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Solving 1D Poisson problem with a Variational Quantum Linear Solver

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper shows that tridiagonal matrices from 1D Poisson problems can be decomposed into $2^{n-1}+n$ unitary terms using SWAP and center-switch gates, enabling the first VQLS runs on such systems.

desk verdict A concrete small-case decomposition for tridiagonal VQLS, but the general-n claim is asserted, not proved. read the letter →

arxiv 2412.04938 v1 pith:YKKN2SIP submitted 2024-12-06 cs.CE cs.NAmath.NA

Giorgio Tosti Balducci , Boyang Chen , Matthias Möller , Roeland De Breuker This is my paper · ORCID
classification cs.CEcs.NAmath.NA
keywords unitarydecompositionvariationalquantumlinearsolvertridiagonalsystemsPoissonproblemcenter-switchgatePaulistringsNISQdevicessystem
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

Variational quantum linear solvers (VQLS) have previously been demonstrated on matrices that are naturally sums of a few unitary terms, such as small-qubit Hamiltonians. This paper targets tridiagonal matrices with constant coefficients, which arise from discretizing the one-dimensional Poisson problem, and shows how to decompose a $2^n\times 2^n$ matrix of this type into $2^{n-1}+n$ unitary terms using Pauli strings together with SWAP and center-switch gates, instead of the $2^n$ Pauli strings required by the standard decomposition. Because VQLS evaluates $O(L^2)$ quantum circuits when the matrix has $L$ terms, cutting $L$ almost in half substantially reduces the cost-circuit count. The paper reports the first simulated and real-hardware VQLS solutions of such tridiagonal systems, with solution fidelities close to 1 even when the cost function estimate is noisy. The tradeoff is that the multi-qubit gates make the cost circuits deeper, illustrated by a $51/99$ maximum-depth ratio for a $4\times4$ example.

What carries the argument

The central object is the center-switch gate $\mathrm{CS}(n-2)$, a permutation matrix on $n$ qubits that swaps the computational basis states $|011\ldots1\rangle$ and $|100\ldots0\rangle$; for $n=2$ it reduces to the standard SWAP gate. The paper decomposes it as a sequence of $2n-1$ multi-controlled Toffoli gates and uses it, together with Pauli strings and $Z$-type tensor products, to reproduce the tridiagonal structure of the matrix $A$ in the VQLS cost function, reducing the number of terms in the expansion from $2^n$ to $2^{n-1}+n$.

What would settle it

For a $16\times 16$ tridiagonal matrix ($n=4$), explicitly construct the paper's claimed 12 unitary terms, sum them, and compare against the original matrix; any missing or extra entry would refute the counting claim, and the same check can be repeated for $n=5$ with the claimed 21 terms.

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Extended reading notes

Core claim

The central claim is that the class of tridiagonal Toeplitz matrices used in 1D Poisson problems admits a unitary decomposition with roughly half as many terms as the standard Pauli-string expansion, by adding multi-qubit permutation gates. The key observation is that the center off-diagonal entries of a tridiagonal matrix are exactly the off-diagonal entries of a SWAP gate for the $4\times4$ case, and of a generalized center-switch gate $\mathrm{CS}(n-2)$ that swaps the bitstrings $011\ldots1$ and $100\ldots0$ for larger $n$. The paper constructs these gates explicitly in terms of Toffoli gates and compensates their diagonal entries with $n$ tensor products of $Z$ operators, leading to the count $2^{n-1}+n$. It then uses this decomposition inside VQLS to solve $2\times2$ and $4\times4$ linear systems, obtaining fidelities close to 1 in simulation and on hardware, and documents the tradeoff between fewer terms and deeper quantum circuits.

Load-bearing premise

The central efficiency claim depends on the unproved assumption that the Pauli strings, SWAP gate, and center-switch gates $\mathrm{CS}(n-2)$ span the space of tridiagonal Toeplitz matrices for every $n$ and that the term count $2^{n-1}+n$ is exact; if the count is off for larger $n$, the advantage over the Pauli decomposition disappears.

Editorial extensions

If this is right

  • For a $2^n\times 2^n$ tridiagonal system, the number of unitary terms in the matrix expansion drops from $2^n$ to $2^{n-1}+n$, reducing the number of quantum circuits needed to estimate the VQLS cost function.
  • The multi-qubit-gate decomposition extends VQLS beyond few-qubit Hamiltonians to a class of engineering matrices derived from finite-element discretizations.
  • The depth penalty is real but not prohibitive at small scale: the $4\times4$ case runs in circuits of depth 99 versus 51, and still reaches high fidelity on hardware.
  • The small hardware runs show that even when the cost estimate is biased by sampling noise and decoherence, the optimizer can land at parameters close to the analytical solution, a resilience the authors attribute to noise tolerance of variational algorithms.

Reading between the lines

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

  • The $O(L^2)$ circuit count means the term reduction from $2^n$ to about $2^{n-1}$ should cut the number of cost-circuit evaluations by roughly a factor of four, a consequence the paper does not state explicitly.
  • The same permutation-gate construction could be adapted to other banded or structured matrices that have off-diagonal blocks resembling SWAP-like permutations, though the diagonal compensation would need to be re-derived.
  • For larger $n$, the center-switch gate requires $2n-1$ multi-controlled Toffoli gates, so one could test whether alternative decompositions of the permutation yield shallower circuits and whether the depth penalty stays acceptable on current hardware.
  • If the term count holds at scale, the main obstacle to using this decomposition becomes circuit depth rather than circuit count, shifting the focus to depth-reduction techniques such as the Hadamard-Overlap Test.
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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 applies the variational quantum linear solver (VQLS) to the one-dimensional Poisson problem, whose discrete matrix is tridiagonal and Toeplitz. It proposes a decomposition of such a matrix into Pauli strings together with SWAP and newly introduced 'center-switch' multi-qubit gates, claiming a reduction in the number of unitary terms from 2^n to 2^{n-1}+n for a 2^n x 2^n matrix. The 4x4 and 8x8 cases are worked out explicitly, and simulated plus real-hardware VQLS results are reported for 2x2 and 4x4 systems.

Significance. If the general-n decomposition is correct, the paper offers a concrete, non-trivial instance of a VQLS decomposition for an engineering-relevant matrix class, with a term-count saving that is exponential in n (from 2^n to 2^{n-1}+n). The trade-off against circuit depth is honestly acknowledged. The 4x4 explicit decomposition is correct and the hardware demonstration, though small, addresses a gap in the VQLS literature, which has mostly focused on physics Hamiltonians. However, the central claim for arbitrary n is asserted rather than proved, and the text contains internal inconsistencies about the gate family and the number of diagonal correction terms. The significance is therefore conditional on a rigorous general construction, which the current manuscript does not supply.

major comments (3)
  1. [Section III.B.2, Eq. (20)] The text states that 'The set of Pauli gates and their tensor products, together with the SWAP and CS(n−2) gates are all the unitary terms required to decompose a tridiagonal 2n × 2n matrix' (Section III.B.2, final paragraph). This is not correct for n>3: the single gate CS(n−2) defined in Eq. (20) only swaps the central pair (2^{n-1}-1, 2^{n-1}). For n=4, Table II also needs I3CS(2−0), i.e., a center-switch gate on fewer qubits, covering the pairs (3,4) and (11,12). The general construction requires a hierarchy of center-switch gates on k+1 qubits for k=1,...,n−1 (with k=1 being SWAP and k=0 being X on the least significant qubit). This hierarchy is visible in Table II but never stated in the text. Please define the full gate family explicitly and prove that the n gates cover each of the 2^n−1 tridiagonal off-diagonal edges exactly once.
  2. [Section III.C] The claim that the diagonal correction 'only requires n additional unitaries' is contradicted by Table II: for n=4 the decomposition contains seven non-identity Z strings (I3I2Z1Z0, I3Z2I1Z0, I3Z2Z1I0, Z3I2I1Z0, Z3I2Z1I0, Z3Z2I1I0, Z3Z2Z1Z0). The correct count of non-identity Z strings is 2^{n-1}-1, so the total term count is n + 1 + (2^{n-1}-1) = 2^{n-1}+n, which matches the stated formula. The explanation in the text must be corrected so that it is consistent with Table II and with the final count.
  3. [Section III.B.2 and III.C] No proof is given that the proposed set of unitaries spans the space of tridiagonal matrices of the form (10) for general n, nor are the expansion coefficients specified for general n. The explicit 4x4 and 8x8 examples are not a substitute, because it is not demonstrated that the same pattern of coefficients and Z corrections works for every n. Please provide a constructive proof (for example, by induction, or by explicitly verifying that the n off-diagonal gates cover all edges of the tridiagonal graph and that the even-parity Z strings fix the diagonal) and state the coefficients in Eq. (3) for arbitrary n.
minor comments (5)
  1. [Abstract and Section I] Minor language issues: 'less terms' should be 'fewer terms', and 'Proabably' is a typo for 'Probably'.
  2. [Section III.C] The formula in the text '2(n−1) + n' should read '2^{n-1} + n'; the superscript formatting appears to have been lost in compilation.
  3. [Section IV] The 51/99 circuit-depth comparison in Section III.C is asserted without details of the compiled circuits; for reproducibility, please include the depth calculation method or a code repository.
  4. [Section IV] The hardware results for the 4x4 case consist of only two runs. Please state the number of shots and any error bars, and consider adding more repetitions or at least an explicit caveat about the limited statistics.
  5. [References] Reference [30] is a master's thesis cited as 'to be published'; please provide a stable reference or archive link.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the new decomposition is a direct mathematical construction, and the VQLS runs are demonstrations rather than fitted predictions.

full rationale

The paper's derivation chain is non-circular. The central contribution in Section III.B is a constructive linear-algebra statement: a tridiagonal 2^n by 2^n matrix is represented as a linear combination of Pauli strings plus SWAP and center-switch unitaries, with Table II giving explicit term lists for n = 2, 3, 4 and a stated general count of 2^(n-1) + n terms. This count is not obtained by fitting parameters to data and then predicting the same data, nor does any result reduce to its input by construction. The VQLS experiments in Section IV are demonstrations of an algorithm; the reported fidelities are measured outputs of a variational optimization, not theoretical predictions derived from fitted quantities. The only remotely self-referential citation is [30], used for the standard fact that a tridiagonal matrix has only N nonzero Pauli coefficients; that fact is elementary and independently checkable, so the citation is not load-bearing. There is a genuine rigor gap: Section III.B.2 asserts without proof that the set of Pauli strings, SWAP gates, and CS(n-2) gates spans the space of tridiagonal Toeplitz matrices for general n, and the text's description of the gate set is under-specified for n > 3, where Table II appears to require a hierarchy of center-switch gates rather than a single CS(n-2) gate. These are unproved mathematical claims and presentation issues, not circular reasoning. Therefore no circular step is present and the score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 1 invented entities

The paper introduces one new mathematical object, the center-switch gate, which is a permutation matrix with an explicit definition and circuit. No free parameters are fitted; the decomposition coefficients derive from the input matrix entries. The main axioms are standard linear algebra facts about Pauli decompositions and the prior VQLS framework, plus the ansatz assumption, which the paper itself acknowledges is only approximately satisfied.

assumptions (4)
  • standard math Every N by N Hermitian matrix can be expressed as a linear combination of N^2 Pauli strings, and for tridiagonal Toeplitz matrices only N terms are non-zero.
    Invoked in Section III.A to define the baseline Pauli decomposition; standard result from Nielsen and Chuang [29] and the referenced thesis [30].
  • standard math The center-switch gate CS(n-2) is a unitary permutation matrix and can be decomposed into 2n-1 multi-controlled gates.
    Permutation theory and Toffoli decomposition, stated in Section III.B.2; unitarity follows from being a permutation matrix.
  • domain assumption The QLSP and VQLS framework, including the Hadamard test and cost function, works as described in Bravo-Prieto et al. [14].
    The paper builds on prior VQLS work, cited in Section II.B.
  • domain assumption The RY single-qubit ansatz V(theta) can represent the solution state well enough for the solver to converge.
    The paper admits in Section IV that for the 4x4 case the RY ansatz cannot perfectly encode the exact solution, so this assumption is only approximately true.
invented entities (1)
  • center-switch gate CS(n-2) independent evidence
    purpose: A permutation gate that reproduces the center off-diagonal entries of a 2^n by 2^n tridiagonal matrix, reducing the number of unitary terms in the VQLS decomposition.
    The gate is explicitly defined as the permutation between bitstrings 011...1 and 100...0, and a circuit implementation is given in Fig. 1; it is a well-defined mathematical object that anyone can verify.

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Pith. "Pith review of Solving 1D Poisson problem with a Variational Quantum Linear Solver." pith.science (2026). https://pith.science/paper/YKKN2SIP

@misc{pith2026241204938,
  author       = {Pith},
  title        = {Pith review of: Solving 1D Poisson problem with a Variational Quantum Linear Solver},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YKKN2SIP}},
  note         = {Machine review of arXiv:2412.04938}
}
read the original abstract

Different hybrid quantum-classical algorithms have recently been developed as a near-term way to solve linear systems of equations on quantum devices. However, the focus has so far been mostly on the methods, rather than the problems that they need to tackle. In fact, these algorithms have been run on real hardware only for problems in quantum physics, such as Hamiltonians of a few qubits systems. These problems are particularly favorable for quantum hardware, since their matrices are the sum of just a few unitary terms and since only shallow quantum circuits are required to estimate the cost function. However, for many interesting problems in linear algebra, it appears far less trivial to find an efficient decomposition and to trade it off with the depth of the cost quantum circuits. A first simple yet interesting instance to consider are tridiagonal systems of equations. These arise, for instance, in the discretization of one-dimensional finite element analyses. This work presents a method to solve a class of tridiagonal systems of equations with the variational quantum linear solver (VQLS), a recently proposed variational hybrid algorithm for solving linear systems. In particular, we present a new decomposition for this class of matrices based on both Pauli strings and multi--qubit gates, resulting in less terms than those obtained by just using Pauli gates. Based on this decomposition, we discuss the tradeoff between the number of terms and the near-term implementability of the quantum circuits. Furthermore, we present the first simulated and real-hardware results obtained by solving tridiagonal linear systems with VQLS, using the decomposition proposed.

Figures

Figures reproduced from arXiv: 2412.04938 by the authors.

Figure 1
Figure 1. Center-switch gate corresponding to the (011, 100) permutation. Qubits are ordered from least to most significant, going from q0 to q2. Each one of the gates corresponds to a Toffoli gate, i.e. gates having two control qubits and one target. An empty dot means that the corresponding qubit must be in state 0 to flip the target qubit, and the analogous holds for a full dot and state 1. This figure was produced using t… view at source ↗
Figure 2
Figure 2. Cost vs. number of iterations for two different tridiagonal quantum linear system problems (QLSPs). Both a simulator (‘qasm [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. A generalized variational quantum linear solver on photonic platform

    quant-ph 2026-07 conditional novelty 4.0 of 10

    A photonic two-qubit experiment runs VQLS on 4D linear systems, adding Tikhonov regularization for singular cases and a modulo-2 cost function for finite-field equations.

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