Quantum Circuit Born Machines for Distribution Learning

Classical generative models often construct an explicit probability distribution or use intricate architectures like GANs or VAEs. Quantum Circuit Born Machines (QCBMs) offer a different approach, harnessing the inherent probabilistic nature of quantum mechanics. The core idea is surprisingly direct: a parameterized quantum circuit (PQC) can be used to prepare a quantum state whose measurement statistics define the probability distribution we want to model.

The Born Rule as a Generative Engine

Recall the Born rule from fundamental quantum mechanics: if a quantum system is in state $|\psi\rangle$, the probability of measuring a specific outcome $x$ (represented by the basis state $|x\rangle$) is given by $P(x) = |\langle x | \psi \rangle|^2$.

A QCBM uses this principle directly. We start with an easily prepared initial state, typically the all-zeros state $|0\rangle^{\otimes n}$ for $n$ qubits. We then apply a PQC, denoted by a unitary operation $U(\theta)$, where $\theta$ represents the set of trainable circuit parameters (like rotation angles). This transforms the initial state into a parameterized state: $|\psi(\theta)\rangle = U(\theta) |0\rangle^{\otimes n}$

Measuring this state $|\psi(\theta)\rangle$ in the computational basis (the basis corresponding to classical bitstrings like $|001\rangle$, $|101\rangle$, etc.) yields an outcome $x$ (a specific bitstring) with probability: $p_\theta(x) = |\langle x | \psi(\theta) \rangle|^2 = |\langle x | U(\theta) |0\rangle^{\otimes n}|^2$

This equation defines the probability distribution $p_\theta(x)$ implicitly modeled by the QCBM. The circuit $U(\theta)$ acts as the generator, and the distribution arises naturally from quantum measurement. The goal of training is to adjust the parameters $\theta$ such that $p_\theta(x)$ closely approximates the target data distribution $p_{\text{data}}(x)$.

Basic workflow of a Quantum Circuit Born Machine (QCBM). Trainable parameters $\theta$ control a Parameterized Quantum Circuit (PQC) $U(\theta)$ applied to an initial state $|0\rangle^{\otimes n}$. Measuring the resulting state $|\psi(\theta)\rangle$ yields samples $x$ according to the probability distribution $p_\theta(x)$.

Training QCBMs

Since the QCBM defines a probability distribution $p_\theta(x)$, we can train it similarly to classical probabilistic models. We need a way to quantify the difference between the distribution generated by the QCBM, $p_\theta(x)$, and the target distribution derived from the training data, $p_{\text{data}}(x)$. Common approaches include minimizing a divergence or distance metric:

1. Kullback-Leibler (KL) Divergence: Measures the difference between two probability distributions. The objective is to minimize $KL(p_{\text{data}} || p_\theta)$. Calculating this often requires estimating both distributions. For discrete data, $p_{\text{data}}$ can be estimated from frequencies in the dataset. Evaluating $p_\theta(x)$ for all possible $x$ can be computationally intensive, especially for many qubits, as it requires calculating $|\langle x | U(\theta) |0\rangle^{\otimes n}|^2$ for potentially $2^n$ outcomes. Approximations or alternative cost functions are often necessary.

2. Maximum Mean Discrepancy (MMD): MMD measures the distance between distributions based on the difference in the mean embeddings of samples in a Reproducing Kernel Hilbert Space (RKHS). A common choice is the Gaussian kernel. The MMD loss compares statistics of samples drawn from $p_{\text{data}}$ (the real data) and samples generated by the QCBM ($p_\theta$). An advantage is that MMD can often be estimated efficiently using batches of samples from both distributions, avoiding the need to explicitly compute $p_\theta(x)$ for all $x$.

3. Maximum Likelihood Estimation (MLE): If we can efficiently evaluate $p_\theta(x)$ for specific data points $x_i$ from the training set, we can aim to maximize the log-likelihood $\sum_i \log p_\theta(x_i)$. However, calculating individual probabilities $p_\theta(x_i)$ can still be demanding.

Regardless of the chosen cost function $C(\theta)$, training involves finding the parameters $\theta^*$ that minimize it: $\theta^* = \arg \min_\theta C(\theta)$ This optimization is typically performed using gradient-based methods. Gradients $\nabla_\theta C(\theta)$ can be estimated using techniques like the parameter-shift rule (discussed in Chapter 4 for VQAs) or finite differences, often requiring multiple circuit executions per gradient component per optimization step. Classical optimizers like Adam or SGD are then used to update the parameters $\theta$.

PQC Architecture and Design

The choice of the PQC architecture $U(\theta)$ is significant for the QCBM's performance.

• Expressibility: The PQC must be capable of generating quantum states $|\psi(\theta)\rangle$ whose measurement statistics can accurately represent the target distribution $p_{\text{data}}(x)$. More complex distributions might require deeper circuits or specific entanglement structures.
• Trainability: As with VQAs, deep or highly unstructured PQCs can suffer from barren plateaus, where gradients vanish exponentially with the number of qubits, making training difficult or impossible. Hardware-efficient ansätze or problem-inspired structures are often preferred.
• Implicit Modeling: It's important to remember that the QCBM provides an implicit model. We don't get an analytical formula for $p_\theta(x)$. Instead, we get a procedure (run the circuit, measure) to sample from $p_\theta(x)$. This is similar to GANs but relies on quantum state preparation and measurement.

Sampling from a Trained QCBM

Once the QCBM is trained (i.e., we have found suitable parameters $\theta^*$), generating new samples that resemble the training data is straightforward:

1. Prepare the initial state $|0\rangle^{\otimes n}$.
2. Apply the trained PQC $U(\theta^*)$.
3. Measure the resulting state $|\psi(\theta^*)\rangle$ in the computational basis.
4. The measurement outcome $x$ is a sample drawn from the learned distribution $p_{\theta^*}(x)$. Repeating steps 1-3 generates multiple samples.

QCBMs represent a fundamental quantum approach to generative modeling. They directly map the structure of a quantum circuit and the physics of measurement to the task of learning and sampling from probability distributions. While challenges in calculating cost functions, estimating gradients, and ensuring trainability exist, they offer a potentially powerful alternative to classical methods, especially for distributions that might have complex correlations naturally captured by quantum states.