Adiabatic quantum optimization with the wrong Hamiltonian

Analog models of quantum information processing, such as adiabatic quantum computation and analog quantum simulation, require the ability to subject a system to precisely specified Hamiltonians. Unfortunately, the hardware used to implement these Hamiltonians will be imperfect and limited in its precision. Even small perturbations and imprecisions can have profound effects on the nature of the ground state. Here we consider an imperfect implementation of adiabatic quantum optimization and show that, for a widely applicable random control noise model, quantum stabilizer encodings are able to reduce the effective noise magnitude and thus improve the likelihood of a successful computation or simulation. This reduction builds upon two design principles: summation of equivalent logical operators to increase the energy scale of the encoded optimization problem, and the inclusion of a penalty term comprising the sum of the code stabilizer elements. We illustrate our findings with an Ising ladder and show that classical repetition coding drastically increases the probability that the ground state of a perturbed model is decodable to that of the unperturbed model, while using only realistic two-body interaction. Finally, we note that the repetition encoding is a special case of quantum stabilizer encodings, and show that this in principle allows us to generalize our results to many types of analog quantum information processing, albeit at the expense of many-body interactions.

Figure 1

Sample QUBO instance susceptible to errors. Top: Hardware graph—The qubit locations are shown by white circles and the allowable couplings with solid lines. Middle: Desired model—Two parallel Ising chains linked with a single antiferromagnetic coupling. Nearest-neighbor couplings of the form $-\alpha Z_i{Z}_j$, with $\alpha$ labeling the edges ($\alpha =0$ couplings not shown). Also indicated is the spin configuration for one of the two degenerate ground states of the model with dark blue (light tan) circles representing qubit state $\left|0\right.〉$ ($\left|1\right.〉$). Bottom: Erred model—The model in the presence of noise on the couplings possesses drastically different ground states. Dashed lines indicate unintended couplings, while thick, red lines indicate violated links. In this case, sufficiently many of the unintended couplings are satisfied to overcome the energetic cost of violating the indicated problem link (thick, red, labeled “0.72”).

Figure 2

Encoding a generic QUBO problem with a repetition code. The QUBO connectivity graph (extreme left) is replicated $K$ times and linked at equivalent vertices with ferromagnetic interactions (shown here with linear connectivity for ease of presentation, but higher connectivity is preferable). This is equivalent to taking a Cartesian product of the original QUBO graph with an Ising ferromagnet (middle). In the code space of the repetition code, all the spins in each code block will align and may be treated as a single dynamical variable under a renormalized Hamiltonian (right).

Figure 3

QUBO problem with various repetition encodings. Violated links are indicated by bold, red lines. Top: 9-bit linear encoding—Errors are slightly reduced in frequency as compared to the unencoded problem, but faults are likely to lie outside of the code space, as shown. Middle: 9-bit grid encoding—Errors are significantly less likely to occur than in the unencoded problem. All observed errors were in the code space, i.e., all physical qubits within each logical qubit are in the same state. Bottom: 9-bit complete encoding—Within each logical qubit, all physical qubits are ferromagnetically linked to each other. Errors are significantly less likely to occur than in the unencoded problem, but results were not significantly better than the simple two-dimensional (2D) grid. All observed errors were in the code space. For clarity, erred couplings are not shown.

Figure 4

Failure probability of Ising chain. Top: Unencoded failure probabilities—Color contour plot of the unencoded failure probabilities for a variable length Ising ladder at $\epsilon \in [0,1]$. Also shown (dashed, unlabeled) are contours of $\sqrt{N}\epsilon$. The close overlap of these with the (solid, labeled) contours of the failure probability implies that the failure probability is a function of $\sqrt{N}\epsilon$, as expected. Computed with 1000 MC iterations per data point. Bottom: Encoded failure probabilities—Solid: Failure probabilities for an Ising ladder of length $N=13$ encoded in four different 2D grid repetition codes. Dashed: Failure probability of the unencoded system with $\epsilon$ reduced by a factor of $\sqrt{K}$. The tendency to disagree at high $\epsilon$ can be attributed to an increased probability of failures lying outside the code space as $\epsilon$ approaches $E_{\max }=1$. These data were computed with 400 MC iterations per data point. The error bars indicate uncertainty in the failure probabilities resulting from the finite MC sample set.