Quantum Probabilistic Modelling

In Machine LearningMachine Learning is an approach on solving problems by deriving the rules from data instead of explicitly programming.
Learn more about Machine Learning, probability is our compass for uncertainty. From Bayesian NetworksA Bayesian Network is a directed acyclic graph where nodes represent random variables and edges represent conditional dependencies between them. Each node has a probability distribution that quantifies how it depends on its parent nodes. The network allows efficient computation of joint, conditional, and marginal probabilities by exploiting these dependencies.
Learn more about Bayesian Network to Boltzmann MachinesA Boltzmann Machine is a type of stochastic (randomized) neural network that learns to represent complex data distributions by adjusting the weights between interconnected neurons. Each neuron has a binary state (on/off), and learning happens by minimizing the difference between observed data patterns and the network’s internal predictions using a probabilistic energy function. It’s mainly used as a building block for deeper models like Restricted Boltzmann Machines and Deep Belief Networks.
Learn more about Boltzmann Machine, we rely on probabilistic inferences to capture structures in data and make predictions. However, as problems grow in size and complexity, classical models reach their limits under the weight of dimensionality. Conclusions become impossible, correlations slow down convergence, and simulation costs explode.

The Weight Of Dimensionality

Coming soon...

What if probability wasn't something we had to laboriously calculate, but something we could derive directly from the physical world?

That is the promise of quantum probabilistic modeling. Quantum SystemsA quantum system is any physical system governed by the rules of quantum mechanics, where quantities like energy or spin are quantized (can take only specific values). Its behavior is described by a wavefunction that encodes probabilities of measurement outcomes rather than definite values. Unlike classical systems, it exhibits superposition and entanglement, meaning components can exist in multiple states simultaneously and be correlated across distance.
Learn more about Quantum System inherently generate probability distributions when measuredIn quantum computing, measurement is the process of extracting classical information from a quantum state. It collapses a qubit’s superposition into one of its basis states (usually or ), with probabilities determined by the amplitudes of those states. After measurement, the qubit’s state becomes definite, destroying the original superposition.
Learn more about Measurement allowing Machine LearningMachine Learning is an approach on solving problems by deriving the rules from data instead of explicitly programming.
Learn more about Machine Learning to draw from physics itselfBiamonte, J., 2017, Nature, Vol. 549, pp. 195-202. And as in the classical world, probability is only the beginning. Once we have distributions, we need structure, and once we have structure, we need causality. Quantum Machine LearningQuantum Machine Learning is the field of research that combines principles from quantum computing with traditional machine learning to solve complex problems more efficiently than classical approaches.
Learn more about Quantum Machine Learning evolves in the same layered way: from probability to Quantum Bayesian NetworksA Quantum Bayesian Network (QBN) is a generalization of a classical Bayesian network where probabilities are replaced by quantum probability amplitudes represented as density matrices. It models dependencies between quantum systems using quantum conditional probabilities instead of classical ones. This allows reasoning about quantum uncertainty and entanglement in a structured, graph-based way similar to how Bayesian networks represent classical uncertainty.
Learn more about Quantum Bayesian Network to Quantum Causal Models.A **Quantum Causal Model (QCM)** generalizes classical causal models (like Bayesian networks) to describe cause–effect relationships between quantum systems. Instead of using probabilities over classical variables, it uses **quantum states and quantum operations** to represent how interventions and influences propagate. This framework captures both **quantum correlations** (like entanglement) and **causal structure**, distinguishing true causation from mere quantum correlation.
Learn more about Quantum Causal Model

Let us consider two binary variables, and . To describe their Joint DistributionA **joint distribution** describes the probability of two or more random variables occurring together. It shows how the outcomes of one variable relate to the outcomes of another, either as a table (for discrete variables) or a function (for continuous ones). From it, you can derive marginal and conditional distributions by summing or integrating over specific variables.
Learn more about Joint Distribution , we need to specify four probabilities as depicted in ?: . A distribution over binary variables, , consists of probabilities. Inference tasks such as calculating the Marginal ProbabilityMarginal probability is the probability of a single event occurring, regardless of the outcomes of other variables. It’s found by summing or integrating joint probabilities over all possible values of the other variables. For example, ( P(A) = \sum_B P(A, B) ) gives the marginal probability of (A) from the joint distribution of (A) and (B).
Learn more about Marginal Probability or the Conditional ProbabilityConditional probability is the probability that an event occurs given that another event has already occurred. It’s written as , assuming . It measures how likely is when we know has happened, narrowing the sample space to cases where is true.
Learn more about Conditional Probability therefore require summing over an exponential number of entries. Approximate algorithms such as Markov Chain Monte CarloMarkov Chain Monte Carlo (MCMC) is a method for sampling from complex probability distributions when direct sampling is difficult. It builds a Markov chain whose long-run behavior matches the target distribution. By running the chain long enough, the collected samples approximate the true distribution, allowing estimation of expectations or probabilities.
Learn more about Markov Chain Monte Carlo can help, but they converge slowly when correlations are strongMurphy, K.P., 2014, MIT Press, .

This curse of dimensionality is not just annoying. It fundamentally limits the capabilities of classical Probability ModelsA **probability model** is a mathematical framework that describes all possible outcomes of a random process and assigns a probability to each one. It consists of a **sample space** (the set of all outcomes) and a **probability rule** (how likely each outcome is). The probabilities must be nonnegative and sum to 1 across all possible outcomes.
Learn more about Probability Model, regardless of how much computing power we provide them with. We need a fundamentally different representation of probability.

These equations show that the power of Quantum SystemsA quantum system is any physical system governed by the rules of quantum mechanics, where quantities like energy or spin are quantized (can take only specific values). Its behavior is described by a wavefunction that encodes probabilities of measurement outcomes rather than definite values. Unlike classical systems, it exhibits superposition and entanglement, meaning components can exist in multiple states simultaneously and be correlated across distance.
Learn more about Quantum System for probability modeling comes from how their state space scales. A single QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit yields two possible outcomes when measuredIn quantum computing, measurement is the process of extracting classical information from a quantum state. It collapses a qubit’s superposition into one of its basis states (usually or ), with probabilities determined by the amplitudes of those states. After measurement, the qubit’s state becomes definite, destroying the original superposition.
Learn more about Measurement, but a systemA quantum system is any physical system governed by the rules of quantum mechanics, where quantities like energy or spin are quantized (can take only specific values). Its behavior is described by a wavefunction that encodes probabilities of measurement outcomes rather than definite values. Unlike classical systems, it exhibits superposition and entanglement, meaning components can exist in multiple states simultaneously and be correlated across distance.
Learn more about Quantum System of QubitsA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit spans possible outcomes. Each outcome is associated with a complex AmplitudeIn quantum computing an amplitude is a complex number that describes the weight of a basis state in a quantum superposition. The squared magnitude of an amplitude gives the probability of measuring that basis state. Amplitudes can interfere, this means adding or canceling, allowing quantum algorithms to bias outcomes toward correct solutions.
Learn more about Amplitude, and the probability of observing that outcome is given by the squared magnitude of its AmplitudeIn quantum computing an amplitude is a complex number that describes the weight of a basis state in a quantum superposition. The squared magnitude of an amplitude gives the probability of measuring that basis state. Amplitudes can interfere, this means adding or canceling, allowing quantum algorithms to bias outcomes toward correct solutions.
Learn more about Amplitude.

Don't worry! The probabilistic perspective of quantum computing doesn't need to be complicated. In fact, it is The Best Explanation Of Quantum Systems To Start With.

The Best Explanation Of Quantum Systems To Start With

Coming soon...

This means that as the number of QubitsA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit grows, the Quantum StateA quantum state is the complete mathematical description of a quantum system, containing all the information needed to predict measurement outcomes. It’s usually represented by a wavefunction or a state vector in a Hilbert space. The state defines probabilities, not certainties, for observable quantities like position, momentum, or spin.
Learn more about Quantum State automatically describes a probability distribution over an exponentially large number of possibilities. Instead of having to explicitly store or compute all probabilities, the Quantum StateA quantum state is the complete mathematical description of a quantum system, containing all the information needed to predict measurement outcomes. It’s usually represented by a wavefunction or a state vector in a Hilbert space. The state defines probabilities, not certainties, for observable quantities like position, momentum, or spin.
Learn more about Quantum State encodes them natively in its AmplitudesIn quantum computing an amplitude is a complex number that describes the weight of a basis state in a quantum superposition. The squared magnitude of an amplitude gives the probability of measuring that basis state. Amplitudes can interfere, this means adding or canceling, allowing quantum algorithms to bias outcomes toward correct solutions.
Learn more about Amplitude. MeasurementIn quantum computing, measurement is the process of extracting classical information from a quantum state. It collapses a qubit’s superposition into one of its basis states (usually or ), with probabilities determined by the amplitudes of those states. After measurement, the qubit’s state becomes definite, destroying the original superposition.
Learn more about Measurement simply reveals samples from that distribution.

Figure 2 shows how a three-QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit Quantum CircuitA quantum circuit is a sequence of quantum gates applied to qubits, representing the operations in a quantum computation. Each gate changes the qubits’ state using quantum mechanics principles like superposition and entanglement. The final qubit states, when measured, yield the circuit’s computational result probabilistically.
Learn more about Quantum Circuit (a) spans a set of complex amplitudes (b) that correspond to a probability distribution of associated outcomes (c).

Quantum Computing As Probabilistic Programming

Coming soon...

Raw probabilities are often not enough for modeling. In practice, it is the structure that matters. Explicit representations of dependencies, conditional independence, and graphical relationships. Classical Bayesian NetworksA Bayesian Network is a directed acyclic graph where nodes represent random variables and edges represent conditional dependencies between them. Each node has a probability distribution that quantifies how it depends on its parent nodes. The network allows efficient computation of joint, conditional, and marginal probabilities by exploiting these dependencies.
Learn more about Bayesian Network achieve this by using graphs to encode the conditional structure.

Quantum Bayesian NetworksA Quantum Bayesian Network (QBN) is a generalization of a classical Bayesian network where probabilities are replaced by quantum probability amplitudes represented as density matrices. It models dependencies between quantum systems using quantum conditional probabilities instead of classical ones. This allows reasoning about quantum uncertainty and entanglement in a structured, graph-based way similar to how Bayesian networks represent classical uncertainty.
Learn more about Quantum Bayesian Network extend the same approach to Quantum SystemsA quantum system is any physical system governed by the rules of quantum mechanics, where quantities like energy or spin are quantized (can take only specific values). Its behavior is described by a wavefunction that encodes probabilities of measurement outcomes rather than definite values. Unlike classical systems, it exhibits superposition and entanglement, meaning components can exist in multiple states simultaneously and be correlated across distance.
Learn more about Quantum System. In a Quantum Bayesian NetworkA Quantum Bayesian Network (QBN) is a generalization of a classical Bayesian network where probabilities are replaced by quantum probability amplitudes represented as density matrices. It models dependencies between quantum systems using quantum conditional probabilities instead of classical ones. This allows reasoning about quantum uncertainty and entanglement in a structured, graph-based way similar to how Bayesian networks represent classical uncertainty.
Learn more about Quantum Bayesian Network,
• Nodes correspond to Quantum SystemsA quantum system is any physical system governed by the rules of quantum mechanics, where quantities like energy or spin are quantized (can take only specific values). Its behavior is described by a wavefunction that encodes probabilities of measurement outcomes rather than definite values. Unlike classical systems, it exhibits superposition and entanglement, meaning components can exist in multiple states simultaneously and be correlated across distance.
  Learn more about Quantum System, represented by density operators, and
• Edges capture conditional dependence through EntanglementEntanglement is a quantum phenomenon where two or more particles become correlated so that measuring one instantly determines the state of the other, no matter how far apart they are. This correlation arises because their quantum states are linked as a single system, not as independent parts. It doesn’t allow faster-than-light communication but shows that quantum systems can share information in ways classical physics can’t explain.
  Learn more about Entanglement.

For two Quantum SystemsA quantum system is any physical system governed by the rules of quantum mechanics, where quantities like energy or spin are quantized (can take only specific values). Its behavior is described by a wavefunction that encodes probabilities of measurement outcomes rather than definite values. Unlike classical systems, it exhibits superposition and entanglement, meaning components can exist in multiple states simultaneously and be correlated across distance.
Learn more about Quantum System and , the joint state factorizes as , where is the marginal state of , is the conditional state of given , and is the Leifer–Spekkens conditional productLeifer, M.S., 2013, Physical Review A, Vol. 88, pp. 052130. This mirrors the classical rule , generalized to density operators. Quantum Bayesian NetworksA Quantum Bayesian Network (QBN) is a generalization of a classical Bayesian network where probabilities are replaced by quantum probability amplitudes represented as density matrices. It models dependencies between quantum systems using quantum conditional probabilities instead of classical ones. This allows reasoning about quantum uncertainty and entanglement in a structured, graph-based way similar to how Bayesian networks represent classical uncertainty.
Learn more about Quantum Bayesian Network thus provide more than a joint distribution. They expose the dependency structure directly in the graph, making quantum probabilistic models both interpretable and amenable to efficient inference when the graph is sparse.

Probabilistic dependence is not the same as causalityPearl, J., 2000, Cambridge University Press, . In classical settings, Pearl's causal models account for this difference by formalizing interventions and counterfactuals through the do calculus. Quantum SystemsA quantum system is any physical system governed by the rules of quantum mechanics, where quantities like energy or spin are quantized (can take only specific values). Its behavior is described by a wavefunction that encodes probabilities of measurement outcomes rather than definite values. Unlike classical systems, it exhibits superposition and entanglement, meaning components can exist in multiple states simultaneously and be correlated across distance.
Learn more about Quantum System with EntanglementEntanglement is a quantum phenomenon where two or more particles become correlated so that measuring one instantly determines the state of the other, no matter how far apart they are. This correlation arises because their quantum states are linked as a single system, not as independent parts. It doesn’t allow faster-than-light communication but shows that quantum systems can share information in ways classical physics can’t explain.
Learn more about Entanglement and resulting nonclassical correlations require a broader frameworkChiribella, G., 2013, Physical Review A, Vol. 88, pp. 022318, such as Quantum Causal ModelsA **Quantum Causal Model (QCM)** generalizes classical causal models (like Bayesian networks) to describe cause–effect relationships between quantum systems. Instead of using probabilities over classical variables, it uses **quantum states and quantum operations** to represent how interventions and influences propagate. This framework captures both **quantum correlations** (like entanglement) and **causal structure**, distinguishing true causation from mere quantum correlation.
Learn more about Quantum Causal ModelAllen, J.M.A., 2017, Physical Review X, Vol. 7, pp. 031021.

A Quantum Causal ModelA **Quantum Causal Model (QCM)** generalizes classical causal models (like Bayesian networks) to describe cause–effect relationships between quantum systems. Instead of using probabilities over classical variables, it uses **quantum states and quantum operations** to represent how interventions and influences propagate. This framework captures both **quantum correlations** (like entanglement) and **causal structure**, distinguishing true causation from mere quantum correlation.
Learn more about Quantum Causal Model represents causal structure using process matrices, quantum Markov networks, or categorical approaches. Its elements can be summarized as:
• Nodes: Quantum SystemsA quantum system is any physical system governed by the rules of quantum mechanics, where quantities like energy or spin are quantized (can take only specific values). Its behavior is described by a wavefunction that encodes probabilities of measurement outcomes rather than definite values. Unlike classical systems, it exhibits superposition and entanglement, meaning components can exist in multiple states simultaneously and be correlated across distance.
  Learn more about Quantum System
• Edges: Quantum ChannelsA **quantum channel** is a mathematical model that describes how a quantum state changes when it’s transmitted or interacts with its environment. It represents any physical process affecting qubits, including noise, measurement, or decoherence. Formally, it’s a **completely positive, trace-preserving (CPTP) map** acting on density matrices.
  Learn more about Quantum Channel (completely positive maps)
• Power: capture both correlations and causal influence, including indefinite causal order, where it is meaningless to say A happens before BOreshkov, O., 2012, Nature Communications, Vol. 3, pp. 1092.

This framework generalizes classical causal reasoning to Quantum SystemsA quantum system is any physical system governed by the rules of quantum mechanics, where quantities like energy or spin are quantized (can take only specific values). Its behavior is described by a wavefunction that encodes probabilities of measurement outcomes rather than definite values. Unlike classical systems, it exhibits superposition and entanglement, meaning components can exist in multiple states simultaneously and be correlated across distance.
Learn more about Quantum System, where interventions are modeled not by conditional probabilities but by Quantum ChannelsA **quantum channel** is a mathematical model that describes how a quantum state changes when it’s transmitted or interacts with its environment. It represents any physical process affecting qubits, including noise, measurement, or decoherence. Formally, it’s a **completely positive, trace-preserving (CPTP) map** acting on density matrices.
Learn more about Quantum Channel.

However, the promising properties of these models are confronted with significant obstacles. Deeply parameterized circuits can represent rich probability distributionsBenedetti, M., 2019, Quantum Science and Technology, Vol. 4, pp. 043001, but their optimization is hampered by barren plateaus where gradients disappear. Real devices cause noise and decoherence, distorting the intended distributionsPreskill, J., 2018, Quantum, Vol. 2, pp. 79.

To make progress, researchers lean on hybrid strategies. Variational Quantum AlgorithmsA Variational Quantum Algorithm is a hybrid quantum–classical algorithm in which a quantum circuit is paramterized by a classical routine. This means, it usually computes the values for rotation angles used inside this parameterized quantum circuit during a classical pre-processing step. Additionally, the measurement results are interpreted during a classical post-processing.
Learn more about Variational Quantum Algorithm provide a practical bridge: Quantum CircuitsA quantum circuit is a sequence of quantum gates applied to qubits, representing the operations in a quantum computation. Each gate changes the qubits’ state using quantum mechanics principles like superposition and entanglement. The final qubit states, when measured, yield the circuit’s computational result probabilistically.
Learn more about Quantum Circuit generate candidate distributions, while classical optimizers tune parameters to minimize divergences.

Born MachinesA **Born Machine** is a quantum-inspired probabilistic model that generates data by sampling from a probability distribution defined by a quantum wavefunction. The probabilities come from the **Born rule**, which states that the likelihood of an outcome equals the squared magnitude of the wavefunction’s amplitude. In practice, it uses quantum states or their simulations to model complex data distributions that are difficult for classical systems to represent.
Learn more about Born Machine have been trained to reproduce the statistics of small image patchesZoufal, C., 2019, npj Quantum Information, Vol. 5, pp. 103. Quantum Boltzman MachinesA **Quantum Boltzmann Machine (QBM)** is a type of probabilistic neural network that uses **quantum states** instead of classical bits to represent and sample from probability distributions. It extends the **classical Boltzmann Machine** by exploiting **quantum superposition and tunneling** to explore the energy landscape more efficiently. In theory, this allows it to learn complex correlations between variables that are difficult for classical models to capture.
Learn more about Quantum Boltzman Machine have modeled molecular datasets with promising efficiency. On the causal side, quantum causal discovery algorithms have successfully reconstructed hidden structures from simulated quantum dataBerry, D.W., 2018, npj Quantum Information, Vol. 4, pp. 22.

Quantum probabilistic modeling transforms how we handle uncertainty. Instead of computing probabilities, we harvest them from physics. Structured models like Quantum Bayesian NetworksA Quantum Bayesian Network (QBN) is a generalization of a classical Bayesian network where probabilities are replaced by quantum probability amplitudes represented as density matrices. It models dependencies between quantum systems using quantum conditional probabilities instead of classical ones. This allows reasoning about quantum uncertainty and entanglement in a structured, graph-based way similar to how Bayesian networks represent classical uncertainty.
Learn more about Quantum Bayesian Network capture dependencies, and causal frameworks like Quantum Causal ModelsA **Quantum Causal Model (QCM)** generalizes classical causal models (like Bayesian networks) to describe cause–effect relationships between quantum systems. Instead of using probabilities over classical variables, it uses **quantum states and quantum operations** to represent how interventions and influences propagate. This framework captures both **quantum correlations** (like entanglement) and **causal structure**, distinguishing true causation from mere quantum correlation.
Learn more about Quantum Causal Model let us reason about interventions and counterfactuals.