Free-Fermionic States Tomography Strengthens Quantum States
Quantum State Characterisation Innovations Lead to Better Quantum Technologies
Advanced methods for learning, benchmarking, and certifying quantum systems are crucial to the rapid development of quantum technology. “Optimal Trace-Distance Bounds for Free-Fermionic States: Testing and Improved Tomography,” published in PRX Quantum, discusses theoretical advances with real-world applications. Lennart Bittel, Antonio Anna Mele, Jens Eisert, and Lorenzo Leone from Freie Universität Berlin's Dahlem Centre for Complex Quantum Systems lead this research on free-fermionic states, a common and easily described class of quantum states. Los Alamos National Laboratory provides assistance.
Understanding Free-Fermion States
Many disciplines of physics require free-fermionic states, also known as fermionic Gaussian states, from condensed matter and analogue quantum simulators to quantum chemistry. These states, often formed by one-dimensional matchgate circuits, are useful for quantum processing since they are non-trivial but classically simulable. The correlation matrix, a mathematical technique that captures all important state information, makes their description unique and effective. This matrix represents all observable two-point correlation functions, hence comprehending it is crucial to completely characterise free-fermionic states.
The Experimental Imperfections Challenge
Real-world investigations can only estimate the correlation matrix with finite accuracy due to measurement precision and sample size limits. The main question is: how does the correlation matrix estimate error affect the quantum states' trace-distance error? The trace distance, which measures the maximum likelihood of distinguishing two quantum states using any quantum measurement, is an important operational parameter. To ensure quantum device reliability and predictability, error propagation must be understood and measured. This latest study addresses this fundamental issue by providing optimal perturbation bounds for free-fermionic states that address this relationship.
New Optimal Bounds and Property Testing
The paper's key achievement is the formulation of new, optimal perturbation restrictions that accurately connect two free-fermionic states' correlation matrices to their trace distance. The authors demonstrate that the quantum state's trace-distance error scales linearly with ε if the correlation matrix has an error ε, and vice versa.
These strict constraints apply to pure and mixed free-fermionic states, even in more sophisticated scenarios where one state is not free-fermionic. These bounds, derived from “exact derivative calculations” and declared “essentially optimal,” are saturated for all pure 3-mode states and single-mode states. This is far better than earlier, less accurate pure state results in the literature.
Based on this, the study provides property testing to determine whether a device's output of an unknown quantum state is “close to” or “far from” the desired free-fermionic states. Quantum device certification and benchmarking require this verification. The work shows a major inefficiency: any algorithm designed to evaluate arbitrary (possibly mixed) free-fermionic states must have a sample complexity that grows exponentially with system size.
The “non-Gaussianity” of a truly random state cannot be efficiently quantified computationally. However, the work presents an effective technique for assessing low-rank free-fermionic states, a common and major subclass, which solves a large number of problems.
Superior Tomography and Noise Resilience
This study boosts quantum state tomography's efficiency in obtaining a comprehensive classical description of an unknown quantum state. Using n fermionic modes (qubit) and ε trace-distance precision, the sample complexity for pure free-fermionic states is significantly reduced from the prior best bound to an improved one. Importantly, the effective approach may be used to mixed states, an issue that requires samples and has yet to be solved.
Both methods use experimental data to estimate the correlation matrix to transfer this precision to the state's trace distance. Applying the new perturbation constraints follows. The authors note that the O(ε-2) scaling for precision ε is ideal for state tomography in the finite-shot measurement zone.
The study also proves their tomography algorithm's noise resistance. The unknown state is reliable even if it is not free-fermionic but close to the ideal set in terms of relative entropy of non-Gaussianity or trace distance. Experimentalists must address natural state preparation and device faults, therefore noise robustness is essential. This ensures that algorithms are implementable and explorable. Additionally, the study computes lower bounds to quantify “non-Gaussianity,” which is crucial for understanding quantum resources.
Future implications and directions
These theoretical advances enhance quantum technology certification, benchmarking, and learning performance guarantees, with significant practical implications. The created tools aim to improve tomography's accuracy and dependability by guaranteeing performance. With readily implementable fermionic temporal evolutions and local measurements, the recommended testing and learning algorithms are empirically viable for digital quantum computing environments and near-term fermionic analogue quantum simulators. Thus, it resolves unresolved fermionic and comparable bosonic difficulties, considerably enhancing free-fermionic pure-state tomography and extending efficient learning to mixed states.
The authors also note tomography and testing for states with a “small Gaussian extent” that can be described as a superposition of a few free-fermionic states as intriguing outstanding issues. An alternative approach is to test if the mixed-state tomography algorithm can be consistently applied to states near the free-fermionic states in trace distance rather than O(1/n) proximity.
