Mean Field Analysis of Quantum Annealing Correction

Quantum annealing correction (QAC) is a method that combines encoding with energy penalties and decoding to suppress and correct errors that degrade the performance of quantum annealers in solving optimization problems. While QAC has been experimentally demonstrated to successfully error correct a range of optimization problems, a clear understanding of its operating mechanism has been lacking. Here we bridge this gap using tools from quantum statistical mechanics. We study analytically tractable models using a mean-field analysis, specifically the $p$-body ferromagnetic infinite-range transverse-field Ising model as well as the quantum Hopfield model. We demonstrate that for $p=2$, where the phase transition is of second order, QAC pushes the transition to increasingly larger transverse field strengths. For $p\ge 3$, where the phase transition is of first order, QAC softens the closing of the gap for small energy penalty values and prevents its closure for sufficiently large energy penalty values. Thus QAC provides protection from excitations that occur near the quantum critical point. We find similar results for the Hopfield model, thus demonstrating that our conclusions hold in the presence of disorder.

Figure 1

The mean field phase diagram for $p=2$ for different $\gamma$ values. The lines represent second order PTs encountered along the anneal from large to small $\mathrm{\Gamma }$ values. For a fixed temperature, the critical point ${\mathrm{\Gamma }}_c$ increases with $\gamma$. At zero temperature, ${\mathrm{\Gamma }}_c=\infty$ for $\gamma >0$.

Figure 2

The mean field phase diagram for $p=4$ for different $\gamma$ values. The lines represent first order PTs. Inset: a magnification of the low temperature region to show the presence of two first order PTs for a particular range of $T$ and $\gamma$. At zero temperature, there exists a value ${\gamma }_c$ such that for $\gamma >{\gamma }_c$, the first order PT is avoided completely, as can be seen by the case $\gamma =1.5$.

Figure 3

Results for $p=4$, $T\to 0$, and $J=1$. (a) The free energy for $\gamma =0$ at the critical point ${\mathrm{\Gamma }}_c=1.185$. The two degenerate global minima are at $m=0$ and 0.943. (b) The free energy for $\gamma =0.5$ at the critical point ${\mathrm{\Gamma }}_c=1.847$. Now the two degenerate global minima are at $m=0.328$ and 0.844. For $\gamma =0.5$, the symmetric point $m=0$ is metastable and the global minimum has nonzero magnetization even for large $\mathrm{\Gamma }$. This minimum continuously moves to $m=0.328$ along the anneal, and then discontinuously jumps to $m=0.844$ at ${\mathrm{\Gamma }}_c=1.847$. (c) The coefficient $C$ associated with the scaling of the gap in the symmetric subspace ($\mathrm{\Delta }\sim C^N$). $C$ increases monotonically towards unity as a function of $\gamma$.

Figure 4

(a) The $q$ value at the free energy extremum for the Hopfield many-patterns case with $p=2$, $R=0.01N$, and $K=3$ under the replica symmetric ansatz. For $\gamma \ne 0$, the system remains in the symmetry-broken phase at least up to $\mathrm{\Gamma }=10$, while for $\gamma =0$ the symmetric phase is present for $\mathrm{\Gamma }\gtrsim 2.2$. (b) The $m$ value at the free energy extremum for the Hopfield many-patterns case with $p=4$, $R=0.01N^3$ and $K=3$. For $\gamma =0$, there is a first order transition around $\mathrm{\Gamma }=1.6$, and the extremum jumps discontinuously from $m=0.86$ to $m=0$. For $\gamma =0.5$, there is again a discontinuous jump in the value of $m$ but it does not reach $m=0$. For $\gamma =1$, 2 a discontinuity is not observed suggesting that the first order PT disappears or is at least weakened considerably by the penalty term.