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arXiv:2509.11074 · Physical Review A

Spectrum measurement of quantum channels and application to Hamiltonian parameter estimation

Yuan-De Jin, Wen-Long Ma

Published Jan 12, 2026• 0 citations
arXivHTMLPDF

Abstract

Quantum channels describe the most general dynamics of open quantum systems. A quantum channel, as a linear map on vectorized quantum states, can be represented by a single matrix, whose spectrum is called the channel spectrum. Here we propose a general method to measure the channel spectrum and apply this method to Hamiltonian parameter estimation. We first demonstrate that the channel spectrum can be measured by tracking the probability of a specific outcome in repeated application of the same channel. Then we construct and analyze {a class of concatenated channels, with each one being a unitary channel followed by a weak-measurement channel induced by a Ramsey sequence of a probe qubit}. We show that the spectrum measurement of such concatenated channels can be utilized for estimating the parameters in the free Hamiltonians generating the unitary channels of the target system. As practical examples, we numerically demonstrate that a probe spin qubit can accurately sense nuclear spin clusters for nanoscale nuclear magnetic resonance.

DOI
10.1103/66yg-qs8x

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Quantum channels describe the most general dynamics of open quantum systems. A quantum channel, as a linear map on vectorized quantum states, can be represented by a single matrix, whose spectrum is called the channel spectrum. Here we propose a general method to measure the channel spectrum and apply this method to Hamiltonian parameter estimation. We first demonstrate that the channel spectrum can be measured by tracking the probability of a specific outcome in repeated application of the same channel. Then we construct and analyze {a class of concatenated channels, with each one being a unitary channel followed by a weak-measurement channel induced by a Ramsey sequence of a probe qubit}. We show that the spectrum measurement of such concatenated channels can be utilized for estimating the parameters in the free Hamiltonians generating the unitary channels of the target system. As practical examples, we numerically demonstrate that a probe spin qubit can accurately sense nuclear spin clusters for nanoscale nuclear magnetic resonance.
Quantum phase estimation (QPE) is a crucial subroutine in many quantum algorithms. It is used to estimate the eigenvalues of a unitary operator acting on a target quantum system. Conventional QPE is often expensive to implement because it requires a large number of ancilla qubits and the ability to perform quantum Fourier transform. Recent developments in iterative QPE reduce the implementation cost by repetitive uses of a single ancilla and classical postprocessing. However, both conventional and iterative schemes often require preparation of the target system in an eigenstate of the unitary operator, while it remains ambiguous to achieve QPE of multiple eigenvalues with no need of initial-state preparation. Here we clarify this issue by developing a theoretical framework based on sequential quantum channels for iterative QPE. We find that QPE of multiple eigenvalues can be efficiently realized for arbitrary initial target system state by actively utilizing the measurement backaction of iterative QPE on the target system with a long coherence time. Specifically, we investigate two iterative QPE schemes based on sequential Ramsey interferometry measurements (RIMs) of an ancilla qubit: (a) the repetitive scheme, which conducts repetitive RIMs to achieve the standard quantum limit in estimating the eigenvalues; (b) the adaptive scheme, which adjusts the parameters of each RIM based on prior measurement outcomes to attain the Heisenberg limit. In both schemes, sequential ancilla measurements generate sequential quantum channels on the target system, gradually steering it to the eigenstates of the estimated unitary operator, while the measurement statistics of the ancilla can reveal the embedded information about its eigenvalues with proper postprocessing. We demonstrate the analysis by simulating a central spin model and evaluate the performance and noise resilience of both schemes.
According to the axioms of quantum mechanics, there exists a quantum measurement corresponding to each observable.Energy is one of such observables, and the operator of the energy is called Hamiltonian.The Hamiltonian specifies the dynamics of closed quantum systems and it also characterizes the property of the equilibrium state in contact with a heat bath.Therefore the energy is an essential quantity in quantum mechanics, which characterizes behavior of a quantum system.On the other hand, since the Hamiltonian is not a local observable in general, a concrete construction of how to apply the quantum measurement of energy is not trivial.Thus various measurement models of energies have been proposed, and also discussions have been made on whether the time is a resource to be consumed to increase the precision of energy measurement in connection with the time-energy uncertainty relationship.Constructing a way to implement the energy measurement on an unknown Hamiltonian system is important, since it avoids constructing a new measurement scheme for each new quantum system.On the other hand, among measurements of energy, the projective measurement of energy has useful properties for applications.The projective measurement is a measurement which satisfies the condition so-called repeatable hypothesis that when we measure the state immediately after obtaining an outcome from the same measurement, we obtain the same outcome again.In this thesis, we consider two schemes to implement the projective measurement of energy for an unknown Hamiltonian system.One is a tomography based method, which is a construction suggested in the paper of Aharanov et.al. (2002).The other is the phase estimation based method, which utilizes the Hamiltonian dynamics of the system as a resource for the quantum algorithm, and realizes the projective measurement of energy on the system without identifying all parameters of the Hamiltonian.In this thesis, we assume there is a finite dimensional quantum system (quantum computer) which is able to apply any interaction between the target system and the finite dimensional system although the interaction should not depend on the target system Hamiltonian.To evaluate the performance of the measurements implemented by the two schemes, we formulate two evaluation functions called fluctuation of measurement value and non-repeatability, which can evaluate how much a measurement is different from the ideal projective measurement of an observable.We calculate a sufficient time to guarantee the fluctuation of measurement value below some small value ε, for each of the two measurement schemes.We find that the tomography based method takes the time proportional to O(d 4 ∆ max /ε), while the phase estimation based method takes O(∆ 3 max /ε 2 ), where d is the dimension of the target system and ∆ max is the difference between the largest energy eigenvalue and the smallest energy eigenvalue.We show that the phase estimation based method performs better for large particle systems.
We propose and experimentally demonstrate an efficient framework for the quantum simulation of quantum channels in nuclear magnetic resonance (NMR). Our approach relies on the suitable decomposition of nonunitary operators in a linear combination of ddd unitary ones, which can be then experimentally implemented with the assistance of a number of ancillary qubits that grows logarithmically in ddd. As a proof-of-principle demonstration, we realize the quantum simulation of three quantum channels for a single-qubit: phase damping, amplitude damping, and depolarizing channels. For these paradigmatic cases, we measure key features, such as the fidelity of the initial state and the associated von Neumann entropy for a qubit evolving through these channels. Our experiments are carried out using nuclear spins in a liquid sample and NMR control techniques.
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.
We investigate a probe state preparation protocol based on two non-selective generalized quantum measurements to enhance parameter estimation in single-qubit systems. By fine-tuning the measurement strengths, we demonstrate the ability to design a broad class of probe states, initially prepared in a thermal state, which can be optimized for specific estimation tasks. We apply this framework to characterize the decay rate and the temperature of a generalized amplitude damping channel. Our results show that the preparation protocol significantly modulates the quantum Fisher information for both parameters. Furthermore, we derive a general analytical relationship between the quantum Fisher information, thermodynamic susceptibilities, and Hamiltonian variance, valid even in the transient regime. This connection highlights the role of energy fluctuations and kinetic response in determining metrological precision. Finally, we briefly discuss a quantum circuit for experimental implementation using nuclear magnetic resonance techniques.