# Error-Correction Features¶

D-Wave quantum computers provide some features to alleviate the effects of the errors described in previous chapters. Additionally, you can mitigate some effects through the techniques described in the D-Wave Problem-Solving Handbook guide.

## Drift Correction¶

Flux noise is described and its effects characterized in the Flux Noise of the Qubits and Characterizing the Effect of Flux Noise sections of the Error Sources for Problem Representation chapter.

By default, the D-Wave system uses the following procedure to measure and correct for the longest drifts once an hour. You can disable the application of any correction by setting the flux_drift_compensation solver parameter to False. If you do so, you should apply flux-bias offsets manually; see the Flux-Bias Offsets section.

1. The number of reads for a given measurement, $N_\mathrm{reads}$, is set to 2000.

2. A measurement of the zero-problem, with all $h_i = J_{i,j} = 0$ is performed, and the average spin computed for the $i$-th qubit according to $\left<s_i\right> = \sum_j{s_i^{(j)}}/N_\mathrm{reads}$, where $s_i^{(j)} \in \{+1,-1\}$ and the sum is performed over the $N_\mathrm{reads}$ independent anneal-read cycles.

3. The flux offset drift of the $i$-th qubit is estimated as $\delta\Phi_i = w_i\left<s_i\right>$, where $w_i$ is the thermal transition width of qubit $i$; defined below.

4. The measured $\delta\Phi_i$ are corrected with an opposing on-QPU qubit flux–bias shift. The magnitude of the shift applied on any given iteration is capped to minimize problems due to (infrequent) large $\delta\Phi_i$ measurement errors.

5. $N_\mathrm{reads}$ is doubled, up to a maximum of 20,000.

6. The procedure repeats from step 2 at least 6 times. It repeats beyond 6 if the magnitude of any of the $\delta\Phi_i$ after the last iteration is significantly larger than the expected variation due to $1/f$ flux noise.

The thermal width, $w_i$, of qubit $i$ is determined during QPU calibration by measuring the isolated qubit ($J_{i,j} = 0$ everywhere) average spin $\left<s_i(\Phi_i^{(x)})\right>$ as a function of applied flux bias $\Phi_i^{(x)}$ for each qubit, and fitting to the expression $\tanh{\left[(\Phi_i^{(x)}-\Phi_i^{(0)})/w_i\right]}$, where $\Phi_i^{(0)}$ and $w_i$ are fit parameters.

For a typical $w_i$ of order 100 $\mu\Phi_0$, statistical error is measured at $100~\mu\Phi_0/\sqrt{20000} \simeq 1~\mu\Phi_0$. This is much smaller than the root mean square (RMS) flux noise, which is on the order of 10 $\mu\Phi_0$ for the relevant time scales.

## Extended $J$ Range¶

Ising minimization problems that the D-Wave system solves may require that the model representing a problem be minor-embedded on the working graph, a process that involves creating qubit chains to represent logical variables, as introduced in the minor-embedding chapter of the Getting Started with D-Wave Solvers guide. In an embedding, intra-chain qubit couplings must be strong compared to the input couplings between the chains.

Most discussions of chain strength involve the ratio of two absolute values:

• Chain coupling strength—Magnitude of couplings between qubits in a chain

• Problem scale—Maximum magnitude of couplings among qubits (physical or logical) in a problem

that is,

$$\frac{chain\_coupling\_strength}{problem\_scale}.$$

For example, if all of the chains have $J$ values of $-1$, and the rescaled1 logical problem has $J$ values of $- \frac{1}{4}$ to $+ \frac{1}{4}$, you say that the chain strength is $4$. Likewise, if the chains have $J$ values of $-2$, and the rescaled logical problem has $J$ values of $- \frac{1}{2}$ to $+ \frac{1}{2}$, again you say that the chain strength is $4$.

1

See the description of the auto_scale solver parameter in the Solver Properties and Parameters Reference guide.

Because the range of coupling strengths available is finite, chaining is typically accomplished by setting the coupling strength to the largest allowed negative value and scaling down the input couplings relative to that. Yet a reduced energy scale for the input couplings may make it harder for the QPU to find global optima for a problem.

To address this issue, some solvers support stronger chain couplings through an extended_j_range solver property. Because embedded problems typically have chain couplings that are at least twice as strong as the other couplings, and standard chain couplings are all negative, this feature effectively doubles the energy scale available for embedded problems; see Figure 100.

Fig. 100 Embedding an input that does not fit directly on a D-Wave QPU. The original problem (A) has a green vertex that is replaced by a chain of two vertices connected with a ferromagnetic coupling (B). With the standard coupling range, the embedded problem must be scaled down for the QPU, which can lead to decreased performance (C). However, with an extended coupling range, no rescaling is necessary (D).

Using the available larger negative values of $J$ increases the dynamic $J$ range. On embedded problems, which use strong chains of qubits to build the underlying graph, this increased range means that the problem couplings are less affected by ICE and other effects. However, strong negative couplings can bias a chain and therefore flux-bias offsets must be applied to recalibrate it to compensate for this effect; see the Flux-Bias Offsets section for more information.

## Flux-Bias Offsets¶

In an optimal QPU calibration, annealing an unbiased chain produces spin-state statistics that are equally split between spin-up and spin-down. When plotted against the $h$ values, this even distribution results in a sigmoid curve that passes through the point of origin (0,0); see Figure 101. However, qubits in a chain with strong negative $J$ values experience a $J$-induced bias—an offset magnetic field that is potentially $s$-dependent.2 This field shifts the sigmoid curve of plotted $h$ values from its ideal path. To compensate, chains using strong negative $J$ couplings must be recalibrated to remove the bias from the chain and reduce the incidence of analog errors that may be associated with minor-embedding.

2

This offset occurs because of the higher susceptibility of the tunable coupler [Har2009].

Recalibration involves applying per-qubit3 flux-bias offsets to nudge the plotted $h$ sigmoid to its ideal position. The optimal offset value for a chain depends on the qubits and couplers involved and on the chain coupling strength.

Note

The applied flux bias is constant in time (and $s$), but it appears in the Hamiltonian shown in equation (3) as a term $I_p \phi_{\rm flux bias} \sigma_z$—the applied energy grows as $\sqrt{A(s)}$. An applied flux bias is different from an applied $h$; do not use one to correct an error in the other.

3

The same flux-bias offset values are used for all qubits in a chain.

Fig. 101 Magnetization (mean spin value) of a single chain as a function of applied per-qubit flux-bias offset. A sigmoid function (dashed line) is fit to the data points, and the optimal per-qubit flux-bias offset is given by the x-intercept of this function.

As stated above, D-Wave provides tools that automatically calculate and apply the appropriate flux-bias offsets when you submit an embedding that has strongly coupled chains. At a high level, however, the process of determining the value for the flux-bias offsets involves:

1. Define the strong chain that you intend to use in your embedding. Define it independently from any problem.

2. Sweeping the per-qubit flux-bias offset from the minimum allowed value to the maximum allowed value, anneal the chain many times (e.g., 1000) and calculate the average spin state of the chain (the magnetization).

3. Plot the results, with the per-qubit flux-bias offset on the x-axis and the chain magnetization on the y-axis. Fit a sigmoid model (in this case, a tanh function) to the data.

4. Measure the x-intercept of the sigmoid model (i.e., the per-qubit flux-bias offset value that results in a perfectly balanced magnetization of 0).

5. Add flux-bias offsets across the qubits in the chain to achieve an equal probability of reading spin up or spin down.

This operation is known as balancing the chain.4

4

While you can also balance a chain by adjusting the $h$ values on each of the chained qubits, this is less effective as a means to improve performance.

## Virtual Graphs¶

The D-Wave virtual graph feature (see the Ocean software VirtualGraphComposite class) simplifies the process of minor-embedding by enabling you to more easily create, optimize, use, and reuse an embedding for a given working graph. When you submit an embedding and specify a chain strength using these tools, they automatically calibrate the qubits in a chain to compensate for the effects of biases that may be introduced as a result of strong couplings. For more information on virtual graphs, see Virtual Graphs for High-Performance Embedded Topologies, D-Wave White Paper Series, no. 14-1020A, 2017. This and other white papers are available from https://www.dwavesys.com/resources/publications.

Virtual graphs make use of Extended J Range and Flux-Bias Offsets features. These controls allow chains to behave more like physical qubits on the working graph, thereby improving the performance of embedded sampling and optimization problems.

Note

Despite the similarity in name, the virtual graphs feature is unrelated to D-Wave’s virtual full-yield chip (VFYC) solver.

Whether virtual graph features are available may vary by solver; check for the extended_j_range property to see if it is present and what the range is. (The j_range property is unchanged.) When using an extended $J$ range, be aware that there are additional limits on the total coupling per qubit: the sum of the $J$ values for each qubit must be in the range specified by the per_qubit_coupling_range solver property.

Flux-biases are set through the flux_biases parameter, which takes a list of doubles (floating-point numbers) of length equal to the number of working qubits. Flux-bias units are $\Phi_0$; typical values needed are $< 10^{-4} \ \Phi_0$. Maximum allowed values are typically $> 10^{-2} \ \Phi_0$. The minimum step size is smaller than the typical levels of intrinsic $1/f$ noise; see the Characterizing the Effect of Flux Noise section.

Note

By default, the D-Wave system automatically compensates for flux drift as described in the Characterizing the Effect of Flux Noise section. If you choose to disable this behavior, you should apply flux-bias offsets manually through the flux_biases parameter.

Be aware of the following points when using virtual graph features:

• Neither spin-reversal transformation (SRT) nor autoscaling is possible with extended $J$ ranges or flux-bias offsets. If you specify one or more SRTs with a problem submission, the problem is scaled down to use $J$ values that fall within the j_range property instead.

• When you use these features, autoscaling is automatically disabled by default, unlike in the usual system behavior.

• The $J$ value of every coupler must fall within the range advertised by the extended_j_range property.

• The sum of all the $J$ values of the couplers connected to a qubit must fall within the per_qubit_coupling_range property. For example, if this property is [-9.0 6.0], the following $J$ values for a six-coupler qubit are permissible:

$1, 1, 1, 1, 1, 1,$

where the sum is $6$, and also

$1, 1, 1, -2, -2, -2,$

where the sum is $-3$. However, the following values, when summed, exceed the range and therefore are impermissible:

$-2, -2, -2, -2, -2, -2.$
• While the extended $J$ range in principle allows you to create almost arbitrarily long chains without breakage, the maximum chain length where embedded problems work well is expected to be in the range of 5 to 7 qubits.

• When embedding logical qubits using the extended $J$ range, limit the degree, $D$, of each node in the logical qubit tree to

$$D = {\rm floor}\bigg[\frac{\min(per\_qubit\_coupling\_range) - num\_couplers\_per\_qubit \times \min(j\_range)}{\min(extended\_j\_range) - \min(j\_range)}\bigg]$$

where $num\_couplers\_per\_qubit = 6$ for the D-Wave 2000Q QPU; see the QPU Architecture section of the Getting Started with D-Wave Solvers guide.