Quantum Science Study Reveals New Method for Estimating Physical Properties of Quantum States

Quantum science researchers have developed a new method for estimating the physical properties of quantum states from measurements, a fundamental task in the field. The method combines the maximum entropy method with tools from the emerging fields of classical shadows and quantum optimal transport. The study also introduces the use of Wasserstein distances in quantum tomography, a significant advancement that could revolutionize the field. The research provides a new perspective on the challenges in quantum tomography, highlighting the need for more physically motivated distinguishability measures and the limitations of existing methods.

What is the Fundamental Task in Quantum Science?

Quantum science is a complex field that revolves around the study of quantum states. One of the most fundamental tasks in this field is estimating the physical properties of quantum states from measurements. This task is crucial as it allows scientists to infer the expectation values of all quasilocal observables of a state from a number of copies that scale polylogarithmically with the system’s size and polynomially on the locality of the target observables.

This task is not straightforward and requires a combination of various methods and tools. In this study, the researchers achieved their results by combining the maximum entropy method with tools from the emerging fields of classical shadows and quantum optimal transport. The latter tool allows the researchers to fine-tune the error made in estimating the expectation value of an observable in terms of how local it is and how well they approximate the expectation value of a fixed set of few-body observables.

The researchers conjecture that their condition holds for all states exhibiting some form of decay of correlations. They establish this for several subsets thereof, including widely studied classes of states such as one-dimensional thermal and high-temperature Gibbs states of local commuting Hamiltonians on arbitrary hypergraphs or outputs of shallow circuits.

What is Quantum Tomography?

Quantum tomography is a subject that aims to devise methods for efficiently obtaining a classical description of a quantum system from access to experimental data. However, all tomographic methods for general quantum states inevitably require resources that scale exponentially in the size of the system. This scaling can be in terms of the number of samples required or the post-processing needed to perform the task.

Most of the physically relevant quantum systems can be described in terms of a quasilocal structure. These range from that of a local interaction Hamiltonian corresponding to a finite-temperature Gibbs state to that of a shallow quantum circuit. Locality is a physically motivated requirement that brings the number of parameters describing the system to a tractable number. Effective tomographic procedures should be able to incorporate this information.

How Does Quantum Tomography Work?

In many cases, one is interested in learning only physical properties of the state on which tomography is being performed. These properties are mostly encoded into the expectation values of quasilocal observables that often only depend on reduced density matrices of subregions of the system. By Helstrom’s theorem, obtaining a good recovery guarantee in trace distance is equivalent to demanding that the expectation value of all bounded observables are close for the two states.

It is desirable to design tomographic procedures that can take advantage of the fact that we wish to only approximate quasilocal observables instead of demanding a recovery in trace distance. Some methods in the literature take advantage of that. For instance, the overlapping tomography or classical shadows methods allow for approximately learning all k-local reduced density matrices of an n-qubit state with failure probability δ using O(eckklognδ1ϵ2) copies without imposing any assumptions on the underlying state.

What are the Challenges in Quantum Tomography?

Despite the advancements in quantum tomography, there are still challenges that need to be addressed. One of the main challenges is devising a tomography protocol that has a sample complexity that is logarithmic in system size and polynomial in the locality of the observables we wish to estimate. This is a tall order as even if we start from the assumption that the underlying state we wish to learn is a high-temperature product state with n qubits, the number of samples required to obtain an estimate that is ϵ close in trace distance from the target state scales like Ω(nϵ2).

To obtain a sample complexity that is logarithmic in system size, we cannot quantify closeness in trace distance and need to resort to more physically motivated distinguishability measures. Moreover, even for product states, the classical shadows protocol will fail to produce a good estimate for k-local observables if the number of samples is not exponential in k.

How Can These Challenges be Overcome?

Despite these challenges, the researchers provide an affirmative answer for the guiding question above for a large class of physically motivated states. They achieve this by combining two insights. First, they observe that recently introduced Wasserstein distances are better suited than the trace distance to estimate by how much the expectation values of physically motivated observables can differ on two states.

The researchers introduce these distances and motivate this claim. But in summary, the Wasserstein distances are a more effective tool for estimating the differences in expectation values of physically motivated observables on two states. This is a significant step forward in the field of quantum tomography and opens up new possibilities for further research and development.

What are the Implications of This Study?

This study has significant implications for the field of quantum science. It provides a new method for estimating the physical properties of quantum states from measurements, which is a fundamental task in quantum science. This method combines the maximum entropy method with tools from the emerging fields of classical shadows and quantum optimal transport, providing a more effective and efficient way to perform this task.

Furthermore, the study also provides a new perspective on the challenges in quantum tomography. It highlights the need for more physically motivated distinguishability measures and the limitations of existing methods such as the classical shadows protocol. This insight can guide future research in this field and lead to the development of more effective tomography protocols.

Finally, the study introduces the use of Wasserstein distances in quantum tomography. This is a significant advancement as it provides a better tool for estimating the differences in expectation values of physically motivated observables on two states. This could potentially revolutionize the field of quantum tomography and pave the way for new discoveries and advancements in quantum science.