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## Quantum Gs

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The term contact resistance refers to the contribution to the total resistance of a system which can be attributed to the contacting interfaces of electrical leads and connections as opposed to the intrinsic resistance. This effect is described by the term electrical contact resistance (ECR) and arises as the result of the limited areas of true contact at an interface and the presence of resistive surface films or oxide layers. ECR may vary with time, most often decreasing, in a process known as resistance creep. The idea of potential drop on the injection electrode was introduced by William Shockley[1] to explain the difference between the experimental results and the model of gradual channel approximation. In addition to the term ECR, interface resistance, transitional resistance, or just simply correction term are also used. The term parasitic resistance is used as a more general term, of which it is usually assumed that contact resistance is a major component.

Sketch of the contact resistance estimation by the transmission line method.

## Experimental characterization

Here we need to distinguish the contact resistance evaluation in two-electrode systems (e.g. diodes) and three-electrode systems (e.g. transistors).

For two electrode systems the specific contact resistivity is experimentally defined as the slope of the I-V curve at V = 0:

${displaystyle r_{c}=left{{frac {partial V}{partial J}}right}_{V=0}}$

where J is the current density, or current per area. The units of specific contact resistivity are typically therefore in ohms-square meter, or ${displaystyle Omega cdot {text{cm}}^{2}}$. When the current is a linear function of the voltage, the device is said to have ohmic contacts.

The resistance of contacts can be crudely estimated by comparing the results of a four terminal measurement to a simple two-lead measurement made with an ohmmeter. In a two-lead experiment, the measurement current causes a potential drop across both the test leads and the contacts so that the resistance of these elements is inseparable from the resistance of the actual device, with which they are in series. In a four-point probe measurement, one pair of leads is used to inject the measurement current while a second pair of leads, in parallel with the first, is used to measure the potential drop across the device. In the four-probe case, there is no potential drop across the voltage measurement leads so the contact resistance drop is not included. The difference between resistance derived from two-lead and four-lead methods is a reasonably accurate measurement of contact resistance assuming that the leads resistance is much smaller. Specific contact resistance can be obtained by multiplying by contact area. It should also be noted that the contact resistance may vary with temperature.

Inductive and capacitive methods could be used in principle to measure an intrinsic impedance without the complication of contact resistance. In practice, direct current methods are more typically used to determine resistance.

The three electrode systems such as transistors require more complicated methods for the contact resistance approximation. The most common approach is the transmission line model (TLM). Here, the total device resistance ${displaystyle R_{text{tot}}}$ is plotted as a function of the channel length:

${displaystyle R_{text{tot}}=R_{text{c}}+R_{text{ch}}=R_{text{c}}+{frac {L}{WCmu left(V_{text{gs}}-V_{text{ds}}right)}}}$

where ${displaystyle R_{text{c}}}$ and ${displaystyle R_{text{ch}}}$ are contact and channel resistances, respectively, ${displaystyle L/W}$ is the channel length/width, ${displaystyle C}$ is gate insulator capacitance (per unit of area), ${displaystyle mu }$ is carrier mobility, and ${displaystyle V_{text{gs}}}$ and ${displaystyle V_{text{ds}}}$ are gate-source and drain-source voltages. Therefore, the linear extrapolation of total resistance to the zero channel length provides the contact resistance. The slope of the linear function is related to the channel transconductance and can be used for estimation of the ”contact resistance-free” carrier mobility. The approximations used here (linear potential drop across the channel region, constant contact resistance, …) lead sometimes to the channel dependent contact resistance.[2]

Beside the TLM it was proposed the gated four-probe measurement[3] and the modified time-of-flight method (TOF).[4] The direct methods able to measure potential drop on the injection electrode directly are the Kelvin probe force microscopy (KFM)[5] and the electric-field induced second harmonic generation.[6]

In the semiconductor industry, Cross-Bridge Kelvin Resistor(CBKR) structures are the mostly used test structures to characterize metal-semiconductor contacts in the Planar devices of VLSI technology. During the measurement process, force the current (I) between contact 1&2 and measure the potential deference between contacts 3&4. The contact resistance Rk can be then calcualted as ${displaystyle Rk=V34/I}$ .[7]

## Mechanisms

For given physical and mechanical material properties, parameters that govern the magnitude of electrical contact resistance (ECR) and its variation at an interface relate primarily to surface structure and applied load (Contact mechanics).[8] Surfaces of metallic contacts generally exhibit an external layer of oxide material and adsorbed water molecules, which lead to capacitor-type junctions at weakly contacting asperities and resistor type contacts at strongly contacting asperities, where sufficient pressure is applied for asperities to penetrate the oxide layer, forming metal-to-metal contact patches. If a contact patch is sufficiently small, with dimensions comparable or smaller than the mean free path of electrons resistance at the patch can be described by the Sharvin mechanism, whereby electron transport can be described by ballistic conduction. Generally, over time, contact patches expand and the contact resistance at an interface relaxes, particularly at weakly contacting surfaces, through current induced welding and dielectric breakdown. This process is known also as resistance creep.[9] The coupling of surface chemistry, contact mechanics and charge transport mechanisms needs to be considered in the mechanistic evaluation of ECR phenomena.

## Quantum limit

When a conductor has spatial dimensions close to ${displaystyle 2pi /k_{text{F}}}$, where ${displaystyle k_{text{F}}}$ is Fermi wavevector of the conducting material, Ohm's law does not hold anymore. These small devices are called quantum point contacts. Their conductance must be an integer multiple of the value ${displaystyle 2e^{2}/h}$, where ${displaystyle e}$ is the elementary charge and ${displaystyle h}$ is Planck's constant. Quantum point contacts behave more like waveguides than the classical wires of everyday life and may be described by the Landauer scattering formalism.[10] Point-contact tunneling is an important technique for characterizing superconductors.

## Other forms of contact resistance

Measurements of thermal conductivity are also subject to contact resistance, with particular significance in heat transport through granular media. Similarly, a drop in hydrostatic pressure (analogous to electrical voltage) occurs when fluid flow transitions from one channel to another.

## Significance

Bad contacts are the cause of failure or poor performance in a wide variety of electrical devices. For example, corroded jumper cable clamps can frustrate attempts to start a vehicle that has a low battery. Dirty or corroded contacts on a fuse or its holder can give the false impression that the fuse is blown. A sufficiently high contact resistance can cause substantial heating in a high current device. Unpredictable or noisy contacts are a major cause of the failure of electrical equipment.

## References

1. ^Shockley, William (September 1964). 'Research and investigation of inverse epitaxial UHF power transistors'. Report No. A1-TOR-64-207.Cite journal requires ` journal=` (help)
2. ^Weis, Martin; Lin, Jack; Taguchi, Dai; Manaka, Takaaki; Iwamoto, Mitsumasa (2010). 'Insight into the contact resistance problem by direct probing of the potential drop in organic field-effect transistors'. Applied Physics Letters. 97 (26): 263304. Bibcode:2010ApPhL.97z3304W. doi:10.1063/1.3533020.
3. ^Pesavento, Paul V.; Chesterfield, Reid J.; Newman, Christopher R.; Frisbie, C. Daniel (2004). 'Gated four-probe measurements on pentacene thin-film transistors: Contact resistance as a function of gate voltage and temperature'. Journal of Applied Physics. 96 (12): 7312. Bibcode:2004JAP..96.7312P. doi:10.1063/1.1806533.
4. ^Weis, Martin; Lin, Jack; Taguchi, Dai; Manaka, Takaaki; Iwamoto, Mitsumasa (2009). 'Analysis of Transient Currents in Organic Field Effect Transistor: The Time-of-Flight Method'. Journal of Physical Chemistry C. 113 (43): 18459. doi:10.1021/jp908381b.
5. ^Bürgi, L.; Sirringhaus, H.; Friend, R. H. (2002). 'Noncontact potentiometry of polymer field-effect transistors'. Applied Physics Letters. 80 (16): 2913. Bibcode:2002ApPhL.80.2913B. doi:10.1063/1.1470702.
6. ^Nakao, Motoharu; Manaka, Takaaki; Weis, Martin; Lim, Eunju; Iwamoto, Mitsumasa (2009). 'Probing carrier injection into pentacene field effect transistor by time-resolved microscopic optical second harmonic generation measurement'. Journal of Applied Physics. 106 (1): 014511–014511–5. Bibcode:2009JAP..106a4511N. doi:10.1063/1.3168434.
7. ^Stavitski, Natalie; Klootwijk, Johan H.; van Zeijl, Henk W.; Kovalgin, Alexey Y.; Wolters, Rob A. M. (February 2009). 'Cross-Bridge Kelvin Resistor Structures for Reliable Measurement of Low Contact Resistances and Contact Interface Characterization'. IEEE Transactions on Semiconductor Manufacturing. 22 (1): 146–152. doi:10.1109/TSM.2008.2010746. ISSN0894-6507. S2CID111829.
8. ^Zhai, Chongpu; Hanaor, Dorian; Proust, Gwénaëlle; Brassart, Laurence; Gan, Yixiang (December 2016). 'Interfacial electro-mechanical behaviour at rough surfaces'(PDF). Extreme Mechanics Letters. 9 (3): 422–429. doi:10.1016/j.eml.2016.03.021.
9. ^Zhai, Chongpu; Hanaor, Dorian A. H.; Proust, Gwenaelle; Gan, Yixiang (2015). 'Stress-Dependent Electrical Contact Resistance at Fractal Rough Surfaces'. Journal of Engineering Mechanics. 143 (3): B4015001. doi:10.1061/(ASCE)EM.1943-7889.0000967.
10. ^Landauer, Rolf (August 1976). 'Spatial carrier density modulation effects in metallic conductivity'. Physical Review B. 14 (4): 1474–1479. Bibcode:1976PhRvB.14.1474L. doi:10.1103/PhysRevB.14.1474.

• Pitney, Kenneth E. (2014) [1973]. Ney Contact Manual - Electrical Contacts for Low Energy Uses (reprint of 1st ed.). Deringer-Ney, originally JM Ney Co. ASINB0006CB8BC.[permanent dead link] (NB. Free download after registration.)
• Slade, Paul G. (February 12, 2014) [1999]. Electrical Contacts: Principles and Applications. Electrical and Computer Engineering. Electrical engineering and electronics. 105 (2 ed.). CRC Press, Taylor & Francis, Inc.ISBN978-1-43988130-9.
• Holm, Ragnar; Holm, Else (June 29, 2013) [1967]. Williamson, J. B. P. (ed.). Electric Contacts: Theory and Application (reprint of 4th revised ed.). Springer Science & Business Media. ISBN978-3-540-03875-7. (NB. A rewrite of the earlier 'Electric Contacts Handbook'.)
• Holm, Ragnar; Holm, Else (1958). Electric Contacts Handbook (3rd completely rewritten ed.). Berlin / Göttingen / Heidelberg, Germany: Springer-Verlag. ISBN978-3-66223790-8.[1] (NB. A rewrite and translation of the earlier 'Die technische Physik der elektrischen Kontakte' (1941) in German language, which is available as reprint under ISBN978-3-662-42222-9.)
• Huck, Manfred; Walczuk, Eugeniucz; Buresch, Isabell; Weiser, Josef; Borchert, Lothar; Faber, Manfred; Bahrs, Willy; Saeger, Karl E.; Imm, Reinhard; Behrens, Volker; Heber, Jochen; Großmann, Hermann; Streuli, Max; Schuler, Peter; Heinzel, Helmut; Harmsen, Ulf; Györy, Imre; Ganz, Joachim; Horn, Jochen; Kaspar, Franz; Lindmayer, Manfred; Berger, Frank; Baujan, Guenter; Kriechel, Ralph; Wolf, Johann; Schreiner, Günter; Schröther, Gerhard; Maute, Uwe; Linnemann, Hartmut; Thar, Ralph; Möller, Wolfgang; Rieder, Werner; Kaminski, Jan; Popa, Heinz-Erich; Schneider, Karl-Heinz; Bolz, Jakob; Vermij, L.; Mayer, Ursula (2016) [1984]. Vinaricky, Eduard; Schröder, Karl-Heinz; Weiser, Josef; Keil, Albert; Merl, Wilhelm A.; Meyer, Carl-Ludwig (eds.). Elektrische Kontakte, Werkstoffe und Anwendungen: Grundlagen, Technologien, Prüfverfahren (in German) (3 ed.). Berlin / Heidelberg / New York / Tokyo: Springer-Verlag. ISBN978-3-642-45426-4.

This section explains what quantum annealing isand how it works, and introduces the underlyingquantum physics that governs its behavior. For morein-depth information on quantum annealing in D-Wave systems, seeTechnical Description of the D-Wave Quantum Processing Unit.

## Applicable Problems¶

Quantum annealing processors naturally return low-energy solutions; someapplications require the real minimum energy (optimization problems) and othersrequire good low-energy samples (probabilistic sampling problems).

Optimization problems. In an optimization problem, you search for the best ofmany possible combinations. Optimization problems include scheduling challenges,such as “Should I ship this package on this truck or the next one?” or “What isthe most efficient route a traveling salesperson should take to visit differentcities?”

Physics can help solve these sorts of problems because you can frame them as energyminimization problems. A fundamental rule of physics is that everything tends toseek a minimum energy state. Objects slide down hills; hot things cool down overtime. This behavior is also true in the world of quantum physics. Quantum annealingsimply uses quantum physics to find low-energy states of a problem and thereforethe optimal or near-optimal combination of elements.

Sampling problems. Sampling from many low-energy states and characterizingthe shape of the energy landscape is useful for machine learning problems whereyou want to build a probabilistic model of reality. The samples give you informationabout the model state for a given set of parameters, which can then be used toimprove the model.

Probabilistic models explicitly handle uncertainty by accounting for gaps inknowledge and errors in data sources. Probability distributions represent theunobserved quantities in a model (including noise effects) and how they relate tothe data. The distribution of the data is approximated based on a finite set ofsamples. The model infers from the observed data, and learning occurs as ittransforms the prior distribution, defined before observing the data, into theposterior distribution, defined afterward. If the training process is successful,the learned distribution resembles the distribution that generated the data,allowing predictions to be made on unobserved data. For example, when training onthe famous MNIST dataset of handwritten digits, such a model can generate imagesresembling handwritten digits that are consistent with the training set.

Sampling from energy-based distributions is a computationally intensive taskthat is an excellent match for the way that the D-Wave system solvesproblems; that is, by seeking low-energy states.

You can see a variety of example problems in the D-Wave Problem-Solving Handbook guide, in D-Wave’scode examples repository on GitHub, and themany user-developed early quantum applications on D-Wave systems shown on theD-Wave website.

## How Quantum Annealing Works in D-Wave Systems¶

The quantum bits—also known as qubits—are the lowest energy states ofthe superconducting loops that make up the D-Wave QPU. These states have a circulatingcurrent and a corresponding magnetic field. As with classical bits, a qubit canbe in state of 0 or 1; see Figure 4. But because the qubit isa quantum object, it can also be in a superposition of the 0 state and the 1 stateat the same time. At the end of the quantum annealing process, each qubit collapsesfrom a superposition state into either 0 or 1 (a classical state).

Fig. 4 A qubit’s state is implemented as a circulating current, shown clockwise for 0 and counter clockwise for 1,with a corresponding magnetic field.

## Image Of 2022 Thor Quantum Gs 27

The physics of this process can be shown (visualized) with an energy diagram asin Figure 5. This diagram changes over time, as shown in(a), (b), and (c). To begin, there is just one valley (a), with a single minimum.The quantum annealing process runs, the barrier is raised, and this turns theenergy diagram into what is known as a double-well potential (b). Here, thelow point of the left valley corresponds to the 0 state, and the low point of theright valley corresponds to the 1 state. The qubit ends up in one of these valleysat the end of the anneal.

Fig. 5 Energy diagram changes over time as the quantum annealing process runs and a bias is applied.

Everything else being equal, the probability of the qubit ending in the 0 or the1 state is equal (50 percent). You can, however, control the probability of itfalling into the 0 or the 1 state by applying an external magnetic field to thequbit (c). This field tilts the double-well potential, increasing the probabilityof the qubit ending up in the lower well. The programmable quantity that controlsthe external magnetic field is called a bias, and the qubit minimizes its energyin the presence of the bias.

The bias term alone is not useful, however. The real power of the qubits comeswhen you link them together so they can influence each other. This is done with adevice called a coupler. A coupler can make two qubits tend to end up in thesame state—both 0 or both 1—or it can make them tend to be in opposite states.Like a qubit bias, the correlation weights between coupled qubits can be programmedby setting a coupling strength. Together, the programmable biases and weights arethe means by which a problem is defined in the D-Wave system.

## Quantum Gis Tutorial

When you use a coupler, you are using another phenomenon of quantum physics calledentanglement. When two qubits are entangled, they can be thought of as a singleobject with four possible states. Figure 6 illustrates thisidea, showing a potential with four states, each corresponding to a differentcombination of the two qubits: (0,0), (0,1), (1,1), and (1,0). The relative energyof each state depends on the biases of qubits and the coupling between them.During the anneal, the qubit states are potentially delocalized in this landscapebefore finally settling into (1,1) at the end of the anneal.

As stated, each qubit has a bias and qubits interact via the couplers. Whenformulating a problem, users choose values for the biases and couplers. The biasesand couplings define an energy landscape, and the D-Wave quantum computer findsthe minimum energy of that landscape: this is quantum annealing.

Systems get increasingly complex as qubits are added: two qubits have fourpossible states over which to define an energy landscape; three qubits haveeight. Each additional qubit doubles the number of states over which you candefine the energy landscape: the number of states goes up exponentially with thenumber of qubits.

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In summary, the systems starts with a set of qubits, each in a superpositionstate of 0 and 1. They are not yet coupled. When they undergo quantum annealing,the couplers and biases are introduced and the qubits become entangled. At thispoint, the system is in an entangled state of many possible answers. By the endof the anneal, each qubit is in a classical state that represents the minimumenergy state of the problem, or one very close to it. All of this happens inD-Wave systems in a matter of microseconds.

## Underlying Quantum Physics¶

This section discusses some concepts essential to understandingthe quantum physics that governs the D-Wave quantum annealing process.

### The Hamiltonian and the Eigenspectrum¶

A classical Hamiltonian is a mathematical description of some physical systemin terms of its energies. You can input any particular state of the system,and the Hamiltonian returns the energy for that state. For most non-convexHamiltonians, finding the minimum energy state is an NP-hard problem thatclassical computers cannot solve efficiently.

As an example of a classical system, consider an extremely simple system of atable and an apple. This system has two possible states: the apple on the table,and the apple on the floor. The Hamiltonian tells you the energies, from which youcan discern that the state with the apple on the table has a higher energy thanthat when the apple is on the floor.

For a quantum system, a Hamiltonian is a function that maps certain states,called eigenstates, to energies. Only when the system is in an eigenstate ofthe Hamiltonian is its energy well defined and called the eigenenergy. When thesystem is in any other state, its energy is uncertain. The collection ofeigenstates with defined eigenenergies make up the eigenspectrum.

For the D-Wave system, the Hamiltonian may be represented as

[{cal H}_{ising} = underbrace{- frac{A({s})}{2} left(sum_i {hatsigma_{x}^{(i)}}right)}_text{Initial Hamiltonian} + underbrace{frac{B({s})}{2} left(sum_{i} h_i {hatsigma_{z}^{(i)}} + sum_{i>j} J_{i,j} {hatsigma_{z}^{(i)}} {hatsigma_{z}^{(j)}}right)}_text{Final Hamiltonian}]

where ({hatsigma_{x,z}^{(i)}}) are Pauli matrices operating on a qubit(q_i), and (h_i) and (J_{i,j}) are the qubit biases and couplingstrengths.[1]

 [1] Nonzero values of (h_i) and (J_{i,j}) are limited to those availablein the working graph; see the D-Wave QPU Architecture: Topologies chapter.

The Hamiltonian is the sum of two terms, the initial Hamiltonian and thefinal Hamiltonian:

• Initial Hamiltonian (first term)—The lowest-energy state of the initialHamiltonian is when all qubits are in a superposition state of 0 and 1.This term is also called the tunneling Hamiltonian.
• Final Hamiltonian (second term)—The lowest-energy state of the finalHamiltonian is the answer to the problem that you are trying to solve. The finalstate is a classical state, and includes the qubit biases and the couplingsbetween qubits. This term is also called the problem Hamiltonian.

In quantum annealing, the system begins in the lowest-energy eigenstate of theinitial Hamiltonian. As it anneals, it introduces the problem Hamiltonian, whichcontains the biases and couplers, and it reduces the influence of the initialHamiltonian. At the end of the anneal, it is in an eigenstate of the problemHamiltonian. Ideally, it has stayed in the minimum energy state throughout thequantum annealing process so that—by the end—it is in the minimum energystate of the problem Hamiltonian and therefore has an answer to the problem youwant to solve. By the end of the anneal, each qubit is a classical object.

## Quantum Gs27

### Annealing in Low-Energy States¶

A plot of the eigenenergies versus time is a useful way to visualize the quantumannealing process. The lowest energy state during the anneal—theground state—is typically shown at the bottom, and any higher excited statesare above it; see Figure 7.

Fig. 7 Eigenspectrum, where the ground state is at the bottom and the higher excited states are above.

As an anneal begins, the system starts in the lowest energy state, which is wellseparated from any other energy level. As the problem Hamiltonian is introduced,other energy levels may get closer to the ground state. The closer they get, thehigher the probability that the system will jump from the lowest energy state intoone of the excited states. There is a point during the anneal where the firstexcited state—that with the lowest energy apart from the groundstate—approaches the ground state closely and then diverges away again. Theminimum distance between the ground state and the first excited state throughoutany point in the anneal is called the minimum gap.

Certain factors may cause the system to jump from the ground state into a higherenergy state. One is thermal fluctuations that exist in any physical system.Another is running the annealing process too quickly. An annealing process thatexperiences no interference from outside energy sources and evolves theHamiltonian slowly enough is called an adiabatic process, and this is where thename adiabatic quantum computing comes from. Because no real-world computationcan run in perfect isolation, quantum annealing may be thought of as thereal-world counterpart to adiabatic quantum computing, a theoretical ideal.In reality, for some problems, the probability of staying in the ground state cansometimes be small; however, the low-energy states that are returned are stillvery useful.

For every different problem that you specify, there is a different Hamiltonianand a different corresponding eigenspectrum. The most difficult problems, in termsof quantum annealing, are generally those with the smallest minimum gaps.

### Evolution of Energy States¶

Figure 8 shows the dependence of the(A) and (B) parameters in the Hamiltonian ons, the normalized anneal fraction, an abstract parameter ranging from 0 to 1.The (A(s)) curve is the tunneling energy and the (B(s)) curve is theproblem Hamiltonian energy at (s). Both are expressed as energies in unitsof Joules as is standard for a Hamiltonian. A linear anneal sets(s = t / t_f), where (t) is time and (t_f) is the total timeof the anneal. At (t=0), (A(0) gg B(0)), which leads to the quantumground state of the system where each spin is in a delocalized combination of itsclassical states. As the system is annealed, (A) decreases and (B)increases until (t_f), when the final state of the qubits representsa low-energy solution.

At the end of the anneal, the Hamiltonian contains the only (B(s)) term.It is a classical Hamiltonian where every possible classical bitstring (that is,list of qubit states that are either 0 or 1) corresponds to an eigenstate and theeigenenergy is the classical energy objective function you have input into thesystem.

Fig. 8 Annealing functions (A(s)), (B(s)). Annealing begins at (s=0) with (A(s) gg B(s)) and ends at (s=1) with (A(s) ll B(s)). Data shown are representative of D-Wave 2X systems.

## Annealing Controls¶

D-Wave continues to pursue a deeper understanding of the fine details of quantumannealing and devise better controls for it. The system includes featuresthat give users programmable control over the annealing schedule, whichenable a variety of searches through the energy landscape. These controlscan improve both optimization and sampling performance for certain types ofproblems, and can help investigate what is happening partway through theannealing process.

## Quantum Gateway Router

For more information about the available annealing controls, see Technical Description of the D-Wave Quantum Processing Unit.