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arXiv2026

Performance evaluation of variational quantum eigensolver and quantum dynamics algorithms on the advection-diffusion equation

Near-term quantum algorithms are a promising route to solving partial differential equations, but gauging their true potential requires separating algorithmic performance from sampling and hardware noise. We benchmark a ground-state variational quantum eigensolver (VQE), cast as a variational quantum linear solver, against the Trotterization, variational quantum imaginary time evolution, and adaptive variational quantum dynamics simulation methods applied to the one-dimensional advection-diffusion equation in the recent quantum-dynamics study by Alipanah et al. [Phys. Rev. Res. 7, 043318 (2025)] at matched grid and problem size. On a noiseless state-vector simulator the $N=4$ VQE drives the final-time infidelity to a numerical floor ($\sim\!10^{-14}$) once the depth reaches $L\approx5$, an \emph{algorithmic ceiling} set by exact expectation values. Evaluating the same solver with a finite number $S$ of measurement shots, still without hardware noise, makes the infidelity sampling limited, following $1-f\approx c/S$ (a best-case readout-sampling estimate, with the solution's signs assumed known), providing a regime-matched comparison with the shot-based emulator of Alipanah \emph{et al.}\ and explaining the gap to their noisy hardware runs ($>10^{-1}$). The benchmark thus decomposes the near-term error budget into algorithmic, sampling, and hardware contributions, with a matched-depth resource comparison. The formulation applies without modification across $N=4,5,6$ qubits and to a two-dimensional (eight-qubit, $16\times16$) problem evolved to $t=1$, where the state-vector VQE holds a $\sim\!10^{-7}$ algorithmic-ceiling infidelity against the sampling-limited $\sim\!10^{-5}$ of the corresponding shot-based simulation, a difference of measurement regime rather than algorithmic superiority.

quant-ph↗

arXiv2026

Quantum matrix arithmetics with Hamiltonian evolution

The efficient implementation of matrix arithmetic operations underpins the speedups of many quantum algorithms. We develop a suite of methods to perform matrix arithmetics -- with the result encoded in the off-diagonal blocks of a Hamiltonian -- using Hamiltonian evolutions of input operators. We show how to maintain this $\textit{Hamiltonian block encoding}$, so that matrix operations can be composed one after another, and the entire quantum computation takes $\leq 2$ ancilla qubits. We achieve this for matrix multiplication, matrix addition, matrix inversion, Hermitian conjugation, fractional scaling, integer scaling, complex phase scaling, as well as singular value transformation for both odd and even polynomials. We also present an overlap estimation algorithm to extract classical properties of Hamiltonian block encoded operators, analogous to the well known Hadamard test, at no extra cost of qubit. Our Hamiltonian matrix multiplication uses the Lie group commutator product formula and its higher-order generalizations due to Childs and Wiebe. Our Hamiltonian singular value transformation employs a dominated polynomial approximation, where the approximation holds within the domain of interest, while the constructed polynomial is upper bounded by the target function over the entire unit interval. We describe a circuit for simulating a class of sum-of-squares Hamiltonians, attaining a commutator scaling in step count, while leveraging the power of matrix arithmetics to reduce the cost of each simulation step. In particular, we apply this to the doubly factorized tensor hypercontracted Hamiltonians from recent studies of quantum chemistry, obtaining further improvements for initial states with a fixed number of particles. We achieve this with $1$ ancilla qubit.

quant-ph↗

arXiv2026

Distributed Variational Quantum Linear Solver

The Variational Quantum Linear Solver (VQLS), a hybrid quantum-classical algorithm for solving linear systems, faces a practical scalability bottleneck: the Linear Combination of Unitaries (LCU) decomposition requires $O(L^2)$ circuit evaluations per optimizer iteration, where $L$ can grow to $4^n$ in the worst case for an $n$-qubit system. We address this computational bottleneck through two complementary strategies. First, we present a distributed VQLS (D-VQLS) framework (https://code.ornl.gov/olcf-qcfd/DVQLS.git), built on NVIDIA CUDA-Q, that enables asynchronous, scalable distribution of the $O(L^2)$ cost evaluations. Second, a fast Walsh--Hadamard transform (FWHT)-based Pauli decomposition with coefficient-amplitude pruning threshold $τ=0.01$ curbs LCU growth for the structured Toeplitz family, reducing $L$ from $O(2^n)$ to 64 for $n>6$ and compressing the circuit complexity per optimizer iteration from $O(n4^n)$ to $O(n)$. We derive the exact top-$L$ Frobenius error and connect it to worst-case solution error. For a 10-qubit tridiagonal Toeplitz system, the $L=64$ pruning yields a $256\times$ reduction---from 23 million to 90k circuits per optimizer iteration. The D-VQLS framework is validated on the NERSC Perlmutter supercomputer using multi-node, multi-GPU ideal state-vector simulations, achieving over $99.99\%$ fidelity against classical solutions on tridiagonal Toeplitz and Hele--Shaw flow benchmarks, with near-ideal strong scaling up to 24 GPUs and $95.3\%$ weak scaling efficiency at 96 GPUs processing more than 360k circuits per optimizer iteration (from larger-$L$ pruning) for the 10-qubit system. Systematic profiling identifies the optimal resource allocation for distributed quantum circuit workloads, yielding a $2.52\times$ speedup for the configurations studied.

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arXiv2026

Essential Unitarity for Higher-Order Quantum Computation

We develop a boundary-centric semantic framework for higher-order quantum computation, building on the Kelly-Laplaza description of compact closure and Abramsky's execution account. In the semantic carrier Perm(C), morphisms are complex-linear combinations of polarized boundary linkings, composed by execution. Finite-family addresses provide coherent control over finite-level quantum registers (qudits) while retaining the multiplicative boundary structure. We identify essential unitarity, a boundary condition extending ordinary unitarity to higher-order interfaces. On positive qudit registers it coincides with ordinary matrix unitarity; at higher order it expresses preservation of information across the full polarized boundary. We define a unit-free coherent quantum core generated by multiplicative wiring, unitary gates on positive qudit registers, and contextual coherent control, and prove that every one of its morphisms is essentially unitary. The framework realizes an applied coherent quantum switch and the unitary stages of equal-ratio one-slot supermap dilations with explicit memory. An extended abstract of this work was accepted for QPL 2026 and is forthcoming in its proceedings.

quant-ph↗

arXiv2026

Phase Retrieval in $\mathbb C^4$ Requires Exactly Eleven Measurements

Determining the minimal number of intensity measurements required for phase retrieval in $\mathbb{C}^4$ has been a long-standing open problem. Prior to this work, the best-known results implied that this minimum was either $10$ or $11$. In this paper, we leverage characteristic classes and cohomology groups from differential topology to prove that no family of $10$ vectors in $\mathbb{C}^4$ possesses the phase retrieval property. Combining our lower bound with Vinzant's explicit eleven-vector construction establishes that the exact minimum is $11$. Our result yields a significant consequence for pure state quantum tomography, namely, a rank-one POVM on $\mathbb{C}^4$ requires exactly $11$ elements to be informationally complete for pure states. This further implies that three orthonormal bases are insufficient to uniquely distinguish all pure states in $\mathbb{C}^4$. Because four orthonormal bases are already known to be sufficient, we conclude that exactly four bases are required, thereby completely resolving the problem left in [C. Carmeli, T. Heinosaari, J. Schultz, A. Toigo, Eur. Phys. J. D].

quant-ph↗

arXiv2026

Exact quantum splitting and the structure of finite algebras

Berlekamp's algorithm factors a squarefree polynomial $f\in\mathbb{F}_q[x]$ by deterministic linear algebra, reducing the problem to splitting an explicit commutative algebra $B\cong\mathbb{F}_q^r$ into its $r$ simple factors. For large odd $q$, the standard efficient splitting step is randomized, while known derandomizations are conditional on the Extended Riemann Hypothesis. We give an unconditional exact quantum implementation in a circuit model permitting single-qubit rotations through efficiently computable angles. The construction uses an unconditional counting argument. For a block containing $s\ge2$ irreducible factors, a quadratic-character test in odd characteristic and an absolute-trace test in characteristic $2$ yield a nonconstant test element with probability $p_{q,s}\ge\tfrac12$, known exactly in advance and depending only on $q$ and $s$, not on the unknown factorization. Exact amplitude amplification therefore converts each randomized test into a procedure succeeding with certainty after one amplification iteration. The resulting algorithm uses exactly $r-1$ quantum splitting rounds and $O(n^3\log q)$ quantum $\mathbb{F}_q$-operations and $O(n^3)$ classical operations, requiring no primitive root, quadratic non-residue, or distinct-degree preprocessing. The method also splits arbitrary finite-dimensional separable commutative $\\mathbb{F}_q$-algebras given by structure constants. Combined with R'onyai's classical structure theory, which computes the radical deterministically and reduces the remaining tasks deterministically to polynomial factorization, it yields the radical and the Wedderburn decomposition of $A/\mathrm{Rad}(A)$ into minimal two-sided ideals, with certainty, for any $n$-dimensional associative $\mathbb{F}_q$-algebra given by structure constants, using $O(n^4\log q)$ quantum $\mathbb{F}_q$-operations.

quant-ph↗

arXiv2026

Gate-Efficient Implementation of the Query-Optimal Time-Dependent Hamiltonian Simulation

The query-optimal algorithm of [CGWZ26] for general time-dependent Hamiltonian simulation uses $$ q = O\left( αT + \frac{\log(1/\varepsilon)}{\log\left(e + \log(1/\varepsilon)/(αT) \right)} \right) $$ queries to $\mathrm{HAM\mbox{-}T}$ within $\varepsilon$ error for a Lipschitz-continuous time-dependent Hamiltonian $H(t)$ on $[0,T]$ satisfying $\left\lVert H(t)\right\rVert\leqα$. However, its direct circuit implementation incurs a substantially larger gate overhead. In this note, we give an implementation of the same algorithm that retains its optimal query complexity and uses $$ O\left[ q \left( a + \log\left(1 + \frac{T(α+ βT)}{\varepsilon} \right) \right) \right] $$ one- and two-qubit gates, where $a$ is the number of block-encoding ancilla qubits and $β$ is the Lipschitz constant of $H$. The main ingredient is an exact dyadic factorization of the ordered update product in the underlying one-query transducer.

quant-ph↗

arXiv2026

Well-conditioned iterative methods for large open quantum systems

Markovian open quantum systems are well modeled by the Lindblad Master Equation (ME) $\frac{\mathrm{d}}{\mathrm{d} t} ρ_t = \mathcal{L} ρ_t$, where $\mathcal{L}$ is a linear (super-)operator and $ρ_t$ is the system state, a positive matrix. When designing or characterizing a quantum system, one is usually interested in the steady state $ρ_\infty$ (such that $\mathcal{L} ρ_\infty = 0$), the first few excited states, and trajectories $t\mapsto ρ_t$. In finite dimension, $ρ_t$ is an $n\times n$ matrix, $\mathcal{L}$ thus typically costs $n^4$ to store explicitly as a dense matrix, and $O(n^6)$ to diagonalize or invert exactly, making standard linear algebraic techniques expensive for large systems. However, $\mathcal{L}$ usually costs only $O(n^3)$ to apply. This makes iterative methods appealing, but they do not work without a good preconditioner. In this article, our main observation is that a part of the Lindblad equation, corresponding to the so-called no-jump evolution $\mathcal{S}$, can be inverted efficiently. Using this inverse map, we introduce an auxiliary completely positive trace-preserving (CPTP) map $Φ$ whose fixed point is directly related to $ρ_\infty$, all the other eigenvalues having smaller magnitude. The map $Φ$ is thus well suited to iterative methods, and $ρ_\infty$ can be found in a few Arnoldi iterations. Using the same inverse map $\mathcal{S}^{-1}$ as preconditioner, we compute the low-lying spectrum efficiently via shift-invert Arnoldi, and, as a proof of concept, build an implicit time integrator that is competitive on stiff systems in the low-precision regime. For the steady-state and low excited states problems, our methods scale like $O(n^3)$ per iteration and offer state-of-the-art performance on CPU and GPU.

quant-ph↗

arXiv2026

Constrained minimax approximation for quantum signal processing

Quantum signal processing (QSP) provides a simple and efficient framework for implementing polynomial transformations using quantum circuits. Its classical design stage leads to a constrained minimax approximation problem: find a polynomial of prescribed parity that approximates a target function uniformly on a fitting set while remaining bounded in magnitude by one on the domain $[0,1]$, which can be viewed as a semi-infinite constraint. Discretization converts the problem into a linear program, but feasibility at a set of finitely many sampled points does not ensure feasibility on the whole domain, especially when an optimal approximant reaches the boundary of the feasible set. We investigate two approaches to address this difficulty. A Remez exchange method combined with active-set constraint enforcement is efficient on many tested instances, but its stability depends on the target and problem geometry. We then introduce nonlinear Fourier retraction, which uses QSP completion and phase synthesis to turn a nearly feasible polynomial into phase factors for a feasible QSP polynomial without increasing the degree. Across representative problems, retraction largely preserves approximation accuracy and remains effective on instances where the Remez heuristic is unstable. The resulting workflow connects classical minimax approximation and semi-infinite optimization with nonlinear Fourier analysis, and is implemented in the qsppack software package.

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arXiv2026

Fast Fault-Tolerant Decoders for Hypergraph Product and Lifted-Product Codes

We design low-complexity, fault-tolerant decoders for quantum low-density parity-check (QLDPC) codes with the goal of reducing decoding latency. We target two major bottlenecks of decoding under the \emph{circuit-level} noise model: (i) post-processing via order-statistics decoding (OSD), and (ii) the large number of auxiliary variable nodes commonly introduced to represent CNOT-induced correlations during syndrome extraction. Our key observation is that propagating CNOT faults (\emph{hook errors}) create \emph{stabilizer-induced} trapping sets (TSs) that are intrinsic to hypergraph-product (HGP) and lifted-product (LP) constructions. Therefore, instead of modeling each such fault with an explicit correlation node and relying on OSD to clean up the resulting failures, we design message-passing decoders that resolve the corresponding \emph{stabilizer-induced} TSs directly. We obtain these decoders by deriving QLDPC decoders from decoders for the parent classical LDPC codes and using them collectively to correct broad families of \emph{stabilizer-induced} TSs. For CNOT faults that manifest primarily as syndrome errors, we show that their effect is equivalent to a data error together with syndrome-bit measurement errors. Consequently, given repeated measurements and a decoding graph that already includes nodes representing syndrome-bit errors, no distinct variable node is needed for each CNOT fault. Using a \emph{phenomenological} Tanner graph with nodes representing only data errors and syndrome-bit errors, simulations on the LP codes show a reduction in, or comparable, logical error rates relative to BP+OSD, at substantially lower decoding complexity.

cs.IT↗

arXiv2026

Unconditional Certified Randomness without Structure

We obtain a certified randomness protocol in the quantum random oracle model. The protocol is non-interactive and publicly verifiable with a classical verifier, and is based on Yamakawa and Zhandry's proof of quantumness [JACM'24]. We prove unconditional security of this protocol against adversaries making subexponentially-many adaptive quantum queries to the random oracle. Prior work on certified randomness relative to a random oracle additionally assumed the Aaronson--Ambainis conjecture or proved security only against low query-depth adversaries.

quant-ph↗

arXiv2026

Towards unsupervised representation learning for quantum data: quantum models with inference and generation

With quantum sensors, simulators and networks emerging, a future of quantum technology may produce quantum states as data---that is, coherently rather than as classical measurement records---thus motivating the study of suitable quantum generalisations of modern machine learning, including the automated, unsupervised extraction of useful representations. Two ingredients are central to the latter: inference, mapping observations to latent representations, and generation, mapping latent states back to synthetic data. Both are related to each other and to joint distributions for training models by the chain-rule of classical probability theory. The fact that quantum states however lack such universal, standard factorisation property thus poses a challenge. Here we develop a conceptual and mathematical framework for unsupervised representation learning from quantum data. Models are joint quantum states over visible and latent systems; state-over-time maps provide a notion of factorisation into a marginal state and inference (generation) channel; models with inference (generation) are ambiguous states---states for which such factorisation obtains---subject to a further consistency condition on extended inference maps as data extension. These stipulations are restrictive: we show that non-trivial models must feature non-linear such maps to the extended space. For three representative state-over-time maps, we completely characterise the ambiguous states, uncovering a hierarchy tied to the positive-partial-transpose (PPT) criterion from entanglement theory. Notably, the Leifer-Spekkens construction supports inference and generation exactly for model classes of PPT states, thus allowing genuinely quantum visible-latent correlations. We also formulate quantum counterparts of exact and approximate inference training, explore weaker notions of data extension and sketch a future research programme.

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arXiv2026

A hybrid quantum-classical neural network for learning to route

This work studies hybrid quantum-classical neural networks for learning routing heuristics. Specifically, this paper asks whether small quantum neural networks can replace parameter-heavy modules inside a competitive attention-based routing model while maintaining solution quality. For the capacitated vehicle routing problem, encoder feed-forward replacement emerges as the most promising design: it reduces the number of model parameters by 56.6% while keeping the hybrid model close to the classical neural baseline at small and medium instance sizes, although the gap grows for larger instances. This work also compares to classical routing algorithms, which remain highly competitive and often superior on the fixed Euclidean test sets. Our results therefore do not indicate quantum advantage or solver dominance, but identify encoder feed-forward replacement as a viable hybrid-module compression strategy for neural combinatorial optimization.

cs.LG↗

arXiv2026

Distributed Quantum Hypothesis Testing under Zero-rate Communication Constraints

The trade-offs between error probabilities in quantum hypothesis testing are by now well-understood in the centralized setting, but much less is known for distributed settings. Here, we study a distributed binary hypothesis testing problem to infer a bipartite quantum state shared between two remote parties, where one of these parties communicates to the tester at (asymptotic) zero-rate, while the other party communicates to the tester at zero-rate or higher. As our main contribution, we derive an efficiently computable single-letter formula for the Stein's exponent of this problem, when the state under the alternative is the product of their marginals. For proving the converse direction of our result, we utilize a novel technique based on reverse hypercontractivity of a quantum markov semigroup combined with the pinching method. For the general case with vanishing type I error probability, we show that the Stein's exponent when (at least) one of the parties communicates classically at zero-rate is given by a multi-letter expression involving regularized measured relative entropy maximized over a sub-class of binary outcome separable measurements. When the state under the alternative commutes with a product eigenbasis of the marginal states under the null and has a larger support, we show that the exponent is characterized as a max-min optimization of regularized measured relative entropy over a sub-class of local binary outcome projective measurements. While this expression becomes single-letter for the fully classical case, we further prove that this already does not happen in the same way for classical-quantum states in general. The converse proof of the max-min characterization relies on an extension of the classical blowing-up lemma to bipartite quantum states satisfying the aforementioned commutativity condition, which could be of independent interest.

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arXiv2026

Simulation-Based Evaluation of Energy-Constrained Quantum-Classical Competition

This paper develops a simulation-based framework for evaluating the energy implications of quantum and classical computing firms competing in a market with limited energy resources. We model providers as differentiated Cournot competitors whose feasible service capacity is induced by technology-specific energy scaling laws: polylogarithmic for quantum algorithms that achieve an equivalent computational target and polynomial for classical emulation. For symmetric groups of quantum and classical firms, the equilibrium reduces to a tractable two-equation system that supports large scenario sweeps over market size, technology mix, and hardware coefficients. We characterize the capacity-constrained Nash equilibrium, prove the existence of a demand scale beyond which quantum service becomes more energy efficient, and report numerical experiments calibrated to trapped-ion and Rydberg platforms. The results identify when quantum energy advantage is only asymptotic and when it becomes operationally relevant.

quant-ph↗

arXiv2026

Mutually unbiased bases as extremal probes of isotropic random Hamiltonians

Mutually unbiased bases (MUBs) are central finite structures in quantum information. We ask whether complete MUB systems have an extremal probing property beyond their projective $2$-design identities. For an isotropic Gaussian traceless Hamiltonian, we prove that, among labeled unions of $d+1$ orthonormal bases, a complete MUB union has the stochastically largest sampled maximum. Each basis induces the same regular-simplex Gaussian block; mutual unbiasedness eliminates cross-block covariance, joint Gaussianity yields independence, and a centered-convex Gaussian correlation inequality makes this independent coupling extremal. We also derive a radial-mixture extension and prove exact MUB-family collapse for fully matched diagonal-cost constructions.

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arXiv2026

Edge-Local and Qubit-Efficient Quantum Graph Learning for the NISQ Era

Graph neural networks (GNNs) are a powerful framework for learning representations from graph-structured data, but their direct implementation on near-term quantum hardware remains challenging due to circuit depth, multi-qubit interactions, and qubit scalability constraints. In this work, we introduce a hybrid quantum graph learning architecture designed explicitly for unsupervised learning in the noisy intermediate-scale quantum (NISQ) regime. Our approach combines a variational quantum feature extraction layer with an edge-local and qubit-efficient quantum message-passing mechanism inspired by the Quantum Alternating Operator Ansatz (QAOA) framework. The message-passing operation is decomposed into pairwise interactions along graph edges using standard single- and two-qubit gates. For a graph with $N$ nodes and $n$-qubit feature registers, this reduces the number of qubits required at one time from $Nn$ to at most $2n$. We train the model using the Deep Graph Infomax objective to perform unsupervised node representation learning. The external class labels are not used during graph construction or training and are used only to evaluate the learned embeddings. Experiments on the Cora citation network and the Phase 3 release of the 1000 Genomes Project show that the quantum edge interaction contributes to the quality of the learned node representations.

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arXiv2026

A Spectral Identifiability Threshold for Dissipative Rate Recovery from Truncated Liouvillian Spectra

Open quantum systems lose energy and phase coherence through different dissipative processes, but these processes can produce overlapping dynamical signatures. The Liouvillian spectrum summarizes how such a system relaxes, yet it is not obvious how much of that spectrum is needed to distinguish the underlying dissipation rates. We study this question for amplitude damping and dephasing in a six-qubit Lindblad model whose spectrum can be derived analytically. We retain only the slowest non-steady spectral modes and ask how many are required before each dissipative rate becomes recoverable. We show that population modes contain no dephasing information, which creates a lower bound of D = 2^n retained modes for uniform dephasing identifiability in the relevant rate regime. The measured recovery threshold reaches this bound at n = 4,5,6, while n = 3 remains above it. At n = 6, least squares achieves a mean joint absolute error of order 10^-9, compared with 4.355 x 10^-4 for four tabular learning methods. Robustness tests show that this advantage weakens when the spectra are perturbed and when a transverse field breaks the commuting structure. These results show that the amount and structure of retained spectral information can determine whether dissipative parameters are recoverable, independently of the estimator used. The present conclusions apply to noise-free simulator spectra rather than measurement-derived spectra.

cs.LG↗