Researchers map qubit states to probability distributions in phase space

Edward Jiang of ICFO, Institute of Photonic Sciences, Timothy Heightman of ICFO, Institute of Photonic Sciences, and Marcin Płodzień of Uniwersytet Jagielloński have developed a new framework for simulating quantum circuits by representing N-qubit states as probability distributions on a 2N-dimensional manifold. The researchers parameterize unitary gates as “normalizing flows” between these distributions, creating a new path for quantum circuit simulation using generative models. Evaluations on three quantum circuit families demonstrate the method’s accuracy exceeds that of current neural network methods, particularly as system sizes increase.

Phase Space Representation of Multi-Qubit States

This framework, detailed in recent work, maps the complex state of multiple qubits onto a more manageable, classically-understandable space, potentially simplifying simulations. The team’s method defines these distributions using the Q function, ensuring a positive probability distribution across the (S^2)^N manifold, a characteristic important for interpretation. Normalizing flows, originally developed for generative modeling, are employed to transform one probability distribution into another, effectively modeling the action of a quantum gate on the qubit state.

Unlike previous applications of normalizing flows to quantum systems, this work focuses on qubits, which possess a compact manifold phase space, and discrete gate operations rather than continuous dynamics. This distinction allows for the assignment of a unique normalizing flow to each gate within a circuit, composing them to simulate the entire quantum process.

The study’s findings suggest a promising avenue for classical simulation of quantum circuits, particularly as system sizes grow beyond the capabilities of conventional methods. The ability to accurately represent quantum states as probability distributions opens up possibilities for using classical machine learning techniques to analyze and predict the behavior of quantum systems.

The paper states, “In this contribution, we introduced a novel framework to learn the output state of a quantum circuit based on normalizing flows in the phase space representation,” and showed numerically that the method is able to scale well to large system sizes. Further research will likely focus on addressing the limitations of flow specificity and exploring the potential for broader applications of this phase space representation.

Normalizing Flows Model Quantum Circuit Evolution

Representing quantum circuit evolution as a series of probability distributions offers a new path toward scaling classical simulations, a challenge that has historically limited the complexity of circuits researchers can accurately model. This work details a method where each unitary gate within a quantum circuit is modeled as a normalizing flow, transforming one probability distribution into another, mirroring the state’s evolution. Unlike prior approaches, this framework operates on a continuous input space, circumventing the need for approximations often required when working with discrete domains.

The core of this advance lies in the phase space representation of multi-qubit systems, where an N-qubit state is depicted as a probability distribution across a 2N-dimensional manifold. This allows researchers to use tools from classical probability and generative modeling, specifically normalizing flows, to represent quantum states and their transformations.

The sequential nature of normalizing flows also provides a natural compatibility with the structure of quantum circuits, a departure from methods requiring retraining of a fixed architecture after each gate application. This composability is a key feature, streamlining the simulation process and potentially reducing computational overhead. Evaluations across three circuit families, 1D cluster state preparation, GHZ state preparation and ground state preparation of a 1D transverse-field Ising model, demonstrated the method’s ability to maintain accuracy as circuit size and depth increased.

Classical fidelity, a measure of how closely the model’s output matches the exact quantum state, exceeded that of two neural network baselines as the number of qubits grew. The framework addresses a critical bottleneck in quantum simulation: the exponential growth of the Hilbert space.

By shifting the challenge from representing the exponentially large quantum state to ensuring sufficient model expressivity within the normalizing flow, the method offers a potential pathway to simulate larger and more complex quantum circuits than previously possible. The researchers’ approach is not simply about achieving higher accuracy. It’s about changing the fundamental nature of the computational task. The team’s method is based on a continuous input space, which avoids the need for continuous relaxations or hand-crafted embeddings.

Future work could explore techniques to transfer knowledge between flows, potentially reducing the computational cost and expanding the method’s applicability to a wider range of quantum circuits and initial conditions. This advancement offers a promising avenue for classical simulation, particularly as quantum computers themselves continue to develop.

Composable Flows Approximate Probability Distributions

The core of this method lies in approximating the quantum state’s Q-function, a phase-space representation, by sequentially applying normalizing flows. Each flow transformation, denoted as $f_k$, maps the Q-function of one state to the next after a gate operation, using the previous state’s Q-function as the initial distribution. Each intermediate state is represented as a probability distribution, and unitary gates are represented as normalizing flow transformations that map from one distribution to the next.

This architecture’s composability is a key advantage, allowing the simulation of complex circuits without retraining the model after each gate application. Previous methods often required a fixed architecture to be variationally retrained following each discrete gate, a process that becomes increasingly demanding as circuit size and depth increase. The researchers state, “Our solution is therefore to represent the Q function of |${\psi }_k$⟩ as a normalizing flow using the Q function of as the initial distribution.”

The approach builds on existing work using normalizing flows for quantum systems, but diverges in its focus on closed, multi-qubit systems and the use of a continuous phase space. Previous applications of normalizing flows often targeted open continuous-variable systems, operating on a different phase space, $R^{\text{2N}}$, and evolving the flow in continuous time under Lindblad dynamics. While flow matching on spin Wigner functions has also been used to learn distributions over multi-qubit states, this work presents a distinct method for approximating quantum gate operations.

Each learned $f_k$ is therefore an approximate transport map from $Q_{k–1}$ to its convolution with $K_{\text{U\_k}}$, but this transport map does not carry over to different input states and would require retraining. This advancement offers a new perspective on tackling the complexities of quantum simulation, potentially opening the way to more efficient and scalable classical approximations of quantum systems.

Classical Fidelity Exceeds Neural Network Baselines

Unlike earlier methods, this framework’s architecture is naturally composable, mirroring the sequential structure of a quantum circuit, and avoids the need to retrain a fixed architecture after each gate operation. The team’s approach differs fundamentally in how it treats qubits, using the compact manifold (S^2)^N as their phase space, and in its handling of gate dynamics, which are modeled as convolutions rather than differential operators.

Instead of a single, evolving flow, the method assigns a normalizing flow to each gate and composes them sequentially. This allows for efficient estimation of fidelity for large system sizes, a challenge that often renders direct quantum fidelity calculations intractable. In experiments with the 1D cluster state and GHZ state, the new method was benchmarked against neural quantum states, which employs the TDVP algorithm to optimize parameters for solving the dynamics.

Comparisons were also drawn against a transformer architecture used to learn the IC-POVM probability vector of a state, evaluating it using a discrete version of classical fidelity. The results show that the classical fidelity between the exact Q function of each intermediate state and the learned distribution of the normalizing flow model remains high as the gate number increases for a 10-qubit VQE circuit preparing a TFIM ground state.

The researchers report, “We demonstrate our method on three common families of quantum circuits, and show that the classical fidelity of our model exceeds that of two neural network baselines.” The framework’s composability and avoidance of retraining further streamline the simulation process, offering a significant advantage over previous techniques.

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