Quantum Circuits for the Metropolis-Hastings Algorithm

Baptiste Claudon; Jean-Philip Piquemal; Pablo Rodenas-Ruiz; Pierre Monmarch\'e

REVIEW 3 major objections 4 minor 1 cited by

Quantum Circuits for the Metropolis-Hastings Algorithm

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A Metropolis-Hastings quantum walk can be built from only the proposal and acceptance oracles, without coherently computing rejection probabilities.

desk verdict Novel dual edge-space quantization of MH that avoids coherent arithmetic, but the main theorem's proof silently switches to the lazy chain, so the gap claim overreaches as stated. read the letter →

arxiv 2506.11576 v6 pith:LVWZWIGI submitted 2025-06-13 quant-ph cond-mat.stat-mechphysics.chem-ph

Baptiste Claudon , Pablo Rodenas-Ruiz , Jean-Philip Piquemal , Pierre Monmarché This is my paper

classification quant-phcond-mat.stat-mechphysics.chem-ph PACS 03.67.Ac03.67.Lx

keywords Metropolis-HastingsalgorithmquantumwalkSzegedyquantizationprojectedunitaryencodingqubitizedMarkovchainMonteCarlogapamplificationMALA

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper claims that a Szegedy quantum walk for the Metropolis-Hastings algorithm can be implemented using only the proposal oracle and the acceptance oracle, with no coherent arithmetic on transition or rejection probabilities. The construction lifts the chain to directed edges of the state space, stores either the previous state or the rejected proposal in a memory register, and thereby makes accept/reject updates reversible. The resulting qubitized walk has an angular gap in $\Omega(\sqrt{\delta})$ whenever the classical chain has spectral gap $\delta$, so the quadratic speedup survives. Because each walk step uses a constant number of oracle calls and only $4m+3$ qubits for $n=2^m$ states, the method removes the main qubit overhead of earlier generic implementations.

What carries the argument

The load-bearing object is the dual edge-space Markov kernel $P = T A$ with its reversible time reversal $P^{\star} = A T$. The proposal $T$ proposes a new head while keeping the tail fixed, and the acceptance $A$ either flips the edge $(x,y)$ to $(y,x)$ or leaves it, so rejection is recorded rather than erased. The argument is carried by a symmetric projected unitary encoding (SPUE), a pair $(U,\boxtimes)$ with $\boxtimes$ a partial isometry and $U$ a symmetric unitary that encodes the discriminant $D = \boxtimes^{\dagger} S \boxtimes$. Hermitianization converts the nonreversible pair into such an encoding, and the qubitized walk $W = (2\boxtimes \boxtimes^{\dagger} - I)(X \otimes S)$ then inherits the quadratic gap through the spectral theorem for qubitized walks.

What would settle it

Compute the spectrum of the exact $4m+3$-qubit unitary $W$ of Theorem 3 for a small Metropolis-Hastings chain with known spectral gap $\delta$ (for example the 64-state MALA example): if the angular gap is not bounded below by a constant multiple of $\sqrt{\delta}$, the theorem's gap claim fails.

Watch

Extended reading notes

Core claim

The central discovery is the dual Metropolis-Hastings kernel $P = T A$ on the edge space $\mathcal{S} = \{(x,y): T(x,y) > 0\}$, with the proposal $T$ and acceptance $A$ both reversible with respect to $\nu(x,y) = \pi(x)T(x,y)$. The first coordinate of this edge process follows exactly the original MH kernel, while the second coordinate stores the previous state on acceptance and the rejected proposal on rejection, which is the information needed to invert the update unitarily. $P$ itself is not reversible, so the paper also constructs its time reversal $P^{\star} = A T$ and combines both into a Hermitianized symmetric projected unitary encoding whose discriminant has spectral gap in $\Omega(\delta)$. By the qubitized-walk spectral theorem, the resulting walk operator $W$ has angular gap in $\Omega(\sqrt{\delta})$ and a unique $1$-eigenvector encoding the stationary distribution in the range of the encoding partial isometry.

Load-bearing premise

Algorithm 1 presupposes an initial state $|\psi_0\rangle$ in the range of $\boxtimes$ with nonzero overlap with the target $\boxtimes |+\rangle |\nu\rangle |0\rangle$, and the paper gives no method to prepare such a state; if that overlap is exponentially small, the projection step can erase the quadratic speedup.

Editorial extensions

If this is right

Each application of the walk $W$ uses two calls to each of $O_T$, $O_T^{\dagger}$, $O_A$, and $O_A^{\dagger}$, so the per-step cost is constant in the state-space size.
Running the walk requires $4m+3$ qubits for $n=2^m$ states, compared with ancilla counts that grow with the arithmetic complexity in earlier reversible-computing constructions.
The first component of the prepared $1$-eigenvector is exactly $|\pi\rangle$, so measuring that register yields samples from the target distribution.
If the phase-estimation projection succeeds with constant probability, the end-to-end sampling cost scales as $\widetilde{O}(1/\sqrt{\delta})$, preserving the quadratic speedup.
The dual step operators constructed here can also be used for quantum singular value transformations, not only for sampling from $\pi$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper leaves the preparation of an initial state with non-negligible overlap with the target eigenvector to the user; if that overlap is exponentially small, the projection step can cancel the quadratic speedup, so an explicit state-preparation method or a warm-start heuristic would be needed for end-to-end advantage.
The edge-space construction suggests a general recipe: any proposal-acceptance chain whose two factors are reversible with respect to a common measure can be quantized by this lifting, not only Metropolis-Hastings kernels.
The single-ancilla controlled-SWAP encoding of Appendix A is even cheaper at $2m+1$ qubits, so a natural test is whether that alternative preserves the gap in settings where the two extra registers are too costly.
A direct numerical benchmark of the projection overhead on small non-log-concave targets could show whether the constant-factor oracle counts survive in practice.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The thing to know: the dual edge-space construction is a genuinely new way to quantize Metropolis-Hastings without coherent computation of transition probabilities, and for the Glauber acceptance choice the spectral analysis is solid. The problem is that Theorem 3, the main theorem, is stated for a general reversible MH kernel but proved only for the lazy chain with acceptance A/2. The circuit in Figure 6 uses the original O_A, not O_{A/2}. For Metropolis with a uniform target and symmetric proposal, A^2=1, so PP* = T^2 = T, which is not ergodic on the edge space; the claimed unique 1-eigenvector and Omega(sqrt(delta)) gap cannot hold for that W. This is not a missing numerical bound; it is a statement/proof mismatch. The fix is straightforward: either restrict the theorem to Glauber (where Corollary 2 gives delta*=delta), or implement the lazy acceptance explicitly (e.g., with an extra coin to realize (1+A)/2) and state the theorem for that construction. I do not see how to get O_{A/2} from O_A in constant calls without additional resources.

The paper does a lot right. The dual kernel P = T A, the reversibility of T and A with respect to nu, and the spectral inclusion in Corollary 1 are clean. The Hermitianization for the non-reversible product is a natural extension of the authors' earlier framework, and the constant oracle-call step operators are interesting independently of the gap question. The numerical MALA example and the open-source code are real evidence of reproducibility at small scale. The alternative SPUE in Appendix A is a simple and elegant construction.

The initial-state overlap in Algorithm 1 is a second, softer caveat. The paper acknowledges the sampling overhead but gives no bound on the overlap between a preparable |psi0> and the target eigenstate. If that overlap is exponentially small, the projection step could cancel the quadratic speedup. That is an incomplete part of the end-to-end claim, not a flaw in the walk operator itself.

The 4m+3 qubit count also hides some overhead: the equality tests in Lemma 3 are multi-controlled Toffolis, and the circuit diagram does not give a complete ancillary count for those. That is a minor accounting issue, not a conceptual one.

This paper deserves a serious referee. The core idea is novel and likely salvageable, but Theorem 3 as written overclaims. A revision that either restricts the theorem or modifies the construction would make it a solid contribution. I would not cite the main theorem in its current form, but I would cite the dual-kernel construction once it is cleaned up.

Referee Report

3 major / 4 minor

Summary. The paper proposes a quantum circuit construction for the Metropolis–Hastings algorithm that avoids coherent computation of transition probabilities. It lifts the MH chain to a dual kernel P = T A on the space of directed edges S, keeps rejected proposals in a memory register, and builds partial isometries for P and its adjoint from the proposal oracle O_T and acceptance oracle O_A. The two encodings are Hermitianized into a symmetric projected unitary encoding whose qubitized walk W is claimed to have angular gap Ω(√δ) and a unique 1-eigenvector encoding the stationary distribution. The paper also gives an alternative controlled-SWAP encoding, resource counts, and numerical simulations for MALA with an open-source implementation.

Significance. If the main theorem were valid as stated, this would be a significant step: a constant-oracle-call Szegedy-type walk for arbitrary MH kernels using only 4m+3 qubits and no reversible arithmetic for rejection probabilities, with quadratic gap amplification. The dual-kernel viewpoint, the variational bound for the lazy chain, and the numerical MALA demonstration with an open-source code are genuine strengths. However, the gap theorem as stated is not established for the walk actually constructed from O_A, and the sampling algorithm has no specified initial-state preparation; these issues substantially qualify the end-to-end claims.

major comments (3)

[Theorem 3 / Sec. V.D.3] Theorem 3 is stated for an arbitrary ergodic reversible Metropolis–Hastings kernel, but its proof invokes Corollary 4, which is proved in Sec. V.D.2 only for the lazy dual kernel with acceptance A/2 ('By replacing A by A/2... P will always denote the dual kernel of (1+P)/2'). The walk W in Theorem 3 and Figure 6 is constructed from the original O_A via Propositions 6 and 7. For the Metropolis choice with uniform target and symmetric proposal, A^2=1 and Eq. (68) gives PP⋆ = T^2 = T on the edge space; the dual proposal T is idempotent and not ergodic on S because it preserves the tail, so the discriminant has singular value 1 with multiplicity. Consequently W has multiple 1-eigenvectors, contradicting the claimed uniqueness and the Ω(√δ) angular gap. The theorem should be restricted to the Glauber case (Cor. 2) or a lazy implementation using O_{A/2} must be explicitly built from O_A and included in the circuit and resource count.
[Algorithm 1, Step 3] Algorithm 1 requires an initial state |ψ0> in Im(⊠) with non-zero overlap with ⊠|+>|ν>|0>. The paper provides no preparation method for such a state, and Eq. (82) shows that |ν> = O_T|π>|0> depends on the unknown target |π>. The text acknowledges a 'sampling overhead due to the initial-state overlap', but if no warm start is available the phase-estimation projection can be exponentially costly and the quadratic speedup is not end-to-end. This limitation should be stated clearly, or a procedure for preparing a suitable |ψ0> with non-negligible overlap should be provided.
[Sec. V.D.2, Prop. 9/Cor. 4] The notation P is overloaded: it denotes the original MH kernel, the dual kernel T A, the lazy dual kernel T((1+A)/2), and the block dilation P in Proposition 8. This makes it difficult to track which spectral gap δ is being used in Proposition 9 and Corollary 4. Please use distinct symbols (e.g., P_cl, P_dual, P_lazy, P_dil) throughout the proof.

minor comments (4)

[Eq. (68)] The identity PP⋆ = T^2 = T uses the fact that the edge-space proposal kernel T is idempotent; state this explicitly so that it is not confused with the original proposal kernel on S.
[Lemma 3 / Table I] Clarify how the equality tests (CNOT copy and multi-controlled Toffoli) in Lemma 3 are counted in the ancilla total of 3 in Table I, and whether the controlled-O_A gate requires an additional ancilla.
[Fig. 2] For the MALA spectrum plot, include the values of τ, β, and the discretization parameters, and draw the theoretical gap cos^{-1}(√(1-δ/2)) to make the comparison quantitative.
[Prop. 10] The notation O^† and O_T^† with implicit tensor factors is terse; specify the registers on which each dagger acts to avoid ambiguity.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the edge-space walk is constructed explicitly from the proposal and acceptance oracles, and the gap proof is a spectral comparison; the only self-citation is a non-load-bearing pointer to the authors' prior work.

full rationale

The central derivation is not circular. The dual edge kernel is defined from the classical proposal and acceptance kernels (Section V.B), the step encodings O and O* are explicit circuits built from O_T and O_A (Propositions 6 and 7), and the claimed gap follows from the spectral comparison in Corollaries 1-4 rather than from re-using the conclusion as an input. No fitted parameter is introduced and renamed as a prediction. The only self-citation, '[20] for more details on quantum algorithms for nonreversible Markov chains' in Proposition 8, is a pointer: Proposition 8 is proved from Propositions 6, 7 and Proposition 1, so the citation is not load-bearing. Algorithm 1's need for an initial state with nonzero overlap with the target eigenstate is explicitly acknowledged as a sampling overhead and is a stated limitation, not a circular step. The real concern is a statement/proof mismatch: Theorem 3 is stated for the original acceptance matrix A, but the gap proof uses Corollary 4, which is proved only for the lazy acceptance A/2 (Section V.D.2, 'By replacing A by A/2'), and for Metropolis choices with A^2=1 one has PP*=T, which is not ergodic on the edge chain. This is a correctness risk, not a circular reduction. The inserted text fragments near the circuit diagrams assert no mathematical claim and do not alter the derivation. The score of 2 reflects the minor, non-load-bearing self-citation rather than any circular step.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted. The paper's claims are conditional on oracle access and on the existence of a suitable initial state for phase estimation; the proofs are otherwise analytic. The dual edge kernel is a mathematical construction, not a new physical entity.

assumptions (5)

domain assumption The target chain P is an ergodic, reversible Metropolis-Hastings kernel on a finite state space S with stationary distribution pi.
Section V.A states this as the setting; all spectral and mixing results rely on finiteness and reversibility (Eq. 12, Theorem 1).
domain assumption Quantum access is given by PUEs (O_T, 1, |0>) for the proposal kernel and (O_A, 1, |0>) for acceptance amplitudes as in Eq. (29).
Section IV.F defines the oracle model; Theorem 3 counts calls to these oracles, so the result is relative to this access model.
domain assumption The proposal kernel is valid: T(x,x)=0 and T(x,y)>0 implies T(y,x)>0.
Section IV.E states this condition, used to ensure S is closed under A and the dual kernel is well behaved.
standard math Standard spectral facts about PUEs, Hermitianization, and qubitized walks are accepted (Thm. 2, Prop. 1).
Section IV.C imports these from the PUE/QSVT literature [19].
domain assumption In the general acceptance case, the chain is replaced by the lazy kernel (1+P)/2 so that PP* is ergodic.
Section V.D.2 makes this replacement explicit; the final spectral statement still gives Omega(sqrt(delta)) because the lazy gap is delta/2.

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Cite this review

Pith. "Pith review of Quantum Circuits for the Metropolis-Hastings Algorithm." pith.science (2026). https://pith.science/paper/LVWZWIGI

@misc{pith2026250611576,
  author       = {Pith},
  title        = {Pith review of: Quantum Circuits for the Metropolis-Hastings Algorithm},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LVWZWIGI}},
  note         = {Machine review of arXiv:2506.11576}
}

read the original abstract

Szegedy's quantization of a reversible Markov chain provides a quantum walk whose spectral gap is quadratically larger than that of the classical walk. Quantum computers are therefore expected to provide a speedup of Metropolis-Hastings (MH) simulations. Existing generic methods to implement the quantum walk require coherently computing the transition probabilities of the underlying Markov kernel. However, reversible computing methods require a number of qubits that scales with the complexity of the computation. This overhead is undesirable in near-term fault-tolerant quantum computing, where few logical qubits are available. In this work, we present a Szegedy quantum walk construction which follows the classical proposal-acceptance logic, and does not require further reversible computing methods. We also compare this construction with an alternative to Szegedy's approach which also provides a quadratic gap amplification. Since each step of the quantum walks uses a constant number of proposal and acceptance steps, we expect the end-to-end quadratic speedup to hold for MH Markov Chain Monte-Carlo simulations.

Figures

Figures reproduced from arXiv: 2506.11576 by the authors.

**Figure 1.** FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 3.** FIG. 3: Graphical representation of a MH step. Step 1. [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: FIG. 4: Graphical representation of a [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

**Figure 5.** Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 6.** Figure 6: FIG. 6: Quantum circuit for the final qubitized walk operator [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]

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Forward citations

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Paquet and H

E. Paquet and H. L. Viktor, Molecular dynamics, monte carlo simulations, and langevin dynamics: A computational review, BioMed Research International2015, 183918 (2015), https://onlinelibrary.wiley.com/doi/pdf/10.1155/2015/183918

work page doi:10.1155/2015/183918 2015

[2] [2]

Kirkpatrick, C

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, Optimization by simulated annealing, Science220, 671 (1983), https://www.science.org/doi/pdf/10.1126/science.220.4598.671

doi:10.1126/science.220.4598.671 1983

[3] [3]

Landau and K

D. Landau and K. Binder,A Guide to Monte Carlo Simulations in Statistical Physics(Cambridge University Press, USA, 2005)

work page 2005

[4] [4]

Lov´ asz and S

L. Lov´ asz and S. Vempala, Simulated annealing in convex bodies and an o*(n4) volume algorithm, Journal of Computer and System Sciences72, 392 (2006), jCSS FOCS 2003 Special Issue

work page 2006

[5] [5]

W. K. Hastings, Monte carlo sampling methods using markov chains and their applications, Biometrika57, 97 (1970), https://academic.oup.com/biomet/article-pdf/57/1/97/23940249/57-1-97.pdf

work page 1970

[6] [6]

Szegedy, Quantum speed-up of markov chain based algorithms, in45th Annual IEEE Symposium on Foundations of Computer Science(2004) pp

M. Szegedy, Quantum speed-up of markov chain based algorithms, in45th Annual IEEE Symposium on Foundations of Computer Science(2004) pp. 32–41

work page 2004

[7] [7]

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[8] [8]

Claudon, A simple algorithm to reflect through eigenspaces of unitaries (2025), arXiv:2412.09320 [quant-ph]

B. Claudon, A simple algorithm to reflect through eigenspaces of unitaries (2025), arXiv:2412.09320 [quant-ph]

arXiv 2025