Quantum Diffusion Models (2311.15444v1)

Abstract: We propose a quantum version of a generative diffusion model. In this algorithm, artificial neural networks are replaced with parameterized quantum circuits, in order to directly generate quantum states. We present both a full quantum and a latent quantum version of the algorithm; we also present a conditioned version of these models. The models' performances have been evaluated using quantitative metrics complemented by qualitative assessments. An implementation of a simplified version of the algorithm has been executed on real NISQ quantum hardware.

• The paper introduces quantum diffusion models that extend classical diffusion techniques to directly generate quantum states using parameterized quantum circuits.
• It presents a dual approach with full and latent models, applying iterative noise addition and learned reverse denoising to approximate target states.
• The study evaluates implementation challenges on NISQ devices, details training objectives via quantum variational methods, and assesses performance using fidelity metrics.

Quantum Diffusion Models (QDMs), as proposed in "Quantum Diffusion Models" (2311.15444), represent an extension of classical generative diffusion models to the quantum domain. The core objective is the direct generation of quantum states, $\rho$, leveraging parameterized quantum circuits (PQCs) in place of classical neural networks within the diffusion framework. This approach aims to harness the principles of diffusion—iterative noise addition followed by learned denoising—for tasks involving quantum state preparation and quantum generative modeling.

Model Architecture and Diffusion Process

The QDM framework adapts the forward and reverse processes characteristic of classical diffusion models.

Forward Process (Noise Addition): A target quantum state $\rho_0$ is gradually corrupted over a sequence of $T$ discrete time steps by applying quantum noise channels, $\mathcal{N}_t$. This process generates a sequence of increasingly noisy states $\{\rho_1, \rho_2, ..., \rho_T\}$, where $\rho_t = \mathcal{N}_t(\rho_{t-1})$. The noise schedule, often defined by parameters $\beta_t$, controls the amount of noise added at each step. The sequence is designed such that the final state $\rho_T$ approximates a tractable distribution, typically the maximally mixed state $\rho_T \approx \frac{I}{2^N}$ for an $N$-qubit system. The choice of quantum channel $\mathcal{N}_t$ is crucial; possibilities include depolarizing channels, amplitude damping, or phase damping channels, applied locally or globally. The overall forward process can be described by the transition kernel $q(\rho_t | \rho_{t-1})$.

Reverse Process (Denoising/Generation): The generative process starts from the maximally mixed state $\rho_T$ and iteratively applies a learned reverse quantum channel $\mathcal{M}_{\theta_t}$ to denoise the state, aiming to recover the original state distribution: $\rho_{t-1} \approx \mathcal{M}_{\theta_t}(\rho_t)$. This reverse process is parameterized by PQCs, denoted $U(\theta_t)$. The PQC takes the noisy state $\rho_t$ as input (potentially encoded via ancilla qubits or specific measurement schemes) and applies a unitary transformation $U(\theta_t)$, possibly followed by partial trace or other non-unitary operations derived from measurements, to approximate the denoised state $\rho_{t-1}$. The learnable parameters $\theta_t$ within the PQC $U(\theta_t)$ are optimized during training to accurately model the reverse transitions $p_\theta(\rho_{t-1} | \rho_t)$.

The PQC $U(\theta_t)$ serves as the quantum analogue of the neural network in classical diffusion models. Its architecture (ansatz) typically consists of layers of single-qubit rotations and multi-qubit entangling gates, where the rotation angles are functions of the parameters $\theta_t$ and potentially the time step $t$.

Full vs. Latent Quantum Diffusion Models

The paper proposes two main variants:

1. Full Quantum Diffusion Model: In this version, both the forward and reverse processes operate directly on quantum states residing in the Hilbert space. * Implementation: The forward process involves applying sequences of quantum channels. The reverse process uses PQCs $U(\theta_t)$ at each step $t$ to approximate the inverse of the noise channel $\mathcal{N}_t$. Training optimizes the parameters $\{\theta_t\}_{t=1}^T$ to maximize the likelihood of generating states from the target distribution, often via a quantum variational lower bound (ELBO) or by minimizing a quantum analogue of the score-matching objective. The input to the PQC at step $t$ is the quantum state $\rho_t$, and the output is intended to be $\rho_{t-1}$. * Challenges: Requires coherent manipulation and storage of quantum states throughout the diffusion process, making it demanding for NISQ hardware. Measurement and state tomography might be needed during training or sampling, adding overhead. Defining and implementing the parameterized reverse channel $\mathcal{M}_{\theta_t}$ based on the unitary $U(\theta_t)$ requires careful consideration (e.g., using ancilla-based methods or quantum channel models).

2. Latent Quantum Diffusion Model: This hybrid approach combines a classical diffusion model operating in a latent space with a quantum generative circuit. * Implementation: A classical diffusion model generates latent vectors $z \in \mathbb{R}^d$. These latent vectors are then mapped to target quantum states $|\psi(z)\rangle$ or $\rho(z)$ using a fixed or parameterized quantum circuit, $G_\phi$. This generator $G_\phi$ could be, for instance, a PQC trained variationally or potentially a form of Quantum Generative Adversarial Network (QGAN) generator. The diffusion process itself (noise addition and denoising) occurs entirely classically on the latent variables $z$. The forward process corrupts latent vectors $z_0$ to $z_T \sim \mathcal{N}(0, I)$, and the reverse process learns to map $z_t$ to $z_{t-1}$. Sampling involves running the classical reverse diffusion to get $z_0$ and then preparing the state $\rho(z_0)$ using $G_\phi$. * Advantages: Leverages mature classical diffusion techniques for the core diffusion dynamics, potentially simplifying training and implementation. The quantum component $G_\phi$ is only needed for the final state generation step. * Challenges: Requires designing an effective mapping from the latent space to the quantum state space. The expressivity and trainability of the quantum generator circuit $G_\phi$ are critical. The entanglement structure and specific properties of the target quantum states must be captured by this mapping.

Conditioned Quantum Diffusion Models

Conditioning allows generating quantum states that satisfy specific properties or belong to a certain class. Similar to classical conditional diffusion, information $c$ (e.g., desired energy level, specific entanglement pattern, classical label) can be incorporated into the QDM.

• Implementation: The conditioning information $c$ can be embedded into the PQC parameters $\theta_t$ or provided as an additional input to the PQC $U(\theta_t, c)$ at each step of the reverse process. For instance, parameters in the PQC could be made functions of $c$, or $c$ could be encoded onto auxiliary qubits that interact with the main system qubits during the unitary evolution $U(\theta_t, c)$. In the latent QDM, conditioning is typically handled within the classical diffusion model operating on the latent space $(z, c)$. The final quantum state $\rho(z_0, c)$ would then be generated based on the sampled conditioned latent vector $z_0$.

Implementation Details and Training

Training Objective: The training typically aims to optimize the parameters $\theta = \{\theta_t\}_{t=1}^T$ of the reverse process PQCs. This can be framed as maximizing a variational lower bound (ELBO) on the log-likelihood of the target quantum state distribution, adapted to the quantum setting. Alternatively, a loss function based on minimizing the difference between the PQC-driven reverse transition $p_\theta(\rho_{t-1} | \rho_t)$ and the true posterior $q(\rho_{t-1} | \rho_t, \rho_0)$ can be formulated. This often simplifies to a form analogous to score matching, possibly involving minimizing the distance (e.g., trace distance or Hilbert-Schmidt distance) between the output of the PQC and the expected denoised state. $$L(\theta) = \sum_{t=1}^T E_{q(\rho_0)} E_{q(\rho_t | \rho_0)} [ D( q(\rho_{t-1} | \rho_t, \rho_0) || p_\theta(\rho_{t-1} | \rho_t) ) ]$$, where $D$ is a suitable distance measure between quantum states or channels.

Gradient Computation: Gradients of the loss function with respect to the PQC parameters $\theta_t$ are required for optimization. The parameter-shift rule or other quantum gradient estimation techniques (e.g., finite differences, linear response theory) can be employed. This typically involves executing the PQC multiple times with shifted parameters and measuring appropriate observables.

Sampling: Generating a new quantum state involves starting from the maximally mixed state $\rho_T$ and iteratively applying the trained reverse PQCs $U(\hat{\theta}_t)$ for $t = T, T-1, ..., 1$.

function sample_qdm(T, trained_pqcs U_theta): // Initialize state to maximally mixed state rho = I / (2**N) for t from T down to 1: // Apply the trained PQC for step t // This might involve encoding rho_t and applying U_theta_t // Or directly applying a learned channel M_theta_t rho = apply_reverse_step(rho, U_theta[t], t) return rho // The generated quantum state rho_0

NISQ Implementation: The paper reports an implementation of a simplified QDM on real NISQ hardware. This likely involved:

• Small number of qubits (e.g., 2-4 qubits).
• Shallow PQC ansätze to mitigate decoherence and gate errors.
• A small number of diffusion steps $T$.
• Simplified noise models (e.g., local depolarizing noise).
• Extensive measurement and error mitigation techniques. The evaluation would compare the experimentally generated states against classically simulated ideal states or target states using metrics like fidelity. Challenges include PQC trainability (barren plateaus), gate infidelity, readout errors, and limited qubit connectivity.

Evaluation Metrics

Performance evaluation combines quantitative and qualitative methods:

• Quantitative:
  • Fidelity: $$F(\rho_{gen}, \rho_{target}) = (\text{Tr}[\sqrt{\sqrt{\rho_{target}} \rho_{gen} \sqrt{\rho_{target}}}])^2$$. Measures the closeness between the generated state $\rho_{gen}$ and the target state $\rho_{target}$.
  • Trace Distance: $$T(\rho_{gen}, \rho_{target}) = \frac{1}{2} ||\rho_{gen} - \rho_{target}||_1$$. Provides another measure of distinguishability.
  • Observable Expectation Values: Comparing $\langle O \rangle_{\rho_{gen}}$ with $\langle O \rangle_{\rho_{target}}$ for relevant observables $O$ (e.g., Hamiltonians, entanglement witnesses).
  • Entanglement Measures: Quantifying entanglement (e.g., concurrence, negativity) in generated states if the target states are entangled.
• Qualitative: Assessing properties of the generated state ensemble, such as the distribution of measurement outcomes in a specific basis or visualizing state representations (e.g., Bloch sphere for single qubits, Q-functions).

Practical Implications and Applications

QDMs offer a potentially powerful framework for generative tasks in the quantum domain:

• Quantum State Preparation: Generating specific ground states of Hamiltonians, preparing resource states for quantum computation (e.g., cluster states), or initializing quantum algorithms. Conditioned QDMs could prepare states with specific energy or entanglement properties.
• Quantum Simulation: Learning distributions of states relevant to physical systems, potentially aiding in the paper of many-body physics or quantum chemistry.
• Quantum Machine Learning: Serving as generative models within broader QML pipelines, for tasks like anomaly detection or data augmentation on quantum datasets.
• Error Mitigation: Potentially learning to reverse the effects of noise channels, although this application requires further investigation.

Compared to other quantum generative models like QGANs or Quantum Boltzmann Machines, QDMs might offer more stable training dynamics, similar to their classical counterparts, although training PQCs remains challenging. The iterative refinement process might allow for generating complex states with high fidelity. However, the computational cost, particularly for the full QDM requiring coherent quantum evolution, can be significant in terms of circuit depth and coherence times. The latent QDM shifts some burden to classical computation but relies heavily on the effectiveness of the classical-to-quantum mapping.

Limitations and Future Directions

Current limitations primarily stem from NISQ hardware constraints: qubit count, coherence times, gate fidelities, and connectivity severely restrict the size ($N$) and depth ($T$ and PQC depth) of implementable QDMs. Barren plateaus can hinder the training of deep PQCs. The theoretical understanding of QDM convergence, expressivity, and the optimal choice of quantum noise channels and PQC ansätze requires further development.

Future research directions include:

• Developing more hardware-efficient PQC ansätze and training methods for QDMs.
• Exploring alternative quantum noise models and diffusion schedules tailored for specific quantum systems.
• Rigorous theoretical analysis of QDM properties and comparison with other quantum generative approaches.
• Applying QDMs to specific problems in physics, chemistry, and materials science.
• Investigating fault-tolerant implementations of QDMs.
• Improving the classical-to-quantum mapping in latent QDMs.

Conclusion

Quantum Diffusion Models (2311.15444) introduce a novel approach to quantum state generation by adapting the classical diffusion paradigm. Utilizing Parameterized Quantum Circuits for the reverse denoising process, QDMs offer pathways (full quantum and latent) to iteratively construct target quantum states from noise. While practical implementation faces significant hurdles on current NISQ devices, the framework presents a promising direction for generative modeling in quantum computation and simulation, with potential applications ranging from state preparation to quantum machine learning. Further research into efficient implementations, theoretical properties, and practical applications will be crucial to realizing their full potential.