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Symmetry

Parametric Symmetries in Architectures Involving Indefinite Causal Order and Path Superposition for Quantum Parameter Estimation of Pauli Channels

Francisco Delgado

Published May 16, 2023• 4 citations
Publisher

Abstract

Parameter estimation for devices containing or supporting quantum systems is a field of quantum metrology using quantum probe states to reach their characterization. Pauli channels are ideal structures where qubits are transmitted or contained, commonly altering them with specific fingerprints. The ultimate limit imposed on such estimation is addressed using the quantum Fisher information, stating a lower bound for it. Although the most simple scheme suggests performing such an estimation directly using the individual channel, other approaches have shown improved outcomes by repeating identical copies of the channel for the characterization, or otherwise those connected inside of specific circuit arrangements. These connections commonly include path superposition or causal indefinite architectures. In addition, other improvements have been observed in concrete channels when complementary unitary controls are included. The current research analyses the complete set of Pauli channels under some of those architectures in a comparative approach to reach a better estimation, thus stating hierarchies. It is observed that the use of those unitary controls notably improves previous outcomes by several orders of magnitude.

DOI
10.3390/sym15051097

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Parameter estimation for devices containing or supporting quantum systems is a field of quantum metrology using quantum probe states to reach their characterization. Pauli channels are ideal structures where qubits are transmitted or contained, commonly altering them with specific fingerprints. The ultimate limit imposed on such estimation is addressed using the quantum Fisher information, stating a lower bound for it. Although the most simple scheme suggests performing such an estimation directly using the individual channel, other approaches have shown improved outcomes by repeating identical copies of the channel for the characterization, or otherwise those connected inside of specific circuit arrangements. These connections commonly include path superposition or causal indefinite architectures. In addition, other improvements have been observed in concrete channels when complementary unitary controls are included. The current research analyses the complete set of Pauli channels under some of those architectures in a comparative approach to reach a better estimation, thus stating hierarchies. It is observed that the use of those unitary controls notably improves previous outcomes by several orders of magnitude.
Quantum Parameter Estimation (QPE) is commonly led using quantum probe states for the characterization of quantum systems. For these purposes, Quantum Fisher Information (QFI) plays a crucial role by imposing a lower bound for the parametric estimation of quantum channels. Several schemes for obtaining QFI lower bounds have been proposed, particularly for Pauli channels regarding qubits. Those schemes commonly employ either the individual channel, multiple copies of it, or arrangements including communication architectures. The present work aims to propose an architecture involving path superposition and causal indefinite order in superposition. Thus, by controlling the symmetry balance of this superposition, it reaches notable improvements in quantum parameter estimation. The proposed architecture has been tested to find the best possible QPE bounds for a representative and emblematic set of Pauli channels. Further, for the most reluctant channels, it was revisited testing the architecture again under a primary path superposition (using double teleportation) and also using entangled probe states to recombine their outputs with the original undisturbed state. Notable outcomes practically near zero were found for the QPE bounds, stating a hierarchy between the approaches, but anyway reaching a perfect theoretical QPE, particularly for the last path superposition including the proposed architecture.
Quantum Fisher Information is considered in Quantum Information literature as the main resource to determine a bound in the parametric characterization problem of a quantum channel by means of probe states. The parameters characterizing a quantum channel can be estimated until a limited precision settled by the Cramér–Rao bound established in estimation theory and statistics. The involved Quantum Fisher Information of the emerging quantum state provides such a bound. Quantum states with dimension d=2, the qubits, still comprise the main resources considered in Quantum Information and Quantum Processing theories. For them, Pauli channels are an important family of parametric quantum channels providing the most faithful deformation effects of imperfect quantum communication channels. Recently, Pauli channels have been characterized when they are arranged in an Indefinite Causal Order. Thus, their fidelity has been compared with single or sequential arrangements of identical channels to analyse their induced transparency under a joint behaviour. The most recent characterization has exhibited important features for quantum communication related with their parametric nature. In this work, a parallel analysis has been conducted to extended such a characterization, this time in terms of their emerging Quantum Fisher Information to pursue the advantages of each kind of arrangement for the parameter estimation problem. The objective is to reach the arrangement stating the best estimation bound for each type of Pauli channel. A complete map for such an effectivity is provided for each Pauli channel under the most affordable setups considering sequential and Indefinite Causal Order arrangements, as well as discussing their advantages and disadvantages.
Indefinite causal order has introduced disruptive procedures to improve the fidelity of quantum communication by introducing the superposition of { orders} on a set of quantum channels. It has been applied to several well characterized quantum channels as depolarizing, dephasing and teleportation. This work analyses the behavior of a parametric quantum channel for single qubits expressed in the form of Pauli channels. Combinatorics lets to obtain affordable formulas for the analysis of the output state of the channel when it goes through a certain imperfect quantum communication channel when it is deployed as a redundant application of it under indefinite causal order. In addition, the process exploits post-measurement on the associated control to select certain components of transmission. Then, the fidelity of such outputs is analysed to characterize the generic channel in terms of its parameters. As a result, we get notable enhancement in the transmission of information for well characterized channels due to the combined process: indefinite causal order plus post-measurement.
Quantum metrology with indefinite causal order (ICO) has attracted intense interest due to its potential to surpass the limitations of conventional fixed-order strategies. A key open question is whether ICO can fundamentally enhance asymptotic precision scaling. In this work, we bridge this gap for the estimation of a single parameter encoded in NNN identical uses of a finite-dimensional quantum channel. We first establish a universal Heisenberg-scaling upper bound for the full general ICO process-matrix class and show that for unitary channels its optimal quantum Fisher information (QFI) coincides exactly with that of parallel strategies. For noisy channels, a structurally refined bound shows that channels restricted to the standard quantum limit (SQL) under parallel strategies remain SQL-limited under general ICO strategies. Most significantly, an asymptotically tight (AT) bound is derived to close the remaining possibility of an asymptotic ICO advantage by showing that general ICO and optimal parallel strategies have exactly the same leading QFI coefficient in both the SQL and Heisenberg regimes. For the operationally motivated class of quantum circuits with quantum control of causal order, we further obtain an iterative constraint on finite-query precision whose asymptotic limit agrees with that of the AT bound. Our results clarify the ultimate role of indefinite causality as a metrological resource for quantum channel estimation.
Optimal probe states and measurements are always required to achieve the best estimation precision in quantum metrology. In the actual physical environment, ubiquitous noise hinders estimation precision and amplifies the challenges of optimizing the probe states and measurements. In this study we extend a theoretical proposal, presented in Phys. Rev. A 103, 032615 (2021), to encompass noisy general Pauli channels for SU(2) phase estimation. By utilizing a superposition of different causal orders of two channels, we establish theoretically a probe-state-independent criterion for the estimation precision. We show that the probe-state-independent property results from the noncommutativity of the Kraus operators of the parameter-encoding channels. Based on this criterion, one can find some parameter-encoding channels without the requirement of precisely preparing the probe state. Moreover, our scheme requires only deterministic projection measurement on the control qubit. In this way, one can simultaneously avoid optimizing both the probe states and measurements. In addition, we also show that the estimation precision and probe-state-independent property of the indefinite causal order scheme are independent of representations of the Kraus operators. This alleviates the experimental challenges in realizing quantum channels. We demonstrate experimentally the advantages of the indefinite causal order for phase estimation of an SU(2) unitary transformation in the three kinds of quantum noise channels. Our result shows that the dynamic evolution of indefinite causal order can outperform the conventional cascaded estimation scenario at high noise levels, exhibiting high robustness and feasibility in practical estimation tasks.
A generic qubit unitary operator affected by quantum noise is duplicated and inserted in a coherently superposed channel, superposing two paths offered to a probe qubit across the noisy unitary, and driven by a control qubit. A characterization is performed of the transformation realized by the superposed channel on the joint state of the probe-control qubit pair. The superposed channel is then specifically analyzed for the fundamental metrological task of phase estimation on the noisy unitary, with the performance assessed by the Fisher information, classical or quantum. A comparison is made with conventional estimation techniques and also with a quantum switched channel with indefinite causal order recently investigated for a similar task of phase estimation. In the analysis here, a first important observation is that the control qubit of the superposed channel, although it never directly interacts with the unitary being estimated, can nevertheless be measured alone for effective estimation, while discarding the probe qubit that interacts with the unitary. This property is also present with the switched channel but is inaccessible with conventional techniques. The optimal measurement of the control qubit here is characterized in general conditions. A second important observation is that the noise plays an essential role in coupling the control qubit to the unitary, and that the control qubit remains operative for phase estimation at very strong noise, even with a fully depolarizing noise, whereas conventional estimation and the switched channel become inoperative in these conditions. The results extend the analysis of the capabilities of coherently controlled channels which represent novel devices exploitable for quantum signal and information processing.