Gabriele Lo Monaco - Università degli Studi di Palermo

The non-stabilizerness cost of quantum state estimation

Accurate estimation of quantum states and their properties is a central task in quantum information, with direct relevance to quantum learning, device calibration, and control. In this context, classical shadow tomography remains the golden standard. However, protocols based on random measurements demand extensive reconfigurability and can incur substantial overhead.  

Efficient reconstruction of quantum states with a fixed measurement setting is highly desirable for near-term experiments, as it avoids the repeated reconfiguration required by randomized protocols. In this work we study which physical resources are necessary for such single-setting schemes to become informationally complete. We consider an n-qubit input state, ancillary qubits, a fixed quantum circuit, and a fixed projective measurement in a stabilizer basis, and ask when the resulting effective POVM allows full state reconstruction.

We show first that stabilizer resources alone are fundamentally insufficient: any protocol based only on Clifford gates, any number of stabilizer ancillas, and fixed-basis readout is informationally equivalent to a projective measurement in a stabilizer basis and therefore cannot be informationally complete. We then quantify the minimal amount of non-stabilizerness needed to overcome this limitation. By doping the Clifford circuits with T-gates, we demonstrate that informational completeness requires at least 2n/log3 T-gates, while explicit constructions show that 2n T-gates are sufficient.  In doing so, we clarify the role of entanglement by observing that universal reconstruction is allowed only when maximal entanglement is generated by certain auxiliary Clifford circuits. 

Our results therefore single out the metrological role of non-stabilizerness in state reconstruction with fixed architectures and, more in general, connect resource theories of magic and entanglement with quantum state estimation, suggesting design principles for single-setting tomography. Following these guiding line, we introduce an algorithm to engineer minimally doped circuits that enable universal state reconstruction and can be efficiently calibrated under reasonable assumptions on device noise.