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arXiv:2011.08228 · Physical Review A

Experimental characterization of quantum processes: A selective and efficient method in arbitrary finite dimensions

Quimey Pears Stefano, Ignacio Perito, J. J. M. Varga, Lorena Rebón, Claudio Iemmi

Published May 28, 2021• 2 citations
arXivHTMLPDF

Abstract

The temporal evolution of a quantum system can be characterized by quantum process tomography, a complex task that consumes a number of physical resources scaling exponentially with the number of subsystems. An alternative approach to the full reconstruction of a quantum channel is the selective and efficient quantum process tomography, a method that allows estimating, individually and up to the required accuracy, each element of the matrix that describes the process, using only a polynomial amount of resources. The implementation of this protocol is closely related to the possibility of building a complete set of mutually unbiased bases (MUBs), whose existence is known only when the dimension of the Hilbert space is the power of a prime number. However, an extension of the method that uses tensor products of maximal sets of MUBs has been recently introduced. Here we explicitly describe how to implement the algorithm for a selective and efficient estimation of a quantum process in a nonprime power dimension and conduct, an experimental verification of the method in a Hilbert space of dimension $d=6$. This is the smallest space for which a complete set of MUBs is not known to exist, but it can be decomposed as a tensor product of two Hilbert spaces of dimensions ${D}_{1}=2$ and ${D}_{2}=3$, in which a complete set of MUBs is already known. The six-dimensional states were codified in the discretized transverse linear momentum of photons. The state preparation and detection stages are dynamically programed with the use of only-phase spatial light modulators, in a versatile experimental setup that allows one to implement the algorithm in any finite dimension.

DOI
10.1103/physreva.103.052438

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The temporal evolution of a quantum system can be characterized by quantum process tomography, a complex task that consumes a number of physical resources scaling exponentially with the number of subsystems. An alternative approach to the full reconstruction of a quantum channel is the selective and efficient quantum process tomography, a method that allows estimating, individually and up to the required accuracy, each element of the matrix that describes the process, using only a polynomial amount of resources. The implementation of this protocol is closely related to the possibility of building a complete set of mutually unbiased bases (MUBs), whose existence is known only when the dimension of the Hilbert space is the power of a prime number. However, an extension of the method that uses tensor products of maximal sets of MUBs has been recently introduced. Here we explicitly describe how to implement the algorithm for a selective and efficient estimation of a quantum process in a nonprime power dimension and conduct, an experimental verification of the method in a Hilbert space of dimension d=6d=6d=6. This is the smallest space for which a complete set of MUBs is not known to exist, but it can be decomposed as a tensor product of two Hilbert spaces of dimensions D1=2{D}_{1}=2D1​=2 and D2=3{D}_{2}=3D2​=3, in which a complete set of MUBs is already known. The six-dimensional states were codified in the discretized transverse linear momentum of photons. The state preparation and detection stages are dynamically programed with the use of only-phase spatial light modulators, in a versatile experimental setup that allows one to implement the algorithm in any finite dimension.
The characterization of quantum processes is a key tool in quantum information processing tasks for several reasons: on one hand, it allows one to acknowledge errors in the implementations of quantum algorithms; on the other, it allows one to characterize unknown processes occurring in nature. Bendersky, Pastawski, and Paz [A. Bendersky, F. Pastawski, and J. P. Paz, Phys. Rev. Lett. 100, 190403 (2008); Phys. Rev. A 80, 032116 (2009)] introduced a method to selectively and efficiently measure any given coefficient from the matrix description of a quantum channel. However, this method heavily relies on the construction of maximal sets of mutually unbiased bases (MUBs), which are known to exist only when the dimension of the Hilbert space is the power of a prime number. In this article, we lift the requirement on the dimension by presenting two variations of the method that work on arbitrary finite dimensions: one uses tensor products of maximal sets of MUBs, and the other uses a dimensional cutoff of a higher prime power dimension.
Alternatively to the full reconstruction of an unknown quantum process, the so-called selective and efficient quantum process tomography (SEQPT) allows estimating, individually and up to the required accuracy, a given element of the matrix that describes such an operation with a polynomial amount of resources. The implementation of this protocol has been carried out with success to characterize the evolution of a quantum system that is well described by a trace-preserving quantum map. Here, we deal with a more general type of quantum process that does not preserve the trace of the input quantum state, which naturally arises in the presence of imperfect devices and system-environment interactions, in the context of quantum information science or quantum dynamics control. In that case, we show with the aid of a priori information on the losses structure of the quantum channel that the SEQPT reconstruction can be adapted to reconstruct the non-trace-preserving map. We explicitly describe how to implement the reconstruction in an arbitrary Hilbert space of finite dimension ddd. The method is experimentally verified on a superconducting quantum processor provided by IBM Quantum services, by estimating several non-trace-preserving quantum processes in dimensions up to d=6d=6d=6. Our results show that it is possible to efficiently reconstruct non-trace-preserving processes, with high precision, and with significantly higher fidelity than when the process is assumed to be trace preserving.
Alternatively to the full reconstruction of an unknown quantum process, the so-called selective and efficient quantum process tomography (SEQPT) allows estimating, individually and up to the required accuracy, a given element of the matrix that describes such an operation with a polynomial amount of resources. The implementation of this protocol has been carried out with success to characterize the evolution of a quantum system that is well described by a trace preserving quantum map. Here, we deal with a more general type of quantum process that does not preserve the trace of the input quantum state, which naturally arises in the presence of imperfect devices and system-environment interactions, in the context of quantum information science or quantum dynamics control. In that case, we show that with the aid of {\it a priori} information on the losses structure of the quantum channel, the SEQPT reconstruction can be adapted to reconstruct the non-trace-preserving map. We explicitly describe how to implement the reconstruction in an arbitrary Hilbert space of finite dimension ddd. The method is experimentally verified on a superconducting quantum processor of the IBM Quantum services, by estimating several non trace-preserving quantum processes in dimensions up to d=6d=6d=6. Our results show that it is possible to efficiently reconstruct non trace-preserving processes, with high precision, and with significantly higher fidelity than when the process is assumed to be trace-preserving.
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.
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 the theory describing a new type of method for quantum process tomography. We describe an 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 main 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 photonic 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.
In this paper we describe in detail and generalize a method for quantum process tomography that was presented by Bendersky et al. [Phys. Rev. Lett. 100, 190403 (2008)]. The method enables the efficient estimation of any element of the $\ensuremath{\chi}$ matrix of a quantum process. Such elements are estimated as averages over experimental outcomes with a precision that is fixed by the number of repetitions of the experiment. Resources required implementing it scale polynomially with the number of qubits of the system. The estimation of all diagonal elements of the $\ensuremath{\chi}$ matrix can be efficiently done without any ancillary qubits. In turn, the estimation of all the off-diagonal elements requires an extra clean qubit. The key ideas of the method, which is based on efficient estimation by random sampling over a set of states forming a 2-design, are described in detail. Efficient methods for preparing and detecting such states are explicitly shown.