Quantum Telescoping: Foundations of Incremental Approximation in Quantum Systems

Joshua Bald

Published Jan 26, 2026• 0 citations

Abstract

Telescoping approximation frameworks have proven effective in a wide range of classicalsettings by exploiting incremental refinement with controlled decay of successive differences.In this paper, we initiate a new research program—quantum telescoping— aimed at extendingtelescoping principles to quantum systems, where non-commutativity, measurement disturbance,and information-theoretic constraints fundamentally alter the nature of approximation.We introduce axiomatic definitions of telescoping for quantum states, quantum channels,and observables, formulate appropriate notions of incremental decay using operational norms,and identify intrinsic obstructions arising from measurement and non-commutativity. Our resultsestablish a foundational framework for hierarchical refinement of quantum informationand delineate the precise sense in which classical telescoping emerges as a commutative, fullyobservable limit.We develop explicit convergence rates, complexity bounds, and constructive examples forquantum state telescoping, channel approximation, and observable estimation. Furthermore,we establish fundamental trade-offs between approximation accuracy, measurement backaction,and information extraction, formalized through quantum information-theoretic inequalities.This paper lays the groundwork for future developments in quantum simulation, quantumalgorithms, quantum machine learning, and quantum information-based complexity theory.

DOI: 10.5281/zenodo.18378769

Related papers

Quantum Telescoping Part VIII: Learned Quantum Telescoping

Joshua Bald

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.

Feb 4, 20260 citations

Quantum Telescoping Part X: Experimental and NISQ Limits of Telescoping Convergence

Joshua Bald

This paper provides a comprehensive analysis of the fundamental limits of quantum telescopingin experimental and noisy intermediate-scale quantum (NISQ) regimes. Building on thetheoretical telescoping framework developed in Parts I–IX, we rigorously study how physicalnoise, finite sampling, device drift, and imperfect control fundamentally cap achievable telescopingorder. We establish that in the presence of non-vanishing noise floors, all telescopingschemes satisfying our stated noise-and-measurement assumptions exhibit saturation beyond afinite refinement depth, regardless of their ideal convergence rate.We develop a complete mathematical framework incorporating:• Detailed noise models for NISQ devices with explicit error bounds• Measurement-induced lower bounds on observable telescoping increments• Statistical hypothesis testing framework for distinguishing telescoping orders• Rigorous conditions under which exponential telescoping becomes experimentally indistinguishableon NISQ devices with finite noise and polynomial resources• Device drift models and their impact on convergence verification• Quantitative trade-offs between circuit depth, sampling budget, and achievable precisionThese results formalize the unavoidable gap between asymptotic algorithmic guarantees andexperimentally realizable convergence, providing essential guidance for practical quantum algorithmdesign in the pre-fault-tolerant era.

Feb 5, 20260 citations

Quantum Telescoping Part VI: Hybrid Classical–Quantum Telescoping

Joshua Bald

This paper studies hybrid classical–quantum telescoping schemes for Hamiltonian simulationand quantum channel approximation. Building on the telescoping framework and fault-tolerantanalysis developed in Parts I–V, we formalize how classical preprocessing, optimization, andadaptivity can reduce constants, improve practical performance, and guide refinement strategies,while leaving fundamental telescoping order unchanged. We prove that classical computationcan optimize telescoping constants and resource allocation but cannot asymptotically improveconvergence rates beyond quantum lower bounds. We establish separation theorems distinguishingconstant-factor improvements from asymptotic scaling, provide explicit hybrid algorithmswith optimized constants, and analyze the interplay between classical learning and quantumexecution in fault-tolerant regimes. These results clarify the precise role of hybrid methodsand explain why classical–quantum co-design is essential for near-term implementations yetasymptotically bounded by quantum information-theoretic limits.

Feb 4, 20260 citations

Quantum Telescoping Part II: Channels and Hamiltonian Simulation

Joshua Bald

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.

Jan 29, 20260 citations

Quantum Telescoping Part III: Query Complexity and Lower Bounds

Joshua Bald

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.

Jan 30, 20260 citations

Quantum Telescoping Part VII: Telescoping for Open Quantum Systems (Lindbladian Dynamics)

Joshua Bald

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.

Feb 4, 20260 citations