. cells of The qubits and couplers in the D-Wave device can be 1 × Theeight D-Wave qubits, connectedOne Rainer by chip programmable consists of 4inductive4 unit couplers cells of thought of as the vertices and edges, respectively, of a 1 The qubits and couplers in the D-Wave device can be eightas shown qubits, by lines. connected by programmable inductive× couplers thoughtbipartite of graph, as the called vertices the “chimera and edges, graph”, respectively, as shown of in a as shown by lines. bipartitefigure 1. graph, This graph called is the built “chimera from unit graph”, cells as containing shown in Graham Norris 2016-04-29 19 figureeight qubits 1. This each. graph Within is built each from unit unit cell cells the qubits containing and tions. The sizes we typically used in our numerical sim- eightcouplers qubits realise each. a complete Within each bipartite unit cell graph theK qubits4,4 where and ulations are L =1,...,8 corresponding to N =8L2 = eachcouplers of the realise four qubitsa complete on the bipartite left is coupled graph toK allwhere of the tions. The sizes we typically used in our numerical sim- 4,4 8, 32, 72, 128, 200, 288, 392 or 512 spins. For the simu-2 eachfour on of the the four right qubits and vice on the versa. left Eachis coupled qubit to on all the of leftthe ulations are L =1,...,8 corresponding to N =8L = lated annealers and exact solvers on sizes of 128 and fouris furthermore on the right coupled and vice to the versa. corresponding Each qubit qubit on the in leftthe 8, 32, 72, 128, 200, 288, 392 or 512 spins. For the simu- above we used a perfect chimera graph. For sizes below isunit furthermore cell above coupled and below, to the while corresponding each of the qubit ones on in the lated annealers and exact solvers on sizes of 128 and 128 where we compare to the device we use the working unitright cell is horizontally above and below, coupled while to the each corresponding of the oneson qubits the above we used a perfect chimera graph. For sizes below 128qubits where within we selections compare to of theL deviceL eight-site we use unit the cells working from rightin the is unit horizontally cells to the coupled left and to the right corresponding (with appropriate qubits × modifications for the boundary qubits). Of the 128 qubits qubitsthe graph within shown selections in figure of L 1. L eight-site unit cells from in the unit cells to the left and right (with appropriate × modificationsin the device,for the the 108 boundary working qubitsqubits). used Of thein our 128 tests qubits of theIn graph references shown [1, in 2] figure it was 1. shown that an optimisation inthe the device device, are the shown 108 in working green, qubits and the used couplers in our between tests of problemIn references on a complete [1, 2] it wasgraph shown with that√N anvertices optimisation can be thethem device are marked are shown as black in green, lines. and the couplers between problemmapped toonan a complete equivalent graph problem with on√N avertices chimera can graph be themFor are our marked scaling as analysis black lines. we follow the standard pro- mappedwith N vertices to an equivalent through problemminor-embedding. on a chimera The graph tree cedureFor our for scaling analysisof finitewe dimensional follow the models standard by con-pro- withwidthN of √verticesN mentioned through in minor-embedding.the main text arises fromThe treethis ceduresidering for the scaling chimera of finitegraph dimensional as an L L modelssquare by lattice con- widthmapping. of √ SeeN mentioned Section VI in A the for main additional text arises details from about this × sideringwith an theeight-site chimera unit graph cell asand an openL boundaryL square latticecondi- mapping.the tree width See Sectionand tree VI decomposition A for additional of a details graph. about × NATUREwith anPHYSICS eight-site | www.nature.com/naturephysics unit cell and open boundary condi- the tree width and tree decomposition of a graph. 1 ǟɥƐƎƏƓɥ !,(++ -ɥ4 +(2'#12ɥ (,(3#"ƥɥ++ɥ1(%'32ɥ1#2#15#"ƥɥ Connectivity Generality

Does this connectivity limit • the problems we can solve? No, as long as each qubit is •

connected to at least three 4 8 3 3 4 8

others 3 1 7 2 1 1 1 7

Map the logical connectivity 2 6 1 6 • (left) onto physical qubits 5 5 (right) Ferromagnetic coupling • G Gemb binds logical qubit Figure 2: Gemb(right) is a minor-embedding of G(left) in the square lattice U.Eachvertex(calledalogicalqubit) of G is mapped to a (connected) subtree of (same color/label) vertices (called physical qubits) of U. G is called Can affect annealing timesa(graph)minorofU. • By reduction through minor-embedding,wemeanthatonecanreducetheoriginalIsingHamiltonianonthe input graph G to the embedded Ising Hamiltonian emb on its minor-embedding G ,i.e.,thesolutiontothe H Graham Norris 2016-04-29emb 20 embedded Ising Hamiltonian gives rise to the solution to the original Ising Hamiltonian. The intuition suggests that the reduction will be correct provided that the ferromagnetic coupler strengths used are sufficiently strong (i.e., large negative number). However, how strong is “strong enough”? In [17, 18], they do not address this question, i.e. what are the required strengths of these ferromagnetic couplers? In Section 4.1, we will show that it is not difficult to give an upper bound for the ferromagneticcouplerstrengthsandthusexplaintheintuition. However, there are indications [2] that too strong ferromagnetic coupler strengths might slow down the adiabatic algorithm. Furthermore, an adiabatic quantum computer is ananalogcomputerandanalogparameterscanonly be set to a certain degree of precision (a condition much more stringent than the setting of digital parameters). Hence, the allowed values of coupler strengths are limited. Therefore, from the computational point of view, it is important to derive as small (in terms of magnitude) as possible sufficient condition for these ferromagnetic coupler strengths. Furthermore, what should the bias for physical qubits be? There are two components to the reduction: embedding and parameter setting.Theembeddingproblemis to find a minor-embedding Gemb of a graph G in U.Thisproblemisinterdependentofthehardwaregraph design problem, which will be discussed in Section 6. The parameter setting problem is to set the corresponding parameters, qubit bias and coupler strengths, of the embedded Ising Hamiltonian. In this paper, we assume that the minor-embedding Gemb is given, and focus on the parameter setting problem (of the final Hamiltonian). Note that there are two aspects of efficiency of a reduction. One is how efficient one can reduce the original problem to the reduced problem. For example, here we are concerned how efficiently we can compute the minor-embedding and how efficiently we can compute the new parameters of the embedded Ising Hamiltonian. The other aspect concerns about the efficiency (in terms the running time) of the adiabatic algorithm for the reduced problem. In general, the latter depends on the former. For example, the running time of the adiabatic algorithm may depend

4 HARRIS et al. PHYSICAL REVIEW B 81, 134510 ͑2010͒

XCO qubits in Fig. 4͑a͒ are coincident with portions of open- Physical Construction circuited interqubit couplers. These latter elements, hereafter q0 q1 q2 q3 referred to as external couplers, allow this eight-qubit unit q4 cell to be tiled on a square grid to create larger arrays of coupled qubits. Tiling the unit cell to the left, right, top, and ICO bottomMultilayer closes the Nb metal loops forming the external couplers q5 • thatsuperconducting mediate interqubit coupling IC process between qubits in different LT unit cells. In such a scenario, any qubit that is not at the q periphery of an array will be coupled to six other qubits, 6 Extremely low noise despite •while those on the periphery will be connected to five other CCJJ qubits.dense, This complex tiling has been circuitry explicitly depicted and is discussed further in Ref. 44. The processor was fabricated from an oxidized Si wafer q7 Superconducting boxes •with Nb/Al/Al2O3 /Nb trilayer junctions and four Nb wiring RO XCO layersaround separated each by planarized qubit plasma enhanced chemical vapor deposited SiO2. A scanning electron micrograph of the (a) processTop: crossOptical section image, is shown qubits, in Fig. 4͑b͒. The Nb metal •layers have been labeled as BASE, WIRA, WIRB, and 500 nm WIRC WIRC.PMM The visible flux qubit wiring was primarily located in WIRB SiO2 WIRB SiO2 and consisted of 2-␮m-wide leads arranged as an approxi- VIA WIRA matelyBottom: 900-␮ SEMm-long image, differential wiring microstrip located 200 nm •above a ground plane in WIRA. CJJ rf-SQUID coupler wir- AlO /Al RESI BASE x inglayers was primarily visible located in WIRC, stacked on top of the (b) qubit wiring to provide inductive coupling. PMM flux storage loops were implemented as stacked spirals of 13–20 turns of 0.25-␮Grahamm-wide Norris wiring 2016-04-29 with 0.25- 21␮m separation in q0 q1 q2 q3 BASE and WIRA ͑WIRB͒. Stored flux was picked up by one-turn washers in WIRB ͑WIRA͒ and fed into transform- q4 ers for flux-biasing devices. External control lines were lo- 100 µm cated in BASE and WIRA. All of these control elements (c) resided below a ground plane in WIRC. The ground plane under the qubits and over both the PMM and external control FIG. 4. Color online a Schematic of the analog components ͑ ͒͑͒ lines were electrically connected using extended vias in of the test device. Qubits ͑light gray͒ are denoted as q ...q . One 0 7 WIRB so as to form a nearly continuous superconducting readout ͑RO͒, CCJJ, and L tuner ͑LT͒ each have been indicated in shield between the analog devices on top and the bias cir- dashed boxes. Internal couplers ͑ICO, black͒ mediate couplings between qubits within this eight-qubit unit cell. External couplers cuitry below. To provide biases to target devices with mini- ͑XCO, red͒ are used to couple qubits in different unit cells in larger mal parasitic crosstalk, transformers for biasing qubits, cou- arrays. ͑b͒ SEM of a cross section of the fabrication profile. Metal plers, QFPs, and dc SQUIDs using bias lines and/or PMM layers denoted as BASE, WIRA, WIRB, and WIRC. Insulating lay- elements were enclosed in superconducting boxes with ers labeled as SiO2. An example via ͑VIA͒, Josephson junction BASE and WIRC forming the top and bottom, respectively, ͑AlOx /Al͒, and resistor ͑RESI͒ are indicated. ͑c͒ Optical image of a and vertical walls formed by extended vias in WIRA and portion of a device completed up to the patterning of WIRB. Por- WIRB. Minimal-sized openings were placed in the vertical tions of qubits q0 ...q3 and the entirety of q4 are visible, as well as walls through which the vias and target device wiring passed much of the external bias wiring and several PMM elements. at opposing ends of each box. An optical image of a portion of a device completed up to mediated by a network of 16 in situ tunable CJJ rf-SQUID WIRB is shown in Fig. 4͑c͒. Qubits are visible as elongated interqubit couplers.45 Each qubit was also coupled to its own objects, WIRB PMM spirals are visible as dark rectangles, dedicated quantum flux parametron ͑QFP͒-enabled readout.46 and WIRB washers are visible as light rectangles with slits. A high-level schematic of the device architecture is shown in The particular chip used in this study had a total of 72 pairs Fig. 4͑a͒. External flux biases were provided to target de- of differentially driven external biases to control the coupled vices using a sparse combination of analog current bias lines eight-qubit system and a variety of on-chip diagnostic tools. to facilitate device calibration and local on-chip sources that For reference, we have also fabricated and are currently test- we refer to as programmable magnetic memory ͑PMM͒. ing a coupled 128-qubit system that uses a total of only 84 These local flux sources consisted of storage inductors that pairs of differentially driven external biases. Much of the could hold an integer number of flux quanta that were pro- inherent scalability of this particular adiabatic quantum in- grammed by a SFQ demultiplexer circuit. By inductively formation processor design is due to the extensive use of coupling the storage loops to their target devices, we were PMM. able to implement on-chip digital-to-analog converters for We have studied the properties of all eight CCJJ rf- applying flux biases. The PMM has been described in detail SQUID flux qubits on this chip in detail and report upon one in Ref. 44. Note that the extrema of the CCJJ rf-SQUID such device herein. The performance of the collective eight-

134510-6 RESEARCH LETTER

energy scales, then the qubit will quickly respond and will be able to process using a quantum mechanical model as discussed in the Sup- evolve into the lower energy well, such that P , the probability of the plementary Information. " LETTER RESEARCH spin being in state æ, is greater than 1/2. The value of P will depend In addition to the experimental results, in Fig. 3b we show the results j" " on both ht and T, as the system will strive to achieve a Boltzmann of three different numerical simulations. In all three cases, the model

29 distribution of its population statistics between æ and æ. However, if parameters were independently measured for the individual devices, coupled together using programmable coupling elements which pro- a j# j" vide a spin–spin coupling energy that is continuously tunable between Classical h is not turned on until after the barrier has been raised sufficiently leaving no free parameters. A simulation, based on a classical model, thermal ferromagnetic and antiferromagnetic coupling. This allows spins to high (td . tfreeze), the system will not be able to follow it and will be treated the flux in the two superconducting loops as the coordinates of annealing favour alignment or anti-alignment, respectively. equally likely to settle into either potential energy well, such that a discrete particle inside the two-dimensional flux qubit potential, and The behaviour of this system can be described with an Ising model P < 1/2. These two situations are illustrated in Fig. 2d. For intermediate then coupled that particle to a thermal bath. The dynamics was simu- Hamiltonian " Quantum values of td,thequbitwillonlypartlysucceedinrespondingtothe lated by numerically solving the Langevin equation, as described in N N annealing ~ h szz J szsz 1 sudden application of the bias h. Supplementary Information. The classically simulated tfreeze value varies HP i i ij i j ð Þ i~1 i,j~1 Example plots of measured P values versus td for a single qubit linearly with T, as expected. The other two simulations involved solving X X " where for spin i sz is the Pauli spin matrix with eigenvectors { æ, æ} b at different temperatures are shown in Fig. 3a. In this case, a quantum mechanical model of a flux qubit coupled to a thermal i j" j# 1 and 2hi is the energy bias; and 2Jij is the coupling energy between the ht 5 2.55 6 0.04 GHz and D 5 9.0 6 0.2 GHz. Experimental para- bath in which only the two or, respectively, four lowest-lying energy spins i and j. Our implementation allows each J and h to be pro- ij i Γ Λ meters controlling the annealing process are discussed in Sup- levels of the flux qubit were kept. The dynamics was simulated by grammed independently within the constraints of the connectivity of 0.5 plementary Information. As expected, P shows an initial (td<0) T numerically solving a non-Markovian density matrix equation of our devices. " The quantum mechanical properties of the individual devices have dependence and then converges to 1/2 at late delay times. These curves motion (Supplementary Information). These two models will be 13 referred to here as the two-level and four-level quantum models. been well characterized , but we are interested in what happens when were numerically fitted to extract tfreeze, the time at the middle of this several of them are coupled together. It is reasonable to ask whether 0 Quantum0 0.2 Bit 0.4 Properties 0.6 LETTER0.8 1.0RESEARCH transition region, which in turn is plotted versus T in Fig. 3b. The curve The experimental results clearly show a saturation of tfreeze below this manufactured, macroscopic (,1 mm) system of artificial spins t/t fnal used for the fitting was obtained by numerical simulations of this 45 mK, inagreement with both the two-level and the four-level quantum behaves quantum mechanically. We report here on an experiment that c models and in disagreement with the classical model. The experimental demonstrates a signature of quantum annealing29 in a coupled set of coupled together using programmable coupling elements which pro- a h eight artificial Ising spins. t data deviate from the two-level model above 45 mK, as the upper energy vide a spin–spin coupling energy that is continuously tunable between Classical a 1 1 Whereas thermal annealing uses progressively weaker thermal fluc- Model levels in the flux qubit start to become thermally occupied. The four- ferromagnetic andtuations antiferromagnetic to allow a system coupling. to explore its This energy allows landscape spins and to arrive at thermalh 22 mK level quantum model describes the behaviour of the system well up to Early Late Measured favour alignment ora anti-alignment, low-energy configuration, respectively. quantum annealing uses progressively annealing

↑ 80 mK, where more energy levels start to be occupied. The experimental weaker quantum fluctuations, mediated by tunnelling. In both thermal 50 mK The behaviour of this system can be described with an Ising model 0.9 P 0.9 and quantum annealing, a system starts with a mixture of all possible 0 data asymptotically approach the classical simulation results at higher Hamiltonian states: a classical mixed state in the former and a coherent superposi- 0 0.2 0.4 0.6 0.8 1.0 temperatures. We propose that if the quantum mechanical modelling t/t Early Quantum fnal tion in the latter. d were extended by keeping even more energy levels, then it would repro- N N 0.8 0.8 Quantum annealingz can be performedz z by slowly changing the sys- annealing duce the data to ever higher temperatures. P~ hisi z Jijsi sj 1 tem Hamiltonian Early H i~1 i,j~1 ð Þ Both the measured and the simulated (four-level quantum model) T N ↑ 20 40 60 80 100 X X P x Start annealing dependence of P for t < 0 are shown in the inset of Fig. 3a. Because z (t)~C(t) Dis zL(t) P b 0.7 Temperature (mK) d where for spin i si is the Pauli spinH matrix with eigenvectorsi H { æ, æ} 1 " i~1 j"• j# this probability has a strong T dependence for td , tfreeze, its measure- and 2hi is the energy bias; and 2Jij is the couplingX energy between the where C decreases from one to zero and L increases from zero to one Late ment provides us with an independent check on the effective temper- spins i and j. Our implementation allows eachD Jij and hi to be pro- Γ monotonically with time, and i parameterizes quantum mechanicalApply well offset term suddenlyΛ 0.6 90 mK ature of the spin system in this regime. Moreover, because the grammed independentlytunnelling within between the constraintsæ and æ. of the connectivity of j" j# 0.5 probability does not saturate at 45 mK, where tfreeze saturates, it is a our devices. At the beginning of the annealing, C 5 1, L 5 0 and• the system is Figure 2 | Quantum annealing. a, Annealing is performed by gradually N x clear indication that saturation of tfreeze is not a result of saturation of fully characterized by the transverse terms, i~1 Disi . The ground raising the energy barrier between states. In thermal annealing, when the The quantum mechanical properties of the individual devices have qubit temperature. The key conclusion we draw from Fig. 3b is that our state13 of this is a superposition of all states in the sz basis. It is straight- barrier becomes much larger than kBT thermal excitation over the barrier 0.5 been well characterized , but we are interested in what happensP whenMeasure probabilityTA of system forward to initialize the system in this state. During quantum anneal- eventually ceases, at some time tfreeze. In quantum annealing, tunnelling qubit dynamics is best characterized as being quantum mechanical in between0 states also will eventually cease, at a time tQA . b, The value of the several of them areing, coupled the transverse together. term isIt gradually is reasonable turned off to (C askR 0) whether and the weight 0 0.2 0.4 0.6freeze 0.8 1.0 60 70 80 90 100 • parameters C and L during annealing are not independent of each other in the nature for T=80 mK. The system evolves to its ground state through a this manufactured,of macroscopic the Ising Hamiltonian, (,1 mm)P,system is increased of ( artificialL R 1) (Fig. spins 2b). If this t/t H flux qubit. The annealing ends at t 5 148fmnals. c, Changing the value of h(t) Delay time (μs) process of quantum annealing. But so far we have shown this only for behaves quantum mechanically.annealing is done We slowly report enough, here onthe an system experiment should remain thatreaching in the the groundfinal state ground state at all times, thus ending up in the ground state of c(see d) at various points during annealing can be used to probe the freeze-out an individual qubit. It remains to be shown whether quantum anneal- P time, t . d, Double-well potential during annealing. If h is turned on early demonstrates a signature(ref. 4). of quantum annealing in a coupled set of H freeze b ing can be performed on several spins coupled together. enoughh (blue line), the system follows the ground state through annealing and 84 eight artificial Ising spins.The above description of quantum annealing is in the language of an reachest the final ground state of equation (1) with high probability. If h(t)is To investigate this, we now configure our array into a chain of eight Whereas thermalideal annealing Ising spin uses system. progressively Let us look more weaker closely thermal at what this fluc- means for turned on too late (red line), the state probabilities are determined by the earlier Calculated freezing time 82 ferromagnetically coupled artificial spins (Fig. 4), with Ji,i11 52J for tuations to allow a systeman individual to explore flux qubit. its Duringenergy annealing, landscape the and energy arrive barrier, at dU(t), hHamiltonian, for which h 5 0. i 5 1, 2, …, 7 along the chain and Jij 5 0 otherwise. In our experiment, between the two wells is gradually raised (Fig. 2a). If thermal fluctua- Early • Late 80 a low-energy configuration,tions are dominant, quantum then the annealing qubit dynamics uses may progressively be viewed as thermal activation or quantum tunnelling is the dominant effect governing we used J 5 12.78 GHz, which is near the maximum available for the weaker quantum fluctuations, mediated by tunnelling. In both thermal{dU=kBT activation over the barrier with a rate that is proportional to eLookqubitat dynamics. temperature dependence 78 couplers. The lowest-energy configurations of this system correspond and quantum annealing,at a temperature a systemT starts(kB, Boltzmann’s with a mixture constant). of This allpossible suggests that the 0Here we modify this annealing procedure to perform a specialized s) 0 0.2 0.4 0.6 0.8 1.0 μ to the two ferromagnetic states æ and æ. Applying dynamics stops when dU?kBT. Because dU is increasing with time, experiment that permits us to distinguish between these two cases, by ( 76 j"""""""" j######## states: a classical mixed state in the former andTA a coherent superposi-TA • t/t strong but opposing biases, h 562J, to the ends of the chain intro- this freezing out happens at t

a +h B h a 1 J t Model 0.8 22 mK Measured 0.8 ↑↑↑↑↑↑↑↓ Look at eight qubits P 0.6 50 mK Early • b Domain 0.6 12h –h 0.4 t wall B Frustrated Ising spin chain 20 40 60 80 100

↑↑↑↑↑↑↑↓ Temperature (mK) • P 0.4 Same results 90 mK 2h 0.2 LETTER RESEARCH • t 0

a +h Figure 4 | Eight-qubit ferromagnetic chain. a, Chainofeight ferromagnetically 0 B h a 1 J coupled qubits with uniform couplingt coefficient Ji,i11 52J , 0fori 5 1, 2, …, 60 70 80 90 Model 100 7. The two end qubits are biased in opposite directions with hB 562J,suchthata Delay0.8 time (μs) domain wall has to form within the chain. All middle qubits are biased with a 22 mK Measured target h 5 0.1J. The configuration depicted is an excited state. The faint grey t 0.8 b 82

arrows indicate the spin biases hi. b,Effectiveenergyofthespinstate ↑↑↑↑↑↑↑↓

P 0.6 corresponding to there being a domain wall at each position along the chain at 80 finite ht. The ground state is the rightmost site.

78 Early b 50 mK Domain –h 0.6 0.4 12ht between these two state distributions (Fig. 5a). As before,B we are able to determine the dominantwall mechanism (thermal or quantum annealing) 76

s) 20 40 60 80 100 μ

by measuring the T dependence of tfreeze. ( ↑↑↑↑↑↑↑↓ 74 Temperature (mK) P

A summary of the experimental results for the eight-qubit chain is freeze

0.4 t shown in Fig. 5b. As with the single-qubit case, the experimentally 72 determined tfreeze values show saturation at low T and a crossover to 70 90 mK near-linear T dependence for T>45 mK. In this case, the classical Quantum model 2-level Quantum model 4-level 2ht model treats the fluxes of all eight qubits as coordinates of a discrete0.2 68 particle in a sixteen-dimensional potential. The classical model does Classical model Measurement 0 not capture the behaviour observed at low T. However, the quantum 66 Figure 4 | Eight-qubit ferromagneticmodels quantitatively chain. a, Chainofeight agree with ferromagnetically the experimental results for 20 40 60 80 100 T=50 mK. At higher temperatures, the classical model and the0 Temperature (mK) coupled qubits with uniform coupling coefficient J 52J , 0fori 5 1, 2, …, 60 70 80 90 100 four-level quantum modeli,i1 are1 both in qualitative agreement with the 56 Figure 5 | Results for the eight-qubit ferromagnetic chain. a, Measured final 7. The two end qubits are biasedexperimental in opposite results. directions with hB 2J,suchthata Delay time (μs) ground-state probability, P , in the eight-qubit chain versus td for domain wall has to form withinThe the saturation chain. All of middlet at qubits low T for are the biased single-spin with a and eight-spin """""""# freeze ht 5 0.1J and T 5 22 mK (blue),Graham 50 mK (green)Norris and 90 2016-04-29mK (red). The solid lines 23 target ht 5 0.1J. The configuration depicted is an excited state. The faint grey systems is a clear signature of quantum annealing. It cannotb be82 are the result of fits used to extract the freeze-out time, tfreeze. Inset, measured arrows indicate the spin biasesexplainedhi. b,Effectiveenergyofthespinstate by an experimental failure to reach lower T,asP , and simulated (four-level quantum model) T dependence of P for """""""# """""""# corresponding to there beingfor atd domain=tfreeze, wall follows at each its expected position temperature along the chain dependence at at low T80 td < 0. b, Measured tfreeze versus T (red points). We also show simulated plots of finite ht. The ground state is(Fig. the 5a, rightmost inset). Nor site. can it be explained by classical thermal activation tfreeze from two-level (dashed blue) and four-level (solid blue) quantum mechanical models and from a classical model of the qubits (black). Error bars, processes, because for these lowering T would always decrease the rate78 1s s.e. between these two stateof distributions thermal activation. (Fig. 5a). This As means before, that, we classically, are able freeze-out to should happen earlier in the evolution, where the barrier is smaller, that is,76 determine the dominantsaturation mechanism is not (thermal possible. This orquantum qualitative argument annealing) is independent of universal quantum computer15, by adding a new type of coupler s)

μ between the qubits, universal quantum computation would become by measuring the T dependencethe detailed of modeltfreeze. used to describe classical dynamics. The low- ( 74 32 temperature behaviour of tfreeze in this system of eight coupled artificial possible .

A summary of the experimental results for the eight-qubit chain is freeze shown in Fig. 5b. As withspins the cannot single-qubit be explained case, by the thermal experimentally activation but is naturallyt 72 explained by quantum tunnelling. This measurement and its result METHODS SUMMARY determined tfreeze valuesare show reminiscent saturation of the atT-dependent low T and escape a crossover rate measurements to in the Sample fabrication. We fabricated samples in a four-niobium-layer supercon- 70 ducting integrated circuit process using a standard Nb/AlO /Nb trilayer, a TiPt near-linear T dependencepioneering for T> works45 mK. on macroscopic In this case, quantum the classical tunnelling30,31, which Quantum modelx 2-level resistor layer and planarized SiO2 dielectricQuantum layers applied model by 4-level plasma-enhanced model treats the fluxesdemonstrated of all eight qubits a clear as signature coordinates of quantum of a discrete tunnelling in current-68 chemical vapour deposition. Design rulesClassical included 0.25-modelmm lines and spaces for particle in a sixteen-dimensionalbiased Josephson potential. junctions. The classical model does wiring layers and a minimum junction diameter of 0.6 mm. Circuit details are Measurement not capture the behaviourThis observed brings us at to low ourT main. However, conclusion: the a quantum programmable artificial66 discussed in ref. 13. spin system manufactured as an integrated circuit can be used to Thermometry. We measured the effective device temperature attained during models quantitativelyimplement agree with a quantum the experimental algorithm. The experiments results for presented here 20these experiments40 in two ways. The60 first is based on80 analysis of the 100 single-qubit T=50 mK. At higher temperatures, the classical model and the macroscopic resonant tunnelling.Temperature The second (mK is based) on measurement of P versus constitute a step between understanding single-qubit annealing and " four-level quantum modelunderstanding are both in the qualitative multi-qubit agreement processes that with could the be used to find W1x at equilibrium and at a fixed barrier height. Both methods are described in ref. Figure 5 Results for the eight-qubit ferromagnetic chain. a, Measured final low-energy configurations in a realistic adiabatic quantum processor.| 33 and in Supplementary Information. Both measurements generally agreed with experimental results. the ruthenium oxide thermometer on the dilution refrigerator mixing chamber to In addition to its problem-solving potential, a system such asground-state this also probability, P , in the eight-qubit chain versus td for The saturation of t at low T for the single-spin and eight-spin within a few millikelvin"""""""# over the range of temperatures used in the experiment. freeze ht 5 0.1J and T 5 22 mK (blue), 50 mK (green) and 90 mK (red). The solid lines systems is a clear signatureprovides of an interestingquantum test annealing. bed for investigating It cannot the be physics of inter- Confirmation of the thermometry comes from the agreement between the acting quantum spins, and is an important step in an ongoingare investi- the result of fits used to extract the freeze-out time, tfreeze. Inset, measured measured T dependence of P:(td=tfreeze) and that predicted by the four-level explained by an experimentalgation into failure much tomore reach complex lower spinT systems,asP realized, usingand this simulated type quantum (four-level model quantum (insets of Fig. model) 3a andT Fig.dependence 5a). This is discussed of P furtherfor in Sup- """""""# """""""# for td=tfreeze, follows itsof expectedarchitecture. temperature Although our dependence manufactured at spin low systemT istd not< 0. yetb, a Measuredplementarytfreeze Information.versus T (red points). We also show simulated plots of (Fig. 5a, inset). Nor can it be explained by classical thermal activation tfreeze from two-level (dashed blue) and four-level (solid blue) quantum mechanical models and from a classical12 MAYmodel 2011 of the | VOL qubits 473 (black). | NATURE Error |bars, 197 processes, because for these lowering T would always decrease©2011 theMacmillan rate Publishers Limited. All rights reserved of thermal activation. This means that, classically, freeze-out should 1s s.e. happen earlier in the evolution, where the barrier is smaller, that is, saturation is not possible. This qualitative argument is independent of universal quantum computer15, by adding a new type of coupler the detailed model used to describe classical dynamics. The low- between the qubits, universal quantum computation would become 32 temperature behaviour of tfreeze in this system of eight coupled artificial possible . spins cannot be explained by thermal activation but is naturally explained by quantum tunnelling. This measurement and its result METHODS SUMMARY are reminiscent of the T-dependent escape rate measurements in the Sample fabrication. We fabricated samples in a four-niobium-layer superconducting integrated circuit process using a standard Nb/AlO /Nb trilayer, a TiPt pioneering works on macroscopic quantum tunnelling30,31, which x resistor layer and planarized SiO2 dielectric layers applied by plasma-enhanced demonstrated a clear signature of quantum tunnelling in current- chemical vapour deposition. Design rules included 0.25-mm lines and spaces for biased Josephson junctions. wiring layers and a minimum junction diameter of 0.6 mm. Circuit details are This brings us to our main conclusion: a programmable artificial discussed in ref. 13. spin system manufactured as an integrated circuit can be used to Thermometry. We measured the effective device temperature attained during implement a quantum algorithm. The experiments presented here these experiments in two ways. The first is based on analysis of the single-qubit macroscopic resonant tunnelling. The second is based on measurement of P versus constitute a step between understanding single-qubit annealing and " understanding the multi-qubit processes that could be used to find W1x at equilibrium and at a fixed barrier height. Both methods are described in ref. low-energy configurations in a realistic adiabatic quantum processor. 33 and in Supplementary Information. Both measurements generally agreed with the ruthenium oxide thermometer on the dilution refrigerator mixing chamber to In addition to its problem-solving potential, a system such as this also within a few millikelvin over the range of temperatures used in the experiment. provides an interesting test bed for investigating the physics of inter- Confirmation of the thermometry comes from the agreement between the acting quantum spins, and is an important step in an ongoing investi- measured T dependence of P:(td=tfreeze) and that predicted by the four-level gation into much more complex spin systems realized using this type quantum model (insets of Fig. 3a and Fig. 5a). This is discussed further in Sup- of architecture. Although our manufactured spin system is not yet a plementary Information.

a 500 b 500 DW SQA 400 400

300 300

200 200

Number of instances 100 Number of instances 100

0 0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Success probability Success probability c 500 d 500 SA SD 400 400

300 300

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Number of instances 100 Number of instances 100

00 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Success probability Success probability

Figure 1 | Success probability distributions. Shown are histograms of the success probabilities of ﬁnding the ground states for N 108 qubits and 1000 Histograms of solution success probabilities of D-Wave,= Simulated dierent spin glass instances. We ﬁnd similar bimodal distributions for the D-Wave results (DW, a) and the simulated quantum annealer (SQA, b), and somewhat similar distributions for spin dynamics (SD, d). The unimodal distribution for the simulated annealer (SA, c) does not match. Error bars indicate 1 statisticalQuantum errors, but doAnnealing, not include systematic errors Simulated due to uncertainties inAnnealing, the annealing schedule and and calibration Spin errors of Dynamics the D-Wave device.

Quantum annealing at positive temperature T starts in the fields h 1). For each of these instances, we performed M 1,000 i =± = limit of a strong transverse field and weak problem Hamiltonian, annealing runs and determinedGraham whether Norris the system 2016-04-29 reached the 24 A(0) max(k T, B(0)), with the system state close to the ground ground state. B state of the transverse field term, the equal superposition state (in the Our strategy is to discover the operating mechanism of the D- computational basis) of all N qubits. Monotonically decreasing A(t) Wave device (DW) by comparing the performance of the device on and increasing B(t), the system evolves towards the ground state of many instances to three models: simulated classical annealing (SA), the problem Hamiltonian, with B(t ) A(t ). simulated quantum annealing (SQA) and classical spin dynamics f f Unlike adiabatic quantum computing15, which has a similar (SD). All three models are described in detail in the Supplementary schedule but assumes fully coherent adiabatic ground state Information. We gain insight into the workings of the device by evolution at zero temperature, quantum annealing4–6,10 is a positive investigating the success probabilities of finding ground states for temperature method involving an open quantum system coupled to dierent instances, and by studying the correlations in these success a thermal bath. Nevertheless, one expects that, similar to adiabatic probabilities between the D-Wave device and the models. quantum computing, small gaps to excited states may thwart finding To start, we counted for each instance the number of runs MGS the ground state. In hard optimization problems, the smallest gaps in which the ground state was reached, to determine the success of avoided level crossings have been found to close exponentially fast probability as s M /M. Plots of the distribution of s over many = GS with increasing problem size16–18, suggesting an exponential scaling dierent instances are shown in Fig. 1, where we see that the DW of the required annealing time tf with problem size N. results match well with SQA, moderately with SD, and poorly with SA. We find a unimodal distribution for the simulated annealer Results model for all schedules, temperatures and annealing times we tried, We performed our tests on a D-Wave One device, comprised of with a peak position that moves to the right as one increases tf superconducting flux qubits with programmable couplings (see (see Supplementary Methods). In contrast, the D-Wave device, the Methods). Of the 128 qubits on the device, 108 were fully simulated quantum annealer and the spin dynamics model exhibit a functioning and were used in our tests. The ‘chimera’ connectivity bimodal distribution, with a clear split into easy and hard instances. graph of these qubits is shown in Supplementary Fig. 1. Instead of Moderately increasing tf in the simulated quantum annealer makes trying to map a specific optimization problem to the connectivity the bimodal distribution more pronounced, as does lowering the graph of the chip19,20, we chose random spin glass problems that can temperature (see Supplementary Methods). be directly realized. For each coupler Jij in the device we randomly A much greater insight is obtained from Fig. 2, which plots assigned a value of either 1 or 1, giving rise to a rough energy the correlation of the success probabilities between the DW data + landscape. Local fields h 0 give a bias to individual spins, tending and the other models. As a reference for the best correlations i 6= to make the problem easier to solve for annealers. We thus set all we may expect, we show in Fig. 2a the correlations between two h 0 for most data shown here and provide data with local fields dierent gauges of the same problem on the device (Methods and i = in the Supplementary Methods. We performed tests for problems of Supplementary Methods): as a result of calibration errors no better sizes ranging from N 8 to N 108. For each problem size N,we correlations than the device with itself can be expected. Figure 2b = = selected 1,000 dierent instances by choosing 1,000 sets of dierent shows a scatter plot of the hardness of instances for the simulated random couplings J 1 (and for some of the data also random quantum annealer and the D-Wave device for a single gauge. The ij =±

NATURE PHYSICS | VOL 10 | MARCH 2014 | www.nature.com/naturephysics 219 ǟɥƐƎƏƓɥ !,(++ -ɥ4 +(2'#12ɥ (,(3#"ƥɥ++ɥ1(%'32ɥ1#2#15#"ƥɥ Future Directions

Possibility of optimizing the problem layout for the chimera • connectivity Increase number of qubits • Expand possible Hamiltonians (Adiabatic quantum optimization • for general Hamiltonians is universal, maps to GMQC)

Graham Norris 2016-04-29 25 Conclusions

D-Wave has built a functioning quantum annealer • Scalable to large numbers of qubits • Extremely well–engineered qubits and infrastructure • Speedup has not been robustly demonstrated • Questions about how quantum the device is (may be thermal • annealer for certain problems)

Graham Norris 2016-04-29 26 References

S. Boixo et. al., Evidence for quantum annealing with more • than one hundred qubits, Nature Physics 10, 218-224 (2014) V. Choi, Minor–embedding in adiabatic quantum computation: I. • The parameter setting problem, Journal of Quantum Information Processing 7, 193-209 (2008) R. Harris et. al., Experimental investigation of an eight–qubit • unit cell in a superconducting optimization processor, Physical Review B 82, 024511 (2010) R. Harris et. al., Experimental demonstration of a robust and • scalable ﬂux qubit, Physical Review B 81, 134510 (2010) M. W. Johnson et. al., Quantum annealing with manufactured • spins, Nature 473, 194-198 (2011)

Graham Norris 2016-04-29 27 Basis EXPERIMENTAL DEMONSTRATION OF A ROBUST AND… PHYSICAL REVIEW B 81, 134510 ͑2010͒

25 ﬁelds of quantum magnetism17 and optimization theory.18 In- |eiterestingly,and |gi are systems energy of coupled ﬂux qubits that are operated 20 • in this mode bear considerable resemblance to Feynman’s eigenstates of the 19 ) original vision of how to build a quantum computer. 15 Hamiltonian

GHz While much seminal work has been done on single junc- (

h e / 10 | ⟩ Longtion coherence and the related three-Josephson-junction rf-SQUID flux g • qubit,20–31 it has been recognized that such devices would be | ⟩ times, but not ideal for Energy 5 impractical in a large-scale quantum information processor qubitas their interactions properties are exceptionally sensitive to fabrication 0 variations. In particular, in the regime E ӷប␻ , ⌬ ϰexp Use persistent current J1 p q ͑ |↓⟩ |↑⟩ −ប␻ E . Thus, it would be unrealistic to expect a large- −5 −2.5 0 2.5 5 • p / J1͒ (Φq Φ0/2)/Lq (µA) basisscale processor involving a multitude of such devices to − yield from even the best superconducting fabrication facility. FIG. 2. ͑Color online͒ Depiction of the two lowest-lying states As such, these devices cannot be viewed as being robust. of an rf SQUID at degeneracy ͑⑀=0͒ with nomenclature for the Moreover, implementation of AQO requires the ability to energy basis ͉͑g͘,͉e͒͘ and flux basis ͉͑ ͘,͉ ͒͘ as indicated. actively tune ⌬ from being the dominant energy scale in the ↓ ↑ Graham Norris 2016-04-29q 28 representation of these states, which is frequently referred to qubit to being essentially negligible during the course of a as either the flux or persistent current basis, can be formed by computation. Thus the single-junction rf-SQUID flux qubit is taking the symmetric and antisymmetric combinations of the of limited practical utility and has passed out of favor as a energy eigenstates: ͉ ͘=͉͑g͘+͉e͒͘/ͱ2 and ͉ ͘=͉͑g͘ prototype qubit. −͉e͒͘/ͱ2, which yield two↓ roughly Gaussian-shaped↑ wave The next step in the evolution of the single-junction flux functions that are centered about each of the wells shown in qubit and related variants was the compound Josephson- Fig. 2. The magnitude of the persistent current used in Eq. junction ͑CJJ͒ rf SQUID, as depicted in Fig. 1͑b͒. This de- p vice was first reported upon by Han et al.32 and was the first ͑3͒ is then defined by ͉Iq͉ϵ͉͗ ͉͑⌽q −⌽0 /2͒/Lq͉ ͉͘. The tun- ↑ ↑ type of flux qubit to display signatures of quantum superpo- neling energy is given by ⌬q =͗e͉ q͉e͘−͗g͉ q͉g͘. The aforementioned dual representationH ofH the states of a sition of macroscopic states.33 The CJJ rf SQUID has been flux qubit allows two distinct modes of operation of the flux used by other research groups13,34,35 and a related four- qubit as a binary logical element with a logical basis defined Josephson-junction device has been proposed.20,21 The CJJ by the states ͉0͘ and ͉1͘. In the first mode, the logical basis is rf-SQUID flux qubit and related variants have reappeared in 14,36,37 mapped onto the energy eigenbasis: ͉0͘ ͉g͘ and ͉1͘ ͉e͘. a gradiometric configuration in more recent history. This mode is useful for optimizing the→ coherence times→ of Here, the single junction of Fig. 1͑a͒ has been replaced by a flux qubits as the dispersion of Hamiltonian ͑3͒ is flat as a flux biased dc SQUID of inductance Lcjj that allows one to x tune the critical current of the rf SQUID in situ. Let the function of ⌽q to first order for ⑀Ϸ0, thus providing some 10 x protection from the effects of low-frequency flux noise. applied flux threading this structure be denoted by ⌽cjj. It is However, this is not a convenient mode of operation for shown in Appendix A that the Hamiltonian for this system implementing interactions between flux qubits.11,12 In the can be written as second mode, the logical basis is mapped onto the persistent 2 x 2 Q ͑␸n − ␸ ͒ current basis: ͉0͘ ͉ ͘ and ͉1͘ ͉ ͘. This mode of opera- = n + U n − U ␤ cos͑␸ − ␸0͒, → ↓ → ↑ H ͚ ͫ 2C n 2 ͬ q eff q q tion facilitates the implementation of interqubit interactions n n via inductive couplings, but does so at the expense of coher- ͑4a͒ ence times. GMQC schemes exist that attempt to leverage 13–15 the benefits of both of the above modes of operation. On where the sum is over n෈͕q,cjj͖, Cq ϵC1 +C2,1/Ccjj the other hand, those interested in implementing AQO ϵ1/C1 +1/C2, and Lq ϵLbody+Lcjj/4. The two-dimensional strictly use the second mode of operation cited above. This, potential energy in Hamiltonian ͑4a͒ is characterized by very naturally, leads to some interesting properties. First and 2 ␸cjj ␤− foremost, in the coherent regime at ⑀=0, the ground state ␤ = ␤ cos 1+ tan͑␸ /2͒ , ͑4b͒ eff + ͩ 2 ͪͱ ͫ ␤ cjj ͬ maps onto ͉g͘=͉͑0͘+͉1͒͘/ͱ2, which implies that it is a su- + perposition state with a fixed phase between components in the logical basis. Second, the logical basis is not coincident ⌽0 ␤ ␸0 ␲ q − ␸ with the energy eigenbasis, except in the extreme limit q ϵ 2 = − arctanͩ tan͑ cjj/2͒ͪ, ͑4c͒ ⌽0 ␤+ ͉⑀͉ /⌬q ӷ1. As such, the qubit should not be viewed as an otherwise free spin- 1 in a magnetic field, rather it maps onto 2 ␤ ϵ 2␲L ͑I Ϯ I ͒/⌽ . ͑4d͒ an Ising spin subjected to a magnetic field with both a lon- Ϯ q 1 2 0 16 gitudinal ͑B ⑀͒ and a transverse ͑B ⌬ ͒ component. Note that if cos͑␸cjj/2͒Ͻ0, then ␤effϽ0 in Hamiltonian ͑4a͒. z → x → q In this case, it is the competition between ⑀ and ⌬q which This feature provides a natural means of shifting the qubit x dictates the relative amplitudes of ͉ ͘ and ͉ ͘ in the ground- degeneracy point from ␸q =␲, as in the single-junction rf- ↓ ↑ x state wave function ͉g͘, thereby enabling logical operations SQUID case, to ␸q Ϸ0. It has been assumed in all that fol- that make no explicit use of the excited state ͉e͘. This latter lows that this ␲-shifted mode of operation of the CJJ rf mode of operation of the flux qubit has connections to the SQUID has been invoked.

134510-3 EXPERIMENTAL INVESTIGATION OF ANTime EIGHT-QUBIT Trace… and ControlPHYSICAL Signals REVIEW B 82, 024511 ͑2010͒

1000 -Φ0/2 m 999 Φccjj x Φccjj(t) b 998 Φccjj

Number Correct -Φ 997 0 ubits 0 50 100 150 200 250 (a) State Q 0.8 Data Ig(t) 0.6 Poisson Stat. 0 } 0.4

0.2 Probability -Φ0/2 x 0 Φlatch(t) 0 1 2 3 4 5 FPs (b) Number of Errors -Φ0 } Q

FIG. 7. ͑Color online͒ Verification of readout fidelity for the ~Φ0/3 x eight-qubit unit cell. ͑a͒ Example results from preparing the eight Φro(t) qubits into each of the 28 =256 possible final spin configurations 0 UIDs Q and recording the number of correct reads for 1000 repetitions. ͑b͒ Probability of observing a given number of errors within 1000 rep- iro(t)} dc S 0 etitions, as taken from the data in ͑a͒. Data have been fit to a 100 µs Poisson distribution with mean number of errors ␭=0.36. t=0 trep Annealing Readout binary word ͉sជ͘ into a decimal number ͉State͘, where FIG. 8. ͑Color online͒ Depiction of the measurement wave form 7 sequence. Only time-dependent biases have been shown. All qubitsGraham Norris 2016-04-29 29 si +1 7−i x State = 2 . are subjected to common ⌽ccjj͑t͒ and Ig͑t͒. All QFPs are subjected ͚ 2 x x i=0 to a common ⌽latch͑t͒. Example dc-SQUID flux ⌽ro and current i ͑t͒ wave forms shown only for device connected to q . In the majority of instances, the readout yielded the expected ro 0 result 1000 times. We never observed more than three errors dings about the center of the unit cell. The result was a test per 1000 repetitions. Multiple iterations of this check reset of 6400 problem instances, each requiring a unique em- vealed that results for those states that appear to be less than bedding, that were then posed to the hardware. Note that perfect in Fig. 7͑a͒, such as State=143, were not reproduc- while other researchers have chosen to focus upon particular ible. Further analysis indicated that these infrequent errors classes of Ising spin problems, such as random 3-SAT,40 were not due to problems in preparing ͉sជ͘. Rather, the errors MAX-clique,41 and exact cover,2,42 we have found our ran- were most likely generated by the statistical nature of the dom ISG problem set to be of great use as it encompasses a dc-SQUID switching times.30 The probability of observing a broad range of problem classes that could be posed to an given number of errors within 1000 repetitions, as obtained AQO processor in practice. by taking a histogram of the data in Fig. 7͑a͒, is shown in The PMM programming required to set up the processor Fig. 7͑b͒. These results are well described by Poisson statis- was rather involved and the details of its implementation are tics with a mean number of errors ␭=0.36 in 1000 repeti- beyond the scope of this paper.31 Nonetheless, it is worth tions. We defined the threshold for a number of counts to be recognizing that, for each problem instance, the interqubit statistically significant as three times the square root of the x p x coupler biases ⌽co,ij and ͉Iq͉-compensator biases ⌽I ,i were variance, 3ͱ␭Ϸ2. One can then deem the fidelity of our p reset to provide the desired Kij and hi, respectively. As a readout of the eight-qubit unit cell in these particular experi- consequence of the changing interqubit coupler susceptibil- ments to have been 1−3ͱ␭/1000=0.998. ity, the L-tuner biases also had to be updated to hold the qubit inductance constant, as described in Ref. 28. Finally, IV. EXPERIMENTAL PROCESSOR PERFORMANCE the effects of junction asymmetry in the CJJ interqubit cou- p plers and ͉Iq͉ compensators led to a small offset of each To test the performance of the eight-qubit unit cell, we 0 qubit’s degeneracy point ␦⌽i , as shown in Ref. 29. These generated a set of 800 unique random ISG problem instances offsets were independently calibrated for each qubit prior to whose hi and Kij were drawn from a distribution of 15 evenly running the processor. To compensate, the PMM controlling spaced values within the set 0 the qubit-flux offset ⌽i was updated. With the details concerning the PMM aside, the analog h , K ෈ ͓− 1 − 6/7...+6/7+1͔. ͑9͒ i ij control signal sequence for implementing the AQO algorithm We then took advantage of the symmetry of the physical introduced in Sec. II was quite simple, as depicted in Fig. 8. layout of the qubits and couplers shown in Fig. 2͑b͒ and During the annealing phase, there were only two time- generated eight permutations of each of the 800 progenitor dependent signals applied to the entire unit cell: the CCJJ x p problems by rotating and reflecting their hi and Kij embed- control signal ⌽ccjj͑t͒ and the ͉Iq͉-compensation signal Ig͑t͒.

024511-7 Phase and Flux

Second Josephson equation ∂δ V = ~ 2e ∂t Faraday’s law of induction ∂Φ E = − ∂t So, we get Φ ϕ = 2π Φ0

Graham Norris 2016-04-29 30