# Error Sources for Problem Representation¶

The dynamic range of \(h\) and \(J\) values may be limited by *integrated
control errors* (ICE). The term *ICE* refers collectively to these sources of
infidelity in problem representation. This chapter provides an overview of ICE,
describes the main sources that contribute to it, and provides guidance on measuring
its effects.

Note

Data given in this chapter are representative of D-Wave systems.

## Overview of ICE¶

Although \(h\) and \(J\) may be specified as double-precision floats, some loss of fidelity occurs in implementing these values in the D-Wave QPU. This fidelity loss may affect performance for some types of problems.

Specifically, instead of finding low-energy states to an optimization problem defined by \(h\) and \(J\) as in equation (1), the QPU solves a slightly altered problem that can be modeled as

where \(\delta h_i\) and \(\delta J_{i,j}\) characterize the errors in parameters \(h_i\) and \(J_{i,j}\), respectively.[1]

[1] | Because \(\delta h\) and \(\delta J\) are summed over \(N\), fidelity limitations tend to have a greater effect on performance for full-QPU sized problems, for a given dynamic range and distribution of \(h\) and \(J\). |

The error \(\delta h_i\) depends on \(h_i\) and on the values of all
incident couplers \(J_{i,j}\) and neighbors \(h_j\), as well as *their*
incident couplers \(J_{j,k}\) and next neighbors \(h_k\). That is, if spin
\(i\) is connected to spin \(j\), and spin \(j\) is connected to spin
\(k\) in the topological graph, then \(\delta h_i\) may depend, to varying
extents, on \(h_i, h_j, h_k\), \(J_{i,j}\), and \(J_{j,k}\). This
dependency holds when the relevant qubits and couplings are present in the graph,
irrespective of whether they are set to nonzero values. Similarly,
\(\delta J_{i,j}\) depends on spins and couplings in the local neighborhood of
\(J_{i,j}\). For example, if a given problem is specified by
\((h_1 = 1 , h_2 = 1, J_{1,2} = -1)\), the QPU might actually solve the
problem \((h_1 = 1.01, h_2 = 0.99, J_{1,2} = -1.01)\). Changing just a single
parameter in the problem could change all three error terms, altering the problem
in different ways.

The *probability distribution* of \(\delta h\) and \(\delta J\) is an
ensemble across a set of problem settings \(h\) and \(J\) and across all
\(i\) and \(i,j\) pairs. The \(\delta h\) and \(\delta J\) values
are Gaussian-distributed with mean \(\mu\) and standard deviation \(\sigma\)
that vary with anneal fraction \(s\) during the anneal.[2] The QPU control
system is calibrated so that there is typically a value of \(s\) for which
\(\mu\) is zero. This point is chosen to be somewhere between the single qubit
freezeout point described in the Freezeout Points section (later in the anneal),
and at the quantum critical point of a one-dimensional Ising chain (earlier in the
anneal). Distributions with nonzero \(\mu\) for some values of \(s\) are
considered to be due to *systematic errors* discussed later in this chapter. Thus,
the expected deviations of \(h\) and \(J\) during operation are the sum
of a systematic contribution \(\mu\) and a random component with standard
deviation \(\sigma.\)

[2] | Assumed for simplicity; distributions seen in actual results are close to this. |

Figure 88 and Figure 89 show example measurements of \(\delta h\) and \(\delta J\) distributions at different fractional times \(s\). See the Using Two-Spin Systems to Measure ICE section for a description of the measurement method.

## Sources of ICE¶

The Ising spins on the D-Wave QPU are intrinsically analog, controlled through spatially local magnetic fields. The controls are a combination of the output of on-QPU DACs and other analog signals shared among groups of spins. A calibration procedure, conducted when the QPU first comes online, determines the mapping between Ising problem specifications \(h\) and \(J\), and the control values used during annealing.

QPU calibration is a significant part of the time required to install a D-Wave system at a site. It involves a series of measurements of the QPU and the refrigerator to obtain data used to build models that achieve the desired Ising spin Hamiltonian.

These models identify several factors that contribute to distortions of \(h\) and \(J\) due to ICE. Understanding these factors, and how to compensate for them, can guide your choices in \(h\) and \(J\) when you specify a problem. Listed below are the dominant sources of ICE. The subsections that follow give additional details.

Background Susceptibility (ICE1)

The on-QPU rf-SQUID qubits behave weakly as couplers, which leads to effective next-nearest-neighbor (NNN) \(J\) interactions and a leakage of applied \(h\) biases from a qubit to its neighbors.

Flux Noise of the Qubits (ICE2)

The qubits experience low-frequency \(1/f\)-like flux noise. This noise contributes an error that varies in time (that is, between runs on the QPU), and also varies with \(s\).

DAC Quantization (ICE3)

The on-QPU DACs that provide the specified \(h\) and \(J\) values have a finite quantization step size.

I/O System Effects (ICE4)

The ratio of \(h/J\) may differ slightly for different annealing parameters such as \(t_f\).

Distribution of h Scale Across Qubits (ICE5)

Qubits cannot be made perfectly identical. As a result, individual spins may have slightly different magnitudes (persistent currents) and tunneling energies. These differences also vary with anneal fraction \(s\).

### Background Susceptibility¶

During the annealing process, every Ising spin has a coupler-like effect on its neighbors (that is, the spins to which it is connected by couplings) that is not captured by the problem Hamiltonian. This effect takes two main forms:

- Spin \(i\) induces
*next-nearest neighbor*(NNN) couplings between pairs of its neighboring spins. - The applied \(h\) bias
*leaks*from spin \(i\) to its neighboring spins.[3]

[3] | A more detailed description of how this arises from rf-SQUID qubits is in [Har2010]. |

The strength of this *background susceptibility* effect is characterized by a
parameter \(\chi\).[4]

[4] | Use of the \(\chi\) symbol is consistent with the usage in physics of \(\chi\) to characterize the susceptibility of a magnetic system. This document uses a normalized \(\chi = M_{\rm AFM} \chi_q\), where \(M_{\rm AFM}\) is the maximum available antiferromagnetic mutual inductance, and \(\chi_q = \frac{dI_p}{d\Phi_q}\) is the physical qubit susceptibility. |

For example, consider the three-spin problem shown in Figure 90. This system is described by the Ising energy function

The energy function solved by the QPU, however, has some extra terms:

The NNN coupling occurs because spin 1 induces an interaction between spins 1 and 3 with magnitude \(J_{1,2} \chi J_{2,3}\). Similarly, \(h_2\) leaks onto spin 1, with magnitude \(h_2 \chi J_{1,2}\), and so on for the other terms.

Figure 91 shows how \(\chi\) typically varies with \(s\), from around -0.04 early in the anneal to near -0.015 late in the anneal, at which point single-spin dynamics freeze out.

### Flux Noise of the Qubits¶

As another component of ICE, each \(h_i\) is subject to an independent (but
time-dependent) error term that comes from the \(1/f\) flux noise of the
qubits.[5] There are fluctuations in the flux noise that have lower frequency
than the typical inverse annealing time, so problems solved in quick succession
have correlated contributions from flux noise. By default, flux drift is
automatically corrected every hour by the D-Wave system so that it is
bounded and approximately Gaussian when averaged across all times; see the
Drift Correction section for the procedure. You can disable this automatic correction
by setting the flux_drift_compensation solver parameter to `False`

. If you
do so, apply flux-bias offsets manually; see the Flux-Bias Offsets section.

[5] | Couplers also have \(1/f\) flux noise, but this effect is insignificant compared to \(\delta h\) and \(\delta J\). |

Because the physical source of noise is flux fluctuations on the qubits, the
*effective* level of noise is larger earlier in the anneal when the persistent
current in qubits is smaller, so that \(h\) is relatively smaller in physical
flux units; see the Hardware: Coupled rf-SQUID Qubits section and the
Characterizing the Effect of Flux Noise section for details.

The Fourier spectrum of the fluctuations of \(\delta h^2\) varies approximately as \(A^2/f^\alpha\), where \(f\) is frequency from 1 mHz to the highest-resolvable frequency, \(A\) is the amplitude of the fluctuations at \(f = 1\) Hz, and \(\alpha\) is the spectral tilt—or frequency dependence—of the fluctuations. Due to the flux drift compensation, the spectrum becomes flat and close to zero below 1 mHz. Integrating the noise from 1 mHz to 1 MHz (the relevant band of interest observable directly), the Gaussian contribution to \(\delta h\) is approximately 0.009 early in the anneal and less later in the anneal. Figure 92 shows typical data for a 64-qubit system.

### DAC Quantization¶

The on-QPU DACs that provide the user-specified \(h\) and \(J\) values have a finite quantization step size. That step size depends on the value of the \(h\) or \(J\) applied because the response to the DAC output is nonlinear.

This random error contribution is described by a uniform distribution centered at 0 and having errors \(\pm b\). Typical errors for \(h\) and \(J\) are shown in Figure 93. Note that these errors are smaller than other contributors to ICE.

### I/O System Effects¶

Several time-dependent analog signals are applied to the QPU during the annealing process. Because the I/O system that delivers these signals has finite bandwidth, the waveforms must be tuned for each anneal to minimize any potential distortion of the signals throughout the annealing process. As a result, the ratio of \(h/J\) may vary slightly with \(t_f\) and with scaled anneal fraction \(s\).

### Distribution of \(h\) Scale Across Qubits¶

Assuming a fixed temperature of the system, and \(J=0\), the expected graph of the relationship between user-specified \(h\) and QPU-realized \(h\) has slope = 1, reflecting a constant mean offset \(\mu\) in the distribution of \(\delta h\). The measured slopes, however, are different for each spin, and are Gaussian-distributed.

This divergence results from small variations in the physical size of each qubit and from imperfections that arise when attempting to homogenize the macroscopic parameters—physical inductance \(L\), capacitance \(C\), and Josephson junction critical current \(I_c\)—across all physical qubits in the QPU.

Assuming a fixed temperature for the system, the ideal relationship, or slope, between qubit population and applied \(h\) is identical across an ensemble of devices. The distribution of measured slopes, however, is Gaussian-distributed with a standard deviation of approximately 1%, which contributes to \(\delta h\) and \(\delta J\). This distribution results from differences in the magnitude of each spin \(s_i\); see the Using Two-Spin Systems to Measure ICE section.

## Measuring ICE¶

This section describes how to measure the effects of \(1/f\) noise as well as how to harness effective two-spin systems to measure the effects of ICE.

### Measuring \(1/f\) Noise¶

A straightforward way to look at the \(1/f\) noise in the \(h\) parameter is to set all \(h\) and \(J\) values equal to 0 on the full QPU and observe how resulting qubit distributions drift over time in repeated tests. This approach identifies the effective \(1/f\) noise error on \(h\) when referenced to the single qubit freezeout point \(s^*_q\). To probe the \(1/f\) noise at earlier anneal times (and to amplify the error signal), create clusters of strongly coupled qubits and measure the time-dependent behavior of the net magnetization of system as described in the next section.

### Using Two-Spin Systems to Measure ICE¶

You can use the results of problems defined on pairs of Ising spins scattered over the topological graph to help to characterize ICE effects.

#### Simple Two-Spin Systems¶

The simplest version of such problems has two Ising spins with biases \(h_1\) and \(h_2\) and a coupling between them of weight \(J\). The energy of this system is

To measure ICE using simple two-spin systems, first find all independent edge sets of the available graph; that is, the sets of two-spin systems that can be manipulated independently. Simultaneously, for each set, ask the QPU to solve the independent two-spin problems at a variety of \(h_1\) and \(h_2\) settings near the phase boundaries indicated in Figure 94. Then fit the resulting data to a thermal model to estimate the deviations, \(\delta h\) and \(\delta J\), from equation (4) above using the correction terms of equation (5) below.

The two-spin phase diagram in Figure 94 characterizes the lowest energy state of the system as a function of \(h_1\) and \(h_2\) for given \(J\), here set to \(J=-0.5\). The dashed line delineates regions of the \(h_1\) and \(h_2\) space where resulting Ising spins are indicated by the up (\(s = 1\)) and down (\(s = -1\)) arrows. Applying a larger magnitude (more negative) \(J\) value increases ferromagnetic interactions, growing the regions of \(\downarrow \downarrow\) and \(\uparrow \uparrow\) while shrinking the regions of \(\downarrow \uparrow\) and \(\uparrow \downarrow\).

The ideal diagram may be compared to one obtained assuming the existence of background susceptibility errors. For this problem, the \(\chi\) correction terms are

The \(h\)-leakage effects are shown in Figure 94: the blue lines denote the boundaries of the different spin ordering regions for the case where \(\chi = -0.05\), versus the ideal case shown by the dashed lines. (The two-spin system does not have NNN effects.)

Given the form of the \(h\)-leakage terms, applying small adjustments to the \(h\) biases on the original problem can compensate somewhat for this error. However, \(\chi\) varies during the anneal (see Figure 91), and this correction corresponds to one specific point during the anneal. The single qubit freezeout point for typical anneal times occurs when \(s \approx 0.8\) and \(A(s)\) is around 100 MHz (see Figure 74). This is the last point in the anneal where any meaningful spin-flip dynamics occur; at that point \(\chi\) is approximately -0.015, so that value can, in principle, compensate for \(h\)-leakage. A better approach, however, is to choose \(\chi\) to correspond to a point earlier in the anneal—at or before the crossing point of \(A(s)\) and \(B(s)\). This is the localization point of a one-dimensional chain of spins.

#### Effective Two-Spin Systems For Larger Problems¶

Similar measurements may be repeated for larger problems made up of *logical spins*
formed by strongly coupled groups of spins with strong intracluster coupling weights
\(J_{cluster}\). Pairs of clusters are connected by the intercluster coupling
weight \(J\), and the \(h_1\) and \(h_2\) weights are assigned to all
spins in each cluster. These clusters freeze out at different anneal times
depending on the number of spins and on coupling strengths \(J\) and \(J_{cluster}\),
as shown in Figure 76 and
Figure 77. These *effective two-spin instances*
make it possible to probe ICE effects on \(h\) and \(J\) at different
times during the anneal.

For each logical qubit size up to 6 (corresponding to 6+6 spins) held together with \(J_{\text{cluster}}=1\), ask the QPU to solve independent problems on the full-sized graph for varying \(J\), \(h_1\) and \(h_2\). Then fit the measured results to a fixed-temperature model of the phase diagram for the ideal problem—appropriately adjusted to the larger qubit counts, including the proper \(\chi\) contributions from the logical qubit components.

The dominant effects taken into account during the fitting to the phase diagram model are:

- The \(J\) realized may vary from the \(J\) requested. ICE effects on couplings characterized by \(\delta J\) are shown in Figure 89.
- With large-enough samples of logical two-spin problems, random \(\delta h\) errors cancel out; what remains is an offset field for each spin. This offset, \(h_{offset}\), characterizes the intrinsic flux offset of the logical qubit that is independent of \(h\).
- The difference between specified \(h\) and realized \(h\) may vary as described in the Distribution of h Scale Across Qubits section. Figure 95 shows the effect of this type of error on the phase diagram.
- Additional warping of the phase diagram occurs as \(\chi\) changes from \(-0.04\) to \(-0.015\) through anneal time \(s\).

## Characterizing the Effect of Flux Noise¶

This section characterizes the effects of flux noise on the quantum annealing process. The Error-Correction Features chapter describes the procedure that D-Wave uses to correct for drift.

Let there be a flux qubit biased at degeneracy \(h=0\) with tunneling energy \(\Delta_q\). Let the qubit be subject to flux noise with noise spectral density \(S_{\Phi}(f)\). If this qubit is subjected to experiments over a time interval \(t_{\rm exp}\), then it is subject to a random flux bias whose flux-flux correlator can be expressed as

where \(f_{\rm min} = 1/t_{\rm exp}\) and \(f_{\rm max} = \Delta_q/h\).[6]

[6] | The derivation of this expression follows the same logic as that for the phase-phase
correlator given in Eq. 10c of Phys. Rev. B 67, 094510 (2003), albeit with an
appropriate low-frequency cutoff. |

The ambiguity with quantum annealing is in defining the appropriate choice of \(\Delta_q\), since this quantity is swept during an experiment. The value of \(\Delta_q\) should be the point in the anneal at which a given qubit localizes. For a single isolated qubit, the appropriate value of \(\Delta_q\) is on the order of the inverse decoherence time \(1/T_2^*\), where \(T_2^*\) is a function of \(\Delta_q\). Thus, \(\Delta_q\) becomes that of the coherent-incoherent crossover. For a system of coupled qubits, localization can occur at much larger values of \(\Delta_q\). In this case, the coupled qubit system can undergo a phase transition earlier in the anneal where the qubits are coherent. D-Wave QPUs realize these phase transitions at \(\Delta_q/h \sim 2\) GHz.

For example, the calculation of the fractional error in the dimensionless 1-local bias \(h_i\) proceeds as follows. Typical qubits experience low-frequency flux noise characterized by a noise spectral density of the form

with amplitude such that \({\sqrt S_\Phi ({\rm 1 Hz}) \sim 2 \mu\Phi_0/\sqrt{\rm Hz}}\) and exponent \(0.75 \leq \alpha < 1\). Given that the uncertainty in \(\alpha\) is large and a lack of experimental evidence that the form given above is valid up to frequencies of order \(\Delta_q/h \sim 2\) GHz, \(\alpha = 1\) is used in these calculations. Integrating this equation from \(f_{\rm min} = 1\) mHz to \(f_{\rm max} = 2\) GHz yields an integrated flux noise

A qubit with \(\Delta_q/h = 2\) GHz also possesses a persistent current \(|I_p| \approx 0.8\) \(\mu\)A. The maximum achievable antiferromagnetic coupling between a pair of qubits is \(M_{\rm AFM} \approx 2 \ {\rm pH}\). Thus, the scale of \(h\) is set by

The relative error in the dimensionless parameter \(h_i\) is then

## Example of ICE Effects on Solution Quality¶

As discussed in the Sources of ICE section, the distributions \(\delta h\) and \(\delta J\) depend on annealing time \(t_f\) and vary with anneal fraction \(s\) during the anneal. Because \(\delta h\) and \(\delta J\) may vary with \(s\) (as well as with any errors in the ratio \(h/J\)), ICE can drive a system across a phase boundary—whether a quantum phase transition or a later classical phase transition.

Consider a simple two-spin problem with biases \(h_1 = h_2 = 0.01\) and \(J = -0.5\). You can expect to see spins \(\downarrow \downarrow\) (that is, \(s_1, s_2 = -1\)) in the problem solution. If \(\delta h\) is such that the QPU sees \(h_1, h_2 = -0.01\) early in the anneal, the system localizes in the \(\uparrow \uparrow\) state. Perhaps later, the effective ICE signal becomes smaller and the system crosses a phase boundary to prefer \(\downarrow \downarrow\). Depending on when this happens in relation to freezeout time, the system may or may not be able to respond:

- If \(\delta h\) changes early enough in the anneal, the system can respond and provide the correct answer.
- If \(\delta h\) occurs too late, the early ICE is effectively locked in and both spins remain up.

This example shows that there are two components to ICE: the final error on the classical Ising spin system defined by equation (1), and the rest of the error contributing to a variation in the anneal path on the way to the final classical Hamiltonian. Depending on problem details, either of these effects may dominate—exchanging roles earlier, later, or at multiple times in the anneal.