Quantum States Reconstructed with Fewer Measurements

Fangjun Hu of QuEra Computing Inc and Harvard University, and colleagues, have achieved polynomial runtime and sample complexity for learning mixed quantum states within the “trivial phase”. Shallow circuits preserving local reversibility prepare these states, and measurement data efficiently learns them. Their algorithm generates a shallow local channel circuit approximating the unknown mixed state, utilising k + 1 steps, and represents a key improvement over previous methods requiring exponential resources. The work demonstrates that certain complex quantum states can be efficiently recreated using only measurement data, addressing a key obstacle in quantum information science.

It focuses on “trivial phase” mixed states, created by specific circuits maintaining local reversibility, simplifying the learning process. This advance establishes a foundation for building quantum generative models and better understanding quantum materials. Fangjun Hu and colleagues have achieved a sharp advance in recreating complex quantum states using only measurement data. Accurately modelling these states previously required exponentially increasing computational resources, but this work circumvents that limitation for “trivial phase” mixed states.

A mixed quantum state can be understood like a blurred photograph, where each possible image of a quantum system has a certain probability of being the true one. This breakthrough relies on algorithms that generate a shallow circuit, a short recipe with only a few steps, to approximate the unknown state, demanding only polynomial computational resources. This establishes a foundation for developing quantum generative models and gaining deeper insights into quantum materials.

Polynomial scaling achieved for learning trivial phase mixed quantum states

Runtime and sample complexity for learning mixed quantum states have now been reduced to polynomial, or quasi-polynomial, in the number of qubits, representing a substantial improvement over previous methods. Previous methods demanded exponential resources, but this advance focused on ‘trivial phase’ mixed states created by shallow circuits preserving local reversibility. Learning these states previously necessitated computational power scaling exponentially with system size, a limitation now overcome. The significance of this reduction lies in its potential to unlock simulations of larger and more complex quantum systems, previously intractable due to computational bottlenecks. The exponential scaling of prior approaches meant that even modest increases in the number of qubits would dramatically increase the computational cost, rendering accurate modelling impossible. This new work, however, offers a pathway to maintain feasibility as system sizes grow.

This establishes an important structural foundation for developing quantum generative models based on these specifically prepared states, opening avenues for simulating complex quantum systems. A new algorithm for classical diffusion models is inspired by this framework, requiring only a polynomial overhead in both training and generation. The algorithm learns to generate mixed states on a finite-dimensional lattice, concentrating on states in the trivial phase where a shallow preparation channel circuit preserves local reversibility. Diffusion models, commonly used in image generation, are adapted here to the quantum realm, leveraging the efficient learning of these trivial phase states to create a quantum analogue. This adaptation demonstrates the broader applicability of the findings beyond simply state reconstruction.

From measurement data alone, any mixed state within this class can be efficiently learned, with the algorithm outputting a shallow local channel circuit that approximately generates the state. Assuming constant or polylogarithmic circuit depth and gate locality, the sample complexity and runtime are polynomial or quasi-polynomial in the number of qubits. This provides a structural foundation for quantum generative models based on shallow channel circuits and offers insights into simplified quantum behaviour. The concept of ‘gate locality’ is crucial here, referring to the physical proximity of quantum gates during circuit execution. Maintaining locality is essential for practical implementation on near-term quantum hardware, as it minimises the impact of decoherence and other noise sources. The algorithm’s efficiency is therefore contingent on both the shallow circuit depth and the locality of the gates.

Efficient learning of locally reversible mixed quantum states facilitates advanced modelling

Recreating quantum states from data offers tantalising prospects for building more powerful quantum computers and simulating complex materials. Now available is a pathway to efficiently learn a specific type of mixed quantum state, those prepared by shallow circuits with a property called local reversibility. Identifying whether a given state actually belongs to this ‘trivial phase’ remains a significant hurdle, as the authors concede. The framework establishes a key foundation for developing more advanced quantum models, enabling the efficient recreation of specific mixed quantum states originating from shallow circuits maintaining local reversibility. Local reversibility ensures that information is not lost during the circuit preparation, simplifying the learning process and allowing for efficient reconstruction from measurement data. This property is not universally present in all quantum states, hence the focus on the ‘trivial phase’.

This simplifies the complex task of modelling quantum systems by bypassing computational limitations previously requiring exponential resources. By achieving polynomial scaling with the number of qubits, represented by k + 1, avenues for exploring the limits of this approach are now open, a significant improvement over previous exponential requirements. The value of k + 1 represents the number of steps in the generated shallow circuit, directly impacting the complexity of the approximation. Lower values of k + 1 lead to simpler circuits and faster learning, but may result in a less accurate representation of the original state. Finding the optimal balance between circuit complexity and accuracy is an ongoing area of research. This work establishes a pathway to efficiently recreate these states, offering a valuable tool for quantum simulation and model development.

The implications extend to materials science, where understanding the behaviour of electrons in complex materials requires accurate modelling of their quantum states. Mixed quantum states are often necessary to describe these systems realistically, as interactions between electrons lead to entanglement and superposition. The ability to efficiently learn and generate these states could accelerate the discovery of new materials with desired properties. Furthermore, the development of quantum generative models based on this framework could enable the design of novel quantum algorithms and protocols, pushing the boundaries of quantum information processing. The algorithm’s reliance on measurement data is particularly advantageous, as it aligns with the natural way quantum systems are probed in experiments. This allows for a direct connection between theoretical models and experimental observations, facilitating validation and refinement of the models.

Researchers demonstrated that mixed quantum states within a defined ‘trivial phase’ can be efficiently learned from measurement data alone. This is significant because it circumvents previous computational limitations requiring exponential resources for modelling these complex systems, instead achieving polynomial scaling with the number of qubits. The algorithm outputs a shallow local channel circuit that approximates the original state in trace distance, without needing prior knowledge of how the state was originally prepared. The authors suggest this framework also inspires an efficient algorithm for classical diffusion models, offering benefits for both quantum and classical computation.