Experimental demonstration of perturbative anticrossing mitigation using nonuniform driver Hamiltonians

Perturbative anticrossings have long been identified as a potential computational bottleneck for quantum annealing. This bottleneck can appear, for example, when a uniform transverse driver Hamiltonian is applied to each qubit. Previous theoretical research sought to alleviate such anticrossings by adjusting the transverse driver Hamiltonians on individual qubits according to a perturbative approximation. Here we apply this principle to a physical implementation of quantum annealing in a D-Wave 2000Q system. We use samples from the quantum annealing hardware and per-qubit anneal offsets to produce nonuniform driver Hamiltonians. On small instances with severe perturbative anticrossings, our algorithm yields an increase in minimum eigengaps, ground-state success probabilities, and escape rates from metastable valleys. We also demonstrate that the same approach can mitigate biased sampling of degenerate ground states.

Figure 1

$A\left(s\right)$ and $B\left(s\right)$ vs $s$ for the QA system under the uniform QA algorithm. A global time-dependent annealing bias tunes both $A\left(s\right)$ and $B\left(s\right)$ simultaneously for all qubits throughout the QA algorithm.

Figure 2

Top: Relationship between $s$ and the tunable parameter $c$. Bottom: $A\left(s\right)$ and $B\left(s\right)$ vs $s$ for the QA hardware and for ${\delta }_i=0.0$, 0.02, and $-0.02$.

Figure 3

Top: 16-qubit instance studied in Ref. [10]; circles represent qubits with fields of value $+1$ (red), 0 (gray), and $-1$ (blue), and lines represent FM (ferromagnetic) couplings with value $-1$. Bottom: A single iteration of our mitigation strategy advances the outer qubits relative to the inner qubits. Observed ground-state probability (bottom left) increases with mitigation magnitude $\alpha$, as does the minimum eigengap (bottom right).

Figure 4

A randomly generated 24-qubit instance shown on a $2\times 2$ grid of unit cells in the D-Wave 2000Q qubit topology. Qubits are represented by circles; couplers are represented by line segments. Each nonzero coupler $J_{ij}$ is $+1$ (red) or $-1$ (blue). This instance has two classical ground states $\uparrow \uparrow \cdots \uparrow$ and $\downarrow \downarrow \cdots \downarrow$, and 424 first excited states.

Figure 5

Algorithmic mitigation improves QA success probability. Shown are ground-state probabilities $p_{\mathrm{GS}}$ after zero, one, and four iterations of mitigation. Median $p_{\mathrm{GS}}$ increases from 62 to 85% over four iterations.

Figure 6

Algorithmic mitigation increases the spectral gap. Shown are the minimum gaps between the instantaneous ground state and third excited state.

Figure 7

Application of anneal offsets causes imbalance in longitudinal fields between qubits. We approximate orthogonal transverse field control by compensating for this imbalance in the Ising Hamiltonian sent to the QA system. Once this compensation is applied, mitigation still achieves a systematic improvement in success probability, albeit smaller than the improvement seen in Fig. 5.

Figure 8

Measurements of escape rates out of the first excited state. For each instance, we prepared the first excited state and measured the escape rate out of this state at the annealing parameter $s_{\mathrm{target}}$ that corresponded to the location of the anticrossing. We show data for baseline $\alpha =0$ and mitigated $\alpha =\pm 0.02$.

Figure 9

Algorithmic mitigation improves uniformity of QA ground-state sampling. Shown are the expected number of samples required to observe every ground state, divided by the expected number for equal sampling so that each instance has minimum value 1. Expectation values for 100 instances are shown after zero, one, and four iterations of mitigation.

Figure 10

Algorithmic mitigation systematically improves uniformity of the zero-temperature quantum Boltzmann distribution over classical ground states at $s=0.30$. To determine $S_{\text{CC}}$, we use the quantum ground state at $s=0.3$, obtained via exact diagonalization, to calculate the probability of a given classical ground state contributing to the quantum ground state. We show $S_{\text{CC}}$ calculated from the mitigated and antimitigated spectra scattered against $S_{\text{CC}}$ calculated from the baseline spectrum.

Figure 11

First, second, and third eigengaps of three exemplary instances using antimitigation (left), no mitigation (middle), and mitigation (right). First eigengap $E_1-E_0$ always goes to zero late in the anneal; the third eigengap gives a clean representation of the desired tunneling event between superpositions of classical ground and excited states.