11. Bayesian Optimizing QAOA Circuit Energy

In this quantum circuit example, we’ll optimize a 54-qubit, \(p=4\) (depth 12), QAOA circuit on a random 3-regular graph, with respect to its energy - a sum of local expectations.

%config InlineBackend.figure_formats = ['svg']

import quimb as qu
import quimb.tensor as qtn

First we instantiate a high quality contraction path finder from cotengra, because each energy term will be a unique contraction, we’ll make a ‘reusable’ optimizer that can be used on multiple different contractions.

import cotengra as ctg

opt = ctg.ReusableHyperOptimizer(
    # use the following for persistently cached paths
    # directory='ctg_path_cache',

11.1. Setting Up the Circuit

Then we generate a random regular graph of conditions to satisfy. Optimizing the antiferromagnetic coupling on this graph is equivalent to trying to solve the MAX-CUT problem.

import networkx as nx

reg = 3
n = 54
seed = 666
G = nx.random_regular_graph(reg, n, seed=seed)

terms = {(i, j): 1 for i, j in G.edges}

quimb has a built-in QAOA circuit ansatz, which takes the dict of couplings to weights, as well as the \(\beta\) and \(\gamma\) parameters describing gate rotations:

p = 4
gammas = qu.randn(p)
betas = qu.randn(p)
circ_ex = qtn.circ_qaoa(terms, p, gammas, betas)

The overall circuit this generates is very complex:

circ_ex.psi.draw(color=['PSI0', 'H', 'RZZ', 'RX'])

But because of the unitary structure of quantum circuits, local quantities can usually be significantly simplified (automatically by quimb). Here, e.g., is the simplfied tensor network describing the reduced density matrix of qubit 0 only:

circ_ex.get_rdm_lightcone_simplified([0]).draw(color=['PSI0', 'H', 'RZZ', 'RX'], highlight_inds=['k0', 'b0'])

11.2. Rehearsing the Computation

Before we actually compute the QAOA energy, its usually worth ‘rehearsing’ it - finding the contraction widths and costs of each energy term to check they are not too big. The contraction paths found (which can take some time), will be cached by the ReusableHyperOptimizer for the actual computation later.

import tqdm

ZZ = qu.pauli('Z') & qu.pauli('Z')

local_exp_rehs = [
    circ_ex.local_expectation_rehearse(weight * ZZ, edge, optimize=opt)
    for edge, weight in tqdm.tqdm(list(terms.items()))
100%|██████████| 81/81 [06:34<00:00,  4.87s/it]

If we plot these we can see that the size of each and the number of ops is very moderate:

import matplotlib.pyplot as plt

fig, ax1 = plt.subplots()
ax1.plot([rehs['W'] for rehs in local_exp_rehs], color='green')
ax1.set_ylabel('contraction width, $W$, [log2]', color='green')
ax1.tick_params(axis='y', labelcolor='green')

ax2 = ax1.twinx()
ax2.plot([rehs['C'] for rehs in local_exp_rehs], color='orange')
ax2.set_ylabel('contraction cost, $C$, [log10]', color='orange')
ax2.tick_params(axis='y', labelcolor='orange')

In fact, the actual contractions will probably not be the most time consuming part of the computation.

11.3. Bayesian Optimizing the Energy

Since we only have \(2 p=8\) variational parameters we can easily apply something like Bayesian optimization to minimize the energy.

We’ll use scikit-optimize here, but many gradient free optimizers could be used instead. Most simply require casting the problem function as taking a single vector of parameters - x:

def energy(x):
    p = len(x) // 2
    gammas = x[:p]
    betas = x[p:]
    circ = qtn.circ_qaoa(terms, p, gammas, betas)

    ZZ = qu.pauli('Z') & qu.pauli('Z')
    ens = [
        circ.local_expectation(weight * ZZ, edge, optimize=opt)
        for edge, weight in terms.items()

    return sum(ens).real

Now we can setup some bounds for our parameters and the optimizer object itself:

from skopt import Optimizer
from skopt.plots import plot_convergence, plot_objective
eps = 1e-6
bounds = (
    [(0.0        + eps, qu.pi / 2 - eps)] * p +
    [(-qu.pi / 4 + eps, qu.pi / 4 - eps)] * p

bopt = Optimizer(bounds)

For the actual minimization, we ask the optimizer to suggest a vector of parameters to sample, then simply report the corresponding energy:

for i in tqdm.trange(100):
    x = bopt.ask()
    res = bopt.tell(x, energy(x))
100%|██████████| 100/100 [1:05:33<00:00, 39.33s/it]

One of the advantages of using Bayesian optimization and scikit-optimize is that we get an estimate of the actual energy landscape we can visualize:

    dimensions=[f'$\\gamma_{i}$' for i in range(p)] + [f'$\\beta_{i}$' for i in range(p)],