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Physical Review A

Variational quantum process tomography of two-qubit maps

Reinaldo O. Vianna, Andrea Crespi, Roberta Ramponi, Roberto Osellame, Linda Sansoni, Giorgio Milani, Paolo Mataloni, Fabio Sciarrino

Published Mar 6, 2013• 5 citations
Publisher

Abstract

Full characterization of quantum states and processes is a fundamental requirement for verification and benchmarking of quantum devices. It has been realized in systems with few components, but for larger systems it becomes unfeasible because of the exponential growing with the system size of the number of measurements and the amount of computational power required to process them. A new approach for quantum state tomography and quantum process tomography, requiring a limited number of measurements, has been recently introduced by Maciel et al. [Quantum Inf. Comput. 12, 0442 (2012); Int. J. Mod. Phys. C 22, 1361 (2011).], namely variational quantum-process tomography (VQT). Here we adopt the VQT approach for the experimental characterization of two-qubit quantum processes and compare the reconstructed maps with those obtained by standard tomographic methods. The results demonstrate the high performance of this approach and propose VQT as a powerful alternative to the standard quantum process tomography.

DOI
10.1103/physreva.87.032304

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Full characterization of quantum states and processes is a fundamental requirement for verification and benchmarking of quantum devices. It has been realized in systems with few components, but for larger systems it becomes unfeasible because of the exponential growing with the system size of the number of measurements and the amount of computational power required to process them. A new approach for quantum state tomography and quantum process tomography, requiring a limited number of measurements, has been recently introduced by Maciel et al. [Quantum Inf. Comput. 12, 0442 (2012); Int. J. Mod. Phys. C 22, 1361 (2011).], namely variational quantum-process tomography (VQT). Here we adopt the VQT approach for the experimental characterization of two-qubit quantum processes and compare the reconstructed maps with those obtained by standard tomographic methods. The results demonstrate the high performance of this approach and propose VQT as a powerful alternative to the standard quantum process tomography.
Quantum process tomography is a fundamental and critical benchmarking and certification tool that is capable of fully characterizing an unknown quantum process. Standard quantum process tomography suffers from an exponentially scaling number of measurements and complicated data post-processing due to the curse of dimensionality. On the other hand, non-unitary operators are more realistic cases. In this work, we put forward a variational quantum process tomography method based on the supervised quantum machine learning framework. It approximates the unknown non-unitary quantum process utilizing a relatively shallow depth parametric quantum circuit and fewer input states. Numerically, we verified our method by reconstructing the non-unitary quantum mappings up to eight qubits in two cases: the weighted sum of the randomly generated quantum circuits and the imaginary time evolution of the Heisenberg XXZ spin chain Hamiltonian. Results show that those quantum processes could be reconstructed with high fidelities (>99%) and shallow depth parametric quantum circuits (d≤8), while the number of input states required is at least two orders of magnitude less than the demands of the standard quantum process tomography. Our work shows the potential of the variational quantum process tomography method in characterizing non-unitary operators.
Quantum process tomography is an experimental technique to fully characterize an unknown quantum process. Standard quantum process tomography suffers from exponentially scaling of the number of measurements with the increasing system size. In this work, we put forward a quantum machine learning algorithm which approximately encodes the unknown unitary quantum process into a relatively shallow depth parametric quantum circuit. We demonstrate our method by reconstructing the unitary quantum processes resulting from the quantum Hamiltonian evolution and random quantum circuits up to eight qubits. Results show that those quantum processes could be reconstructed with high fidelity, while the number of input states required are at least two orders of magnitude less than required by the standard quantum process tomography.
Several methods, known as quantum process tomography, are available to characterize the evolution of quantum systems, a task of crucial importance. However, their complexity dramatically increases with the size of the system. Here we present a new method for quantum process tomography. We describe a new algorithm that can be used to selectively estimate any parameter characterizing a quantum process. Unlike any of its predecessors this new quantum tomographer combines two virtues: it requires investing a number of physical resources scaling polynomially with the number of qubits and at the same time it does not require any ancillary resources. We present the results of the first implementation of this quantum device, characterizing quantum processes affecting two qubits encoded in heralded single photons. Even for this small system our method displays clear advantages over the other existing ones.
ventional full state tomography reaches its limit already for a few qubits and hence novel methods for the verification and benchmarking of quantum devices are called for. We show how the complete reconstruction of density matrices is possible even if one relies only on local information about the state. This results in an experimental effort that is linear in the number of qubits and efficient post-processing – in stark contrast to the exponential scaling of standard tomography. Whenever full tomography is not needed but instead less information required, one would expect that even fewer measurements suffice. Taking entanglement content of solid state samples and bosons in lattices as an example, we show how it may be quantified unconditionally using already routinely performed measurements only.
Abstract Quantum process tomography conventionally uses a multitude of initial quantum states and then performs state tomography on the process output. Here we propose and study an alternative approach which requires only a single (or few) known initial states together with time-delayed measurements for reconstructing the unitary map and corresponding Hamiltonian of the time dynamics. The overarching mathematical framework and feasibility guarantee of our method is provided by the Takens embedding theorem. We explain in detail how the reconstruction of a single-qubit Hamiltonian works in this setting and provide numerical methods and experiments for general few-qubit and lattice systems with local interactions. In particular, the method allows to find the Hamiltonian of a two qubit system by observing only one of the qubits.