# Flux Noise and Quantum Annealing¶

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

## Effects of Flux Noise¶

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

$$\left<\Phi_n(t)\Phi_n(0)\right> = \int_{f_{\rm min}}^{f_{\rm max}} df S_\Phi(f) \frac{\sin^2 2\pi f t}{(\pi f)^2}$$

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

 [1] 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

$$S_\Phi(f) = \frac{A}{f^\alpha}$$

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 that we have no experimental evidence that the form given above is valid up to frequencies of order $\Delta_q/h \sim 2$ GHz, we have used $\alpha = 1$ in our calculations. Integrating this equation from $f_{\rm min} = 1$ mHz to $f_{\rm max} = 2$ GHz yields an integrated flux noise

$$\delta \Phi_n = \sqrt{\int_{f_{\rm min}}^{f_{\rm max}} df S_\Phi(f)} \approx 10\ \mu\Phi_0.$$

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

$$\Phi_h \equiv M_{\rm AFM} |I_p| \approx 0.8 \ {\rm m} \Phi_0.$$

The relative error in the dimensionless parameter $h_i$ is then

$$\delta h_i \equiv \frac{\delta \Phi_n}{\Phi_h} \approx 0.01.$$

## Drift Correction¶

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 parameter to false. If you do so, we recommend that you 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.