Researchers Recover Hamiltonians from Time Evolution with High Probability

Constantin Cedillo Vayson de Pradenne and colleagues at Harvard University, in collaboration with Caltech, have found that a unique approximate conserved observable exists and can be efficiently recovered for many local Hamiltonians. The findings show that any normalised, local observable orthogonal to the Hamiltonian exhibits a minimal commutator with the time-evolved operator, guaranteeing its identifiability. This provides a pathway to learning Hamiltonians even at extended time scales and offers a new understanding of thermalisation in quantum systems.

Improved Commutator Gap Enables Efficient Hamiltonian Recovery from Single Quantum Observations

Physicists have demonstrated a significant improvement in the lower bound of the commutator between a time-evolved operator, U(t) = exp(-iHt), and any normalised local observable, A, orthogonal to the Hamiltonian, H. This commutator, a measure of disagreement between quantum operations, is shown to scale from 1/2ⁿ to exceeding 1/poly(n), where ‘n’ represents the number of qubits. This scaling represents a key threshold, enabling efficient Hamiltonian recovery where previously only limited or specific cases were possible. The commutator, mathematically defined as [U(t), A] = U(t)A, AU(t), quantifies the extent to which the observable A fails to commute with the time evolution generated by the Hamiltonian. A larger commutator indicates a greater degree of change in the expectation value of A under the time evolution. The Frobenius norm squared of this commutator, ‖[U(t), A]‖_F², provides a measure of this disagreement. This advance allows scientists to determine the complete set of rules governing a quantum system’s energy and interactions from a single observation, even after a prolonged period of change. The ability to accurately determine the Hamiltonian is fundamental to understanding the system’s dynamics and predicting its future behaviour.

The ability to characterise quantum systems with fewer measurements is crucial for scaling up quantum technologies. Further analysis confirms that the infinite-temperature autocorrelation of any local observable, excluding the Hamiltonian, decays by at least an inverse-polynomial amount, demonstrating its distinct behaviour. This decay rate highlights the unique conservation properties of the Hamiltonian compared to other observables. An unknown n-qubit Hamiltonian is uniquely identifiable as the only approximately conserved local observable, enabling its recovery via data constructed from random inputs and ‘classical shadows’; a technique that efficiently estimates quantum properties with limited measurements. This dramatically reduces the computational burden, making it feasible to analyse systems even with many qubits and extended evolution times, opening new avenues for complex quantum simulations. The implications extend to areas such as quantum materials’ discovery, where understanding the Hamiltonian is essential for predicting material properties, and quantum error correction, where accurate Hamiltonian identification is crucial for designing robust quantum codes.

Hamiltonian identification via compressed sensing using random quantum projections

Classical shadows underpinned this work, serving as a clever way to sidestep the challenge of fully measuring a quantum system’s state. Rather than exhaustively probing every possible configuration, the technique generates numerous random ‘shadows’ of the quantum state, each a relatively simple measurement outcome. These shadows are essentially projections of the quantum state onto a set of randomly chosen basis states. Combining these shadows with classical computation allows scientists to efficiently reconstruct an approximate representation of the original quantum state, and, crucially, to estimate the properties needed to identify the Hamiltonian. The efficiency stems from the fact that only a few shadows are needed to accurately represent the quantum state, a principle known as compressed sensing. This is in contrast to traditional methods that require exponentially many measurements to fully characterise a quantum state.

The method circumvents the need for complete state measurement by generating numerous random projections, or shadows. It efficiently recovers the Hamiltonian, up to a scaling factor, from data built using these shadows and random product-state inputs, chosen to reduce computational demands compared to exhaustive state probing. The use of random product states further simplifies the computation by allowing for efficient estimation of expectation values. The work focused on local Hamiltonians, systems where interactions are limited to nearby qubits, and demonstrated the unique approximate conservation of observables orthogonal to the Hamiltonian, providing a mathematical basis for the technique’s success. Local Hamiltonians are particularly relevant in many physical systems, such as spin systems and quantum materials, making this result broadly applicable. The mathematical proof relies on establishing a lower bound on the commutator between the time-evolved operator and any observable orthogonal to the Hamiltonian, ensuring that the Hamiltonian can be uniquely identified from the observed data.

Single-shot Hamiltonian identification advances quantum system characterisation

Determining the rules governing a quantum system is vital for advances in both quantum computing and materials science. This research offers a pathway to identify a Hamiltonian, the complete description of a system’s energy and interactions, from observing its evolution just once, a significant improvement over methods demanding repeated measurements or precise timing. The ability to achieve this with a single observation dramatically reduces the experimental effort and time required to characterise a quantum system. However, the technique currently applies to ‘broad families of local Hamiltonians’, raising an important question regarding its scalability to more complex systems.

Acknowledging that this technique currently functions best with specific types of quantum systems, those described as ‘local Hamiltonians’, does not diminish its importance. A ‘Hamiltonian’ defines a system’s energy and interactions, and identifying it is key for building better quantum computers and designing new materials. This research provides a strong step towards doing so from a single observation, bypassing the need for multiple, precise measurements, and opening doors for future expansion to more complex scenarios. This achievement relies on a mathematical ‘gap’ ensuring the Hamiltonian stands out from other potential explanations of the system’s behaviour, a concept quantified by the commutator, measuring disagreement between quantum operations. Future work will likely focus on extending this technique to more general Hamiltonians, potentially by incorporating additional measurements or developing more sophisticated data analysis methods. The development of such methods would significantly broaden the applicability of this approach and unlock new possibilities for quantum system characterisation and control.

The researchers demonstrated that an unknown Hamiltonian governing an n-qubit system can be uniquely identified from observing its evolution only once. This is significant because determining a Hamiltonian is crucial for progress in quantum computing and materials science, and this method reduces the experimental effort needed for characterisation. The study proves that any observable differing from the Hamiltonian will exhibit a measurable response, allowing for its efficient recovery from data. The authors suggest future work will focus on extending this technique to a wider range of more complex Hamiltonians.