Denny Dahl gave some formulas that allowed the audience to compute the exact number of qubits in a system as a function of the so-called geometry parameters. The geometry parameters provide the size of a square grid of the unit cells and each unit cell has 8 qubits on it so this is the chimeric topology. The fact that D-Wave has been able to introduce more qubits and more couplers with each generation has been very exciting because it has allowed people to map larger problems onto the system.
Denny Dahl explained some features that D-Wave has introduced with the latest generation system, allowing the user to have more precise control over the annealing cycle. The speaker showed the time-dependent Hamiltonian that describes the evolution of the system as it goes through the annealing cycle. The first term, the A(s) multiplied by the summation of the Sigma X matrices represents the quantum fluctuations that allow the system to toggle from state to state. The second term that's multiplying B(s) is the energy landscape. This energy landscape is controlled by the programmer. The programmer controls the parameters A and B that determine the individual biases on the qubits and the strengths of the couplers. The graph that Denny Dahl showed, represents the two curves. The A curve represents the strength of the quantum fluctuations at a point in time and the B curve represents the strength of the energy landscape at a point in time, as one goes through the annealing cycle which has a duration of around 20 microseconds. The reason D-Wave architects the system this way is so that the optimization problem can be solved by using an adiabatic process. This is the fundamental working model of the system.
Denny Dahl introduced some new features in the annealing control. They are called anneal offsets, anneal pause, and anneal quench. Anneal offset means that, rather than each qubit following the given trajectory, one can individually and independently retard or advance individual qubits as they progress through the annealing cycle. There are very good reasons for wanting to do that. The initial motivation for providing this additional level of control was to allow D-Wave to more precisely calibrate the system so that this physical device behaved more closely to the idealized model that D-Wave had written down. In addition to calibration it turns out that this provides the users some power to improve the performance of the machine.
As second and third features Denny Dahl explained anneal pause and anneal quench. The idea behind these features is that midway through the annealing process one can freeze the annealing parameters, the A and B curves. One can allow the system to stay fixed with a particular mixture of the quantum fluctuation terms, the Sigma X and the energy landscape terms, the Sigma Z terms. At that fixed portion in the pause region the system is behaving as a full quantum mechanical system because one has both Sigma X and Sigma Z terms present, providing the dynamics of the system.
Denny Dahl showed a sample problem which is an integer factor. One of the big drivers of interest in the field in the early 90s was Peter Shor's algorithm for factoring integers on a circuit model or gate model quantum computer. D-Wave built a quantum annealer, not a gate model system. Shor's algorithm does not run as it stands on the D-Wave machine but it is still possible to factor numbers on the machine. You just do it in a different way. This work was done in late 2016 by half a dozen people at the company headquarters. The way you do factoring on the D-Wave system is pretty simple to describe. In the example that Denny Dahl gave one is taking two 4-bit binary integers. One multiplies them in exactly the same way that one is taught to multiply by writing down the products of the individual digits and then summing down the columns. The product of these two 4-bit binary integers gives you an 8-bit number. What one does to program this on the D-Wave system is one effectively takes this logic circuit to embed it into the fabric of qubits on the system.
Denny Dahl gave an example of what that looks like. One is using a five by six array of unit cells in the system. That is 30 unit cells, eight qubits per unit cell, so this is using a total of 240 qubits in order to do this factoring problem. One writes down a multiplication circuit for p = A*B. Then, one connects up unit cells to create the entire multiplication circuit. Once the multiplication circuit is embedded in the chip, any ground state of this problem will be a valid multiplication. In other words, if one is multiplying 2 4-bit integers to get an 8-bit integer, this means that 7 times 6 equaling 42 is a valid ground state and that 3 times 5 equaling 15 is a valid ground state. All possible multiplies are equal energy-valid generated ground states of the Hamiltonian that one has crafted.
Next, one takes the output bits to bias them. If one is interested in factoring the number of 15, one would bias the 8 output bits to represent the binary representation 15 and then out of all the possible ground states, the ones that remain at the bottom of the energy well represent the factors of 15 so one runs the factoring circuit backwards to get the answers.
Without using anneal offsets one thing that the team discovered is that as it tries to factor larger and larger numbers, its ability to hit the ground state reduces. In the blue bar Denny Dahl showed sizes of 6, 8 and 10 bits and the audience could see that as one goes from 6 to 8 to 10, the number of the success rate is measured by the height of the bar. The success rate is dropping. At 6 bits, one is able to factor every time; at 8 bits the factoring rate drops around 60 percent. When one gets to 10 bits, one is well under 10 percent. This means one would have to pull hundreds of samples from the system in order to be guaranteed of getting a valid answer. Using the anneal offset feature, one can tune the trajectory of individual qubits as they move through the annealing cycle. The audience could see in the maroon bar that there is much less degradation in performance as one goes from 6 to 8 bits. One is reliably able to factor at about 90%. Denny Dahl was not claiming that this trick of using anneal offsets is generally applicable to all problems but it is a proof by example. There are certain categories of problems whose performance seems to dramatically improve through the use of some of these advanced annealing features. D-Wave hasn't really explored a wide range of problems so it was very hard for Denny Dahl to say whether the annealing offset technique will work all the time.
The second example described an anneal pause/quench and 3D Ising process. Richard Harris from D-Wave Systems performed a simulation of a 3-dimensional transverse Ising system with a D-Wave quantum annealing processor. He applied a transverse magnetic field that induces an energy splitting. The chimera topology is two-dimensional. If one embeds the third dimension, one gets a 2x2 grid of unit cells with 8 qubits a piece. Four red qubits are linked together to form a single logical qubit. This is likewise for blue, aqua, green, and so on. These chains are strongly coupled to one another. A chain of four physical qubits behaves like a single logical qubit in the three-dimensional Ising model. The picture that Denny Dahl showed, illustrates a pair of unit cells on top and a pair of unit cells on the bottom. These are four qubit cells. Each one of the unit cells has eight qubits on it, so 8 qubits per unit cell times 4 unit cells gives 32 physical qubits. Richard Harris has grouped together these 32 physical qubits into 8 groups of 4. The audience could see a horizontal red line and another vertical red line that together represent a set of 4 physical qubits that are strongly coupled together. They all behave as a single logical qubit in the three-dimensional lattice. There is also a dark blue, an aqua, a green, a brown colour; another blue, and another green colour. Together, these colours represent 8 logical qubits in the vertical dimension of the three-dimensional lattice. This is a set of unit cells. The chip is a total of 16 by 16 unit cells. This is the vertical dimension and the other dimensions provide the two horizontal dimensions. In essence, this is how Richard Harris mapped the three-dimensional qubit lattice into the two-dimensional structure.
With regard to the experimental annealing parameters that Richard Harris worked with, the audience could see that in the middle of the annealing cycle, one is holding the system fixed. Normally, an annealing cycle has a default duration of 20 microseconds but what Richard Harris did, is to hold the system at an intermediate point during the middle of the anneal for one millisecond. That is 50 times longer than one would normally be at that point. During that 1 millisecond interval, the system is behaving as a three-dimensional nearest neighbour-coupled Ising model with a transverse magnetic field. It is a quantum system during that part of the anneal.
What Richard Harris was able to see very clearly is in effect the simplest result that he came up with. The diagram that Danny Dahl showed, represents on the left the paramagnetic phase and on the right, the audience could see that the order parameter which represents the magnetization of the spins in the lattice is turning on. Richard Harris was able to see very clearly a signal for the phase transition that one expects in a three-dimensional Ising model of the transverse magnetic field.
The last feature that Denny Dahl mentioned, was reverse annealing. D-Wave hasn't actually released this in its software but the idea is that one can allow the user to pre-initialize the values of all the qubits and then go through the annealing cycle in a reverse direction which looks like it holds some very interesting algorithmic potentials for maybe doing a quantum parallel tempering as D-Wave tries to do a better job of looking for ground states in these complicated energy landscapes.
To finish up, Denny Dahl concluded that the point that he wanted to get across is that at the same time D-Wave as a company is trying to increase the number of qubits and the number of couplers in its system, the company is also trying to increase the kind of controls that the user has over the annealing process that the system goes through. The hope is that by improving the control that the user has over the anneal, he will be able to increase the system's ability to hit ground states and also simulate genuine full quantum systems.