System identification and inversion are two closely related problems, which lead to various configurations. In the quantum framework, system identification was only studied in the non-blind (or supervised) mode, i.e. with known input states (in addition to known output states). It is then called quantum process tomography. We here develop its blind (or unsupervised) version, operating with unknown (unentangled) input states (which are here not controlled by an automated loop). Our approach takes advantage of the blind quantum source separation (i.e. blind multi-qubit system inversion) methods based on disentanglement that we recently introduced. It is detailed for Heisenberg spin coupling and could be extended to other classes of processes. It estimates the parameter values of the Heisenberg Hamiltonian.
System identification and inversion are two closely related problems, which lead to various configurations. In the quantum framework, system identification was only studied in the non-blind (or supervised) mode, i.e. with known input states (in addition to known output states). It is then called quantum process tomography. We here develop its blind (or unsupervised) version, operating with unknown (unentangled) input states (which are here not controlled by an automated loop). Our approach takes advantage of the blind quantum source separation (i.e. blind multi-qubit system inversion) methods based on disentanglement that we recently introduced. It is detailed for Heisenberg spin coupling and could be extended to other classes of processes. It estimates the parameter values of the Heisenberg Hamiltonian.
Quantum process tomography (QPT) methods aim at identifying, i.e., estimating, a given quantum process. QPT is a major quantum information processing tool, since it especially allows one to characterize the actual behavior of quantum gates, which are the building blocks of quantum computers. However, usual QPT procedures are complicated, since they set several constraints on the quantum states used as inputs of the process to be characterized. In this paper, we extend QPT so as to avoid two such constraints. On the one hand, usual QPT methods require one to know, hence to very precisely control (i.e., prepare), the specific quantum states used as inputs of the considered quantum process, which is cumbersome. We therefore propose a blind, or unsupervised, extension of QPT (i.e., BQPT), which means that this approach uses input quantum states whose values are unknown and arbitrary, except that they are requested to meet some general known properties (and this approach exploits the output states of the considered quantum process). On the other hand, usual QPT methods require one to be able to prepare many copies of the same (known) input state, which is constraining. In contrast, we propose ``single-preparation BQPT methods'' (SBQPT) , i.e., methods which can operate with only one instance of each considered input state. These two concepts are here illustrated with practical (S)BQPT methods which are numerically validated, in the case when (i) random pure states are used as inputs and their required properties are especially related to the statistical independence of the random variables that define them and (ii) the considered quantum process is based on cylindrical-symmetry Heisenberg spin coupling. As a benchmark, we moreover introduce nonblind QPT methods dedicated to the considered Heisenberg process, we analyze their theoretical behavior (this requires the tools developed in this paper for random input states), and we numerically test their sensitivity to systematic and nonsystematic errors, which are most likely to occur in practice. This shows that, even for very low preparation errors (especially systematic ones), these nonblind QPT methods yield much lower performance than our SBQPT methods. Our blind and single-preparation QPT concepts may be extended, e.g., to a much wider class of processes and to SBQPT methods based on other quantum state properties, as outlined in this paper.
System identification, system inversion and its multiple-signal extension to source separation, including non-blind and blind (i.e. unsupervised) configurations, are major topics in classical, i.e. nonquantum, signal processing. Their application fields especially include communications, where they are often referred to as (possibly blind and MIMO) channel identification / estimation and channel equalization. Some of their configurations have also been extended to quantum channels. In this paper, we focus on such channels and on the most challenging configurations, namely blind and multiplequbit ones, hence with main emphasis on source separation. In this framework, two classes of methods were previously proposed for two-qubit configurations and a specific type of channels. We here first define their extensions to any number of qubits and general channels. For that configuration too, we then propose a third class of methods and we analyze its relationships with the previous two. This leads us to provide new general insight into the connections between the unentanglement of a pure quantum state and the statistical independence of associated measurement components.
Our recent investigations of blind quantum source separation and process tomography methods for Heisenberg-coupled quantum bits (qubits) were focused on introducing a new separation principle, based on output disentanglement. We here extend them by proposing a more advanced implementation of their cost function and optimization algorithm. This leads us to move from a feedback to a feedforward adapting block, which avoids potential issues related to feedback in quantum circuits. The number of quantum source state preparations required to blindly adapt the separating system is thus strongly decreased (roughly from 107to 104), yielding much faster adaptation.
Classical, i.e. non-quantum, blind source separation (BSS) methods estimate unknown source signals by using only mixed signals obtained by transferring these source signals through a mixing transform, which typically has unknown parameter values. We developed quantum versions of BSS (called BQSS) and we extended them to the blind estimation of the unknown mixing/coupling parameters, thus achieving blind quantum process tomography (BQPT). In these investigations, the mixing parameter values remained fixed over time. We here analyze configurations where they vary. We show that the operation of our previous system may then be extended to handle BQPT and simple BQSS configurations. BQSS with rapidly evolving mixtures is more complex, because the fundamental quantum no-cloning theorem entails that each quantum state available at the output of our separating system cannot be copied so as to be used as the inputs of both (i) the subsequent quantum system which exploits it and (ii) the block of our separating system which continuously uses these states to update the separating system parameter values. We avoid this issue by introducing a quite different BQSS system, which is based on a master-slave structure and performs parallel adaptation and source state restoration. The proposed approach is thus based on spin electronics for implementing the considered quantum states, and on an original control loop in the proposed master-slave structure, which requires a specific measurement procedure, due to the quantum nature of the signals to be processed. This paper therefore bridges the gap between advanced information processing functions (especially self-learning algorithms) and the physical devices, defined at the atomic level, required for implementing such computations.
Quantum process tomography (QPT) methods aim at identifying a given quantum process. QPT is a major quantum information processing tool, since it especially allows one to characterize the actual behavior of quantum gates, which are the building blocks of quantum computers. The present paper focuses on the estimation of a unitary process. This class is of particular interest because quantum mechanics postulates that the evolution of any closed quantum system is described by a unitary transformation. Unitary processes have significantly fewer parameters than general quantum processes (22nqb vs. 24nqb−22nqb real independent parameters for nqb qubits). By assuming that the process is unitary we develop two methods that scale better with the size of the system. In the present paper, we stay as close as possible to the standard setup of QPT: the operator has to prepare copies of different input states. The properties those states have to satisfy in order for our method to achieve QPT are very mild. Therefore, we choose to operate with copies of 2nqb initially unknown pure input states. In order to perform QPT without knowing the input states, we perform measurements on half the copies of each state, and let the other half be transformed by the system before measuring them (each copy is only measured once). This setup has the advantage of removing the issue of systematic (i.e., same on all the copies of a state) errors entirely because it does not require the process input to take predefined values. We develop a straightforward analytical solution that first estimates the states from the averaged measurements and then finds the unitary matrix (representing the process) coherent with those estimates by using our analytical solution to an extended version of Wahba’s problem. This estimate may then be used as an initial point for a fine tuning algorithm that maximizes the likelihood of the measurements. Simulation results show the effectiveness of the proposed methods.
We have developed an enhanced technique for characterizing quantum optical processes based on probing unknown quantum processes only with coherent states. Our method substantially improves the original proposal (Lobino et al 2008 Science 322 563), which uses a filtered Glauber–Sudarshan decomposition to determine the effect of the process on an arbitrary state. We introduce a new relation between the action of a general quantum process on coherent state inputs and its action on an arbitrary quantum state. This relation eliminates the need to invoke the Glauber–Sudarshan representation for states; hence, it dramatically simplifies the task of process identification and removes a potential source of error. The new relation also enables straightforward extensions of the method to multi-mode and non-trace-preserving processes. We illustrate our formalism with several examples, in which we derive analytic representations of several fundamental quantum optical processes in the Fock basis. In particular, we introduce photon-number cutoff as a reasonable physical resource limitation and address resource versus accuracy trade-off in practical applications. We show that the accuracy of process estimation scales inversely with the square root of photon-number cutoff.