pith. sign in

arxiv: 2604.15179 · v2 · submitted 2026-04-16 · 🪐 quant-ph

Quantum Metropolis-Hastings via Penalised Qubitized Walks: Spectral Filtering and Circuit Implementation

Miguel Carrasco-Arango , Rosa M. Badia , Artur Garcia-Saez This is my paper

Pith reviewed 2026-05-10 10:41 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum metropolis-hastingsqubitized walksspectral filteringquantum circuitsmarkov chain monte carlostationary distributionquantum algorithmssampling methods
0
0 comments X

The pith

Quantum Metropolis-Hastings via penalized qubitized walks recovers the correct stationary distribution only after spectral filtering in the circuit model.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper presents an explicit circuit implementation of a quantum Metropolis-Hastings algorithm that relies on penalized qubitized walks. The authors detail a full workflow including several modifications to adapt the approach to realistic quantum circuits. Simulations reveal that these modifications, particularly spectral filtering, are required to achieve the intended stationary distribution. A sympathetic reader would care because such quantum algorithms could offer faster mixing for sampling problems in computational physics and machine learning. The results also underscore the remaining challenges before these methods become useful in fault-tolerant quantum computing.

Core claim

The paper establishes that constructing a circuit-level quantum Metropolis-Hastings algorithm from penalized qubitized walks necessitates the inclusion of spectral filtering and other adjustments to the walk dynamics. When these changes are incorporated, the simulated quantum circuit produces the same stationary distribution as the classical algorithm; without them, the distribution is incorrect.

What carries the argument

The spectral-filtered penalized qubitized walk, which serves as the quantum operator that encodes the Metropolis-Hastings transition probabilities while operating within the constraints of a quantum circuit.

If this is right

  • The stationary distribution of the quantum algorithm matches the classical one when filtering is applied.
  • Quantum walks can be used for accelerated Monte Carlo sampling in circuit-based quantum computers.
  • The approach highlights specific engineering steps needed for practical quantum Markov chain methods.
  • Limitations remain in scaling to large systems due to circuit complexity.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Techniques like spectral filtering could be generalized to enhance other quantum walk algorithms for sampling.
  • Implementation on current quantum simulators would allow verification of the distribution matching beyond ideal cases.
  • This could open paths to quantum-enhanced Bayesian inference in machine learning models.

Load-bearing premise

The modifications to the qubitized walk, including penalization and spectral filtering, can be realized in a quantum circuit without altering the target stationary distribution of the Metropolis-Hastings algorithm.

What would settle it

Performing a circuit simulation of the quantum walk without applying spectral filtering and checking whether the output state probabilities differ from the analytically computed Metropolis-Hastings stationary distribution.

Figures

Figures reproduced from arXiv: 2604.15179 by Artur Garcia-Saez, Miguel Carrasco-Arango, Rosa M. Badia.

Figure 1
Figure 1. Figure 1: FIG. 1: Scheme of the circuit implementing the Quantum Metropolis-Hastings. A QPE filtering is applied to an initial state [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Flowchart of the quantum Metropolis-Hastings sampling procedure based on qubitization and Quantum Phase Esti [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Visualization of the double well energy landscape. [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Stationary distribution prepared by the penalised [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Comparison of the probability distribution over computational basis states [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Fidelity between the filtered quantum state and the [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Estimated energy as a function of the inverse temper [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Absolute error in the number of domain walls as a [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Quantum circuit implementing the proposal oracle [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Quantum circuit implementing the qubitized walk operator [PITH_FULL_IMAGE:figures/full_fig_p017_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: Quantum circuit implementing the acceptance op [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: Quantum circuit implementing the acceptance op [PITH_FULL_IMAGE:figures/full_fig_p017_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13: Quantum circuit implementing the Quantum Phase [PITH_FULL_IMAGE:figures/full_fig_p018_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14: Quantum circuit implementing the final transforma [PITH_FULL_IMAGE:figures/full_fig_p018_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15: Semiclassical quantum circuit implementing the [PITH_FULL_IMAGE:figures/full_fig_p020_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16: Energy level probability mass of the filtered state as a function of the inverse temperature [PITH_FULL_IMAGE:figures/full_fig_p021_16.png] view at source ↗
read the original abstract

The Metropolis-Hastings algorithm is a cornerstone of Markov Chain Monte Carlo methods, underpinning a wide range of applications in computational physics, Bayesian inference, and machine learning. Quantum variants of Metropolis-Hastings promise accelerated mixing through quantum walks, but their practical realisation remains challenging. In this work, we construct and simulate an explicit circuit level implementation of a quantum Metropolis-Hastings algorithm based on the framework introduced by Claudon \emph{et al.} (arXiv:2506.11576). We present the full quantum workflow required to prepare a stationary distribution, including a number of modifications required to make the algorithm implementable in a realistic quantum circuit model. Our results demonstrate that these modifications are essential to recover the correct stationary behaviour and highlight both the potential and current limitations of quantum Metropolis-Hastings algorithms, which are expected to become practically relevant in the fault tolerant quantum computing regime.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript constructs an explicit circuit-level implementation of a quantum Metropolis-Hastings algorithm by adapting the penalized qubitized walk framework of Claudon et al., incorporating spectral filtering and other modifications to enable realization in a realistic quantum circuit model. Numerical simulations on small instances are presented to demonstrate that these modifications are essential for recovering the target stationary (Boltzmann) distribution.

Significance. If the modifications preserve the intended stationary distribution, the work supplies a concrete workflow and identifies practical obstacles for quantum MCMC algorithms, which could become relevant in the fault-tolerant regime. The emphasis on circuit adaptations and their necessity offers guidance for future implementations, though the absence of analytical guarantees restricts the result's immediate generality.

major comments (1)
  1. No derivation is given showing that the effective walk operator (after penalization, spectral filtering, and circuit approximations) continues to satisfy detailed balance or has the target Boltzmann distribution as its unique stationary state. The central claim that the modifications recover the correct stationary behaviour therefore rests entirely on numerical simulations of small systems; any perturbation to the null space would shift the distribution without necessarily being visible in those instances.
minor comments (2)
  1. Simulation results lack error bars, full circuit diagrams, and quantitative metrics (e.g., fidelity or distance to target distribution), limiting independent verification of the claim that the modifications are essential.
  2. Notation and operator definitions carried over from the cited prior work would benefit from explicit re-statement to improve standalone readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for recognizing the practical value of the circuit workflow and identified obstacles. We address the major comment point by point below.

read point-by-point responses

  1. Referee: No derivation is given showing that the effective walk operator (after penalization, spectral filtering, and circuit approximations) continues to satisfy detailed balance or has the target Boltzmann distribution as its unique stationary state. The central claim that the modifications recover the correct stationary behaviour therefore rests entirely on numerical simulations of small systems; any perturbation to the null space would shift the distribution without necessarily being visible in those instances.

    Authors: We agree that the manuscript provides no complete analytical derivation proving that the effective walk operator after penalization, spectral filtering, and circuit approximations exactly satisfies detailed balance or possesses the target Boltzmann distribution as its unique stationary state. The central claim is indeed supported by numerical simulations on small systems, where the effective operator can be exactly diagonalized to verify recovery of the stationary distribution (to high accuracy) only when the modifications are included. We will revise the manuscript by adding a dedicated subsection that supplies a construction-based argument: the penalization is engineered to suppress components outside the target subspace while spectral filtering projects onto the low-energy eigenspace, and the circuit approximations are controlled (small-angle or perturbative). A brief perturbation analysis will be included to bound the shift in the stationary state for the error levels considered, addressing visibility concerns by showing that deviations scale with the approximation error and remain detectable even in the simulated regimes. This strengthens the presentation without claiming a full rigorous proof, which lies outside the current scope focused on explicit circuit realization. revision: partial

Circularity Check

0 steps flagged

No circularity detected; construction relies on external framework plus independent numerical validation

full rationale

The paper adapts the penalized qubitized walk framework from the externally cited Claudon et al. (arXiv:2506.11576) and introduces circuit-level modifications whose effect on the stationary distribution is checked via explicit small-system simulations. No step reduces a claimed result to a fitted parameter, self-citation, or definitional renaming; the central demonstration that the modifications recover the target Boltzmann distribution is presented as an empirical outcome rather than an algebraic identity with the inputs. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed from abstract only; no explicit free parameters, invented entities, or non-standard axioms are stated in the provided text.

axioms (1)
  • domain assumption Standard quantum circuit model and fault-tolerant quantum computing assumptions hold for the target implementation.
    Implicit in the discussion of circuit implementation and the fault-tolerant regime.

pith-pipeline@v0.9.0 · 5460 in / 1160 out tokens · 54154 ms · 2026-05-10T10:41:09.530517+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

39 extracted references · 39 canonical work pages

  1. [1]

    Instead, it realises a Dirichlet kernel [22], corresponding to a sharply peaked response at phase 0 with small side lobes

    How- ever, when a finite number of ancilla qubits is used, the resulting filter is not exact. Instead, it realises a Dirichlet kernel [22], corresponding to a sharply peaked response at phase 0 with small side lobes. These side lobes allow eigenstates with nearby phases to pass the filter with a probability that depends on the number of ancillas used: P r...

    work page

  2. [2]

    Model and target distribution To illustrate the behaviour of the quantum Metropolis- Hastings algorithm in the presence of multimodality, we consider a simple but representative double well energy landscape defined on a finite two-dimensional lattice (see Fig. 3). The figure provides a continuous visualization of the underlying landscape for geometric int...

    work page

  3. [3]

    The parametermcontrols the spectral resolution of the filtering step and thus di- rectly affects how accurately the stationary distribution can be reproduced

    Results and analysis We begin by analysing how the quality of the distri- bution prepared by the penalised quantum Metropolis- Hastings algorithm depends on the numbermof precision qubits used in the QPE filter. The parametermcontrols the spectral resolution of the filtering step and thus di- rectly affects how accurately the stationary distribution can b...

    work page

  4. [4]

    Its simplicity, combined with temper- ature dependent correlations, makes it a natural testbed for analysing both classical and quantum Markov chain Monte Carlo methods

    Model and target distribution The Ising model is a paradigmatic system in statisti- cal physics and serves as a standard benchmark for sam- pling algorithms. Its simplicity, combined with temper- ature dependent correlations, makes it a natural testbed for analysing both classical and quantum Markov chain Monte Carlo methods. State space and proposal geom...

    work page

  5. [5]

    For smallβ(high temperature),π β is close to uniform, and the associated Metropolis-Hastings chain onEexhibits a relatively large spectral gapδ(β)

    Performance versus spectral gap Under the above choices, the inverse temperatureβis the sole parameter controlling the structure of the Gibbs distribution. For smallβ(high temperature),π β is close to uniform, and the associated Metropolis-Hastings chain onEexhibits a relatively large spectral gapδ(β). Asβ increases, correlations strengthen and the spectr...

    work page

  6. [6]

    Energy estimation The obtained distributions can be used to estimate physical observables in direct analogy with classical Monte Carlo methods [3]. Given an observableOand an output probability distributionp X extracted from the filtered state, we compute expectation values as ⟨O⟩pX = X x∈E pX(x)O(x).(54) In particular, we consider the energy observable d...

    work page

  7. [7]

    hysteresis type phenomenon

    Basin structure In realistic implementations, resource limitations may prevent the algorithm from operating in the fully re- solved regime where we have enough precision ancillas to filter the target distribution. Although in the present work this limitation arises from classical simulation con- straints, similar restrictions are expected in larger scale ...

    work page doi:10.13039/501100011033 2026

  8. [8]

    E.W. Larsen. An overview of neutron transport problems and simulation techniques. InComputational Methods in Transport, pages 513–534, 2006

    work page 2006

  9. [9]

    Kalos and P.A Whitlock.Monte Carlo Methods

    M.H. Kalos and P.A Whitlock.Monte Carlo Methods. WILEY-VCH Verlag GmbH & Co. KGaA, second edi- tion, 2008

    work page 2008

  10. [10]

    K. Choo, A. Mezzacapo, and G. Carleo. Fermionic neural-network states for ab-initio electronic structure. Nature Communications, 11(1), 2020

    work page 2020

  11. [11]

    J. W. T. Keeble, M. Drissi, A. Rojo-Franc` as, B. Juli´ a- D´ ıaz, and A. Rios. Machine learning one-dimensional spinless trapped fermionic systems with neural-network quantum states.Physical Review A, 108(6), 2023

    work page 2023

  12. [12]

    Kaipio and E

    J.P. Kaipio and E. Somersalo.Statistical and Computa- tional Inverse Problems. Springer, first edition, 2004

    work page 2004

  13. [13]

    Beskos, A

    A. Beskos, A. Jasra, E.A. Muzaffer, and A.M. Stuart. Se- quential monte carlo methods for bayesian elliptic inverse problems.Statistics and Computing, 25, 2015

    work page 2015

  14. [14]

    Roberts and J

    G. Roberts and J. Rosenthal. General state space markov chains and mcmc algorithms.Probability Surveys, 1, April 2004

    work page 2004

  15. [15]

    K. Wu, S. Schmidler, and Y. Chen. Minimax mixing time of the metropolis-adjusted langevin algorithm for log-concave sampling.Journal of Machine Learning Re- search, 23, 2022

    work page 2022

  16. [16]

    Beskos, M

    A. Beskos, M. Girolami, S. Lan, P. Farrell, and A. Stuart. Geometric mcmc for infinite-dimensional inverse prob- lems.J. Comput. Phys., 335, 2016

    work page 2016

  17. [17]

    Zhou and H

    Q. Zhou and H. Chang. Complexity analysis of bayesian learning of high-dimensional dag models and their equiv- alence classes.The Annals of Statistics, 51, 2021

    work page 2021

  18. [18]

    M. Szegedy. Quantum speed-up of markov chain based algorithms. In45th Annual IEEE Symposium on Foun- dations of Computer Science, pages 32–41, 2004

    work page 2004

  19. [19]

    Levin and Y

    D. Levin and Y. Peres.Markov Chains and Mixing Times. American Mathematical Society, second edition, 2009

    work page 2009

  20. [20]

    Lemieux, B

    J. Lemieux, B. Heim, D. Poulin, K. Svore, and M. Troyer. Efficient quantum walk circuits for metropolis-hastings algorithm.Quantum, 4, June 2020

    work page 2020

  21. [21]

    Temme, T.J

    K. Temme, T.J. Osborne, K.G.H. Vollbrecht, D. Poulin, and F. Verstraete. Quantum metropolis sampling.Na- ture, 471(7336), 2011

    work page 2011

  22. [22]

    Montanaro

    A. Montanaro. Quantum speedup of monte carlo meth- ods.Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 471(2181), 2015

    work page 2015

  23. [23]

    Claudon, P

    B. Claudon, P. Rodenas-Ruiz, J.P. Piquemal, and P. Monmarch´ e. Quantum circuits for the metropolis- hastings algorithm, 2025. 16

    work page 2025

  24. [24]

    Alonso and O

    D.S. Alonso and O. Al-Ghattas. A first course in monte carlo methods, 2024

    work page 2024

  25. [25]

    S¨ underhauf

    C. S¨ underhauf. Generalized quantum singular value transformation, 2023

    work page 2023

  26. [26]

    Magniez, A

    F. Magniez, A. Nayak, J. Roland, and M. Santha. Search via quantum walk.SIAM Journal on Computing, 40(1), 2011

    work page 2011

  27. [27]

    Cleve, A

    R. Cleve, A. Ekert, C. Macchiavello, and M. Mosca. Quantum algorithms revisited.Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 454(1969), 1998

    work page 1969

  28. [28]

    Martyn, Z.M

    J.M. Martyn, Z.M. Rossi, A.K. Tan, and I.L. Chuang. Grand unification of quantum algorithms.PRX Quan- tum, 2(4), 2021

    work page 2021

  29. [29]

    Bruckner, J.B

    A.M. Bruckner, J.B. Bruckner, and B.S. Thomson.Real Analysis. Prentice-Hall, first edition, 1997

    work page 1997

  30. [30]

    Parker and M.B

    S. Parker and M.B. Plenio. Efficient factorization with a single pure qubit and logn mixed qubits.Physical Review Letters, 85(14), 2000

    work page 2000

  31. [31]

    Carrasco-Arango

    M. Carrasco-Arango. Quantum metropolis hastings: Circuit implementation, 2026.https://github.com/ bsc-wdc/q-mcmc. Appendix A: Circuit implementations In this appendix we present the quantum circuits implementing the penalised version of the quantum Metropolis-Hastings algorithm. The full implementation, including circuit generation and simulation code, is...

    work page 2026

  32. [32]

    Proposal oracleO T The proposal oracleO T prepares, for each current state x, a coherent superposition over neighbouring statesy according to the chosen proposal kernelT(x, y). Its im- plementation is topology dependent: the circuit structure encodes the adjacency relation of the proposal graph and produces a clean coin register that is uncomputed at the ...

    work page

  33. [33]

    11: Quantum circuit implementing the acceptance op- erator ofP,O A

    Acceptance oracleO A |a⟩=|0⟩ OA |0⟩ |x⟩ |y⟩ |z⟩ FIG. 11: Quantum circuit implementing the acceptance op- erator ofP,O A. The circuits implementing the acceptance oracle, shown in FIG. 11 and FIG. 12, are constructed follow- ing the procedure introduced in [16]. They are composed of CNOT and SWAP gates together with a controlled implementation of the class...

    work page

  34. [34]

    As discussed in the main text, however,W is not used directly

    Penalised qubitized walkV Given the proposal and acceptance oracles, we can con- struct the PUEs required to implement the qubitized walkW. As discussed in the main text, however,W is not used directly. Instead, we work with a penalised versionV= Π⊠ +e iφ(Id−Π ⊠) W, as introduced in section III. Since projectors cannot be implemented directly in a quantum...

    work page

  35. [35]

    Quantum Phase Estimation The eigenstate filtering step is implemented using the Quantum Phase Estimation algorithm, whose circuit is shown in Fig. 13. The QPE is applied to the penalised |Ψ⟩= ∑ jαj |λj⟩ V V 2 V 4 V 8 ∑ jαj |λj⟩ |ϕj⟩ |0⟩ H QFT † |0⟩ H |0⟩ H |0⟩ H FIG. 13: Quantum circuit implementing the Quantum Phase Estimation (QPE) algorithm. qubitized ...

    work page

  36. [36]

    14: Quantum circuit implementing the final transforma- tion on the ancillaband the four main registers

    Dual to vertex decoding transformation m m m m ⊠|+⟩ |ν1,ν2⟩ |0,0⟩ |+⟩ |π,0⟩ |0,0⟩ O† A O† T O† T FIG. 14: Quantum circuit implementing the final transforma- tion on the ancillaband the four main registers. After the QPE based filtering step, the resulting quan- tum state is close to the 1−eigenstate of the penalised qubitized walk operatorV. When the pena...

    work page

  37. [37]

    The classical fidelity is symmet- ric, insensitive to small local fluctuations, and provides a smooth measure of overall agreement between distribu- tions

    Classical fidelity To quantify the global overlap between the prepared distributionp X and the target distributionπ, we use the classical fidelity F(p X , π) = X x∈E p pX(x)π(x) !2 .(B1) This quantity takes values in [0,1], withF(p X , π) = 1 if and only ifp X =π. The classical fidelity is symmet- ric, insensitive to small local fluctuations, and provides...

    work page

  38. [38]

    Unlike the fidelity, it is sensitive to local discrepancies and therefore highlights misallocation of probability mass

    Total variation distance As a complementary metric, we consider the total vari- ation distance dTV(pX , π) = 1 2 X x∈E |pX(x)−π(x)|.(B2) The total variation distance has a clear operational inter- pretation as the maximum probability of distinguishing the two distributions with a single sample. Unlike the fidelity, it is sensitive to local discrepancies a...

    work page

  39. [39]

    Given a minimumx ⋆ = (i ⋆, j⋆)∈E and a radiusr≥0, we define a local basin as theL ∞ neighbourhood B∞(x⋆, r) ={(i, j)∈E| |i−i ⋆| ≤r,|j−j ⋆| ≤r}

    Local basin mass In order to analyse the structure of deviations between pX andπin Section III, we additionally consider the probability mass assigned to local neighbourhoods of the energy minima. Given a minimumx ⋆ = (i ⋆, j⋆)∈E and a radiusr≥0, we define a local basin as theL ∞ neighbourhood B∞(x⋆, r) ={(i, j)∈E| |i−i ⋆| ≤r,|j−j ⋆| ≤r}. (B3) For a set o...

    work page