From c933c138edd2d70912a57b0ae39e7e89211b722f Mon Sep 17 00:00:00 2001 From: Alejandro Montanez Date: Wed, 28 Feb 2024 22:10:12 +0100 Subject: [PATCH] Qubo tutorial (#733) [![Unitary Fund](https://img.shields.io/badge/Supported%20By-UNITARY%20FUND-brightgreen.svg?style=for-the-badge)](http://unitary.fund) ### Before submitting Please complete the following checklist when submitting a PR: - [x] Ensure that your tutorial executes correctly, and conforms to the guidelines specified in the [README](../README.md). - [x] Add a thumbnail link to your tutorial in `beginner.rst`, or if a QML implementation, in `implementations.rst`. - [x] All QML tutorials conform to [PEP8 standards](https://www.python.org/dev/peps/pep-0008/). To auto format files, simply `pip install black`, and then run `black -l 100 path/to/file.py`. When all the above are checked, delete everything above the dashed line and fill in the pull request template. ------------------------------------------------------------------------------------------------------------ **Title:** The Quadratic Unconstrained Binary Optimization (QUBO) **Summary:** A tutorial to learn how to represent combinatorial optimization problems using the QUBO formulation. The method is used to represent the Knapsack problem as a QUBO and it is solved using the quantum approximate optimization algorithm. **Relevant references:** **Possible Drawbacks:** **Related GitHub Issues:** --------- Co-authored-by: Guillermo Alonso-Linaje <65235481+KetpuntoG@users.noreply.github.com> Co-authored-by: Rashid N H M <95639609+rashidnhm@users.noreply.github.com> Co-authored-by: Alvaro Ballon <91897656+alvaro-at-xanadu@users.noreply.github.com> Co-authored-by: Ludmila Botelho <39570941+ludmilaasb@users.noreply.github.com> --- _static/authors/alejandro_montanez.jpg | Bin 0 -> 412486 bytes _static/authors/alejandro_montanez.txt | 4 + .../QUBO/dwave_results_slack.json | 1 + .../QUBO/dwave_results_unbalanced.json | 1 + .../demonstration_assets/QUBO/items_QUBO.png | Bin 0 -> 412705 bytes .../QUBO/thumbnail_QUBO_2024-02-06.png | Bin 0 -> 29271 bytes .../thumbnail_large_QuBO_2024-02-06.png | Bin 0 -> 125121 bytes demonstrations/QUBO/dwave_results_slack.json | 1 + .../QUBO/dwave_results_unbalanced.json | 1 + demonstrations/tutorial_QUBO.metadata.json | 32 + demonstrations/tutorial_QUBO.py | 744 ++++++++++++++++ poetry.lock | 803 +++++++++++++++++- pyproject.toml | 4 +- 13 files changed, 1581 insertions(+), 10 deletions(-) create mode 100644 _static/authors/alejandro_montanez.jpg create mode 100644 _static/authors/alejandro_montanez.txt create mode 100644 _static/demonstration_assets/QUBO/dwave_results_slack.json create mode 100644 _static/demonstration_assets/QUBO/dwave_results_unbalanced.json create mode 100644 _static/demonstration_assets/QUBO/items_QUBO.png create mode 100644 _static/demonstration_assets/QUBO/thumbnail_QUBO_2024-02-06.png create mode 100644 _static/large_demo_thumbnails/thumbnail_large_QuBO_2024-02-06.png create mode 100644 demonstrations/QUBO/dwave_results_slack.json create mode 100644 demonstrations/QUBO/dwave_results_unbalanced.json create mode 100644 demonstrations/tutorial_QUBO.metadata.json create mode 100644 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z&$GOOxs=EO#mPwN>7Rb8tF?D_>Xs&B^)LqFsrF)dzPJ&<&;R5wMn+o%9Wvf3YXy5U zMG!zqAV!({jO7=l;=YiGi2@o0c z9vZ4d@gJijFYHzad#_W**NMLtQ3p_J!)8o%c`7NKO2-s0eA`Z-T-by=5r5gs5FVjI zX_NZ-R5=)P-)S(3##cNNSyzQN?%f{f#`zbQpT(IZH}yMjYb@F`*;vn62N;VkyA50VNHpg07Z8nIDI*_1vPv;!TvoMn`O)@Nl5~Vw7kR zf+d*RnPHB;uz}nBDw-MxTW-KFc-FxrkNgK5nqn|#{vH4K&;J{T|C max_value + and combinations[case_i]["weight"] <= maximum_weight + ): + max_value = combinations[case_i]["value"] + optimal_solution = { + "items": combinations[case_i]["items"], + "value": combinations[case_i]["value"], + "weight": combinations[case_i]["weight"], + } + + +print( + f"The best combination is {optimal_solution['items']} with a total value: {optimal_solution['value']} and total weight {optimal_solution['weight']} " +) + +###################################################################### +# That was easy, right? But what if we have larger cases like 10, 50, or 100? Just to see how this +# scales, suppose it takes 1 ns to try one case. +# + +def time_to_solution(n, time_single_case): + """ + n (int): number of variables + time_single_case (float): time to solve a single case + """ + return time_single_case * 2 ** n + +time_per_case = 1e-9 # time to execute a single case in seconds +sec_day = 3600 * 24 # seconds in a day +sec_year = sec_day * 365 # seconds in a year + +print( + f"- For 10 items, 2^10 cases, we need {time_to_solution(2, time_per_case)} seconds." +) +print( + f"- For 50 items, 2^50 cases, we need {round(time_to_solution(50, time_per_case) / sec_day)} days." +) +print( + f"- For 100 items, 2^100 cases, we need {round(time_to_solution(100, time_per_case) / sec_year)} years." +) + + +###################################################################### +# Guess we don’t have the time to try all the possible solutions for 100 items 😅! Thankfully, +# we don't need to — there are algorithms to find good solutions to combinatorial +# optimization problems, and maybe one day we will show that one of these algorithms is quantum. So +# let’s continue with our quest 🫡. +# +# Our next step is to represent our problem mathematically. First, we represent our items by binary +# variables :math:`x_i` that take the value :math:`1` if we bring the :math:`i`-th item and :math:`0` otherwise. Next, we know that we want to **maximize** +# the value of the items carried, so let’s create a function :math:`f(\mathrm{x})` with these +# characteristics. To do so, we assign the variables :math:`x_i` to each of the items +# :math:`\mathrm{x} = \{x_0:⚽️ , x_1:💻, x_2:📸, x_3:📚, x_4:🎸\},` multiply each variable by the corresponding item value, and define a function that calculates the weighted sum +# value of the item: +# +# .. math:: \max_x f(\mathrm{x}) = \max_x \left(8x_0 + 47x_1 + 10x_2 + 5x_3 + 16x_4\right) \tag{1} +# +# This function, called the ``objective function``, represents the value of the items we can +# transport. Usually, solvers `minimize functions `__, so a simple trick in our case is to minimize the +# negative of our function (which ends up being maximizing our original function 🤪) +# +# .. math:: \min_x -(8x_0 + 47x_1 + 10x_2 + 5x_3 + 16x_4 ) \tag{2} +# +# We can write our equation above using the general form of the `QUBO +# representation `__, i.e., +# using an upper triangular matrix :math:`Q \in \mathbb{R}^{n \ \mathrm{x} \ n}`: +# +# .. math:: \min_x \mathrm{x}^TQ \mathrm{x} = \min_x \left(\sum_i \sum_{j\ge i} Q_{ij} x_i x_j\right) = \min_x \left(\sum_i Q_{ii} x_i + \sum_i\sum_{j>i} Q_{ij}x_i x_j\right) \tag{3} +# +# where :math:`\mathrm{x}` is a vector representing the items of our problem. Note that +# :math:`x_i x_i = x_i` for binary variables. Let's look at an example of how to calculate the :math:`\mathrm{x}^TQ \mathrm{x}` above: +# + +Q = -np.diag(list(items_values.values())) # Matrix Q for the problem above. +x_opt = np.array( + [[1 if i in optimal_solution["items"] else 0] for i in items_values.keys()] +) # Optimal solution. +opt_str = "".join(str(i[0]) for i in x_opt) +min_cost = (x_opt.T @ Q @ x_opt)[0, 0] # using Equation 3 above +print(f"Q={Q}") +print(f"The minimum cost is {min_cost}") + +###################################################################### +# But just with this function, we cannot solve the problem. We also need the weight restriction. Based +# on our variables, the weight list (items_weight = {“⚽️”: :math:`3`, “💻”: :math:`11`, “📸”: :math:`14`, “📚”: :math:`19`, “🎸”: :math:`5`}), and +# the knapsack maximum weight (maximum_weight :math:`W = 26`), we can construct our constraint +# +# .. math:: 3x_0 + 11x_1 + 14x_2 + 19x_3 + 5x_4 \le 26 +# +# Here comes a crucial step in the solution of the problem: we need to find a way to combine our +# *objective function* with this *inequality constraint*. One common method is to include the +# constraint as a **penalty** term in the objective function. This penalty term should be +# zero when the total weight of the items is less or equal to 26 and large otherwise. So to make them +# zero in the range of validity of the constraint, the usual approach is to use *slack variables*. +# There is an alternative method that has shown to perform better, called +# `unbalanced penalization `_ , but we present this method +# later 😉. +# +# The slack variable is an auxiliary variable that allows us to convert inequality constraints into equality +# constraints. The slack variable :math:`S` represents the amount by which the left-hand side of the +# inequality falls short of the right-hand side. If the left-hand side is less than the right-hand +# side, then :math:`S` will be positive and equal to the difference between the two sides. In our case +# +# .. math:: 3x_0 + 11x_1 + 14x_2 + 19x_3 + 5x_4 + S = 26\tag{4} +# +# where :math:`0 \le S \le 26`. But let’s take this slowly because we can get lost here, so let’s see +# this with some examples: +# +# - Imagine this case. No item is selected {:math:`x_0`: :math:`0`, :math:`x_1`: :math:`0`, :math:`x_2`: :math:`0`, +# :math:`x_3`: :math:`0`, :math:`x_4`: :math:`0`}, so the overall weight is zero (a valid solution) and the equality +# constraint Eq.(4) must be fulfilled. So we select our slack variable to be 26. +# +# - Now, what if we bring ⚽️ and 📚 {:math:`x_0`: :math:`1`, :math:`x_1`: :math:`0`, :math:`x_2`: :math:`0`, :math:`x_3`: :math:`1`, +# :math:`x_4`:0}. In this case, the overall weight is :math:`3 + 19 = 22` (a valid solution) and the equality +# constraint is fulfilled if :math:`22 + S = 26 \rightarrow S = 4`. +# +# - Finally, what if we try to bring all the items {:math:`x_0`: :math:`1`, :math:`x_1`: :math:`1`, :math:`x_2`: :math:`1`, +# :math:`x_3`: :math:`1`, :math:`x_4`: :math:`1`}, the total weight, in this case, is :math:`3+11+14+19+5=52` (not a valid +# solution), to fulfill the constraint, we need :math:`52 + S = 26 \rightarrow S=-26` but the slack +# variable is in the range :math:`(0,26)` in our definition, so, in this case, there is valid solution for :math:`S.` +# +# Excellent, now we have a way to represent the inequality constraint. Two further steps are needed. +# First, the slack variable has to be represented in binary form so we can cast it as a sum +# +# .. math:: S = \sum_{k=0}^{N-1} 2^k s_k, +# +# where :math:`N = \lfloor\log_2(\max S)\rfloor + 1`. In our case +# :math:`N = \lfloor\log_2(26)\rfloor + 1 = 5`. We need 5 binary variables to represent the range +# of our :math:`S` variable. +# +# .. math:: S = 2^0 s_0 + 2^1 s_1 + 2^2 s_2 + 2^3 s_3 + 2^4 s_4 +# +# To compact our equation later, let’s rename our slack variables by :math:`s_0=x_5`, :math:`s_1=x_6`, +# :math:`s_3=x_7`, :math:`s_4=x_8`, and :math:`s_5=x_9`. Then we have +# +# .. math:: S = 1 x_5 + 2 x_6 + 4 x_7 + 8 x_8 + 16 x_9. +# +# For example, if we need to represent the second case above (⚽️, 📚), +# :math:`S = 4 \rightarrow\{x_5:0, x_6:0,x_7:1,x_8:0, x_9:0\}`. +# +# We are almost done in our quest to represent our problem in such a way that our quantum computer can +# manage it. The last step is to add the penalty term, a usual choice for it is to use a +# quadratic penalty +# +# .. math:: p(x,s) = \lambda \left(3x_0 + 11x_1 + 14x_2 + 19 x_3 + 5x_4 + x_5 + 2 x_6 + 4x_7 + 8 x_8 + 16 x_9 - 26\right)^2. \tag{5} +# +# Note that this is simply the difference between the left- and right-hand sides of equation :math:`(4).` With this +# expression, the condition is satisfied only when the term inside the parentheses is zero. +# :math:`\lambda` is a penalty coefficient that we must tune to make that the constraint will always +# be fulfilled. +# +# Now, the objective function can be given by: +# +# .. math:: \min_{x,s} f(x) + p(x,s) = \min_{x,s} -(8x_0 + 47x_1 + 10x_2 + 5x_3 + 16x_4) + +# .. math:: \lambda \left(3x_0 + 11x_1 + 14x_2 + 19x_3 + 5x_4 + x_5 + 2 x_6 + 4x_7 + 8 x_8 + 16 x_9 - 26\right)^2 \tag{6} +# +# or, compacted, +# +# .. math:: \min_{x,s} \left(f(x) + p(x,s) = -\sum_i v_i x_i +\lambda \left(\sum_i w_i x_i - W\right)^2\right), \tag{7} +# +# where :math:`v_i` and :math:`w_i` are the value and weight of the :math:`i`-th item. Because of +# the square in the second term, some :math:`x_i x_i` terms show up. We can apply the property +# :math:`x_i x_i = x_i` (if :math:`x_i = 0 \rightarrow x_ix_i = 0\cdot0 = 0` or +# :math:`x_i = 1 \rightarrow x_ix_i = 1\cdot1 = 1`). +# +# The quadratic term on the right-hand side of equation :math:`(7)` can be rewritten as +# +# .. math:: \left(\sum_i w_i x_i - C\right)^2 = \left(\sum_i w_i x_i - C\right)\left(\sum_j w_j x_j - C\right) +# .. math:: = \sum_i \sum_j w_i w_j x_i x_j - 2C \sum_i w_i x_i + C^2 +# .. math:: = 2\sum_i \sum_{j>i} w_i w_j x_i x_j - \sum_i w_i(2C - w_i) x_i + C^2 \tag{8} +# +# where :math:`w_i` represent the weights for the items and :math:`2^k` for the slack variables. We +# can combine equations :math:`(7)` and :math:`(8)` to get the terms of the matrix :math:`Q`. So we end up with +# +# .. math:: Q_{ij} = 2\lambda w_i w_j,\tag{9} +# +# .. math:: Q_{ii} = - v_i + \lambda w_i(w_i - 2W).\tag{10} +# +# The term :math:`\lambda W^2` is only an offset value that does not affect the optimization result +# and can be added after the optimization to represent the right cost. Let's see how it looks +# like in our particular example. +# + +N = round(np.ceil(np.log2(maximum_weight))) # number of slack variables +weights = list(items_weight.values()) + [2**k for k in range(N)] + +QT = np.pad(Q, ((0, N), (0, N))) # adding the extra slack variables at the end of the Q matrix +n_qubits = len(QT) +lambd = 2 # We choose a lambda parameter enough large for the constraint to always be fulfilled +# Adding the terms for the penalty term +for i in range(len(QT)): + QT[i, i] += lambd * weights[i] * (weights[i] - 2 * maximum_weight) # Eq. 10 + for j in range(i + 1, len(QT)): + QT[i, j] += 2 * lambd * weights[i] * weights[j] # Eq. 9 +offset = lambd * maximum_weight**2 +print(f"Q={QT}") +# optimal string slack string +slack_string = np.binary_repr(maximum_weight - optimal_solution["weight"], N)[::-1] +x_opt_slack = np.concatenate( + (x_opt, np.array([[int(i)] for i in slack_string])) +) # combining the optimal string and slack string +opt_str_slack = "".join(str(i[0]) for i in x_opt_slack) +cost = (x_opt_slack.T @ QT @ x_opt_slack)[0, 0] + offset # Optimal cost using equation 3 +print(f"Cost:{cost}") + +# At this point, we have encoded the problem in a format that we can use to solve it on quantum +# computers. Now it only remains to solve it using quantum algorithms! +###################################################################### +# +# QAOA +# ------- +# +# We use QAOA `[1] `__ to find the solution to our Knapsack +# Problem (`read this demo `__ for a more detailed explanation +# of the QAOA algorithm). In this case, the cost Hamiltonian, :math:`H_c(Z)`, obtained from the QUBO +# formulation is translated into a parametric unitary gate given by +# +# .. math:: +# +# +# U(H_c, \gamma_i)=e^{-i \gamma_i H_c},\tag{11} +# +# .. math:: +# +# +# U(H_c, \gamma_i)=e^{-i \gamma_i \left( \sum_{i`__. We +# start in the ground state :math:`|+\rangle ^{\otimes n}` of the mixer Hamiltonian :math:`X` and move +# to the ground state of the cost Hamiltonian :math:`H_c` slowly enough to always be close to the +# ground state of the Hamiltonian. How slow? In our case the rate is determined by the number of layers +# :math:`p`. We can adopt this principle and initialize the :math:`\beta_i` and :math:`\gamma_i` in +# this way, moving :math:`\beta_i` from :math:`1` to :math:`0` and :math:`\gamma_i` from :math:`0` to :math:`1`. With this approach, +# we can skip the optimization part in QAOA. +# + +import matplotlib.pyplot as plt +# Annealing schedule for QAOA +betas = np.linspace(0, 1, 10)[::-1] # Parameters for the mixer Hamiltonian +gammas = np.linspace(0, 1, 10) # Parameters for the cost Hamiltonian (Our Knapsack problem) + +fig, ax = plt.subplots() +ax.plot(betas, label=r"$\beta_i$", marker="o", markersize=8, markeredgecolor="black") +ax.plot(gammas, label=r"$\gamma_i$", marker="o", markersize=8, markeredgecolor="black") +ax.set_xlabel("i", fontsize=18) +ax.legend() +fig.show() + +###################################################################### +# +# This Figure shows the annealing schedule we will use in our QAOA protocol. The y-axis represents the +# angle in radians and the x-axis represents the i-th layer of QAOA, from :math:`0` to :math:`9` for a total of :math:`p=10` layers. + + +###################################################################### +# +# I know this is a lot of information so far, but we are almost done! The last step to represent the +# QUBO problem on QPUs is to change the :math:`x_i\in \{0, 1\}` variables to spin variables +# :math:`z_i \in \{1, -1\}` via the transformation :math:`x_i = (1 - z_i) / 2`. We also want to set +# the penalty term, so a value of :math:`\lambda = 2` will be enough for our problem. In +# practice, we choose a value for :math:`\lambda` and, if after the optimization the solution does not +# fulfill the constraints, we try again using a larger value. On the other hand, if the solution is suspected +# to be a valid but suboptimal, then we will reduce :math:`\lambda` a little. Eq.(3) can be +# represented by an Ising Hamiltonian with quadratic and linear terms plus a constant :math:`O`, namely +# +# .. math:: +# +# +# H_c(\mathrm{z}) = \sum_{i, j > i}^{n} J_{ij} z_i z_j + \sum_{i=1}^n h_{i}z_i + O. \tag{13} +# +# Here, :math:`J_{ij}` are interaction terms and :math:`h_i` are linear terms, all of them depending +# on the combinatorial optimization problem. +# + + +def from_Q_to_Ising(Q, offset): + """Convert the matrix Q of Eq.3 into Eq.13 elements J and h""" + n_qubits = len(Q) # Get the number of qubits (variables) in the QUBO matrix + # Create default dictionaries to store h and pairwise interactions J + h = defaultdict(int) + J = defaultdict(int) + + # Loop over each qubit (variable) in the QUBO matrix + for i in range(n_qubits): + # Update the magnetic field for qubit i based on its diagonal element in Q + h[(i,)] -= Q[i, i] / 2 + # Update the offset based on the diagonal element in Q + offset += Q[i, i] / 2 + # Loop over other qubits (variables) to calculate pairwise interactions + for j in range(i + 1, n_qubits): + # Update the pairwise interaction strength (J) between qubits i and j + J[(i, j)] += Q[i, j] / 4 + # Update the magnetic fields for qubits i and j based on their interactions in Q + h[(i,)] -= Q[i, j] / 4 + h[(j,)] -= Q[i, j] / 4 + # Update the offset based on the interaction strength between qubits i and j + offset += Q[i, j] / 4 + # Return the magnetic fields, pairwise interactions, and the updated offset + return h, J, offset + + +def energy_Ising(z, h, J, offset): + """ + Calculate the energy of an Ising model given spin configurations. + + Parameters: + - z: A dictionary representing the spin configurations for each qubit. + - h: A dictionary representing the magnetic fields for each qubit. + - J: A dictionary representing the pairwise interactions between qubits. + - offset: An offset value. + + Returns: + - energy: The total energy of the Ising model. + """ + if isinstance(z, str): + z = [(1 if int(i) == 0 else -1) for i in z] + # Initialize the energy with the offset term + energy = offset + # Loop over the magnetic fields (h) for each qubit and update the energy + for k, v in h.items(): + energy += v * z[k[0]] + # Loop over the pairwise interactions (J) between qubits and update the energy + for k, v in J.items(): + energy += v * z[k[0]] * z[k[1]] + # Return the total energy of the Ising model + return energy + + +# Our previous example should give us the same result +z_exp = [ + (1 if i == 0 else -1) for i in x_opt_slack +] # Converting the optimal solution from (0,1) to (1, -1) +h, J, zoffset = from_Q_to_Ising(QT, offset) # Eq.13 for our problem +energy = energy_Ising( + z_exp, h, J, zoffset +) # Caluclating the energy (Should be the same that for the QUBO) +print(f"Minimum energy:{energy}") + +samples_slack = samples_dict(qaoa_circuit(gammas, betas, h, J, num_qubits=len(QT)), n_qubits) +values_slack = { + sum_values(sample_i, values_list): count + for sample_i, count in samples_slack.items() + if sum_weight(sample_i, weights_list) <= maximum_weight +} # saving only the solutions that fulfill the constraint +print( + f"The number of optimal solutions using slack variables is {samples_slack[opt_str_slack]} out of {shots}" +) + +###################################################################### +# As you can see, only a few samples from the 5000 shots give us the right answer, there are only +# :math:`2^5 = 32` options. Randomly guessing the solution will give us on average :math:`5000/32 \approx 156` +# optimal solutions. Why don’t we get good results using QAOA? Maybe we can blame the algorithm +# or we look deeper— it turns out our encoding method is really bad. Randomly guessing using the whole set of +# variables (:math:`5` items + :math:`5` slack) :math:`2^{10} = 1024` options, :math:`5000/1024 \approx 5`. So in fact we have a +# tiny improvement. +# + +###################################################################### +# Unbalanced penalization (An alternative to slack variables) +# ----------------------------------------------------------- +# +# Unbalanced penalization is a function characterized by a larger penalty when the inequality +# constraint is not achieved than when it is. So we have to modify Eq. 7 to include a linear term in +# the following way: +# +# .. math:: \min_{x,s} \left(f(x) + p(x,s)\right) = \min_{x,s} \left(-\sum_i v_i x_i - \lambda_1 \left(\sum_i w_i x_i - W\right) + \lambda_2 \left(\sum_i w_i x_i - W\right)^2\right)\tag{14}. +# +# where :math:`\lambda_{1,2}` are again penalty coefficients. Here `[2] `__ and `[3] `__ some details about unbalanced penalization. +# The method is already implemented in `OpenQAOA `__ and `D-Wave Ocean `__ so we don't have to code it ourselves. **The cliffnotes are +# that you don’t need slack variables for the inequality constraints anymore using this approach**. +# + +from openqaoa.problems import FromDocplex2IsingModel +from docplex.mp.model import Model + + +def Knapsack(values, weights, maximum_weight): + """Create a docplex model of the problem. (Docplex is a classical solver from IBM)""" + n_items = len(values) + mdl = Model() + x = mdl.binary_var_list(range(n_items), name="x") + cost = -mdl.sum(x[i] * values[i] for i in range(n_items)) + mdl.minimize(cost) + mdl.add_constraint(mdl.sum(x[i] * weights[i] for i in range(n_items)) <= maximum_weight) + return mdl + + +# Docplex model, we need to convert our problem in this format to use the unbalanced penalization approach +mdl = Knapsack(values_list, weights_list, maximum_weight) +lambda_1, lambda_2 = ( + 0.96, + 0.0371, +) # Parameters of the unbalanced penalization function (They are in the main paper) +ising_hamiltonian = FromDocplex2IsingModel( + mdl, + unbalanced_const=True, + strength_ineq=[lambda_1, lambda_2], # https://arxiv.org/abs/2211.13914 +).ising_model + +h_new = { + tuple(i): w for i, w in zip(ising_hamiltonian.terms, ising_hamiltonian.weights) if len(i) == 1 +} +J_new = { + tuple(i): w for i, w in zip(ising_hamiltonian.terms, ising_hamiltonian.weights) if len(i) == 2 +} + +samples_unbalanced = samples_dict( + qaoa_circuit(gammas, betas, h_new, J_new, num_qubits=n_items), n_items +) +values_unbalanced = { + sum_values(sample_i, values_list): count + for sample_i, count in samples_unbalanced.items() + if sum_weight(sample_i, weights_list) <= maximum_weight +} # saving only the solutions that fulfill the constraint + +print( + f"The number of solutions using unbalanced penalization is {samples_unbalanced[opt_str]} out of {shots}" +) + +###################################################################### +# We have improved the QAOA solution by encoding our QUBO wisely, with almost 2000 out of the 5000 samples +# being the optimal solution. Below, we compare the two different methods to encode the problem. The +# x-axis is the value of the items we bring based on the optimization (the larger the better) and the +# y-axis is the number of samples with that value (in log scale to observe the slack variables +# approach). In this sense, QAOA is pointing to the optimal and suboptimal solutions. +# + +fig, ax = plt.subplots() +ax.hist( + values_unbalanced.keys(), + weights=values_unbalanced.values(), + bins=50, + edgecolor="black", + label="unbalanced", + align="right", +) +ax.hist( + values_slack.keys(), + weights=values_slack.values(), + bins=50, + edgecolor="black", + label="slack", + align="left", +) +ax.vlines(-min_cost, 0, 3000, linestyle="--", color="black", label="Optimal", linewidth=2) +ax.set_yscale("log") +ax.legend() +ax.set_ylabel("counts") +ax.set_xlabel("values") +fig.show() + +###################################################################### +# Quantum Annealing Solution +# -------------------------- +# + +###################################################################### +# `Quantum +# annealing `__ +# is a process that exploits quantum mechanical effects to find low energy states of Ising +# Hamiltonians. We will use the quantum annealer D-Wave Advantage, a quantum computing system +# developed by D-Wave Systems Inc that has more than 5000 qubits. +# + +from dwave.system import DWaveSampler, EmbeddingComposite +from dwave.cloud import Client +import dimod +import pandas as pd + +bqm = {} +# BQM - Binary Quadratic Model +# This creates the QUBO model of our Knapsack problem using the slack variables approach +# offset is the constant term in our QUBO formulation +# ----------- SLACK METHOD ----------- +bqm["slack"] = dimod.BQM.from_qubo(QT, offset=lambd * maximum_weight**2) +bqm["slack"].relabel_variables({i: f"x_{i}" for i in range(bqm["slack"].num_variables)}) +# ----------- UNBALANCED METHOD ----------- +lagrange_multiplier = [0.96, 0.0371] # Again values from the paper +bqm["unbalanced"] = dimod.BQM.from_qubo(Q) # This adds the objective function to the model +bqm["unbalanced"].add_linear_inequality_constraint( + [(n, i) for n, i in enumerate(weights_list)], # This adds the constraint + lagrange_multiplier, + "unbalanced", + ub=maximum_weight, + penalization_method="unbalanced", +) +bqm["unbalanced"].relabel_variables({i: f"x_{i}" for i in range(bqm["unbalanced"].num_variables)}) + +# If you have an account you can execute the following code, otherwise read the file. +account = False +df = {} +if account: + # Replace with your client information + sampler = DWaveSampler(region="eu-central-1") + sampler_qpu = EmbeddingComposite(sampler) + for method in ["slack", "unbalanced"]: + samples = sampler_qpu.sample(bqm[method], num_reads=5000) # Executing on real hardware + df[method] = ( + samples.to_pandas_dataframe().sort_values("energy").reset_index(drop=True) + ) # Converting the sampling information and sort it by cost + df[method].to_json(f"QUBO/dwave_results_{method}.json") # save the results +else: + df = {} + for method in ["slack", "unbalanced"]: + df[method] = pd.read_json(f"QUBO/dwave_results_{method}.json") + # Loading the data from an execution on D-Wave Advantage + + +samples_dwave = {} +values = {} +for method in ["slack", "unbalanced"]: + samples_dwave[method] = defaultdict(int) + for i, row in df[method].iterrows(): + # Postprocessing the information + sample_i = "".join(str(round(row[q])) for q in bqm[method].variables) + samples_dwave[method][sample_i] += row["num_occurrences"] + values[method] = { + sum_values(sample_i, values_list): count + for sample_i, count in samples_dwave[method].items() + if sum_weight(sample_i, weights_list) <= maximum_weight + } + +###################################################################### +# The histogram below shows the results of both encodings on D-Wave Advantage. Once again, we prove +# that depending on the encoding method for our problem, we get good or bad results. +# + +fig, ax = plt.subplots() +bins = {"unbalanced": 5, "slack": 40} +for method in ["unbalanced", "slack"]: + ax.hist( + values[method].keys(), + weights=values[method].values(), + bins=bins[method], + edgecolor="black", + label=method, + align="right", + ) +ax.vlines(-min_cost, 0, 5000, linestyle="--", color="black", label="Optimal", linewidth=2) +ax.set_yscale("log") +ax.legend() +ax.set_ylabel("counts") +ax.set_xlabel("value") +fig.show() + +###################################################################### +# Conclusion +# ------------ +# +# We have come to the end of this demo. We have covered the definition of combinatorial optimization problems +# and how to formulate one of them, the Knapsack Problem, using QUBO, and two different encodings: slack +# variables and unbalanced penalization. Then, we solved them using optimization-free QAOA and QA. Now, +# it’s your turn to experiment with QAOA! If you need some inspiration: +# +# - Look at the `OpenQAOA `__ set of problems. There are plenty of them like bin packing, +# traveling salesman, and maximal independent set, among others. +# +# - Play around with larger problems. +# +# References +# ----------- +# +# [1] Farhi, E., Goldstone, J., & Gutmann, S. (2014). A Quantum Approximate Optimization Algorithm. +# http://arxiv.org/abs/1411.4028 +# +# [2] Montanez-Barrera, A., Willsch, D., A., Maldonado-Romo, & Michielsen, K. (2022). Unbalanced +# penalization: A new approach to encode inequality constraints of combinatorial problems for quantum +# optimization algorithms. http://arxiv.org/abs/2211.13914 +# +# [3] Montanez-Barrera, J. A., Heuvel, P. van den, Willsch, D., & Michielsen, K. (2023). Improving +# Performance in Combinatorial Optimization Problems with Inequality Constraints: An Evaluation of the +# Unbalanced Penalization Method on D-Wave Advantage. https://doi.org/10.1109/QCE57702.2023.00067 +# + +###################################################################### +# About the author +# ---------------- +# .. include:: ../_static/authors/alejandro_montanez.txt diff --git a/poetry.lock b/poetry.lock index bd4e7def33..711e41424d 100644 --- a/poetry.lock +++ b/poetry.lock @@ -1,4 +1,4 @@ -# This file is automatically @generated by Poetry 1.6.1 and should not be changed by hand. +# This file is automatically @generated by Poetry 1.7.1 and should not be changed by hand. 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