Log-Concavity and Tunneling: Quantum Adiabatic Algorithm for Convex Functions (With a Spike)

Quantum tunneling is expected to provide a computational speedup in quantum computing, a phenomenon that Adiabatic Quantum Optimization (AQO) aims to leverage. While some academic proofs of concept have been studied, such as the "Hamming weight with a spike" (HWS) problem, the algorithmic gains of this effect remain underexplored. In this work we extend the analysis underlying HWS to more general potentials. In the first half of the work, we establish (discrete) log-concavity of the ground state as a key structural property in this context. We devise a framework for establishing log-concavity of the ground state for a large family of discrete, 1-dimensional Schrödinger operators. The family includes convex potentials, but also certain potentials with local minima. In the convex case, this provides a discrete version of a continuous result by Brascamp and Lieb ('76). We demonstrate the utility of our result by establishing new spectral gap bounds, going beyond related results by Jarret and Jordan ('14) for convex potentials. In the second half of the work, we use our results on log-concavity to extend the perturbative analysis of HWS by Reichardt ('04) to the larger family of potentials with log-concave ground state. As a concrete instantiation, we use our result to extend the HWS analysis from a linear potential (which is exactly solvable) to a quadratic potential (which is no longer solvable). Our result strongly suggests the broader applicability of tunneling to convex potentials with spikesQuantum tunneling is expected to provide a computational speedup in quantum computing, a phenomenon that Adiabatic Quantum Optimization (AQO) aims to leverage. While some academic proofs of concept have been studied, such as the "Hamming weight with a spike" (HWS) problem, the algorithmic gains of this effect remain underexplored. In this work we extend the analysis underlying HWS to more general potentials. In the first half of the work, we establish (discrete) log-concavity of the ground state as a key structural property in this context. We devise a framework for establishing log-concavity of the ground state for a large family of discrete, 1-dimensional Schrödinger operators. The family includes convex potentials, but also certain potentials with local minima. In the convex case, this provides a discrete version of a continuous result by Brascamp and Lieb ('76). We demonstrate the utility of our result by establishing new spectral gap bounds, going beyond related results by Jarret and Jordan ('14) for convex potentials. In the second half of the work, we use our results on log-concavity to extend the perturbative analysis of HWS by Reichardt ('04) to the larger family of potentials with log-concave ground state. As a concrete instantiation, we use our result to extend the HWS analysis from a linear potential (which is exactly solvable) to a quadratic potential (which is no longer solvable). Our result strongly suggests the broader applicability of tunneling to convex potentials with spikes

quant-pharXiv:2606.23614Jun 23, 20260 citations

The performance of the quantum adiabatic algorithm on spike Hamiltonians

Linghang Kong, Elizabeth Crosson

Spike Hamiltonians arise from optimization instances for which the adiabatic algorithm provably out performs classical simulated annealing. In this work, we study the efficiency of the adiabatic algorithm for solving the “the Hamming weight with a spike” problem by analyzing the scaling of the spectral gap at the critical point for various sizes of the barrier. Our main result is a rigorous lower bound on the minimum spectral gap for the adiabatic evolution when the bit-symmetric cost function has a thin but polynomially high barrier, which is based on a comparison argument and an improved variational ansatz for the ground state. We also adapt the discrete WKB method for the case of abruptly changing potentials and compare it with the predictions of the spin coherent instanton method which was previously used by Farhi, Goldstone and Gutmann. Finally, our improved ansatz for the ground state leads to a method for predicting the location of avoided crossings in the excited energy states of the thin spike Hamiltonian, and we use a recursion relation to understand the ordering of some of these avoided crossings as a step towards analyzing the previously observed diabatic cascade phenomenon.Spike Hamiltonians arise from optimization instances for which the adiabatic algorithm provably out performs classical simulated annealing. In this work, we study the efficiency of the adiabatic algorithm for solving the “the Hamming weight with a spike” problem by analyzing the scaling of the spectral gap at the critical point for various sizes of the barrier. Our main result is a rigorous lower bound on the minimum spectral gap for the adiabatic evolution when the bit-symmetric cost function has a thin but polynomially high barrier, which is based on a comparison argument and an improved variational ansatz for the ground state. We also adapt the discrete WKB method for the case of abruptly changing potentials and compare it with the predictions of the spin coherent instanton method which was previously used by Farhi, Goldstone and Gutmann. Finally, our improved ansatz for the ground state leads to a method for predicting the location of avoided crossings in the excited energy states of the thin spike Hamiltonian, and we use a recursion relation to understand the ordering of some of these avoided crossings as a step towards analyzing the previously observed diabatic cascade phenomenon.

arXiv:1511.06991International Journal of Quantum InformationFeb 7, 201717 citations

Quantum Speedups for Log-Concave Sampling from Local Structure

Chenghua Liu, Qisheng Wang, Zhengfeng Ji

For a convex function $f \colon \mathbb{R}^d \to \mathbb{R}$ , the problem of sampling from a distribution proportional to $e^{-f(x)}$ is called log-concave sampling. In many practical scenarios, the function $f(x)$ turns out to admit a local decomposition $f(x) = \sum_{a=1}^R ψ_a(x_{S_a})$ . In this paper, we consider log-concave sampling using local queries, i.e., evaluation and gradient queries to each clause $ψ_a(\cdot)$ , which can be computationally much cheaper than the queries to $f(x)$ itself. We show that if each coordinate appears in only a small number of clauses, there is a quantum algorithm for strongly log-concave sampling using $\widetilde{O}(\sqrtκd)$ local queries, where $κ$ is the condition number. This improves the prior best classical result $\widetilde{O}(κd)$ due to Ascolani, Lavenant, and Zanella (Ann. Probab. 2026) and the quantum result $\widetilde{O}(\sqrtκ d^2)$ implied by Childs et al. (NeurIPS 2022). Our quantum sampler applies to a broad class of locally structured models from statistical computing and machine learning, with representative examples including Gaussian Markov random fields, finite-element latent Gaussian models, and sparse generalized linear models. These results demonstrate that local structure is not merely an implementation detail, but a quantum algorithmic resource for high-dimensional sampling.For a convex function $f \colon \mathbb{R}^d \to \mathbb{R}$, the problem of sampling from a distribution proportional to $e^{-f(x)}$ is called log-concave sampling. In many practical scenarios, the function $f(x)$ turns out to admit a local decomposition $f(x) = \sum_{a=1}^R ψ_a(x_{S_a})$. In this paper, we consider log-concave sampling using local queries, i.e., evaluation and gradient queries to each clause $ψ_a(\cdot)$, which can be computationally much cheaper than the queries to $f(x)$ itself. We show that if each coordinate appears in only a small number of clauses, there is a quantum algorithm for strongly log-concave sampling using $\widetilde{O}(\sqrtκd)$ local queries, where $κ$ is the condition number. This improves the prior best classical result $\widetilde{O}(κd)$ due to Ascolani, Lavenant, and Zanella (Ann. Probab. 2026) and the quantum result $\widetilde{O}(\sqrtκ d^2)$ implied by Childs et al. (NeurIPS 2022). Our quantum sampler applies to a broad class of locally structured models from statistical computing and machine learning, with representative examples including Gaussian Markov random fields, finite-element latent Gaussian models, and sparse generalized linear models. These results demonstrate that local structure is not merely an implementation detail, but a quantum algorithmic resource for high-dimensional sampling.

quant-pharXiv:2609.20253Aug 17, 20260 citations

Mechanisms for Quantum Advantage in Global Optimization of Nonconvex Functions

Dylan Herman, Guneykan Ozgul, Anuj Apte +4 more

We present new theoretical mechanisms for quantum speedup in the global optimization of nonconvex functions, expanding the scope of quantum advantage beyond traditional tunneling-based explanations. As our main building-block, we demonstrate a rigorous correspondence between the spectral properties of Schrödinger operators and the mixing times of classical Langevin diffusion. This correspondence motivates a mechanism for separation on functions with unique global minimum: while quantum algorithms operate on the original potential, classical diffusions correspond to a Schrödinger operators with a WKB potential having nearly degenerate global minima. We formalize these ideas by proving that a real-space adiabatic quantum algorithm (RsAA) achieves provably polynomial-time optimization for broad families of nonconvex functions. First, for block-separable functions, we show that RsAA maintains polynomial runtime while known off-the-shelf algorithms require exponential time and structure-aware algorithms exhibit arbitrarily large polynomial runtimes. These results leverage novel non-asymptotic results in semiclassical analysis. Second, we use recent advances in the theory of intrinsic hypercontractivity to demonstrate polynomial runtimes for RsAA on appropriately perturbed strongly convex functions that lack global structure, while off-the-shelf algorithms remain exponentially bottlenecked. In contrast to prior works based on quantum tunneling, these separations do not depend on the geometry of barriers between local minima. Our theoretical claims about classical algorithm runtimes are supported by rigorous analysis and comprehensive numerical benchmarking. These findings establish a rigorous theoretical foundation for quantum advantage in continuous optimization and open new research directions connecting quantum algorithms, stochastic processes, and semiclassical analysis.We present new theoretical mechanisms for quantum speedup in the global optimization of nonconvex functions, expanding the scope of quantum advantage beyond traditional tunneling-based explanations. As our main building-block, we demonstrate a rigorous correspondence between the spectral properties of Schrödinger operators and the mixing times of classical Langevin diffusion. This correspondence motivates a mechanism for separation on functions with unique global minimum: while quantum algorithms operate on the original potential, classical diffusions correspond to a Schrödinger operators with a WKB potential having nearly degenerate global minima. We formalize these ideas by proving that a real-space adiabatic quantum algorithm (RsAA) achieves provably polynomial-time optimization for broad families of nonconvex functions. First, for block-separable functions, we show that RsAA maintains polynomial runtime while known off-the-shelf algorithms require exponential time and structure-aware algorithms exhibit arbitrarily large polynomial runtimes. These results leverage novel non-asymptotic results in semiclassical analysis. Second, we use recent advances in the theory of intrinsic hypercontractivity to demonstrate polynomial runtimes for RsAA on appropriately perturbed strongly convex functions that lack global structure, while off-the-shelf algorithms remain exponentially bottlenecked. In contrast to prior works based on quantum tunneling, these separations do not depend on the geometry of barriers between local minima. Our theoretical claims about classical algorithm runtimes are supported by rigorous analysis and comprehensive numerical benchmarking. These findings establish a rigorous theoretical foundation for quantum advantage in continuous optimization and open new research directions connecting quantum algorithms, stochastic processes, and semiclassical analysis.

arXiv:2510.03385Oct 3, 20250 citations

Spectral-gap analysis for efficient tunneling in quantum adiabatic optimization

Lucas T. Brady, Wim van Dam

We investigate the efficiency of quantum adiabatic optimization when overcoming potential barriers to get from a local to a global minimum. Specifically we look at $n$ qubit systems with symmetric cost functions $f:{{0,1}}^{n}\ensuremath{\rightarrow}\mathbb{R}$, where the ground state must tunnel through a potential barrier of width ${n}^{\ensuremath{\alpha}}$ and height ${n}^{\ensuremath{\beta}}$. By the quantum adiabatic theorem the time delay sufficient to ensure tunneling grows quadratically with the inverse spectral gap during this tunneling process. We analyze barrier sizes with $1/2\ensuremath{\le}\ensuremath{\alpha}+\ensuremath{\beta}$ and $\ensuremath{\alpha}<1/2$ and show that the minimum gap scales polynomially as ${n}^{1/2\ensuremath{-}\ensuremath{\alpha}\ensuremath{-}\ensuremath{\beta}}$ when $2\ensuremath{\alpha}+\ensuremath{\beta}\ensuremath{\le}1$ and exponentially as ${n}^{\ensuremath{-}\ensuremath{\beta}/2}exp(\ensuremath{-}C{n}^{(2\ensuremath{\alpha}+\ensuremath{\beta}\ensuremath{-}1)/2})$ when $1<2\ensuremath{\alpha}+\ensuremath{\beta}$. Our proof uses elementary techniques and confirms and extends an unpublished folklore result by Goldstone from 2002, which used large spin and instanton methods. Parts of our result also refine recent results by Kong and Crosson [arXiv:1511.06991] and Jiang et al. [arXiv:1603.01293] about the exponential gap scaling.We investigate the efficiency of quantum adiabatic optimization when overcoming potential barriers to get from a local to a global minimum. Specifically we look at $n$ qubit systems with symmetric cost functions $f:{{0,1}}^{n}\ensuremath{\rightarrow}\mathbb{R}$, where the ground state must tunnel through a potential barrier of width ${n}^{\ensuremath{\alpha}}$ and height ${n}^{\ensuremath{\beta}}$. By the quantum adiabatic theorem the time delay sufficient to ensure tunneling grows quadratically with the inverse spectral gap during this tunneling process. We analyze barrier sizes with $1/2\ensuremath{\le}\ensuremath{\alpha}+\ensuremath{\beta}$ and $\ensuremath{\alpha}<1/2$ and show that the minimum gap scales polynomially as ${n}^{1/2\ensuremath{-}\ensuremath{\alpha}\ensuremath{-}\ensuremath{\beta}}$ when $2\ensuremath{\alpha}+\ensuremath{\beta}\ensuremath{\le}1$ and exponentially as ${n}^{\ensuremath{-}\ensuremath{\beta}/2}exp(\ensuremath{-}C{n}^{(2\ensuremath{\alpha}+\ensuremath{\beta}\ensuremath{-}1)/2})$ when $1<2\ensuremath{\alpha}+\ensuremath{\beta}$. Our proof uses elementary techniques and confirms and extends an unpublished folklore result by Goldstone from 2002, which used large spin and instanton methods. Parts of our result also refine recent results by Kong and Crosson [arXiv:1511.06991] and Jiang et al. [arXiv:1603.01293] about the exponential gap scaling.

arXiv:1601.01720Physical Review ASep 9, 201617 citations

Quantum Algorithms for Sampling Log-Concave Distributions and Estimating Normalizing Constants

Andrew M. Childs, Tongyang Li, Jin-Peng Liu +2 more

Given a convex function $f\colon\mathbb{R}^{d}\to\mathbb{R}$ , the problem of sampling from a distribution $\propto e^{-f(x)}$ is called log-concave sampling. This task has wide applications in machine learning, physics, statistics, etc. In this work, we develop quantum algorithms for sampling log-concave distributions and for estimating their normalizing constants $\int_{\mathbb{R}^d}e^{-f(x)}\mathrm{d} x$ . First, we use underdamped Langevin diffusion to develop quantum algorithms that match the query complexity (in terms of the condition number $κ$ and dimension $d$ ) of analogous classical algorithms that use gradient (first-order) queries, even though the quantum algorithms use only evaluation (zeroth-order) queries. For estimating normalizing constants, these algorithms also achieve quadratic speedup in the multiplicative error $ε$ . Second, we develop quantum Metropolis-adjusted Langevin algorithms with query complexity $\widetilde{O}(κ^{1/2}d)$ and $\widetilde{O}(κ^{1/2}d^{3/2}/ε)$ for log-concave sampling and normalizing constant estimation, respectively, achieving polynomial speedups in $κ,d,ε$ over the best known classical algorithms by exploiting quantum analogs of the Monte Carlo method and quantum walks. We also prove a $1/ε^{1-o(1)}$ quantum lower bound for estimating normalizing constants, implying near-optimality of our quantum algorithms in $ε$ .Given a convex function $f\colon\mathbb{R}^{d}\to\mathbb{R}$, the problem of sampling from a distribution $\propto e^{-f(x)}$ is called log-concave sampling. This task has wide applications in machine learning, physics, statistics, etc. In this work, we develop quantum algorithms for sampling log-concave distributions and for estimating their normalizing constants $\int_{\mathbb{R}^d}e^{-f(x)}\mathrm{d} x$. First, we use underdamped Langevin diffusion to develop quantum algorithms that match the query complexity (in terms of the condition number $κ$ and dimension $d$) of analogous classical algorithms that use gradient (first-order) queries, even though the quantum algorithms use only evaluation (zeroth-order) queries. For estimating normalizing constants, these algorithms also achieve quadratic speedup in the multiplicative error $ε$. Second, we develop quantum Metropolis-adjusted Langevin algorithms with query complexity $\widetilde{O}(κ^{1/2}d)$ and $\widetilde{O}(κ^{1/2}d^{3/2}/ε)$ for log-concave sampling and normalizing constant estimation, respectively, achieving polynomial speedups in $κ,d,ε$ over the best known classical algorithms by exploiting quantum analogs of the Monte Carlo method and quantum walks. We also prove a $1/ε^{1-o(1)}$ quantum lower bound for estimating normalizing constants, implying near-optimality of our quantum algorithms in $ε$.

arXiv:2210.06539neural information processing systemsJan 1, 20220 citations

For a convex function

f \colon \mathbb{R}^d \to \mathbb{R}

, the problem of sampling from a distribution proportional to

e^{-f(x)}

is called log-concave sampling. In many practical scenarios, the function

f(x)

turns out to admit a local decomposition

f(x) = \sum_{a=1}^R ψ_a(x_{S_a})

. In this paper, we consider log-concave sampling using local queries, i.e., evaluation and gradient queries to each clause

ψ_a(\cdot)

, which can be computationally much cheaper than the queries to

f(x)

itself. We show that if each coordinate appears in only a small number of clauses, there is a quantum algorithm for strongly log-concave sampling using

\widetilde{O}(\sqrtκd)

local queries, where

κ

is the condition number. This improves the prior best classical result

\widetilde{O}(κd)

due to Ascolani, Lavenant, and Zanella (Ann. Probab. 2026) and the quantum result

\widetilde{O}(\sqrtκ d^2)

implied by Childs et al. (NeurIPS 2022). Our quantum sampler applies to a broad class of locally structured models from statistical computing and machine learning, with representative examples including Gaussian Markov random fields, finite-element latent Gaussian models, and sparse generalized linear models. These results demonstrate that local structure is not merely an implementation detail, but a quantum algorithmic resource for high-dimensional sampling.

We investigate the efficiency of quantum adiabatic optimization when overcoming potential barriers to get from a local to a global minimum. Specifically we look at

n

qubit systems with symmetric cost functions $f:{{0,1}}^{n}\ensuremath{\rightarrow}\mathbb{R}$, where the ground state must tunnel through a potential barrier of width ${n}^{\ensuremath{\alpha}}$ and height ${n}^{\ensuremath{\beta}}$. By the quantum adiabatic theorem the time delay sufficient to ensure tunneling grows quadratically with the inverse spectral gap during this tunneling process. We analyze barrier sizes with $1/2\ensuremath{\le}\ensuremath{\alpha}+\ensuremath{\beta}$ and $\ensuremath{\alpha}<1/2$ and show that the minimum gap scales polynomially as ${n}^{1/2\ensuremath{-}\ensuremath{\alpha}\ensuremath{-}\ensuremath{\beta}}$ when $2\ensuremath{\alpha}+\ensuremath{\beta}\ensuremath{\le}1$ and exponentially as ${n}^{\ensuremath{-}\ensuremath{\beta}/2}exp(\ensuremath{-}C{n}^{(2\ensuremath{\alpha}+\ensuremath{\beta}\ensuremath{-}1)/2})$ when $1<2\ensuremath{\alpha}+\ensuremath{\beta}$. Our proof uses elementary techniques and confirms and extends an unpublished folklore result by Goldstone from 2002, which used large spin and instanton methods. Parts of our result also refine recent results by Kong and Crosson [arXiv:1511.06991] and Jiang et al. [arXiv:1603.01293] about the exponential gap scaling.

Given a convex function

f\colon\mathbb{R}^{d}\to\mathbb{R}

, the problem of sampling from a distribution

\propto e^{-f(x)}

is called log-concave sampling. This task has wide applications in machine learning, physics, statistics, etc. In this work, we develop quantum algorithms for sampling log-concave distributions and for estimating their normalizing constants

\int_{\mathbb{R}^d}e^{-f(x)}\mathrm{d} x

. First, we use underdamped Langevin diffusion to develop quantum algorithms that match the query complexity (in terms of the condition number

κ

and dimension

d

) of analogous classical algorithms that use gradient (first-order) queries, even though the quantum algorithms use only evaluation (zeroth-order) queries. For estimating normalizing constants, these algorithms also achieve quadratic speedup in the multiplicative error

ε

. Second, we develop quantum Metropolis-adjusted Langevin algorithms with query complexity

\widetilde{O}(κ^{1/2}d)

and

\widetilde{O}(κ^{1/2}d^{3/2}/ε)

for log-concave sampling and normalizing constant estimation, respectively, achieving polynomial speedups in

κ,d,ε

over the best known classical algorithms by exploiting quantum analogs of the Monte Carlo method and quantum walks. We also prove a

1/ε^{1-o(1)}

quantum lower bound for estimating normalizing constants, implying near-optimality of our quantum algorithms in

ε