Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2605.23442

Ancilla-Efficient QSAMPLE Preparation for Reversible Markov Chains

Nicholas Zhao

quant-phcs.DSSubmitted May 22, 2026• Announced May 25, 2026• 0 citations• Revised Sep 7, 2026 (v2)
v1 · May 22, 2026v2 · Sep 7, 2026
arXivHTMLPDF

Abstract

Preparing quantum samples (QSAMPLES), coherent encodings of stationary distributions of reversible Markov chains, is a fundamental primitive in quantum sampling, particularly for quantum simulated annealing. A central limitation of existing phase-estimation-based frameworks is the ancilla qubit overhead. In this work, we present a new end-to-end framework requiring only one ancilla qubit in the working register. The key technical ingredient is a selective phase compiler circuit using one ancilla qubit, built from a generalized quantum signal processing (GQSP)-based projector onto the 1-eigenspace of the qubitized Szegedy walk. Embedding these selective phase compilers into the fixed-point amplitude amplification (FPAA) procedure and iterating yields a quantum algorithm that, given an initial state, oracle access, lower bounds on the overlaps between adjacent states, and lower bounds on the phase gaps, outputs a QSAMPLE within any desired trace distance and thus total variation distance. The query complexity scales inversely with the square roots of both the minimum overlap and the minimum spectral gap of the Markov chains across the cooling schedule, up to polylogarithmic factors. We also perform simulations to verify how our qubit and query complexity evolve with the trace distance, and how this work compares to the previous framework. These results establish two improvements over the previous framework by Wocjan and Abeyesinghe. First, the working-register ancilla cost is reduced to one. Second, by inserting our GQSP-based selective phase compiler into the FPAA procedure, we improve the QSAMPLE transport overlap dependence from inverse minimum overlap to inverse square-root minimum overlap, relative to their Grover pi-over-three fixed-point method. Finally, as a direct application, we apply the quantum algorithm to prepare a Gibbs QSAMPLE and obtain a rigorous complexity analysis.

Comments
9 pages, 3 figures. Accepted to IEEE QCE 2026 (QALG track)

Related papers

Faster quantum sampling of Markov chains in nonregular graphs with fewer qubits

Xinying Li, Yun Shang

Sampling from the stationary distribution is one of the fundamental tasks of Markov-chain-based algorithms and has important applications in machine learning, combinatorial optimization, and network science. For the quantum case, quantum sampling from Markov chains corresponds to preparing quantum states with amplitudes arbitrarily close to the square root of a stationary distribution instead of classical sampling from a stationary distribution. A different quantum sampling algorithm for all reversible Markov chains is constructed by discrete-time quantum walks. We build a quantum sampling algorithm that not only accelerates nonregular graphs, but also keeps the speed up of existing quantum algorithms for regular graphs. In nonregular graphs, the invocation of the quantum fast-forwarding algorithm accelerates previous state-of-the-art quantum sampling algorithms for both discrete-time and continuous-time cases, especially on sparse graphs. Compared to existing algorithms, we decrease the runtime by a factor of $logn$, where $n$ is the number of graph vertices. In regular graphs, our results match other quantum algorithms and the reliance on the gap of Markov chains achieves quadratic speedup compared with classical cases. For both cases, we reduce the number of ancilla qubits required compared with the existing results. In some widely used graphs and a series of sparse graphs where stationary distributions are difficult to reach quickly, our algorithm achieves complete quadratic acceleration (without log factor) over the classical case without any limit. In addition, we construct a different reflection around the stationary state with fewer ancilla qubits and believe it may have an independent application.

arXiv:2205.06099•Physical Review A•Feb 27, 2023•2 citations

Adaptive Quantum Simulated Annealing for Bayesian Inference and Estimating Partition Functions

Aram W. Harrow, Annie Y. Wei

Markov chain Monte Carlo algorithms have important applications in counting problems and in machine learning problems, settings that involve estimating quantities that are difficult to compute exactly. How much can quantum computers speed up classical Markov chain algorithms? In this work we consider the problem of speeding up simulated annealing algorithms, where the stationary distributions of the Markov chains are Gibbs distributions at temperatures specified according to an annealing schedule. We construct a quantum algorithm that both adaptively constructs an annealing schedule and quantum samples at each temperature. Our adaptive annealing schedule roughly matches the length of the best classical adaptive annealing schedules and improves on nonadaptive temperature schedules by roughly a quadratic factor. Our dependence on the Markov chain gap matches other quantum algorithms and is quadratically better than what classical Markov chains achieve. Our algorithm is the first to combine both of these quadratic improvements. Like other quantum walk algorithms, it also improves on classical algorithms by producing “qsamples” instead of classical samples. This means preparing quantum states whose amplitudes are the square roots of the target probability distribution. In constructing the annealing schedule we make use of amplitude estimation, and we introduce a method for making amplitude estimation nondestructive at almost no additional cost, a result that may have independent interest. Finally we demonstrate how this quantum simulated annealing algorithm can be applied to the problems of estimating partition functions and Bayesian inference.

arXiv:1907.09965•Society for Industrial and Applied Mathematics eBooks•Dec 23, 2019•20 citations

Sequential quantum mixing for slowly evolving sequences of Markov chains.

Vedran Dunjko, Hans J. Briegel

In this work we consider the problem of preparation of the stationary distribution of irreducible, time-reversible Markov chains, which is a fundamental task in algorithmic Markov chain theory. For the classical setting, this task has a complexity lower bound of $\Omega(\delta^{-1})$, where $\delta$ is the spectral gap of the Markov chain, and other dependencies contribute only logarithmically. In the quantum case, the conjectured complexity is $O(\sqrt{\delta^{-1}})$, with other dependencies contributing only logarithmically. However, this bound has only been achieved for a few special classes of Markov chains. In this work, we provide a method for the sequential preparation of stationary distributions for sequences of time-reversible $N$-state Markov chains, akin to the setting of simulated annealing methods. The complexity of preparation we achieve is $O(\sqrt{\delta^{-1}} \sqrt[4]{N})$, neglecting logarithmic factors. While this result falls short of the conjectured optimal time, it provides a quadratic improvement over na\{i}ve approaches. Moreover, for the case when the output distributions are required to be encoded in pure quantum states we identify the settings where our algorithm is strictly optimal. The settings of slowly evolving sequences of Markov chains naturally appear in reinforcement learning, and consequently our results can be readily applied in quantum machine learning as well.

•Mar 4, 2015•4 citations

Eigenstate Preparation Through Near-Optimal Eigenprobability Filtering

Po-Wei Huang, Bence Bakó, Bálint Koczor

Quantum simulation is expected to be a main application of quantum computers with realistic utility in quantum chemistry, materials science and beyond. However, preparing excited or general eigenstates is a central challenge, particularly when the desired eigenvalue is not known in advance, or when the overlap with the initial state is insufficient. We introduce the Dominant Eigenstate Filtering via Eigenprobability Amplification and Thresholding (DEFEAT) algorithm that identifies and filters the eigenstate with the largest overlap with the supplied initial state. Our key observation is that we do not need prior knowledge of the target eigenvalue, as we construct efficient twirling superoperators that map initial states to eigenprobability density operators $ρ$, diagonal in the Hamiltonian eigenbasis and encoding its spectral weights, alongside its block-encoding implementation. Our crucial innovation is the quadratic amplification of probabilities via the factorisation $ρ=ρ_{\rm sqrt}^\daggerρ_{\rm sqrt}$, analogous to the recently introduced sum-of-squares spectral amplification (SOSSA), which we use here to amplify the separation between dominant and subdominant components. Thresholding then yields the dominant-eigenstate projector. Compared with conventional phase estimation, DEFEAT improves the dependence on the overlap with the initial state while requiring substantially fewer ancillary qubits. We prove that the query complexity of the filtering step is optimal up to logarithmic factors and establish a complementary lower bound for eigenstate preparation under purified query access to $ρ$. We validate in numerical simulations that the convergence rate of DEFEAT matches our theoretical results. Our results provide a general eigenvalue-agnostic primitive for dominant eigenstate filtering and preparation, and for estimating properties of dominant eigenstates.

quant-pharXiv:2608.12297•Aug 13, 2026•0 citations

Adaptive quantum simulated annealing for Bayesian inference and estimating partition functions

Harrow, AW, Wei, AY

Copyright © 2020 by SIAM Markov chain Monte Carlo algorithms have important applications in counting problems and in machine learning problems, settings that involve estimating quantities that are difficult to compute exactly. How much can quantum computers speed up classical Markov chain algorithms? In this work we consider the problem of speeding up simulated annealing algorithms, where the stationary distributions of the Markov chains are Gibbs distributions at temperatures specified according to an annealing schedule. We construct a quantum algorithm that both adaptively constructs an annealing schedule and quantum samples at each temperature. Our adaptive annealing schedule roughly matches the length of the best classical adaptive annealing schedules and improves on nonadaptive temperature schedules by roughly a quadratic factor. Our dependence on the Markov chain gap matches other quantum algorithms and is quadratically better than what classical Markov chains achieve. Our algorithm is the first to combine both of these quadratic improvements. Like other quantum walk algorithms, it also improves on classical algorithms by producing “qsamples” instead of classical samples. This means preparing quantum states whose amplitudes are the square roots of the target probability distribution. In constructing the annealing schedule we make use of amplitude estimation, and we introduce a method for making amplitude estimation nondestructive at almost no additional cost, a result that may have independent interest. Finally we demonstrate how this quantum simulated annealing algorithm can be applied to the problems of estimating partition functions and Bayesian inference.

•Jan 1, 2020•26 citations

Accelerating quantum Gibbs sampling without quantum walks

Jiaqi Leng, Jiaqing Jiang, Lin Lin

Szegedy's quantum walk quadratically accelerates reversible classical Markov chains, but extending this mechanism to efficiently implementable quantum Gibbs samplers has remained challenging beyond special classes. We present a walk-free quantum algorithm for preparing purified Gibbs states with a quadratic improvement in spectral-gap dependence for a broad class of quantum Gibbs samplers that satisfy exact Kubo-Martin-Schwinger detailed balance. Our main structural result is an explicit factorization of the corresponding parent Hamiltonian into noncommutative first-order operators. This turns purified Gibbs-state preparation into a singular-value filtering problem and enables a quantum singular value transformation algorithm with quadratically improved gap dependence under standard coherent-access and warm-start assumptions. The framework applies to several efficiently implementable Gibbs samplers beyond the Davies setting. Moreover, we construct noncommuting Hamiltonians with efficiently preparable warm starts whose refinement under quantum Gibbs samplers requires exponential time. For these models, our algorithm achieves an end-to-end quantum speedup over the worst-case mixing scale.

quant-pharXiv:2604.22996•Apr 28, 2026•0 citations
Preparing quantum samples (QSAMPLES), coherent encodings of stationary distributions of reversible Markov chains, is a fundamental primitive in quantum sampling, particularly for quantum simulated annealing. A central limitation of existing phase-estimation-based frameworks is the ancilla qubit overhead. In this work, we present a new end-to-end framework requiring only one ancilla qubit in the working register. The key technical ingredient is a selective phase compiler circuit using one ancilla qubit, built from a generalized quantum signal processing (GQSP)-based projector onto the 1-eigenspace of the qubitized Szegedy walk. Embedding these selective phase compilers into the fixed-point amplitude amplification (FPAA) procedure and iterating yields a quantum algorithm that, given an initial state, oracle access, lower bounds on the overlaps between adjacent states, and lower bounds on the phase gaps, outputs a QSAMPLE within any desired trace distance and thus total variation distance. The query complexity scales inversely with the square roots of both the minimum overlap and the minimum spectral gap of the Markov chains across the cooling schedule, up to polylogarithmic factors. We also perform simulations to verify how our qubit and query complexity evolve with the trace distance, and how this work compares to the previous framework. These results establish two improvements over the previous framework by Wocjan and Abeyesinghe. First, the working-register ancilla cost is reduced to one. Second, by inserting our GQSP-based selective phase compiler into the FPAA procedure, we improve the QSAMPLE transport overlap dependence from inverse minimum overlap to inverse square-root minimum overlap, relative to their Grover pi-over-three fixed-point method. Finally, as a direct application, we apply the quantum algorithm to prepare a Gibbs QSAMPLE and obtain a rigorous complexity analysis.
Sampling from the stationary distribution is one of the fundamental tasks of Markov-chain-based algorithms and has important applications in machine learning, combinatorial optimization, and network science. For the quantum case, quantum sampling from Markov chains corresponds to preparing quantum states with amplitudes arbitrarily close to the square root of a stationary distribution instead of classical sampling from a stationary distribution. A different quantum sampling algorithm for all reversible Markov chains is constructed by discrete-time quantum walks. We build a quantum sampling algorithm that not only accelerates nonregular graphs, but also keeps the speed up of existing quantum algorithms for regular graphs. In nonregular graphs, the invocation of the quantum fast-forwarding algorithm accelerates previous state-of-the-art quantum sampling algorithms for both discrete-time and continuous-time cases, especially on sparse graphs. Compared to existing algorithms, we decrease the runtime by a factor of lognlognlogn, where nnn is the number of graph vertices. In regular graphs, our results match other quantum algorithms and the reliance on the gap of Markov chains achieves quadratic speedup compared with classical cases. For both cases, we reduce the number of ancilla qubits required compared with the existing results. In some widely used graphs and a series of sparse graphs where stationary distributions are difficult to reach quickly, our algorithm achieves complete quadratic acceleration (without log factor) over the classical case without any limit. In addition, we construct a different reflection around the stationary state with fewer ancilla qubits and believe it may have an independent application.
Markov chain Monte Carlo algorithms have important applications in counting problems and in machine learning problems, settings that involve estimating quantities that are difficult to compute exactly. How much can quantum computers speed up classical Markov chain algorithms? In this work we consider the problem of speeding up simulated annealing algorithms, where the stationary distributions of the Markov chains are Gibbs distributions at temperatures specified according to an annealing schedule. We construct a quantum algorithm that both adaptively constructs an annealing schedule and quantum samples at each temperature. Our adaptive annealing schedule roughly matches the length of the best classical adaptive annealing schedules and improves on nonadaptive temperature schedules by roughly a quadratic factor. Our dependence on the Markov chain gap matches other quantum algorithms and is quadratically better than what classical Markov chains achieve. Our algorithm is the first to combine both of these quadratic improvements. Like other quantum walk algorithms, it also improves on classical algorithms by producing “qsamples” instead of classical samples. This means preparing quantum states whose amplitudes are the square roots of the target probability distribution. In constructing the annealing schedule we make use of amplitude estimation, and we introduce a method for making amplitude estimation nondestructive at almost no additional cost, a result that may have independent interest. Finally we demonstrate how this quantum simulated annealing algorithm can be applied to the problems of estimating partition functions and Bayesian inference.
In this work we consider the problem of preparation of the stationary distribution of irreducible, time-reversible Markov chains, which is a fundamental task in algorithmic Markov chain theory. For the classical setting, this task has a complexity lower bound of Ω(δ−1)\Omega(\delta^{-1})Ω(δ−1), where δ\deltaδ is the spectral gap of the Markov chain, and other dependencies contribute only logarithmically. In the quantum case, the conjectured complexity is O(δ−1)O(\sqrt{\delta^{-1}})O(δ−1​), with other dependencies contributing only logarithmically. However, this bound has only been achieved for a few special classes of Markov chains. In this work, we provide a method for the sequential preparation of stationary distributions for sequences of time-reversible NNN-state Markov chains, akin to the setting of simulated annealing methods. The complexity of preparation we achieve is O(δ−1N4)O(\sqrt{\delta^{-1}} \sqrt[4]{N})O(δ−1​4N​), neglecting logarithmic factors. While this result falls short of the conjectured optimal time, it provides a quadratic improvement over na\{i}ve approaches. Moreover, for the case when the output distributions are required to be encoded in pure quantum states we identify the settings where our algorithm is strictly optimal. The settings of slowly evolving sequences of Markov chains naturally appear in reinforcement learning, and consequently our results can be readily applied in quantum machine learning as well.
Quantum simulation is expected to be a main application of quantum computers with realistic utility in quantum chemistry, materials science and beyond. However, preparing excited or general eigenstates is a central challenge, particularly when the desired eigenvalue is not known in advance, or when the overlap with the initial state is insufficient. We introduce the Dominant Eigenstate Filtering via Eigenprobability Amplification and Thresholding (DEFEAT) algorithm that identifies and filters the eigenstate with the largest overlap with the supplied initial state. Our key observation is that we do not need prior knowledge of the target eigenvalue, as we construct efficient twirling superoperators that map initial states to eigenprobability density operators ρρρ, diagonal in the Hamiltonian eigenbasis and encoding its spectral weights, alongside its block-encoding implementation. Our crucial innovation is the quadratic amplification of probabilities via the factorisation ρ=ρsqrt†ρsqrtρ=ρ_{\rm sqrt}^\daggerρ_{\rm sqrt}ρ=ρsqrt†​ρsqrt​, analogous to the recently introduced sum-of-squares spectral amplification (SOSSA), which we use here to amplify the separation between dominant and subdominant components. Thresholding then yields the dominant-eigenstate projector. Compared with conventional phase estimation, DEFEAT improves the dependence on the overlap with the initial state while requiring substantially fewer ancillary qubits. We prove that the query complexity of the filtering step is optimal up to logarithmic factors and establish a complementary lower bound for eigenstate preparation under purified query access to ρρρ. We validate in numerical simulations that the convergence rate of DEFEAT matches our theoretical results. Our results provide a general eigenvalue-agnostic primitive for dominant eigenstate filtering and preparation, and for estimating properties of dominant eigenstates.
Copyright © 2020 by SIAM Markov chain Monte Carlo algorithms have important applications in counting problems and in machine learning problems, settings that involve estimating quantities that are difficult to compute exactly. How much can quantum computers speed up classical Markov chain algorithms? In this work we consider the problem of speeding up simulated annealing algorithms, where the stationary distributions of the Markov chains are Gibbs distributions at temperatures specified according to an annealing schedule. We construct a quantum algorithm that both adaptively constructs an annealing schedule and quantum samples at each temperature. Our adaptive annealing schedule roughly matches the length of the best classical adaptive annealing schedules and improves on nonadaptive temperature schedules by roughly a quadratic factor. Our dependence on the Markov chain gap matches other quantum algorithms and is quadratically better than what classical Markov chains achieve. Our algorithm is the first to combine both of these quadratic improvements. Like other quantum walk algorithms, it also improves on classical algorithms by producing “qsamples” instead of classical samples. This means preparing quantum states whose amplitudes are the square roots of the target probability distribution. In constructing the annealing schedule we make use of amplitude estimation, and we introduce a method for making amplitude estimation nondestructive at almost no additional cost, a result that may have independent interest. Finally we demonstrate how this quantum simulated annealing algorithm can be applied to the problems of estimating partition functions and Bayesian inference.
Szegedy's quantum walk quadratically accelerates reversible classical Markov chains, but extending this mechanism to efficiently implementable quantum Gibbs samplers has remained challenging beyond special classes. We present a walk-free quantum algorithm for preparing purified Gibbs states with a quadratic improvement in spectral-gap dependence for a broad class of quantum Gibbs samplers that satisfy exact Kubo-Martin-Schwinger detailed balance. Our main structural result is an explicit factorization of the corresponding parent Hamiltonian into noncommutative first-order operators. This turns purified Gibbs-state preparation into a singular-value filtering problem and enables a quantum singular value transformation algorithm with quadratically improved gap dependence under standard coherent-access and warm-start assumptions. The framework applies to several efficiently implementable Gibbs samplers beyond the Davies setting. Moreover, we construct noncommuting Hamiltonians with efficiently preparable warm starts whose refinement under quantum Gibbs samplers requires exponential time. For these models, our algorithm achieves an end-to-end quantum speedup over the worst-case mixing scale.