We study the problem of estimating frequency response functions of systems of coupled, classical harmonic oscillators using a quantum computer. The functional form of these response functions can be mapped to a corresponding eigenproblem of a Hermitian matrix H , thus suggesting the use of quantum phase estimation. Our proposed quantum algorithm operates in the standard s -sparse, oracle-based query access model. For a network of N oscillators with maximum norm ∥ H ∥ max , and when the eigenvalue tolerance ɛ is much smaller than the minimum eigenvalue gap, we use O ( log ( N s ∥ H ∥ max / ɛ ) algorithmic qubits and obtain a rigorous worst-case query complexity upper bound O ( s ∥ H ∥ max / ( δ 2 ɛ ) ) up to logarithmic factors, where δ denotes the desired precision on the coefficients appearing in the response functions. Crucially, our proposal does not suffer from the infamous state preparation bottleneck and can as such potentially achieve large quantum speedups compared to relevant classical methods. As a proof-of-principle of exponential quantum speedup, we show that a simple adaptation of our algorithm solves the random glued-trees problem in polynomial time. We discuss practical limitations as well as potential improvements for quantifying finite-size, end-to-end complexities for application to relevant instances.
We study the problem of estimating frequency response functions of systems of coupled, classical harmonic oscillators using a quantum computer. The functional form of these response functions can be mapped to a corresponding eigenproblem of a Hermitian matrix H , thus suggesting the use of quantum phase estimation. Our proposed quantum algorithm operates in the standard s -sparse, oracle-based query access model. For a network of N oscillators with maximum norm ∥ H ∥ max , and when the eigenvalue tolerance ɛ is much smaller than the minimum eigenvalue gap, we use O ( log ( N s ∥ H ∥ max / ɛ ) algorithmic qubits and obtain a rigorous worst-case query complexity upper bound O ( s ∥ H ∥ max / ( δ 2 ɛ ) ) up to logarithmic factors, where δ denotes the desired precision on the coefficients appearing in the response functions. Crucially, our proposal does not suffer from the infamous state preparation bottleneck and can as such potentially achieve large quantum speedups compared to relevant classical methods. As a proof-of-principle of exponential quantum speedup, we show that a simple adaptation of our algorithm solves the random glued-trees problem in polynomial time. We discuss practical limitations as well as potential improvements for quantifying finite-size, end-to-end complexities for application to relevant instances.
We study the problem of estimating frequency response functions of systems of coupled, classical harmonic oscillators using a quantum computer. The functional form of these response functions can be mapped to a corresponding eigenproblem of a Hermitian matrix 𝐻, thus suggesting the use of quantum phase estimation. Our proposed quantum algorithm operates in the standard 𝑠-sparse, oracle-based query access model. For a network of 𝑁 oscillators with maximum norm ∥𝐻∥max, and when the eigenvalue tolerance ɛ is much smaller than the minimum eigenvalue gap, we use 𝒪(log(𝑁𝑠∥𝐻∥max/ɛ) algorithmic qubits and obtain a rigorous worst-case query complexity upper bound 𝒪(𝑠∥𝐻∥max/(𝛿2ɛ)) up to logarithmic factors, where 𝛿 denotes the desired precision on the coefficients appearing in the response functions. Crucially, our proposal does not suffer from the infamous state preparation bottleneck and can as such potentially achieve large quantum speedups compared to relevant classical methods. As a proof-of-principle of exponential quantum speedup, we show that a simple adaptation of our algorithm solves the random glued-trees problem in polynomial time. We discuss practical limitations as well as potential improvements for quantifying finite-size, end-to-end complexities for application to relevant instances.
We study the problem of simulating the time evolution of a system of 2nclassical coupled oscillators (e.g., 2nballs connected by springs) on a quantum computer. We map Newton’s equation for harmonic potentials to Schrödinger’s equation, such that the amplitudes of an O(n)-qubit quantum state encode the momenta and displacements of the 2nclassical oscillators. Given oracle access to the masses and spring constants, we describe a quantum algorithm with query and time complexity poly (n) that solves this problem when certain parameters are polynomially bounded and the initial state is easy to prepare. As an example application, we apply our quantum algorithm to efficiently estimate the normalized kinetic energy of an oscillator at any time. We then show that any classical algorithm solving the same problem must make 2Ω(n) queries to the oracle and we also show that when the oracles are instantiated by poly (n)-size circuits, the problem is BQP-complete. Thus, our approach solves a potentially practical application with an exponential speedup over classical computers.
We study hybrid quantum-classical algorithms in the query model. The computation consists of quantum subroutines that make at most q oracle queries; between subroutines, all qubits are measured and discarded. We prove matching upper and lower bounds for several problems in this model:
(i) (Unbiased) Phase estimation up to a precision ε requires Θ(qε21) queries;
(ii) (Unbiased) Amplitude estimation up to a precision ε requires Θ(qε2p(1−p)) queries;
(iii) Search among N items requires Θ(qN) queries;
(iv) Two level AND-OR tree (AND of m ORs, with n inputs to each OR) requires Θ(qnm) queries.
All of these bounds are optimal up to a constant factor, for all q from 1, corresponding to the classical complexity, to Q(f), the unrestricted quantum query complexity of the respective problem. We also derive a hybrid lower bound for distinguishing two distributions specified by a state-preparation unitary; this bound is tight up to a logarithmic factor.
Besides specific lower bounds, an important contribution is developing methods for proving lower bounds on hybrid quantum algorithms (which have been very ad-hoc up to now). The first two bounds follow from a common framework: we analyze probability distributions over measurement transcripts and bound a progress measure that captures their distinguishability. The AND-OR bound requires a more delicate argument, combining two progress measures that track the information available to the algorithm classically and in quantum form, respectively.
We present a quantum algorithm for simulating the classical dynamics of 2 n coupled oscillators (e.g., 2 n masses coupled by springs).Our approach leverages a mapping between the Schrödinger equation and Newton's equation for harmonic potentials such that the amplitudes of the evolved quantum state encode the momenta and displacements of the classical oscillators.When individual masses and spring constants can be efficiently queried, and when the initial state can be efficiently prepared, the complexity of our quantum algorithm is polynomial in n, almost linear in the evolution time, and sublinear in the sparsity.As an example application, we apply our quantum algorithm to efficiently estimate the kinetic energy of an oscillator at any time.We show that any classical algorithm solving this same problem is inefficient and must make 2 ΩðnÞ queries to the oracle, and when the oracles are instantiated by efficient quantum circuits, the problem is bounded-error quantum polynomial time complete.Thus, our approach solves a potentially practical application with an exponential speedup over classical computers.Finally, we show that under similar conditions our approach can efficiently simulate more general classical harmonic systems with 2 n modes.
Quantum phase estimation (QPE) is fundamental to quantum algorithms for eigenvalue problems, chemistry simulation, and number-theoretic applications. However, canonical QPE implementations employ controlled unitary operations and quantum Fourier transform, incurring substantial circuit depth and control overhead that severely limit applicability on nearterm quantum hardware. This work develops a novel class of control-free phase estimation algorithms that eliminate both controlled unitaries and QFT by leveraging signal processing principles. Our approach applies uncontrolled powers of the unitary U = e −iHτ to an eigenstate or eigenstate-rich state, measures in multiple bases to capture interference patterns, and reconstructs eigenphases via classical post-processing techniques including Prony's method, compressed sensing, and harmonic inversion. We establish that for eigenstates or states with sparse eigendecomposition, our algorithms achieve phase estimation precision ϵ using Õ(1/ϵ) applications of U , matching standard QPE query complexity up to logarithmic factors while dramatically reducing circuit depth and eliminating control overhead. We provide detailed error analysis characterizing robustness to measurement noise, finite sampling, and state preparation errors. Applications to electronic structure estimation in small molecules and variational quantum eigensolvers demonstrate the practical viability of control-free QPE on both NISQ and early fault-tolerant quantum computers. Our results suggest that signal-processing-based phase estimation may offer more efficient implementations of quantum algorithms on realistic hardware.
We present a quantum algorithm for simulating classical oscillator networks characterized by non-Markovian dissipation and time-varying material properties, extending recent speedups for undamped harmonic systems to viscoacoustic and viscoelastic media. We embed the history-dependent dynamics into a Markovian state space governed by a non-Hermitian operator by approximating memory kernels through a Prony series. We then use linear combination of Hamiltonian simulation to estimate the instantaneous kinetic and potential energy for a subset of oscillators at time t within error ε using a number of queries to the oscillator system that scales as O(αtott/ε), where αtot is polynomial in the strength of the dissipation, spring constants, inverse masses, and sparsity of the connections in the network. We further show that this energy estimation task is in the worst-case classically hard (i.e., a corresponding decision problem is BQP-complete), even in the presence of strong dissipation. For time-dependent materials, we show that changes in material properties appear as effective dissipation or growth in the energy representation. Additionally, we prove the infeasibility of exponential quantum advantages in locally coupled topologies through a novel form of Lieb-Robinson-like bounds that apply to differential equations. This allows our quantum algorithms to provide quartic speedups for locally coupled damped oscillator systems in three dimensions, raising the possibility of practical quantum speedups for realistically damped oscillator networks and approximated wave equations.