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arXiv:2010.00632 · Physical Review Letters

Robust and Efficient High-Dimensional Quantum State Tomography

Markus Rambach, Mahdi Qaryan, Michael J. Kewming, Christopher Ferrie, Andrew G. White, Jacquiline Romero

Published Mar 10, 2021• 77 citations
arXivHTMLPDF

Abstract

The exponential growth in Hilbert space with increasing size of a quantum system means that accurately characterizing the system becomes significantly harder with system dimension d. We show that self-guided tomography is a practical, efficient, and robust technique of measuring higher-dimensional quantum states. The achieved fidelities are over 99.9% for qutrits (d=3) and ququints (d=5), and 99.1% for quvigints (d=20)-the highest values ever realized for qudit pure states. We also show excellent performance for mixed states, achieving average fidelities of 96.5% for qutrits. We demonstrate robustness against experimental sources of noise, both statistical and environmental. The technique is applicable to any higher-dimensional system, from a collection of qubits through to individual qudits, and any physical realization, be it photonic, superconducting, ionic, or spin.

DOI
10.1103/physrevlett.126.100402

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Abstract High-dimensional time-frequency encodings have the potential to significantly advance quantum information science; however, practical applications require precise knowledge of the encoded quantum states, which becomes increasingly challenging for larger Hilbert spaces. Self-guided tomography (SGT) has emerged as a practical and scalable technique for this purpose in the spatial domain. Here, we apply SGT to estimate time-frequency states using a multi-output quantum pulse gate. We achieve fidelities of more than 99% for 3- and 5-dimensional states without the need for calibration or post-processing. We demonstrate the robustness of SGT against statistical and environmental noise, highlighting its efficacy in the photon-starved regime typical of quantum information applications.
Quantum tomography—characterising a quantum system by assigning it a quantum state—–is an essential building block for future technologies like quantum metrology or quantum computation. Ideal quantum tomography is accurate and robust to noise, while simultaneously being quick and efficient. The standard technique used for decades, full quantum tomography (FQT), only fulfills parts of these criteria and quickly becomes unfeasible, even for medium sized systems. As the number of possible states increases, the requirements on the amount of measurements to characterise the system and the storage of the results grow with it exponentially. The very feature that gives quantum mechanics its immense possibilities, hinders our capacity to describe it. Additionally, FQT is neither robust to noise nor to experimental errors, and generally fails in the presence of these environmental perturbations unless additional resources are used at the cost of efficiency.
Quantum state tomography is an important tool for quantum communication, computation, metrology, and simulation. Efficient quantum state tomography on a high dimensional quantum system is still a challenging problem. Here, we propose a novel quantum state tomography method, auxiliary Hilbert space tomography, to avoid pre-rotations before measurement in a quantum state tomography experiment. Our method requires projective measurements in a higher dimensional space that contains the subspace that includes the prepared states. We experimentally demonstrate this method with orbital angular momentum states of photons. In our experiment, the quantum state tomography measurements are as simple as taking a photograph with a camera. We experimentally verify our method with near-pure- and mixed-states of orbital angular momentum with dimension up to d=13d=13d=13, and achieve greater than 95 % state fidelities for all states tested. This method dramatically reduces the complexity of measurements for full quantum state tomography, and shows potential application in various high dimensional quantum information tasks.
Quantum state tomography is a process for estimating an unknown quantum state; which is innately probabilistic. The exponential growth of unknown parameters to be estimated is a fundamental difficulty in realizing quantum state tomography for higher dimensions. Iterative optimization algorithms like self-guided quantum tomography have been effective in robust and accurate ascertaining a quantum state even with exponential growth in Hilbert space. We propose a faster convergent simultaneous perturbation stochastic approximation algorithm which is more practical in a resource-deprived situation for determining the underlying quantum states by incorporating the Barzilai-Borwein two-point step size gradient method with minimal loss of accuracy.