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Multivariate Quantum Signal Processing

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Guang Hao Low

Quantum Physics · Submitted 1 Oct 2026 · 23 references

Pith's verdict: The framework is real and the weighted-cost Hamiltonian result is genuine, but the headline complexities rest on per-application analytic certificates in unshipped appendices, and the advertised Lean 4 proofs come with no artifact.

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CitationGuang Hao Low. "Multivariate Quantum Signal Processing." arXiv:2610.01125 (2026). https://pith.science/paper/PAG3FVD6
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The paper

The paper claims

A signal-processing framework designs quantum algorithms by specifying analytic transfer functions of multiple block-encoded operators, and the authors claim that every finite causally ordered query circuit admits such a…

The reading

This paper develops Multivariate Quantum Signal Processing (MQSP), a method for designing quantum algorithms that transform several noncommuting, nonnormal matrices at once by giving each oracle query an analytic variable and treating the whole computation as a multiport infinite-impulse-response (IIR) filter. The central claim is that a transfer function describing the query algorithm's action can be unrolled into a finite quantum circuit with logarithmic error dependence and near-unit success probability, provided the function continues analytically through a neighborhood of the evaluation point with uniform growth bounds across the promised oracle family. If the claim holds, a single design language yields per-oracle query costs that are optimal or near-optimal for Hamiltonian simulation with nonuniform oracle costs, sparse simulation, linear systems, inhomogeneous linear differential equations with nonnormal matrices, ground-state preparation and energy estimation, and generalized eigenvalue problems. The paper is significant because it moves quantum algorithm design to the level of choosing a transfer function and allocating queries among differently priced oracles, rather than compiling each transformation separately.

What carries the argument

The central object is the analytic transfer function $F(z;O)$ of a quantum transducer, defined by feedback equations on a public space and a direct sum of private oracle ports, with analytic variables $z_j$ appended to each oracle. The workhorse construction is quditization: a unitary Toeplitz lift $W_N[G]$ block-encodes the finite impulse response $\tilde{G}_N = \sum_{n<N} c_n G_n$, and clock-state optimization reduces normalization to a nuclear-norm problem on an $N\times N$ matrix. Clock shaping converts uniform analytic bounds on the group-delay operator $W = G^\dagger G'$ and radial growth of $G$ near $z=1$ into explicit feasible clocks with logarithmic error dependence. Delay allocation over pairwise costs leads to the paired half-quasinorm $\langle C,\lambda\rangle_{1/2} = (\sum_j \sqrt{C_j\lambda_j})^2$, which controls the leading weighted query cost.

Extended reading notes

Core claim

The paper claims that MQSP provides a complete design methodology for multi-oracle quantum algorithms by equipping quantum transducers with the analytic structure of multiport IIR filters. Its central result is a two-part machinery: quditization, which unrolls a transfer function $F(z)$ into a finite circuit via a unitary Toeplitz lift $W_N[G]$ and input/output clock states, and clock shaping, which chooses finite horizons and clocks so that approximation error depends logarithmically on precision rather than polynomially. The key theorem (Theorem 1.1, refined as Theorem 3.9) states that if the delayed transfer function is contractive in the unit disk, unitary at $z=1$, and continues analytically through $|z-1|\le\zeta$ with group-delay bounds $aI\preceq W\preceq wI$ and radial growth bounds, then an explicit clock of dimension $N = w + O((w-a+h_+ + h_- + \zeta^{-1}\log(M_{\mathrm{loc}}/\epsilon))/\sqrt{\delta})$ achieves normalization $1+\delta$ and error $\epsilon$. On this basis the authors prove that every finite causally ordered quantum query circuit has a finite analytic MQSP realization preserving oracle order and per-oracle query counts (Theorem 4.4), characterize the full achievable error-normalization frontier for a fixed schedule (Theorem 4.8), and derive optimal or near-optimal per-oracle query complexities for six problem classes, including weighted Hamiltonian simulation with cost about $t\langle C,\lambda\rangle_{1/2} + O(\|C\|_1\log(1/\epsilon))$.

Load-bearing premise

The claimed logarithmic error dependence and near-unit clock normalization require the transfer function to continue analytically through a fixed neighborhood of the evaluation point $z=1$, with uniform group-delay bounds $aI\preceq W\preceq wI$ and radial growth bounds holding across the entire promised oracle family; each application must prove these bounds for its own transfer function.

Editorial extensions

If this is right

  • Hamiltonian simulation under $H=\sum_j H_j$ with block-encodings of cost $C_j$ and normalization $\lambda_j$ runs at weighted cost about $t\langle C,\lambda\rangle_{1/2} + O(\|C\|_1\log(1/\epsilon))$, which can be much smaller than the LCU-qubitization cost $t\|\lambda\|_1\|C\|_1$ when expensive terms are weakly weighted.
  • In the $d$-sparse access model, simulation achieves optimal time-norm dependence $O(t\sqrt{d}\|H\|_{1\to 2} + \sqrt{d}\|H\|_{\max}t\log(1/\epsilon) + \log(1/\epsilon))$.
  • Linear systems can be solved with simultaneously optimal query counts $\Theta(\kappa\log(1/\epsilon))$ to the matrix block-encoding and $\Theta(1/\sqrt{p})$ to state preparation, closing a gap left by prior methods that were optimal in only one oracle.
  • Solution states of inhomogeneous linear differential equations for arbitrary time-dependent nonnormal matrices are prepared with $O(C/\sqrt{p})$ state-preparation queries and $O(\lambda T C g \sqrt{p}\log(1/\epsilon))$ matrix-block-encoding queries, which is optimal in the contractive case.
  • The same machinery yields per-oracle complexities for ground-state preparation and energy estimation under gap, threshold, and gapless promises, and for the three-oracle generalized eigenvalue problem that generalize Hermitian phase-estimation costs.
  • Exactly simulating downfolded (Schur-complement) Hamiltonians costs about $t\langle \vec{\lambda}_{\mathrm{eff}},\vec{C}_{\mathrm{eff}}\rangle_{1/2}$ with correlation-weighted normalizations, extendable to multiscale graph classification via Kron reduction.
  • An online generalization updates the IIR as events arrive, exemplified by an anisotropic Kalman filter that queries each transport and observation oracle once per event in event order.

Reading between the lines

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

  • The framework recasts quantum query complexity as coherent analog filter design: any classical lossless IIR template with suitable analytic continuation near $z=1$ becomes a candidate quantum algorithm, which suggests that the seventy years of classical filter-synthesis theory is a large untapped source of quantum constructions.
  • Because cost enters through delay allocation, nonuniform oracle pricing becomes a first-class design parameter rather than a post-hoc accounting detail; this is likely to change how resource estimates are quoted for Hamiltonian and linear-systems problems with heterogeneous terms.
  • If the analytic-continuation hypotheses fail uniformly across a promise set, the logarithmic error dependence and near-unit clock normalization break simultaneously; a natural test is to search for oracle families whose transfer functions have poles approaching $z=1$ or unbounded group-delay spread, where the claimed complexity must degrade.
  • The achievability theorem suggests that any finite fixed-schedule query algorithm can be profitably re-expressed as a transfer function, which could be used as an optimization surface for gate counts and success probability after the fact.
Original abstract

We develop Multivariate Quantum Signal Processing for designing matrix-valued transformations of multiple noncommuting nonnormal block-encoded matrices. Our framework equips quantum transducers with the analytic structure of multiport Infinite-Impulse-Response (IIR) filters and establishes a transfer-function methodology for designing quantum query algorithms. Efficiently computable analytic bounds that hold uniformly across the oracle promise yield efficient quantum circuits with logarithmic error dependence and near-unity success probability, paralleling the approximation-theoretic guarantees of univariate QSP/QSVT. We provide a complete characterization of achievable transfer functions and a principled library of analytic IIR components. Our modular designs greatly improve or resolve the per-oracle query complexity of open multi-oracle problems: (1) Hamiltonian simulation under $H=\sum_j H_j$ with weighted quasinorm cost $\approx t\langle\vec\lambda,\vec C\rangle_{1/2}+\mathcal{O}(\|\vec C\|_1\log(1/\epsilon))$ for block-encoding normalizations $\vec\lambda$ and costs $\vec C$; (2) sparse simulation with optimal time--norm $\Theta(\sqrt{d}\|H\|_{\rm 1\to2}t)$ and additive error dependence; (3) solving linear systems and (4) differential equations under minimal matrix assumptions with optimal weighted state-preparation and multi-term cost; (5) ground-state preparation and energy estimation; (6) generalized eigenvalue problems. We apply these to exactly simulate downfolded Hamiltonians with leading cost $t\langle\vec\lambda_{\rm eff},\vec C_{\rm eff}\rangle_{1/2}$ further reduced by fractional dynamic correlation energy contributions from coarse-grained Hamiltonians, and show similar classical-data-input-reduction results for multiscale graph classification via Kron reduction. We also show the online generalization with anisotropic Kalman filtering of streaming quantum data.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-02: 1 request, 6,040 input tokens.

What the abstract itself states, as probabilities: claims first 0.07 · states limitation 0.10 · human participants 0.02 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.95 · compares with prior results 0.67 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We develop Multivariate Quantum Signal Processing for designing matrix-valued transformations of multiple noncommuting nonnormal block-encoded matrices. Our framework equips quantum transducers with the analytic structure of multiport Infinite-Impulse-Response (IIR) filters and establishes a transfer-function methodology for designing quantum query algorithms. Efficiently computable analytic bounds that hold uniformly across the oracle promise yield efficient quantum circuits with logarithmic error dependence and near-unity success probability, paralleling the approximation-theoretic guarantees of univariate QSP/QSVT. We provide a complete characterization of achievable transfer functions and a principled library of analytic IIR components. Our modular designs greatly improve or resolve the per-oracle query complexity of open multi-oracle problems: (1) Hamiltonian simulation under $H=\sum_j H_j$ with weighted quasinorm cost $\approx t\langle\vec\lambda,\vec C\rangle_{1/2}+\mathcal{O}(\|\vec C\|_1\log(1/\epsilon))$ for block-encoding normalizations $\vec\lambda$ and costs $\vec C$; (2) sparse simulation with optimal time--norm $\Theta(\sqrt{d}\|H\|_{\rm 1\to2}t)$ and additive error dependence; (3) solving linear systems and (4) differential equations under minimal matrix assumptions with optimal weighted state-preparation and multi-term cost; (5) ground-state preparation and energy estimation; (6) generalized eigenvalue problems. We apply these to exactly simulate downfolded Hamiltonians with leading cost $t\langle\vec\lambda_{\rm eff},\vec C_{\rm eff}\rangle_{1/2}$ further reduced by fractional dynamic correlation energy contributions from coarse-grained Hamiltonians, and show similar classical-data-input-reduction results for multiscale graph classification via Kron reduction. We also show the online generalization with anisotropic Kalman filtering of streaming quantum data.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We develop Multivariate Quantum Signal Processing for designing matrix-valued transformations of multiple noncommuting nonnormal block-encoded matrices. contribution 1.00 0.96 0.21 asserted 1.00
s1 Our framework equips quantum transducers with the analytic structure of multiport Infinite-Impulse-Response (IIR) filters and establishes a transfer-function methodology for designing quantum query algorithms. contribution 0.62 0.92 0.13 asserted 1.00
s2 Efficiently computable analytic bounds that hold uniformly across the oracle promise yield efficient quantum circuits with logarithmic error dependence and near-unity success probability, paralleling the approximation-theoretic guarantees of univariate QSP/QSVT. result 0.82 0.89 0.17 asserted 1.00
s3 We provide a complete characterization of achievable transfer functions and a principled library of analytic IIR components. contribution 0.99 0.95 0.11 asserted 1.00
s4 Our modular designs greatly improve or resolve the per-oracle query complexity of open multi-oracle problems: (1) Hamiltonian simulation under $H=\sum_j H_j$ with weighted quasinorm cost $\approx t\langle\vec\lambda,\vec C\rangle_{1/2}+\mathcal{O}(\|\vec C\|_1\log(1/\epsilon))$ for block-encoding normalizations $\vec\lambda$ and costs $\vec C$; (2) sparse simulation with optimal time--norm $\Theta(\sqrt{d}\|H\|_{\rm 1\to2}t)$ and additive error dependence; (3) solving linear systems and (4) differential equations under minimal matrix assumptions with optimal weighted state-preparation and multi-term cost; (5) ground-state preparation and energy estimation; (6) generalized eigenvalue problems. result 0.69 0.93 0.46 asserted 1.00
s5 We apply these to exactly simulate downfolded Hamiltonians with leading cost $t\langle\vec\lambda_{\rm eff},\vec C_{\rm eff}\rangle_{1/2}$ further reduced by fractional dynamic correlation energy contributions from coarse-grained Hamiltonians, and show similar classical-data-input-reduction results for multiscale graph classification via Kron reduction. result 0.60 0.93 0.10 asserted 1.00
s6 We also show the online generalization with anisotropic Kalman filtering of streaming quantum data. contribution 0.91 0.90 0.04 asserted 1.00

What kind of paper the classifier reports this is: method or tool 0.99 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
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  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. One oracle port with delay r = 2 and horizon N = 5. At each time, the system matrix couples the corresponding public clock subspace to one private position. These subspaces are |k⟩clk ⊗ P and |k mod 2⟩j ⊗ Lj . Each oracle rectangle denotes a controlled call on both private positions. The calls occur at times 2 and 4, before the corresponding system operation. An amplitude written at time k is read at k + 2 after one oracle call. The lines denote direct-sum subspaces represented by the binary address register. The input consists of the public amplitudes (u0, . . . ,u4) and two…
Figure 2. Circuit for quditization on the binary address register and clock label. The system operation
Figure 3. Examples of shared processing modules. An arrow points from a construction to a module it
Figure 4. The same tap and preparation reflection define an IIR and a finite preparation algorithm. The lift receives the prepared state in its private subspace, initially empty public slots retain every transmitted amplitude, and the final success flag selects their direct sum. Proposition 5.3 (FPAA with the sharp leading preparation coefficient). In the parameter range 0 < λ0 < 1 and 0 < ϵ < e−2 , choose N = ⌈log(χ/ϵ)/ log(1/c)⌉. The finite construction has failure probability at most ϵ. Its preparation counts are QP = N + 1 and QP† = N, with QP + QP† ≤ log(1/ϵ) + log log(1/ϵ) 2 artanh λ0 +…
Figure 5. The sign lattice is a path of unitary sections. The two endpoints are public. A known boundary swap closes every internal coordinate, and the feedback connection applies the even and odd matchings. The marker z counts a whole two-query round. The public space has two endpoint copies of H. Its known address map is J
Figure 6. The reciprocal is block-encoded by a two-port unitary cascade. Each section retains its feedback connection and its additional public channel. The final clock acts on the connected network. For an effective Hamiltonian Heff ⪰ aI, substitute u = 2Heff/a. The resulting entry approximates (a/2)H −1 eff . A clock with normalization two then selects (a/4)H −1 eff . The degree is O(log(1/ϵ)); the oracle cost additionally depends on a common analytic neighborhood of the connected sections. Corollary 7.13 proves that neighborhood from the reduced model’s response and perturbation bounds.…
Figure 7. The weighted-cost example uses the same exponential transform, with a weighted Cayley junction and unequal port delays. Each delay is Delay; the weighted preparation uses known gates whose cost is included. Choose positive time and an integer delay with t > 0, 0 < ϵ ≤ δ ≤ 1/4, and R ≥ 1. Set the normalizations and costs to (λ1, λ2) = (1, R−1 ) and (C1, C2) = (1, R). With delays r = (1, R), use G(z) = Ft(z, zR). Direct differentiation of Equation (5.49) gives G†G′ = 2tI. For | log z| ≤ s/R, the calculation in Equation (5.38) bounds the Hermitian part of each load. The inequality
Figure 8. StatePrep is Hamiltonian simulation with weighted query cost on the span of the reference and target states. The output-sector selection is Project; its unnormalized successful output vector is b ≃ |Ψ⟩/2. To count quditization resources, let each output state have n qubits and use D = N Schur modes. The complete space of oracle Vj , including its sector flag and padding, has dimension at most 2 n+aj . The public space has dimension 1 + 2n ≤ 2 n+1. With integer delays rj from Theorem 6.1, qj = ⌊(N − 1)/rj⌋, Qj = 2qj , dc ≤ 2N + N X j rj2 aj , bc = ⌈log2 dc⌉, (5.60a) ntotal ≤ n +
Figure 9. The weighted Cayley transfer function in the feedback connection of the exponential transform. Solid wires are public and dashed wires are private to the complete system. The inner feedback connection keeps one separate port zjVj per term. Replacing Exp by its finite realization gives Equation (6.11). At the evaluation point z = 1, the load and its derivative are M = iH and ∂zjM = −λjI. The unitarity identity in Theorem 2.1 therefore gives F † ∂zjF = tλjI. (6.12) By the unitarity identity, these operators give the squared norms of the private query amplitudes on each input. They the
Figure 10. The projector Cayley transformation and its reflected feedback operation. Public wires are
Figure 11. The finite cascade and its circuit. Each system matrix has a distinct feedback connection, with aggregate feedback operation IJ ⊗ Ob (z). Clock preparation, the unitary Toeplitz lift, output-clock unpreparation, and known attenuation produce the compression shown below the circuit. All output coordinates are retained by the unitary implementation. The fixed J0 depends on δ; increasing precision leaves the target walk unchanged. Lemma B.3 implements the complete known cascade with a cost polynomial in log J. Assume the input has support only on energies satisfying 0 ≤ E ≤ Γ ≤ Λ and…
Figure 12. The feedback connection at tick k with a shared SELECT. Only the sectors scheduled at that step Dk are active. A preparation and its inverse are counted for each such sector, while SELECT is applied once to their union. The identity on all inactive sectors is retained. The batched operation is the same feedback operation on the full Hilbert space, so the system matrix and transfer function of the preceding construction apply
Figure 13. Sparse Hamiltonian simulation. Solid arrows are external connections; dashed arrows
Figure 14. The average of the incoming and outgoing scalar vectors lies on the constraint line (left).
Figure 15. shows the oracle connections. The line in S0 separates the open load coordinates from the private oracle spaces. Closing its Y coordinate leaves public space H ⊕ C. The inverse defining R is known: C † C = diag κI, κ/ρ,(κ + κ/ρ)I  . (6.125) Thus constructing R requires only known scalar inverses. Each matrix-feedback operation uses one call to OH. Each preparation-feedback operation uses one controlled call to B and one to B† . Lemma 5.2 proves the junction identity. If the solution norm were available, matching the reference scale to it would transmit all probability into the s
Figure 16. The scalar output of each scale system matrix drives the next system matrix. The vertical
Figure 17. shows the public and terminal outputs of the finite lift. The terminal-buffer map Lρ,s sums the unmeasured private amplitudes using known coordinate operations and a constant number of original-oracle calls (Lemma D.1). A matrix-only block JH satisfying ∥JH − H−1/(4κ)∥ ≤ η removes this residual. If L is the direct sum of the maps in Equation (6.130) and ℓ ≥ ∥L∥, a known row selection implements vN0 −p iβJH(L/ℓ)eN0 1 + β 2 , β = 8ℓ √ N0 . (6.132) With the exact inverse, every scale component is a scalar multiple of H−1 |b⟩. Choose a sufficiently large constant in N0 = O(κ) so th
Figure 18. The factor-access load has two public end coordinates and three closed auxiliary coordinates. The bidirectional arrows denote Hermitian couplings: both signal edges use x, and closure gives x 2 ξ = c/ρ. In the vector-valued realization, the signal edges use the same block encoding of T, and the last edge couples to |b⟩. In Lemma 5.2, the − √ D identity coupling belongs to the known K0. The two signal couplings occupy separate copies of the oracle space, so one controlled OT call acts on both at a due event, leaving the edge label unchanged. Equations (D.21) and (D.24) give the maps
Figure 19. The summed constraint has one scalar public input and separate private connections for the
Figure 20. Two ways to use given classical information. The left branch changes parameters of the
Figure 21. Top: the common residual bj − DjY j determines both interval outputs, with the dashed feedback path imposing the Volterra stage constraint. Arrows denote linear relations; the sum node is a row of the encoded coefficient matrix. Bottom: conditioned on its success flag, QLSP solves all interval constraints and prepares |Ψ⟩ = A−1 |c⟩/∥A−1 |c⟩∥. Selecting the final-state output gives the final state with constant probability. Its kth row is √ N(Y j )k = Xj + δ P l<k[b(tj,l) − Hj,l √ N(Y j )l ]. The Neumann series gives ∥E −1 j ∥ ≤ (1 − hα∥JN ∥) −1 ≤ 2, independently of N. The…
Figure 22. The scalar IIR generating the phases 1, q, q2 , . . .; the dashed path is feedback. The Kaiser amplitudes are prepared in the phase register. Controlled queries attach the phase q k to the corresponding clock amplitude. Equation (6.254) encodes these operations in one linear system. For the matrix problem, define the ordered Cayley transforms V = (I − iH/R) −1 (I + iH/R), W = (B − iA/R) −1 (B + iA/R) = B −1/2V B1/2 . (6.246) On the transformed eigenstate, the unitary V has eigenphase θ = 2 arctan(λ/R) ∈ [−π/2, π/2]. The similarity relation gives W|ψ⟩ = e iθ|ψ⟩. Prepare the kno
Figure 23. The two preparation algorithms. Each dashed pair connects an outer module to an inserted module and back. Expanding the inserted modules into their unitary realizations gives the original oracle feedback connections. The connected coarse stage is compiled once per sample. Solid arrows between the coarse stage and Filter show the actual finite-circuit state flow. The gap algorithm retains every sample’s labels and histories; the dotted arrow controls the filter by their reversibly computed median. The final filter uses the Hamiltonian oracle alone. We assume controlled access to an…
Figure 24. Normalize is Sign applied to the average of two reflections. The choice qubit c selects Rω or RΠ and is recombined by Hadamards, as specified by Equation (6.273). The left diagram expands each matching layer of Sign into these reflections. On the prepared span, A2 = aI, so its boundary action is Π|ω⟩/ √ a. The reflection about the prepared state is implemented by V † , the known zero-state reflection, and V , in execution order. The sign gap on this span is √ a. By Equation (6.269), the choice weight is O(a −1/2 ). Each reflection about the prepared state uses V and V † , so the…
Figure 25. From the Hamiltonian oracle to the ground reflection. The upper row shows the walk and
Figure 26. The accurate filter. The flag Hadamards in
Figure 27. shows both parts of this construction. The two neighboring amplitudes vary continuously as the phase crosses a bin. Their Fourier coefficients decay cubically, which makes both the total branch weight and its first moment finite. 0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1 Fractional position t in the bin Amplitude in R(ϑ)|0⟩ Two neighboring bin amplitudes Bin k: cos2 (πt/2) Bin k + 1: sin2 (πt/2) 1 2 4 8 16 32 10−8 10−6 10−4 10−2 100 Fourier index magnitude |ℓ| Larger coefficient at ℓ and −ℓ Fourier coefficient tails φ = π/4 φ = π/2 φ = 3π/4 Bound |ℓ| −3
Figure 28. The positive two-bin rounder. In each known momentum block,
Figure 29. One selector step and its closed response. The left IIR has the system matrix in
Figure 30. Finite implementation after the sampler weights have been bounded. The clock horizon and delays determine the oracle counts. The Fourier cutoff then bounds the accumulated change in the known branch-preparation reflections. Finally, sign paths are truncated beyond the sites reachable in the executed matching layers. The five-layer example shades a containing region for every possible path, including later injections at site zero. Its points indicate allowed sites, independently of their actual amplitudes. Finite history storage, Fourier truncation, and path truncation provide…
Figure 31. Median control and the final success test. Median is a known unitary on public registers,
Figure 32. The two energy-estimation routes. In the upper circuit, t is the integer time label of the phase-estimation window. The public Hilbert space throughout the upper row contains both time and the already prepared system registers; the dashed pair opens the controlled walk powers into Hamiltonian queries. This finite sequence adds the phase-estimation cost after preparation succeeds. The lower route reads the coarse sampler’s labels at the desired energy resolution and takes their median, as proved in Proposition 6.24. It stops before the accurate state filter. Proposition 6.23 (Energy…
Figure 33. Online state flow. The upper and lower branches are public and private direct-sum
Figure 34. The rational metric reweighting, shown as a classical signal-flow relation for each

Figures reproduced from arXiv:2610.01125 by the authors.

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