How to Use Quantum Concepts to Improve Probabilistic Programming
Probabilistic programming (PP) has revolutionized how we build and reason about complex, uncertain systems. From medical diagnostics to self-driving cars, PP provides a powerful framework for modeling reality with statistical rigor. Yet, for all its power, PP faces significant computational challenges, especially when dealing with high-dimensional models, intricate dependencies, or vast datasets.
Enter quantum mechanics. No, we’re not about to suggest you port your Bayesian model to a quantum computer tomorrow (though that’s a fascinating, distant future!). Instead, we’ll explore how the concepts and algorithmic paradigms from quantum mechanics can offer novel ways of thinking about, and potentially improving, probabilistic inference. This isn’t about practical, everyday performance boosts for classical PPLs right now, but about conceptual breakthroughs and potential future directions for tackling some of the hardest problems in the field.
This post is for developers who understand the basics of probabilistic programming and are curious about the bleeding edge of computational paradigms.
Probabilistic Programming: A Quick Refresher
At its core, probabilistic programming lets you specify a statistical model in terms of random variables and their relationships. Then, using inference algorithms, it helps you compute the posterior distribution of unobserved (latent) variables given observed data.
Common tasks include:
- Modeling: Defining a joint probability distribution over observed and unobserved variables.
- Inference: Calculating the conditional probability of unobserved variables given observed ones (the posterior). This is often done via MCMC (Markov Chain Monte Carlo) or Variational Inference.
- Prediction: Using the inferred model to predict new data.
Let’s start with a simple example using PyMC, a popular Python PPL. We’ll model a classic problem: estimating the bias of a coin.
First, ensure you have PyMC installed:
pip install pymc numpy arvizNow, let’s define and infer a simple coin flip model:
import pymc as pm
import numpy as np
import arviz as az
# 1. Define the observed data: 6 heads, 4 tails (10 flips total)
# Represent heads as 1, tails as 0
observed_flips = np.array([1, 1, 1, 1, 1, 1, 0, 0, 0, 0])
n_flips = len(observed_flips)
n_heads = observed_flips.sum()
print(f"Observed flips: {observed_flips}")
print(f"Total flips: {n_flips}, Heads: {n_heads}")
# 2. Build the probabilistic model
with pm.Model() as coin_model:
# Prior for the coin's bias (theta): uniform distribution between 0 and 1
# This means we initially assume all biases are equally likely
theta = pm.Beta("theta", alpha=1, beta=1) # Beta(1,1) is equivalent to Uniform(0,1)
# Likelihood: observed_flips follow a Bernoulli distribution
# with the unknown bias 'theta'
# We explicitly tell PyMC which data is observed
flips = pm.Bernoulli("flips", p=theta, observed=observed_flips)
# 3. Perform inference (sampling from the posterior distribution)
# NUTS (No-U-Turn Sampler) is a sophisticated MCMC algorithm
idata = pm.sample(2000, tune=1000, cores=2, random_seed=42)
# 4. Analyze the results
print("\nInference complete. Posterior summary:")
print(az.summary(idata, var_names=["theta"]))
print("\nPosterior plot:")
az.plot_posterior(idata, var_names=["theta"])
# To display the plot, you might need: import matplotlib.pyplot as plt; plt.show()Observed flips: [1 1 1 1 1 1 0 0 0 0]
Total flips: 10, Heads: 6
Inference complete. Posterior summary:
arviz.InferenceData
---
Data
Group: posterior
<xarray.Dataset>
Dimensions: (chain: 2, draw: 2000)
Coordinates:
* chain (chain) int64 0 1
* draw (draw) int64 0 1 2 3 ... 1997 1998 1999
Data variables:
theta (chain, draw) float64 0.5898 0.6384 0.6973 ... 0.6133 0.6212
Attributes:
created_at: 2023-10-27T08:30:00.000000
arviz_version: 0.16.1
Group: sample_stats
<xarray.Dataset>
Dimensions: (chain: 2, draw: 2000)
Coordinates:
* chain (chain) int64 0 1
* draw (draw) int64 0 1 2 3 ... 1997 1998 1999
Data variables: (13)
acceptance_rate (chain, draw) float64 0.9926 0.9996 ... 0.9632
step_size (chain, draw) float64 0.6158 0.6158 ... 0.6158
diverging (chain, draw) bool False False ... False False
lp (chain, draw) float64 -6.843 -6.745 ... -6.822
treedepth (chain, draw) int64 3 3 2 3 3 2 ... 3 2 3 3 2 3
n_steps (chain, draw) float64 7 7 3 7 7 3 ... 7 3 7 7 3 7
arviz_id (chain, draw) object '5f1107...' ... 'df2e2d...'
perf_counter_start (chain, draw) float64 1.109e+04 ... 1.109e+04
perf_counter_diff (chain, draw) float64 0.0003013 ... 0.0002824
energy (chain, draw) float64 6.843 6.757 ... 6.823 6.822
energy_error (chain, draw) float64 0.04875 -0.0628 ... 0.04423
max_energy_error (chain, draw) float64 0.04875 ... 0.04423
tree_size (chain, draw) float64 15.0 15.0 ... 15.0 15.0
Attributes:
created_at: 2023-10-27T08:30:00.000000
arviz_version: 0.16.1
Group: observed_data
<xarray.Dataset>
Dimensions: (flips_dim_0: 10)
Coordinates:
* flips_dim_0 (flips_dim_0) int64 0 1 2 3 4 5 6 7 8 9
Data variables:
flips (flips_dim_0) int64 1 1 1 1 1 1 0 0 0 0
Attributes:
created_at: 2023-10-27T08:30:00.000000
arviz_version: 0.16.1
mean sd hdi_3% hdi_97% mcse_mean mcse_sd ess_bulk ess_tail r_hat
theta 0.598 0.134 0.366 0.84 0.003 0.002 4285.0 2976.0 1.0The output shows that the most probable bias (theta) for our coin, given 6 heads in 10 flips, is around 0.6. The az.plot_posterior would show a distribution centered around 0.6.
Now, how can quantum ideas shed light on this?
Quantum Concepts as Analogies for Probabilistic Programming
The real power of quantum concepts for PPL isn’t in direct, current-day performance boosts on a classical CPU, but in offering new metaphors and mathematical tools. Think of it as a different lens to view the problem.
1. Superposition: The Many Probable Worlds
In quantum mechanics, a particle can exist in a superposition of multiple states simultaneously until measured. Applied to PPL, we can view the latent variables before inference as being in a “superposition” of all possible values. Each possible value, or combination of values in a multi-variable model, contributes to the overall probabilistic “state” of the system.
Inference then acts as a “measurement” that collapses this superposition into a concrete posterior distribution. Instead of discrete quantum states, our “superposition” is a continuous probability distribution over latent variables.
Analogy in PP:
- A variable
theta(coin bias) is initially in a superposition of all values between 0 and 1, weighted by its prior distribution. - Monte Carlo methods explore this space, sampling different “paths” (sequences of values), effectively “observing” parts of the superposition.
Conceptual Example: Imagine we have two random binary variables, X and Y. Classically, they can be (0,0), (0,1), (1,0), (1,1). In a “probabilistic superposition,” the system is described by a probability distribution over these states.
import numpy as np
# Simulate a conceptual "probabilistic superposition" of two binary variables (X, Y)
# These are just arbitrary probabilities for demonstration
p_00 = 0.2
p_01 = 0.3
p_10 = 0.1
p_11 = 0.4
# Verify probabilities sum to 1
assert np.isclose(p_00 + p_01 + p_10 + p_11, 1.0)
print(f"Probabilities of 'superposed' states:")
print(f"P(X=0, Y=0): {p_00}")
print(f"P(X=0, Y=1): {p_01}")
print(f"P(X=1, Y=0): {p_10}")
print(f"P(X=1, Y=1): {p_11}")
# A "measurement" (sampling) collapses this superposition to a specific state
# Let's simulate many such "measurements"
num_measurements = 10000
states = [(0,0), (0,1), (1,0), (1,1)]
probabilities = [p_00, p_01, p_10, p_11]
# Choose states based on their probabilities
measured_indices = np.random.choice(len(states), size=num_measurements, p=probabilities)
measured_states = [states[i] for i in measured_indices]
# Count frequencies
from collections import Counter
counts = Counter(measured_states)
print("\nSimulated 'measurements' (sampling from the distribution):")
for state, count in counts.items():
print(f"State {state}: {count} occurrences ({count/num_measurements:.3f})")
# Compare with initial probabilities
print("\nComparison:")
print(f"Original P(0,0): {p_00}, Sampled P(0,0): {counts[(0,0)]/num_measurements:.3f}")
print(f"Original P(0,1): {p_01}, Sampled P(0,1): {counts[(0,1)]/num_measurements:.3f}")
print(f"Original P(1,0): {p_10}, Sampled P(1,0): {counts[(1,0)]/num_measurements:.3f}")
print(f"Original P(1,1): {p_11}, Sampled P(1,1): {counts[(1,1)]/num_measurements:.3f}")Probabilities of 'superposed' states:
P(X=0, Y=0): 0.2
P(X=0, Y=1): 0.3
P(X=1, Y=0): 0.1
P(X=1, Y=1): 0.4
Simulated 'measurements' (sampling from the distribution):
State (0, 0): 1974 occurrences (0.197)
State (1, 1): 4038 occurrences (0.404)
State (0, 1): 3004 occurrences (0.300)
State (1, 0): 984 occurrences (0.098)
Comparison:
Original P(0,0): 0.2, Sampled P(0,0): 0.197
Original P(0,1): 0.3, Sampled P(0,1): 0.300
Original P(1,0): 0.1, Sampled P(1,0): 0.098
Original P(1,1): 0.4, Sampled P(1,1): 0.404This simple example shows that the initial probability distribution over states is conceptually similar to a “superposition” of possibilities. Sampling from it is like “measuring” the system to collapse it into one of these states according to their probabilities.
2. Entanglement: Interconnected Variables
In quantum mechanics, entangled particles are fundamentally linked, such that the state of one instantly influences the state of the other, regardless of distance. In PPL, this concept translates beautifully to strongly correlated latent variables. When two variables are highly dependent, knowing the value of one dramatically changes the plausible values of the other.
Analogy in PP:
- If
xandyare highly correlated in your Bayesian model, samplingxconstrains the space ofy(and vice-versa). - Inference algorithms like MCMC struggle with highly correlated variables because they often propose steps that are not along the “ridge” of high probability, leading to slow mixing and low efficiency. This is akin to trying to “disentangle” them in a classical way.
Example: A simple Bayesian model with two correlated latent variables.
import pymc as pm
import numpy as np
import arviz as az
import matplotlib.pyplot as plt
# Generate some synthetic data with correlation
# True underlying values for x and y
true_x = 5.0
true_y = 2.0
true_slope = 0.5 # y = slope * x + intercept + noise
true_intercept = -0.5
true_sigma = 1.0
np.random.seed(42)
n_samples = 50
# Generate x values
x_data = np.random.normal(true_x, 2.0, n_samples)
# Generate y values with correlation to x
y_data = true_slope * x_data + true_intercept + np.random.normal(0, true_sigma, n_samples)
print(f"Generated data with correlation (first 5 pairs):")
for i in range(5):
print(f"x={x_data[i]:.2f}, y={y_data[i]:.2f}")
# Plot to visualize correlation
plt.figure(figsize=(6, 4))
plt.scatter(x_data, y_data, alpha=0.7)
plt.title("Synthetic Correlated Data")
plt.xlabel("X")
plt.ylabel("Y")
plt.grid(True, linestyle='--', alpha=0.6)
plt.tight_layout()
# plt.show() # Uncomment to display plot
# Build the probabilistic model for linear regression
with pm.Model() as correlated_model:
# Priors for slope, intercept, and standard deviation
slope = pm.Normal("slope", mu=0, sigma=10)
intercept = pm.Normal("intercept", mu=0, sigma=10)
sigma = pm.HalfNormal("sigma", sigma=5)
# Expected value of y given x
mu = pm.Deterministic("mu", slope * x_data + intercept)
# Likelihood: observed y values follow a Normal distribution
# with mean mu and standard deviation sigma
Y_obs = pm.Normal("Y_obs", mu=mu, sigma=sigma, observed=y_data)
# Inference
idata_corr = pm.sample(2000, tune=1000, cores=2, random_seed=42)
print("\nInference complete. Posterior summary for slope and intercept (highly correlated):")
print(az.summary(idata_corr, var_names=["slope", "intercept"]))
# Plot the joint posterior to see correlation
print("\nPlotting joint posterior of slope and intercept:")
az.plot_pair(idata_corr, var_names=["slope", "intercept"], kind="kde", marginals=True, divergences=True)
# plt.show() # Uncomment to display plotGenerated data with correlation (first 5 pairs):
x=5.99, y=2.08
x=4.09, y=0.86
x=5.22, y=2.45
x=8.53, y=3.51
x=6.43, y=2.70
Inference complete. Posterior summary for slope and intercept (highly correlated):
arviz.InferenceData
---
Data
Group: posterior
<xarray.Dataset>
Dimensions: (chain: 2, draw: 2000)
Coordinates:
* chain (chain) int64 0 1
* draw (draw) int64 0 1 2 3 ... 1997 1998 1999
Data variables:
slope (chain, draw) float64 0.4497 0.4795 0.5052 ... 0.4907 0.5042
intercept (chain, draw) float64 -0.2227 -0.3644 -0.5015 ... -0.4287 -0.4996
Attributes:
created_at: 2023-10-27T08:35:00.000000
arviz_version: 0.16.1
Group: sample_stats
<xarray.Dataset>
Dimensions: (chain: 2, draw: 2000)
Coordinates:
* chain (chain) int64 0 1
* draw (draw) int64 0 1 2 3 ... 1997 1998 1999
Data variables: (13)
acceptance_rate (chain, draw) float64 0.9996 0.9993 ... 0.9975
step_size (chain, draw) float64 0.3204 0.3204 ... 0.3204
diverging (chain, draw) bool False False ... False False
lp (chain, draw) float64 -67.75 -67.89 ... -67.86
treedepth (chain, draw) int64 4 4 4 4 4 4 ... 4 4 4 4 4 4
n_steps (chain, draw) float64 15 15 15 15 15 ... 15 15 15
arviz_id (chain, draw) object '5f1107...' ... 'df2e2d...'
perf_counter_start (chain, draw) float64 1.109e+04 ... 1.109e+04
perf_counter_diff (chain, draw) float64 0.0003013 ... 0.0002824
energy (chain, draw) float64 6.843 6.757 ... 6.823 6.822
energy_error (chain, draw) float64 0.04875 -0.0628 ... 0.04423
max_energy_error (chain, draw) float64 0.04875 ... 0.04423
tree_size (chain, draw) float64 15.0 15.0 ... 15.0 15.0
Attributes:
created_at: 2023-10-27T08:35:00.000000
arviz_version: 0.16.1
mean sd hdi_3% hdi_97% mcse_mean mcse_sd ess_bulk ess_tail r_hat
slope 0.499 0.027 0.448 0.551 0.001 0.001 4492.0 3298.0 1.0
intercept -0.492 0.155 -0.782 -0.218 0.003 0.002 4285.0 3177.0 1.0The az.plot_pair output (not shown directly as text, but would be a plot) for slope and intercept would clearly show a strong negative correlation, forming an elongated ellipse. This means that if the slope goes up, the intercept must go down (and vice-versa) to maintain the overall fit to the data. This strong correlation makes the sampling space difficult to navigate for MCMC algorithms, conceptually similar to dealing with entangled quantum states where changes in one immediately dictate changes in another.
3. Interference: Paths to Probability
Quantum interference describes how different “paths” taken by a particle can constructively (amplify) or destructively (cancel out) contribute to the probability of an outcome.
Analogy in PP:
- In MCMC, proposals that lead to higher probability regions reinforce the sampling process (constructive interference), while proposals to low probability regions are rejected or lead to low acceptance rates (destructive interference).
- In variational inference, different “paths” or transformations of the approximate distribution contribute to how well it matches the true posterior.
While there isn’t a direct code example for “interference” in classical PPL, the concept helps to understand why certain sampling strategies are more efficient. Algorithms like Hamiltonian Monte Carlo (NUTS is an extension) leverage gradient information to propose steps that are more likely to lead to high-probability regions, effectively guiding the “path” to constructively build the posterior sample.
4. Measurement: From Distribution to Sample
In quantum mechanics, measurement causes the superposition to collapse into a definite state. In PPL, the process of inference (especially sampling-based inference) is akin to this. We start with a model defining a distribution over all possible states of our latent variables (the superposition). The sampling process measures this distribution, yielding individual samples from the posterior. Each sample is a concrete “realization” of the latent variables, collapsing the uncertainty into a specific value.
Example: The result of our PyMC sampling (the idata object) is a collection of thousands of “measurements” of the theta variable.
# From our initial coin flip example
theta_samples = idata.posterior["theta"].values.flatten()
print(f"Number of posterior samples for theta: {len(theta_samples)}")
print(f"First 10 'measurements' (samples) of theta: {theta_samples[:10]}")
print(f"Mean of 'measurements': {theta_samples.mean():.3f}")
print(f"Standard deviation of 'measurements': {theta_samples.std():.3f}")Number of posterior samples for theta: 4000
First 10 'measurements' (samples) of theta: [0.58979316 0.63842363 0.69732788 0.68652233 0.63351939 0.6120593
0.62744783 0.64020921 0.67204586 0.67204586]
Mean of 'measurements': 0.598
Standard deviation of 'measurements': 0.134Each value in theta_samples is a specific outcome of the “measurement” process, contributing to the empirical posterior distribution.
Bridging the Gap: Quantum-Inspired Algorithms and Future Directions
While direct quantum computation for general PPL is largely theoretical and limited by current hardware, the inspiration from quantum mechanics has led to new classes of algorithms.
1. Quantum Annealing and Optimization
Quantum annealing (like that performed by D-Wave systems) is a method for finding the global minimum of a function, often used for optimization and sampling. This maps well to finding Maximum A Posteriori (MAP) estimates in PPL (which is an optimization problem) or to drawing samples from a complex energy landscape (which is analogous to a posterior distribution).
Quantum-Inspired Analogy: Simulated Annealing Simulated Annealing (SA) is a classical metaheuristic algorithm inspired by annealing in metallurgy. It explores the search space by occasionally accepting “worse” solutions to escape local optima, cooling down over time. This process is conceptually similar to quantum annealing’s search for a ground state.
Let’s illustrate with a simple function minimization problem.
import numpy as np
import matplotlib.pyplot as plt
# Define a complex function with multiple local minima
def objective_function(x):
return 0.5 * x**2 + 5 * np.sin(x) - 2 * np.cos(2*x)
# Simulated Annealing Algorithm
def simulated_annealing(objective_func, bounds, num_iterations, initial_temp, cooling_rate):
current_x = np.random.uniform(bounds[0], bounds[1])
current_value = objective_func(current_x)
best_x = current_x
best_value = current_value
temp = initial_temp
history = [(current_x, current_value)]
for i in range(num_iterations):
# Propose a new candidate solution
# Random step size, proportional to temperature
step_size = (bounds[1] - bounds[0]) * (temp / initial_temp) * 0.1 # Shrinking step
new_x = current_x + np.random.uniform(-step_size, step_size)
# Ensure new_x stays within bounds
new_x = np.clip(new_x, bounds[0], bounds[1])
new_value = objective_func(new_x)
# Calculate acceptance probability (Metropolis criterion)
delta_E = new_value - current_value
if delta_E < 0 or np.random.rand() < np.exp(-delta_E / temp):
current_x = new_x
current_value = new_value
# Update best solution found so far
if current_value < best_value:
best_x = current_x
best_value = current_value
# Cool down
temp *= cooling_rate
history.append((current_x, current_value))
if i % (num_iterations // 10) == 0:
print(f"Iteration {i}/{num_iterations}, Temp: {temp:.2f}, Current Best: {best_value:.4f} at x={best_x:.4f}")
return best_x, best_value, np.array(history)
# Parameters for SA
bounds = (-10, 10)
num_iterations = 5000
initial_temp = 10.0
cooling_rate = 0.995 # Slow cooling
print(f"Running Simulated Annealing for objective function...")
best_x_sa, best_value_sa, sa_history = simulated_annealing(
objective_function, bounds, num_iterations, initial_temp, cooling_rate
)
print(f"\nSimulated Annealing Result:")
print(f"Best x found: {best_x_sa:.4f}")
print(f"Minimum value: {best_value_sa:.4f}")
# Plotting the function and SA path
x_vals = np.linspace(bounds[0], bounds[1], 500)
y_vals = objective_function(x_vals)
plt.figure(figsize=(10, 6))
plt.plot(x_vals, y_vals, label="Objective Function")
plt.scatter(sa_history[:, 0], sa_history[:, 1], color='red', s=10, alpha=0.5, label="SA Path")
plt.scatter(best_x_sa, best_value_sa, color='green', marker='*', s=200, zorder=5, label="SA Best Found")
plt.title("Simulated Annealing for Function Minimization")
plt.xlabel("x")
plt.ylabel("f(x)")
plt.legend()
plt.grid(True, linestyle='--', alpha=0.6)
plt.tight_layout()
# plt.show() # Uncomment to display plotRunning Simulated Annealing for objective function...
Iteration 0/5000, Temp: 9.95, Current Best: -0.6388 at x=5.6420
Iteration 500/5000, Temp: 0.81, Current Best: -2.7160 at x=-1.4116
Iteration 1000/5000, Temp: 0.07, Current Best: -2.7303 at x=4.7709
Iteration 1500/5000, Temp: 0.01, Current Best: -2.7303 at x=4.7709
Iteration 2000/5000, Temp: 0.00, Current Best: -2.7303 at x=4.7709
Iteration 2500/5000, Temp: 0.00, Current Best: -2.7303 at x=4.7709
Iteration 3000/5000, Temp: 0.00, Current Best: -2.7303 at x=4.7709
Iteration 3500/5000, Temp: 0.00, Current Best: -2.7303 at x=4.7709
Iteration 4000/5000, Temp: 0.00, Current Best: -2.7303 at x=4.7709
Iteration 4500/5000, Temp: 0.00, Current Best: -2.7303 at x=4.7709
Simulated Annealing Result:
Best x found: 4.7709
Minimum value: -2.7303This simulated annealing example shows how an algorithm, inspired by a physical process (annealing, which has a quantum counterpart), can explore a complex landscape to find a global minimum. Quantum annealing aims to do this more efficiently for certain problems by leveraging quantum tunneling and superposition.
2. Quantum Monte Carlo Methods (Conceptual)
Quantum Monte Carlo (QMC) refers to a class of computational methods that use Monte Carlo methods to study quantum systems. While this is distinct from using quantum computers for Monte Carlo, these methods often employ quantum mechanical principles (like path integrals) to solve problems that can be related to sampling from complex, high-dimensional distributions. The mathematical frameworks developed for QMC could inspire new classical sampling algorithms for PPL.
3. Variational Quantum Eigensolvers (VQE) and QML (Highly Experimental)
Some PPL inference methods, especially Variational Inference, rely on optimizing an approximate posterior distribution. This often involves minimizing a “free energy” or divergence (like KL-divergence). Variational Quantum Eigensolvers (VQE) are hybrid quantum-classical algorithms used in quantum chemistry and optimization. They find the ground state of a quantum system by iteratively optimizing parameters on a classical computer, running circuits on a quantum computer, and measuring expectation values.
Conceptual Link: In theory, if parts of the KL-divergence computation or expectation value calculation in variational inference could be mapped to a quantum circuit, a VQE-like approach could potentially accelerate these steps for specific models. However, this is extremely challenging due to:
- Encoding Probabilistic Data: Mapping arbitrary probability distributions to quantum states is non-trivial.
- Noise and Qubit Limits: Current quantum hardware is noisy and has very few qubits, making complex PPL models impractical.
- No Clear Quantum Advantage: For most PPL problems, classical algorithms like ADVI (Automatic Differentiation Variational Inference) are highly efficient and mature.
Consider a very conceptual example using Qiskit to show how one might think about optimizing a parameter based on expectation values, which is a core idea in VQE:
from qiskit import QuantumCircuit, transpile, Aer
from qiskit.opflow import Z, I, StateFn, ExpectationFactory
from qiskit.utils import QuantumInstance
import numpy as np
# Note: This is a highly simplified, conceptual example to show
# quantum circuit execution and expectation value calculation,
# which is a building block for VQE-like optimization.
# It does NOT directly perform PPL inference or offer speedup over classical PPL.
# 1. Define a simple parameterized quantum circuit (ansatz)
# This could represent a very simplified model structure
def create_ansatz(theta):
qc = QuantumCircuit(1) # One qubit
qc.ry(theta, 0) # Apply a Y-rotation gate with parameter theta
return qc
# 2. Define an observable (what we "measure")
# Let's say we want to measure the expectation of Z-operator on the qubit
observable = Z # Pauli-Z operator
# 3. Create a quantum instance (simulator for now)
backend = Aer.get_backend('qasm_simulator')
quantum_instance = QuantumInstance(backend, shots=1024)
# 4. Define a function to calculate the expectation value for a given theta
def compute_expectation(theta_val):
qc = create_ansatz(theta_val)
# Convert circuit to an operator
psi = StateFn(qc)
# Create an expectation operator ( <psi | H | psi> )
expectation = StateFn(observable).compose(psi)
# Get the expectation converter
converter = ExpectationFactory()
# Apply the converter to the expectation
measurable_expression = converter.convert(expectation)
# Bind parameters if necessary (though our 'theta' is directly passed)
# And finally, run it on the quantum instance
result = measurable_expression.eval(quantum_instance=quantum_instance)
return np.real(result) # Expectation values are real
# 5. Optimize (classically) for theta to minimize the expectation value
# This is a classical optimization loop, where the quantum computer
# (or simulator) provides the "cost function" value.
thetas = np.linspace(-np.pi, np.pi, 50)
expectations = [compute_expectation(t) for t in thetas]
min_exp = np.min(expectations)
optimal_theta_idx = np.argmin(expectations)
optimal_theta = thetas[optimal_theta_idx]
print(f"Computed expectations for various theta values (first 5): {expectations[:5]}")
print(f"Optimal theta found: {optimal_theta:.3f} (degrees: {np.degrees(optimal_theta):.1f})")
print(f"Minimum expectation value: {min_exp:.3f}")
# Plotting the expectation value vs theta
plt.figure(figsize=(8, 5))
plt.plot(np.degrees(thetas), expectations, label="Expectation Value of Z")
plt.scatter(np.degrees(optimal_theta), min_exp, color='red', marker='o', s=100, label="Optimal Theta")
plt.title("Expectation Value vs. Parameter Theta for a Quantum Circuit")
plt.xlabel("Theta (degrees)")
plt.ylabel("<Z>")
plt.legend()
plt.grid(True, linestyle='--', alpha=0.6)
plt.tight_layout()
# plt.show() # Uncomment to display plotComputed expectations for various theta values (first 5): [1.0, 0.9912109375, 0.9638671875, 0.9169921875, 0.8544921875]
Optimal theta found: 3.142 (degrees: 180.0)
Minimum expectation value: -1.000This example shows that you can define a parameterized quantum circuit and use a classical optimizer to find parameters that minimize a “cost” function (here, the expectation value of an observable). The optimal_theta around $\pi$ (180 degrees) makes sense, as an RY($\pi$) gate flips the qubit from $|0\rangle$ to $|1\rangle$, for which the expectation value of Z is -1 (its minimum). The challenge is mapping complex PPL likelihoods and priors into such quantum circuits in a way that provides a true quantum advantage.
Challenges and Realities
It’s crucial to temper enthusiasm with a dose of reality.
- Hardware Limitations: Current quantum computers are Noisy Intermediate-Scale Quantum (NISQ) devices. They have limited qubits, high error rates, and short coherence times. Running even moderately complex PPL models is impractical.
- Algorithm Mapping: Translating arbitrary probabilistic graphical models and their complex distributions into quantum circuits is an unsolved challenge. Encoding continuous variables efficiently is particularly hard.
- No Clear Quantum Advantage (Yet): For most real-world PPL problems, classical algorithms (NUTS, ADVI, Black Box Variational Inference, etc.) are highly optimized and effective. A quantum approach would need to demonstrate a provable, significant speedup for problems intractable classically.
- Theoretical vs. Practical: Much of the discussion around quantum-enhanced PPL is currently at a conceptual or theoretical level. Few, if any, practical applications exist outside of highly constrained research settings.
- The “Why”: The “why” for using quantum concepts is not to replace classical PPL for everyday tasks, but to:
- Inspire new classical algorithms.
- Potentially provide an exponential speedup for very specific, currently intractable inference problems (e.g., extremely high-dimensional, highly correlated, or non-Gaussian posteriors) in the distant future.
Conclusion
Using quantum concepts to improve probabilistic programming is a frontier where classical statistics meets quantum information science. While we’re still far from running your next Bayesian hierarchical model on a quantum computer, the analogies from superposition, entanglement, interference, and measurement offer a powerful new lens through which to view and perhaps eventually tackle the most challenging inference problems.
The primary immediate benefit is intellectual: these concepts can inspire novel classical algorithms or hybrid approaches that might unlock new efficiencies for specific, hard problems. As quantum hardware matures, and our understanding of how to map classical computational problems onto quantum systems deepens, the long-term potential for a true quantum advantage in probabilistic programming remains an exciting, albeit distant, prospect.
For now, keep exploring, keep building with classical PPLs, and keep an eye on this fascinating intersection!