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

\begin{equation} \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} \end{equation}

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

\begin{equation} S_\Phi(f) = \frac{A}{f^\alpha} \end{equation}

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

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

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

\begin{equation} \Phi_h \equiv M_{\rm AFM} |I_p| \approx 0.8 \ {\rm m} \Phi_0. \end{equation}

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

\begin{equation} \delta h_i \equiv \frac{\delta \Phi_n}{\Phi_h} \approx 0.01. \end{equation}

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.