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arXiv:1909.08401 · Physical Review A

Quantum process tomography with unknown single-preparation input states: Concepts and application to the qubit pair with internal exchange coupling

Yannick Deville, Alain Deville

Published Apr 27, 2020• 13 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.101.042332

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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.
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.
Quantum process tomography (QPT) methods aim at identifying a given quantum process. QPT is a major quantum information processing tool, since it 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. The standard approach of QPT is to measure copies of a particular set of predetermined (generally pure) states after they have been modified by the process to be identified. The main problem with this setup is that preparing an input state and setting it precisely to a predetermined value is challenging and thus yields errors. These errors can be decomposed into a sum of centered errors (i.e., whose average on all the copies is zero) and systematic errors that are the same for all the copies. The latter is often the main source of error in QPT. The algorithm we introduce works for any input states that make QPT theoretically possible (i.e., unless there are several solutions due to, e.g., a lack of diversity in the input states). The fact that we do not require the input states to be precisely set to predetermined values means that we can use a trick to remove the issue of systematic errors by considering that some states' copies are unknown but measured before they go through the process to be identified. We achieve this by splitting the copies of each input state into several groups and measuring the copies of the kthk\mathrm{th}kth group after they have successively been transferred through kkk instances of the process to be identified (each copy of each input state is used for only a single measurement). Using this approach, we can compute estimates of the measured states before and after they go through the process without using the knowledge we might have about the initial states. We test our algorithm with simulated data, and we assess its performance with a cnot gate on a trapped-ions qubit quantum computer.
Quantum Process Tomography (QPT) methods aim at identifying, i.e. estimating, a quantum process. QPT is a major quantum information processing tool, since it especially allows one to experimentally characterize the actual behavior of quantum gates, that may be used as the building blocks of quantum computers. We here consider unitary (possibly dense, i.e. without sparsity constraints) processes, which corresponds to isolated systems. Moreover, we develop a QPT method applicable to a significant number of qubits and hence to a high state space dimension, which allows one to tackle more complex problems. Using the unitarity of the process allows us to propose a method that first achieves part of QPT by performing an eigenanalysis of the estimated density matrix of a process output. We then complete the QPT by using another state differently. The relevance of our method is validated by means of simulations.
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 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 two-qubit 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. Usual QPT methods can be hard to implement in practice because they require the user to prepare copies of a set of known quantum states. In contrast, the proposed methods only require the user to be able to provide copies of a single unknown input state which can be estimated along with the process. However, we need to be able to make measurements of the state of the system after 5 different time delays ∆t, 2∆t, 3∆t, 4∆t, 5∆t(one measurement per copy). We detail a least square approach which exploits tools used in the aerospace community. A slower but more precise maximum likelihood method is also introduced. It uses a simple parametrization of unitary matrices.
A fundamental task in photonics is to characterise an unknown optical process, defined by properties such as birefringence, spectral response, thickness and flatness. Amongst many ways to achieve this, single-photon probes can be used in a method called quantum process tomography (QPT). Furthermore, QPT is an essential method in determining how a process acts on quantum mechanical states. For example for quantum technology, QPT is used to characterise multi-qubit processors and quantum communication channels; across quantum physics QPT of some form is often the first experimental investigation of a new physical process, as shown in the recent research into coherent transport in biological mechanisms. However, the precision of QPT is limited by the fact that measurements with single-particle probes are subject to unavoidable shot noise---this holds for both single photon and laser probes. In situations where measurement resources are limited, for example, where the process is rapidly changing or the time bandwidth is constrained, it becomes essential to overcome this precision limit. Here we devise and demonstrate a scheme for tomography which exploits non-classical input states and quantum interferences; unlike previous QPT methods our scheme capitalises upon the possibility to use simultaneously multiple photons per mode. The efficiency---quantified by precision per photon used---scales with larger photon number input states. Our demonstration uses four-photon states and our results show a substantial reduction of statistical fluctuations compared to traditional QPT methods---in the ideal case one four-photon probe state yields the same amount of statistical information as twelve single probe photons.