Quantum Tomography Sets Fast, Robust, and Efficient State Characterisation Records. The exponential growth of unknown parameters with increasing quantum system dimensions has long hindered quantum state tomography (QST), the crucial process of accurately characterising an unknown quantum state. This “curse of higher dimensionality” is nearly impossible for bigger systems due to the vast experimental resources needed, including multiple measurement settings and massive ensembles of identical quantum states. However, current advances in numerous fields of study are fundamentally transforming QST and enabling more advanced quantum technology by increasing robustness, speed, and efficiency.

New methods boost quantum state reconstruction efficiency. Novel strategies are addressing the resource barrier in QST, making it easier to characterise complex quantum states: Improved Compressive Threshold Tomography for Qudits Giovanni Garberoglio, Maurizio Dapor, Diego Maragnano, Marco Liscidini, and Daniele Binosi developed an effective quantum state tomography technique inspired by threshold quantum state tomography and compressed sensing. This method considerably reduces the number of measurement settings needed to reconstruct an N-qudit system's density matrix. Demonstrations on IBMQ confirmed successful reconstruction of GHZ, W, and random states with O(1), O(N^2), and O(N) settings for N < 7 qubit systems. This study covers quantum circuits, tomography, and verification.

Gaussian State Tomography Double-Exponential Gain

A novel tomography method for Gaussian quantum state reconstruction by Lennart Bittel, Francesco A. Mele, Jens Eisert, and Antonio A. Mele improves current methods by a twofold exponential factor.

An key shortcoming of older approaches is that this algorithm's accuracy is strikingly independent of state energy and photon count. High-energy states used in quantum computing, communication, sensing, and basic physics require energy independence.

Generalised heterodyne measurement produces a set number of measurements regardless of state energy and improves if the transposed state is accessible. The protocols scale better to larger, multi-mode systems.

Barzilai-Borwein Self-Guided Tomography Optimisation Syed Tihaam Ahmad, Ahmad Farooq, and Hyundong Shin enhanced Self-Guided Quantum Tomography (SGQT), an iterative optimisation technique beneficial in the noisy intermediate-scale quantum (NISQ) era due to its robustness and adaptable measurement bases.

While SGQT is efficient, its convergence may be slow as dimensions rise, requiring more quantum states. The researchers construct a quicker convergent SPSA algorithm using the Barzilai-Borwein (BB) two-point step size gradient approach. Unlike SGQT, which relies on empirically set hyper-parameters, the BB method adaptively decides its step size using first-order and implicit second-order cost function knowledge, speeding convergence in resource-constrained circumstances.

Numerical simulations reveal that BB-SGQT is robust to depolarising noise and converges faster than SPSA for less iterations, especially in bigger dimensions. Despite its superior resource efficiency, it may lose accuracy compared to SGQT over time when resources are abundant. The authors also note its advantages over post-processing methods like conjugate gradient-descent (CGD) and projected gradient-descent (PGD), which are incompatible with experiments and saturated in noisy environments.

Leveraging Auxiliary Information and Characterising Dynamic Systems

Researchers are studying dynamic quantum systems and new measuring methods in addition to static state reconstruction:

Parametrized Quantum State Tomography

Franz J. Schreiber, Jens Eisert, and Johannes Jakob Meyer introduced a paradigm for quantum state tomography that includes “parametrised quantum states” that vary constantly or depend on changing control parameters. They treat the entire family of states as a continuous entity rather than each state separately, which improves data usage. The concept uses compressed sensing to exploit structure in parameter dependence by combining signal processing and tomography.

This “plug-and-play” method can be utilised for parametrised quantum channels by replacing state tomography with process tomography. NMR shadow tomography of time-evolved states and free-fermionic Hamiltonians are examples.

Tomography with Auxiliary Systems: Efficiency

Wenlong Zhao and colleagues developed auxiliary system-based state tomography to address the exponential increase in measurement settings and sampling needs in large quantum systems. This method uses correlation or entanglement with a probabilistic classical or quantum auxiliary system. It helps measure the combined system to better understand the goal quantum state. This technique simplifies experimental procedures with only two measurement settings and a total sample complexity of O(d^2). It also offers purity measurement equipment with Heisenberg limit precision.

Quantum Technology Future Implications