This paper develops a telescoping framework for inverse problems in quantum dynamics, focusingon parameter recovery from quantum channels. We study families of channels {Φ(θ)}θ∈Θgenerated by unknown Hamiltonian or Lindbladian parameters and analyze how telescopingrefinement controls the resolution of parameter estimation. We show that telescoping orderbounds the achievable rate of inverse convergence and establish lower bounds linking parameteridentifiability to channel distinguishability. In particular, we prove that exponential telescopingof forward simulation is necessary to achieve logarithmic resolution in inverse problems, whilepower-law telescoping implies polynomial sample and query complexity. These results providefundamental limits on parameter estimation accuracy as a function of computational resources,unifying quantum simulation, quantum metrology, and inverse problems under a single telescopingperspective. We establish concrete complexity bounds for Hamiltonian learning andLindbladian tomography, showing that the latter is intrinsically harder due to the absence ofexponential telescoping in dissipative dynamics. Numerical simulations validate the predictedscaling laws for both power-law and exponential telescoping regimes.
This paper develops a telescoping framework for inverse problems in quantum dynamics, focusingon parameter recovery from quantum channels. We study families of channels {Φ(θ)}θ∈Θgenerated by unknown Hamiltonian or Lindbladian parameters and analyze how telescopingrefinement controls the resolution of parameter estimation. We show that telescoping orderbounds the achievable rate of inverse convergence and establish lower bounds linking parameteridentifiability to channel distinguishability. In particular, we prove that exponential telescopingof forward simulation is necessary to achieve logarithmic resolution in inverse problems, whilepower-law telescoping implies polynomial sample and query complexity. These results providefundamental limits on parameter estimation accuracy as a function of computational resources,unifying quantum simulation, quantum metrology, and inverse problems under a single telescopingperspective. We establish concrete complexity bounds for Hamiltonian learning andLindbladian tomography, showing that the latter is intrinsically harder due to the absence ofexponential telescoping in dissipative dynamics. Numerical simulations validate the predictedscaling laws for both power-law and exponential telescoping regimes.
This paper extends the theory of quantum telescoping to open quantum systems governedby Lindblad generators. We develop a comprehensive channel-level telescoping framework forcompletely positive trace-preserving (CPTP) semigroups and conduct a rigorous analysis ofachievable convergence rates in Lindbladian simulation. We show that dissipative dynamicswith non-normal generators admit at best power-law telescoping under physically reasonableoracle models, in sharp contrast with the exponential telescoping possible for Hamiltonian evolution.We identify structural conditions—such as normality, commutativity, or unitary dilation—under which faster convergence may occur, and we establish that the absence of efficientquantum signal processing mechanisms obstructs exponential telescoping for typical non-normalLindbladians. We provide detailed convergence analysis, explicit error bounds, computationalcomplexity characterizations, and concrete examples including a non-normal toy model illustratingpseudospectral obstructions. These results establish fundamental limits on open-systemsimulation and clarify the distinction between coherent and dissipative quantum dynamics froma telescoping perspective, with implications for quantum algorithm design, noise modeling, andquantum thermodynamics.
This paper establishes structural lower bounds on query complexity and refinement cost forquantum telescoping schemes. Building on the channel-level telescoping framework developed inParts I and II, we show that the decay rate of telescoping increments directly constrains the minimumnumber of oracle queries required to achieve a given simulation accuracy within the class ofrefinement-structured algorithms. We prove general lower bounds relating power-law telescopingorder to polynomial query complexity, and show that exponential telescoping is necessaryto achieve logarithmic dependence on the error tolerance. These results explain, from a structuralperspective, why quantum signal processing (QSP) and related eigenvalue-transformationmethods attain near-optimal ε-dependence in standard block-encoding models [3, 7, 8], whileproduct-formula and randomized product-formula methods remain algebraically convergent inε [10, 11]. Our analysis applies to unitary channels, CPTP maps, and block-encoded oraclemodels, and provides a unifying framework relating algorithmic structure to query complexity.We clarify the relationship between our results and known information-theoretic lower bounds,positing our theorems as structural constraints on algorithmic approaches rather than fundamentaloracle lower bounds.
This paper introduces learned quantum telescoping, a framework in which the refinementsteps of a quantum telescoping scheme are selected using data-driven or learning-based methods.Building on the channel-level telescoping theory developed in Parts I–VII, and complementingexisting adaptive product-formula methods [21, 22], we formalize learning as an oracle thatproposes refinement channels based on training data, prior simulations, or experimental measurements.We prove that learning can optimize telescoping constants and enable data-drivenrefinement strategies that improve typical-case performance in structured regimes, while remainingsubject to the same fundamental lower bounds on telescoping order and query complexity.Our results clarify one precise role of machine learning in quantum simulation: learning canguide which quantum channels to use, but cannot overcome information-theoretic limits inherentto the target dynamics. We provide explicit constructions, sample complexity bounds, andconnections (with explicit metric caveats) to quantum signal processing, variational algorithms,and shadow tomography. This work develops a principled framework for integrating machinelearning into quantum simulation while respecting quantum information-theoretic constraints.
This paper develops the theory of quantum telescoping for quantum channels, with a particularfocus on Hamiltonian simulation. Building on the axiomatic framework introduced in QuantumTelescoping Part I [11], we formalize channel-level telescoping schemes using the diamondnorm, analyze convergence and error propagation under composition, and establish rigorous connectionsbetween telescoping order and simulation complexity. We show that product-formulamethods exhibit power-law telescoping with polynomial increment decay, while quantum signalprocessing induces exponential telescoping with increment decay e−ad, yielding the optimalO(t+log(1/ε)) complexity. LCU implementations of truncated series achieve super-polynomial(factorial) decay. This perspective yields transparent complexity bounds, clarifies the role of incrementalrefinement in quantum simulation, and provides a unifying framework for comparingclassical and quantum simulation strategies.
This paper addresses the problem of verifying telescoping order in quantum simulation usinglimited measurements. Building on the telescoping framework and lower bounds developed inParts I–III, we study how increment decay rates can be certified without full process tomography.We show that power-law and exponential telescoping can be distinguished via a witness-familyseparation test using Ntotal = O((log n - log (d^2/δ))/ε^2)measurements for d2 Pauli witness observables,n telescoping levels, error tolerance ε, and confidence 1−δ, where d = 2m is the dimension for mqubits. We further analyze three structural barriers: (i) the refinement cost axiom underlyingquery lower bounds, (ii) the apparent obstruction to exponential telescoping in open quantumsystems, and (iii) intrinsic limits imposed by time-dependent Hamiltonians. Our results clarifywhich aspects of telescoping behavior are observable, which are algorithmic, and which reflectfundamental physical constraints.
The goal of quantum channel discrimination and estimation is to determine the identity of an unknown channel from a discrete or continuous set, respectively. The query complexity of these tasks is equal to the minimum number of times one must call an unknown channel to identify it within a desired threshold on the error probability. In this paper, we establish lower bounds on the query complexities of channel discrimination and estimation, in both the parallel and adaptive access models. We do so by establishing new or applying known upper bounds on the squared Bures distance and symmetric logarithmic derivative Fisher information of channels. Phrasing our statements and proofs in terms of isometric extensions of quantum channels allows us to give conceptually simple proofs for both novel and known bounds. We also provide alternative proofs for several established results in an effort to present a consistent and unified framework for quantum channel discrimination and estimation, which we believe will be helpful in addressing future questions in the field.