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Fast Laplace transforms on quantum computers

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Pith's one-line read This paper introduces a Quantum Laplace Transform that block-encodes the $N\times N$ discrete Laplace transform on $\lceil \log_2 N \rceil$ qubits, with depth $O(\log\log N)$ and size $O(\log N)$ when the input diagonals are shallow.

desk verdict First quantum circuit for the discrete Laplace transform, with a clean block-encoding construction and honest caveats; the headline speedup is conditional on input-structure assumptions that the paper only partially quantifies. read the letter →

arxiv 2412.05173 v2 pith:553IKTQ3 submitted 2024-12-06 quant-ph math-phmath.MPphysics.comp-ph

Julien Zylberman This is my paper · ORCID
classification quant-phmath-phmath.MPphysics.comp-ph MSC 81P68 PACS 03.67.Lx
keywords quantumLaplacetransformblock-encodinglinearcombinationofunitariesChebyshevseriesdiagonaloperatorscircuitdepthZ-transformsignalprocessing
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

The paper introduces the Quantum Laplace Transform (QLT), a gate-based circuit that implements the $N\times N$ discrete Laplace transform with entries $e^{x_i y_j}/N$ on states encoded in $\lceil \log_2 N \rceil$ qubits. The construction expands each exponential in a truncated Taylor or Chebyshev series, block-encodes each term as a product of diagonal operators and a uniform matrix, and sums the terms with a linear combination of block-encodings. When the input-dependent diagonal operators admit shallow block-encodings, the paper argues the full circuit has depth $O(\log\log N)$ and size $O(\log N)$, exponentially fewer operations and double-exponentially less time than classical evaluation. This would make the discrete Laplace transform, previously missing from quantum routines, available as a building block for transform-based simulation, signal processing, machine learning, and finance applications.

What carries the argument

The construction is carried by three mechanisms. First, each exponential $e^{x_i y_j}/N$ is expanded in a Taylor series or a Chebyshev/Jacobi-Anger series truncated at $K$ terms; Lemma 2.1 gives exponentially fast convergence with the stated $K$. Second, each series term factorizes into a left diagonal operator, the uniform matrix $(1/N)_{i,j}=1/N$, and a right diagonal operator; the uniform matrix is block-encoded in depth 3 as the element-wise product of two Hadamard towers using the element-wise product lemma. Third, the $K$ term-block-encodings are combined through a linear combination of block-encodings, with SELECT and PREPARE routines, while copy/unary-control techniques parallelize the controls so the extra cost is only $O(K)$ controlled diagonal operations and $\widetilde{O}(K\log n)$ depth. The overall procedure yields Theorem 2.5, and the bottleneck is explicitly delegated to the block-encodings of the diagonal operators $\hat{X}_k$, $\hat{Y}_k$ (or $\hat{I}_k$, $\hat{T}_k$ for the Chebyshev variant).

What would settle it

Take $N=2^n$ with $x_i,y_j$ drawn independently and uniformly from $[0,1]$. Since a generic $n$-qubit diagonal operator cannot be block-encoded exactly without exponentially many gates, the diagonal subroutines in Theorem 2.5 would dominate the circuit, and the total depth would grow with $N$ rather than as $O(\log\log N)$; a reader can verify this by counting the diagonal subroutines for such a random instance, or alternatively by implementing a structured family from Appendix B and checking that the depth stays $O(\log\log N)$ for $N$ up to $2^{20}$.

Watch

Extended reading notes

Core claim

The central claim is that the discrete Laplace transform can be block-encoded efficiently: for $N=2^n$ and real or complex coefficient vectors $x,y$ with $|x_i|\leq x_{\max}$, $|y_j|\leq y_{\max}$, the $n$-qubit operation $[\mathrm{QLT}]_{i,j}=e^{x_i y_j}/N$ admits an $(e^{x_{\max} y_{\max}}, b, \varepsilon)$-block-encoding using $O(K)$ controlled diagonal operators and an additional circuit of size $\widetilde{O}(Kn)$ and depth $\widetilde{O}(K\log n)$, where $K=\Theta(x_{\max}y_{\max} + \ln(1/\varepsilon)/\ln(e+\ln(1/\varepsilon)/(x_{\max}y_{\max})))$. Because $K$ depends only on the range of the exponent and on $1/\varepsilon$, not on $N$, and because the uniform matrix is implemented by the element-wise product of two Hadamard towers in constant depth, the $N$-dependence of the depth can be as small as $O(\log\log N)$ when the diagonal operators are implementable in depth independent of $N$. The paper frames this as the first circuit implementation of the discrete Laplace transform on gate-based quantum computers.

Load-bearing premise

The advertised depth and size scalings assume that the non-unitary diagonal operators $\hat{X}_k$ and $\hat{Y}_k$ admit block-encodings of polylogarithmic depth and size; for arbitrary input vectors, exact implementations of such diagonal operators need resources exponential in $n$, as the paper itself states.

Editorial extensions

If this is right

  • If the diagonal subroutines are shallow, the QLT gives exponentially fewer primitive gates and double-exponentially less time than the classical $O(N)$ discrete Laplace transform; the paper is explicit that this is a routine speedup, not a decision-problem speedup.
  • The Z-transform of a sequence encoded in a quantum state can be computed as a QLT with coefficients $x_i=i$ and $y_j=-\ln(z_j)$, which gives a route to generating functions of finite probability distributions.
  • The QLT can approximate the continuous Laplace transform and its inverse on quantum states, with truncation $M=O(\log(1/\varepsilon))$, and the success probability can be improved by amplitude amplification.
  • For complex coefficients, the Taylor variant keeps the same $K$ scaling, while the Chebyshev variant uses $O(K^2)$ diagonal operators, a quadratic overhead.
  • Because $K$ is set by $x_{\max}y_{\max}$ and $1/\varepsilon$ rather than by $N$, the circuit remains logarithmic in $N$ even when high accuracy requires many series terms.

Reading between the lines

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

  • The paper leaves the diagonal subroutines as an oracle-style assumption; a concrete next step is to instantiate them for a fixed input family such as $x_i=i/N$, $y_j=\cos(2\pi j/N)$ using the Fourier/GQSP construction of Appendix B, then count the resulting total depth and size for $N$ up to $2^{20}$ to turn the scaling claim into an explicit circuit benchmark.
  • Because the QLT is a routine rather than a standalone algorithm, its practical payoff depends on downstream tasks that can consume a transformed quantum state; amplitude amplification on the block-encoding flag appears necessary except in smooth-input regimes where the paper shows the success probability is already $\Theta(1)$.
  • A testable extension is to use the QLT for Laplace-domain filtering without post-selection by estimating expectation values of the form $\langle\psi|\mathrm{QLT}^\dagger \hat{O}\,\mathrm{QLT}|\psi\rangle$; the block-encoding formalism supports observable-based usage even when the success probability of preparing the transformed state is low.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper introduces the Quantum Laplace Transform (QLT), a block-encoding of the N×N discrete Laplace transform matrix [QLT]_{i,j} = e^{x_i y_j}/N on n = ⌈log_2 N⌉ qubits. The construction expands e^{x_i y_j} in a truncated Taylor or Chebyshev series, writes each series term as a product of two diagonal operators and the uniform all-ones matrix, block-encodes the uniform matrix via an element-wise product of two Hadamard towers (Lemma 2.3), and combines the terms with a linear combination of block-encodings (Lemma 2.2). Theorem 2.5 states an (e^{xmax ymax}, O(n), ε)-block-encoding using O(K) controlled diagonal block-encodings plus an additional circuit of size O~(Kn) and depth O~(K log n), with K = Θ(xmax ymax + ln(1/ε)/ln(e + ln(1/ε)/(xmax ymax))). Section 3 extends the construction to complex coefficients, Section 4 discusses the dependence on diagonal-operator subroutines and applications, and the appendices contain error proofs, control-parallelization lemmas, a GQSP-based example for a single diagonal operator, and an approximation of the continuous Laplace transform and its inverse.

Significance. The core block-encoding reduction is technically sound and, conditional on access to efficient diagonal-operator block-encodings, provides a genuine circuit construction for the discrete Laplace transform on a gate-based quantum computer—something the paper correctly identifies as previously open. The element-wise product lemma (Lemma 2.3) is an elegant and efficient way to implement the uniform matrix in constant depth, and the error accounting in Lemma 2.1, Lemma 2.2, and Lemma 2.4 is explicit and standard. The main advertised advantage (O(log log N) depth and O(log N) size) is, however, conditional on the cost of the diagonal subroutines, and the paper's examples do not yet close that gap for the specific families of diagonal operators required by the QLT. The paper is honest about the general exponential cost of arbitrary diagonal operators, but the 'in many cases' claim needs an end-to-end demonstration.

major comments (3)
  1. [Appendix B, Lemma B.1] The headline claim that 'in many cases' the QLT has depth O(log log N) and size O(log N) requires a concrete family of input data (x_i, y_j) for which the entire set of diagonal operators in Eq. (10) can be block-encoded within polylogarithmic resource bounds. Appendix B analyzes only a single diagonal operator D_g for one function g; it does not analyze the k-dependence of the family g_k(s) = z(s)^k / ymax^k (or the Chebyshev variants I_k(ymax x_i) and T_k(y_j/ymax)). The Fourier degree M_k required for an ε-approximation generally grows with k, and the derivative-bound condition R > 1 in Eq. (33) can fail for high powers of a fixed analytic function. Consequently, the total cost Σ_k s_{U,k} and Σ_k d_{U,k} is not bounded by the results presented, and the abstract's 'in many cases' is an assertion rather than a proven consequence. Please provide an explicit end-to-end example with bounds on the k-dependence, or restate the main theorem as an oracle reduction and correspondingly qualify the abstract and conclusion claims.
  2. [Appendix B, Lemma B.1] The lemma statement claims that a diagonal operator D_g can be block-encoded with a quantum circuit of depth O(log(1/ε)), but the proof's final paragraph concludes a depth of O(log(n) log(1/ε)) after using n−1 ancilla qubits to parallelize the controls of the n single-qubit phase gates in U_ω. Since n = log_2 N, this depth is O(log log N), not independent of N as stated. The statement must be corrected to match the proof, or an alternative control scheme that removes the log n factor must be supplied. This directly affects the claim in Section 2 that the diagonal operators can be implemented with depth independent of N.
  3. [Section 4, 'Probability of success'] The displayed formula for the limiting success probability contains e^{-∥f∥∞∥g∥∞} as the exponential factor, but for a block-encoding with normalization α = e^{xmax ymax}, the post-selection probability is ∥QLT|h⟩∥²/α², which carries e^{-2∥f∥∞∥g∥∞}. The Θ(1) conclusion with respect to N is unaffected for fixed f and g, but the formula should be corrected.
minor comments (5)
  1. [Throughout] There are several typos: 'Chebytshev' for Chebyshev, 'Additionnally' for Additionally, 'demonstatre' for demonstrate, 'Futhermore' for Furthermore, 'constaints' for constraints, and 'proove' for prove.
  2. [Theorem 2.5] The tilde-O notation is used in the theorem statement but defined only inside the proof; please define it in the statement for readability.
  3. [Theorem 2.5 and Section 2] The ancilla count in Theorem 2.5 is 2n + 2a_be + ⌈log2(K+1)⌉, but the uniform-matrix block-encoding already uses n ancillae and the control-parallelization uses n−1 additional ancillae; the counting should be made explicit so that the total matches the described circuit.
  4. [Eq. (15)] The SELECT decomposition in Eq. (15) is standard, but the action of σ_X^k on the control register should be spelled out to avoid ambiguity in the product over k.
  5. [Abstract and Conclusion] The abstract states 'size O(log N)' for the QLT circuit, but Theorem 2.5 gives the additional circuit as O~(Kn); please state explicitly that this scaling assumes fixed xmax ymax and ε, with K constant in N.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the QLT block-encoding is assembled from standard LCU and product lemmas, with diagonal subroutines treated as explicit oracles whose costs are tracked.

full rationale

The paper's central derivation is self-contained in the relevant sense. Equation (6) expands each truncated series term as a product of two diagonal operators and the uniform matrix; Lemma 2.3 block-encodes the uniform matrix via element-wise products of Hadamard towers; Lemmas 2.2 and 2.4 assemble the LCU and product constructions; Theorem 2.5 states the resulting complexity in terms of O(K) controlled diagonal block-encodings plus an explicit O~(nK)-size, O~(K log n)-depth auxiliary circuit. No fitted parameter is later relabeled as a prediction, and no quantity is defined in terms of the target QLT. The truncation order K is obtained from the standard bound in [40], not from the block-encoding result. The advertised O(log log N) depth and O(log N) size scalings are explicitly conditional on the diagonal subroutines having polylogarithmic or N-independent depth, as stated after Eq. (10), in Section 4, and in Appendix B; Section 4 even warns that arbitrary diagonal operators require exponentially many resources. That is a stated scope condition, not a circular reduction. The self-citations, mainly [38], [44], and [51], support technical subroutines or limitations; Appendix B independently provides a logarithmic-depth diagonal block-encoding construction via GQSP, so the cited prior work is not load-bearing for the advertised scaling. I find no circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The QLT construction introduces no new particles or fitted constants. Its complexity claims rest on the assumed efficiency of block-encoding diagonal operators, which is an input-structure condition rather than a free parameter.

assumptions (4)
  • domain assumption Access to (1, a_be, ε')-block-encodings of diagonal operators X_k, Y_k, I_k, T_k (Eq. 10) with controllable error and polylogarithmic depth for the cases of interest.
    The efficiency of the QLT depends on these subroutines; Section 2 assumes their existence, Section 4 discusses cases where they are efficient.
  • domain assumption Quantum computer has full connectivity, is fault-tolerant, and gates are single- and two-qubit without gate-set specification.
    Stated in Section 1; the depth and size claims rely on idealized hardware.
  • standard math Taylor and Chebyshev truncation error bounds (Lemma 2.1), Bessel function bounds from Abramowitz-Stegun, and LCU lemmas from Gilyén et al. (Lemma 52/53).
    Standard results used to bound truncation and block-encoding errors.
  • domain assumption For the continuous Laplace transform (Appendix C), f is continuously differentiable, of exponential type, and its Laplace transform is smooth on the integration curve C.
    Needed for the truncation and discretization error bounds in Theorem C.3.

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Cite this review

Pith. "Pith review of Fast Laplace transforms on quantum computers." pith.science (2026). https://pith.science/paper/553IKTQ3

@misc{pith2026241205173,
  author       = {Pith},
  title        = {Pith review of: Fast Laplace transforms on quantum computers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/553IKTQ3}},
  note         = {Machine review of arXiv:2412.05173}
}
abstract

While many classical algorithms rely on Laplace transforms, it has remained an open question whether these operations could be implemented efficiently on quantum computers. In this work, we introduce the Quantum Laplace Transform (QLT), which enables the implementation of $N\times N$ discrete Laplace transforms on quantum states encoded in $\lceil \log_2(N)\rceil$-qubits. In many cases, the associated quantum circuits have a depth that scales with $N$ as $O(\log(\log(N)))$ and a size that scales as $O(\log(N))$, requiring exponentially fewer operations and double-exponentially less computational time than their classical counterparts. These efficient scalings open the possibility of developing a new class of quantum algorithms based on Laplace transforms, with potential applications in physics, engineering, chemistry, machine learning, and finance.

Figures

Figures reproduced from arXiv: 2412.05173 by the authors.

Figure 1
Figure 1. Scheme of the quantum circuit associated with the block-encoding of the Quantum Laplace [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗

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