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Joseph Tindall

Publications and source records attributed to Joseph Tindall.

At least 19 recordsLinked to original sources
arXiv2026

Using topology to reveal all mutually unbiased bases in dimension six

Mutually unbiased bases (MUBs) describe complementary quantum measurements: certainty in one basis gives no information about the others. In dimension six, triples of MUBs are known, but whether a fourth basis could exist remained unresolved for over three decades. OpenAI recently reported a proof that none exists. Here we use the topology of the space of complex Hadamard matrices, which describe transitions between MUBs, to characterize all sets in dimension six. We prove that every mutually unbiased triple belongs to a two-parameter family or is a single isolated triple built from the Fourier matrix. Since none of them extend to a fourth basis, this also proves that no four MUBs exist. Our proof rests on a new topological result that all order-six complex Hadamard matrices except Tao's form a connected space. Connectedness reduces the classification of triples to finitely many overlapping four-dimensional phase balls. We certify with interval arithmetic that these balls cover the space and for every pair of MUBs represented in a ball, we then determine all third bases that extend it. Dimension six is now the first non-prime-power dimension in which all sets of MUBs are classified.

quant-ph↗
arXiv2026

Compressing multivariate functions with tree tensor networks

Tensor networks are a compressed format for multi-dimensional data. One dimensional tensor networks -- often referred to as tensor trains (TT) or matrix product states (MPS) -- are increasingly being used as a numerical ansatz for continuum functions by ``quantizing'' the inputs into discrete binary digits. Here we demonstrate the power of more general tree tensor networks (TTNs) for this purpose. We provide direct constructions of a number of elementary functions as generic tree tensor networks and interpolative constructions for more complicated functions via a generalization of the tensor cross interpolation algorithm. For a range of multi-dimensional functions we show how more structured tree tensor networks offer a significantly more efficient ansatz than the commonly used tensor train. Finally, we demonstrate how the methods introduced in this work can be used to realize a TTN-based solver for multi-dimensional, non-linear Fredholm equations.

quant-ph↗
arXiv2026

Emergent quantum algebras in dissipative time crystals

Strong dynamical symmetries provide an algebraic description of persistent oscillations in open quantum systems governed by the Lindblad equation. For finite-dimensional systems with a full-rank stationary state, such symmetries are necessary and sufficient for non-decaying oscillatory modes to appear in the Liouvillian's spectrum. These modes can be directly constructed by acting on stationary states with the identified symmetry operators. Here we establish an asymptotic counterpart of this algebra for collective spins. Taking a single spin $S$ in a magnetic field, subject to dissipation through linear jump operators, we identify the exact conditions on the field and dissipative couplings under which non-decaying oscillatory modes emerge as $S \rightarrow \infty$. We then write down a simple, closed form for the corresponding `symmetry' operator which can be applied to the steady state to construct these modes, which are eigenvectors of the Liouvillian, with imaginary eigenvalues, up to $O(1/S)$ corrections. A second operator, an emergent conserved quantity, completes the algebra, and its slow relaxation determines how quantum noise selects the stationary state. The construction can be directly extended to a pair of interacting spins sharing a common reservoir, with the resulting quantum algebra making the entanglement during a quench analytically accessible.

quant-ph↗
arXiv2026

More mutually unbiased bases

Mutually unbiased bases (MUBs) describe quantum measurements for which certainty in one basis gives uniform outcomes in the others. Their maximum number remains unknown in dimensions that are not powers of a prime. We introduce an ansatz that, in many dimensions, allows us to construct more MUBs than have previously been found. Each basis in the ansatz consists of a diagonal phase matrix applied to a fixed unitary, which is a tensor product of Fourier matrices and, optionally, any real Hadamard matrix. This construction yields five MUBs in dimension 12, six in dimensions 48, 96, and 192, seven in dimensions 36 and 108, and ten in dimension 648. These bases can then be extended to exceed the tensor product bound, asymptotically, in every eighteenth dimension. Moreover, we show that Paley's real Hadamard matrix can be used to construct $q+1$ bases in dimension $d = q(q+1)$ for every prime power $q\equiv3\pmod4$. This count grows as $\sqrt d$ and cannot be exceeded by tensor products of smaller sets.

quant-ph↗
arXiv2026

Matrix Product Belief Propagation

We introduce "matrix product belief propagation" (MP-BP) as a controlled method for the approximate contraction of two-dimensional tensor networks and graphical models, which generically leads to a quadratic reduction of errors relative to existing methods: at similar computational effort, the number of accurate digits is asymptotically doubled. For infinite systems, MP-BP can be understood as a practical implementation of Baxter's corner-transfer-matrix method; for reflection-symmetric networks, it further coincides with the boundary matrix-product-state and corner-transfer-matrix renormalization group algorithms as conventionally implemented. In the limit of unit matrix-product rank, MP-BP is equivalent to belief propagation, and more generally, can be understood as a generalization of BP in which the "messages" live on surfaces and are assumed to have a matrix-product factorization. We further extend the quadratic improvement to the computation of expectation values, and numerically demonstrate the qualitatively improved convergence for a variety of classical and quantum 2D tensor network problems.

quant-ph↗
arXiv2026

A Complete Classification of Complex Hadamard Matrices of Order Six

Complex Hadamard matrices encode perfectly balanced unitary transformations. Their classification is complete through order five, but order six -- the first dimension in which several continuous families coexist with an isolated solution -- has remained open for decades. Here, we give a complete and exact finite-incidence classification of order-six complex Hadamard matrices up to standard equivalence. We supply the global step missing from Szöllősi's dilation method, which allows us to prove an even stronger version of his conjecture: every complex Hadamard matrix of order six can be recovered algebraically from a suitable, dephased $3 \times 3$ corner defined by four initial phases. We then describe the geometry of the reconstruction from these phases and show that, except for Tao's isolated matrix and a single explicit Karlsson matrix, every class admits a representative obtained by solving one quadratic and one cubic equation in both the horizontal and vertical directions. Our work resolves the classification problem and provides a rigorous framework for further investigating order-six Hadamards, with applications to balanced six-mode interferometers and the study of mutually unbiased bases.

quant-ph↗
arXiv2026

Exploring the Relaxation Landscape of a 2D Quantum Magnet on a 256-Qubit Processor

How quantum matter relaxes far from equilibrium is a central open problem in many-body physics, and one for which analog quantum simulators are well positioned to move from confirming theory to discovering new physics. Here, we use a two-dimensional Rydberg atom array of 256 qubits to map the relaxation landscape of the two-dimensional transverse-field Ising model across its phase diagram. Beyond the expected rapid thermalization, we identify two further regimes. The first is a prethermal regime whose dynamics are governed by an effective XY model. The second, and most unexpected, is a crossover regime characterized by a slowdown in relaxation. This slowdown occurs precisely where state-of-the-art classical tensor-network methods lose control at late times, whereas the quantum simulation remains consistent across system sizes. These results establish Rydberg atom arrays as a platform for scientific discovery in nonequilibrium quantum many-body dynamics.

quant-ph↗
arXiv2026

Observable Estimation in the Absence of Classical Verification

The predictive success of quantum mechanics underpins many areas of modern science, even as the exact simulation of large, interacting quantum systems remains beyond the reach of classical computation. This success has been enabled by the remarkable advancement of scalable numerical approximation methods, which often demonstrate practical accuracy despite the absence of formal guarantees. As quantum simulation pushes into regimes where these approximations struggle, a fundamental challenge arises: How can quantum outcomes be trusted when reliable classical benchmarks are unavailable? Here, we establish a framework for the independent validation of quantum estimates in this setting and present evidence that they provide the most credible result among several considered methods, in the absence of an immediately accessible ground-truth solution. We apply our framework to the semi-scrambling dynamics of a physical model that strains several leading classical simulation methods yet remains experimentally accessible, in part through our introduction of the \textit{operator Loschmidt echo}. We systematically design a series of experiments using quantum heuristics that, taken together, test the underlying assumptions and provide strong confidence in the observable estimates obtained from the quantum computer. We then show how this framework can be extended to place accuracy bounds on quantum estimates via careful characterization and manipulation of the device noise, transforming the problem of validating the observable estimation to validating the noise model. These results establish a route towards trusted quantum computation for scientific discovery, independent of classical verification.

quant-ph↗
arXiv2026

Microscopic mechanism for resonant light-enhanced pair correlations in K$_3$C$_{60}$

Recent experiments on K3C60 revealed a giant enhancement of the light-induced superconducting-like optical response for pump frequencies near 10 THz, with an efficiency roughly two orders of magnitude larger than for off resonant excitation. Here we show that a resonant enhancement of pair correlations arises naturally in a driven purely electronic model of K$_3$C$_{60}$ with ab initio parameters. The underlying mechanism is a symmetry constrained two-photon pathway: the first photon drives the system from the even-parity ground state to an intermediate odd-parity manifold, while the second photon induces a transition to an even-parity excited state with enhanced pair correlations. Larger-cluster calculations show that the corresponding resonance energy is strongly renormalized downward with system size and connectivity, reflecting the kinetic-energy gain of delocalized photo-excited doublon-holon configurations. A simplified single-orbital model reproduces this scaling trend and reaches a 14-site fcc cluster, where the resonant peak is pushed to 30 THz, with a trend compatible with a further reduction toward the experimental 10 THz scale in larger systems. Varying the Hubbard coupling strength, we find that the resonance is lowest and the pairing enhancement strongest near intermediate couplings, where doublon-holon excitations are both well defined and mobile. Our results establish a purely electronic mechanism for resonant light-enhanced pair correlations in K$_3$C$_{60}$ and support the interpretation of the experimental 10 THz resonance as optical access to a paired many-body excited state rather than improved metallicity. More broadly, they suggest that related resonant pathways may arise in other intermediate-coupling Hubbard materials.

cond-mat.supr-con↗
arXiv2026

Finite-temperature formation of magnetic plateaus and simplex liquid states on the frustrated ruby lattice

Geometric frustration in quantum systems can stabilize unconventional phases of matter that avoid traditional magnetic ordering at low temperatures. Here, we observe this phenomenon while mapping out the finite temperature phase diagram of the spin-1/2 Heisenberg antiferromagnet on the ruby lattice with next-nearest-neighbor interactions. Using an infinite tensor network state (iTNS) optimized and measured with belief propagation (BP) and corrections to BP, we observe the low temperature formation of stable magnetic plateaus at various magnetic field strengths. We find these plateaus host a novel `simplex liquid state' -- a disordered phase involving strongly paired spin simplices that retains non-zero residual entropy due to an exponentially large subspace of crystalline configurations. We accurately quantify the energy gap associated with these states and show that, as the temperature of the system is lowered, it does not go through a phase transition to reach them: the heat capacity remains finite and continuous at all observed temperatures. Our work demonstrates how BP-based tensor network techniques provide a powerful route to understanding frustrated quantum magnets at finite temperature.

cond-mat.str-el↗
arXiv2026

Dynamics of disordered quantum systems with two- and three-dimensional tensor networks

Large scale quantum annealing dynamics of Ising spin glasses were recently implemented on D-Wave's Advantage$2$ system on a range of lattices. Following extensive comparison to existing numerical methods, these experiments were claimed to be beyond the reach of classical computation. Here, we simulate these spin glass models with lattice-specific tensor networks, using belief propagation (BP) to keep up with the entanglement generated during the time evolution and then extracting expectation values with more sophisticated variants of BP. We find that state-of-the-art accuracies can be achieved with modest computational resources. Moreover, our results are scalable in both two and three dimensions, which we leverage to verify universal Kibble-Zurek physics on systems involving hundreds of qubits.

quant-ph↗
arXiv2026

Contracting Tensor Networks with Generalized Belief Propagation

Recent years have seen a growing interest in the use of belief propagation - an algorithm originally introduced for performing statistical inference on graphical models - for approximate, but highly efficient, tensor network contraction. Here, we detail how to apply generalized belief propagation (GBP) - where messages are passed within a hierarchy of overlapping regions of the tensor network - to approximately contract tensor networks and obtain accurate results. The original belief propagation algorithm is a corner case of this approach, corresponding to a particularly simple choice of regions of the tensor network. We implement the GBP algorithm for a number of different region choices on a range of two- and three-dimensional, infinite and finite tensor networks, solving the corresponding fixed point equations both numerically and, in certain tractable cases, analytically. Our examples include calculating the partition function of the fully frustrated Ising model, computing the ground state degeneracy of three-dimensional ice models, measuring observables on the deformed AKLT quantum state and evaluating the norm of randomly generated tensor network states.

quant-ph↗
arXiv2026

Tensor network surrogate models for variational quantum computation

We adopt a two-dimensional tensor-network (TN) ansatz to simulate variational quantum algorithms on two-dimensional qubit architectures, demonstrating its capability to accurately simulate deep circuits through the Quantum Approximate Optimization Algorithm (QAOA) applied to Ising spin-glass problems on heavy-hexagonal and square lattices. For heavy-hexagonal problems with up to three-body interactions, parameters trained on small instances and transferred to systems an order of magnitude larger improve the sampled energy distribution only up to intermediate depths, indicating a fundamental limit of parameter concentration as a transfer strategy. By extending the training itself with TN simulations on larger system sizes, we avoid local minima and obtain lower-energy samples. Analyses of entanglement growth and importance sampling show that the simulation remains classically feasible with moderate bond dimension. We find that parameter concentration also persists on square lattices, albeit at substantially higher computational cost to perform reliable sampling. Overall, our TN framework not only provides an efficient and controlled framework for benchmarking variational quantum algorithms on two-dimensional lattices, but also serves as an effective surrogate model for training variational algorithms.

quant-ph↗
arXiv2026

Belief Propagation and Tensor Network Expansions for Many-Body Quantum Systems: Rigorous Results and Fundamental Limits

Belief propagation (BP) provides a scalable heuristic for contracting tensor networks on loopy graphs, but its success in quantum many-body settings has largely rested on empirical evidence. Developing upon a recently introduced cluster-expansion framework for tensor networks, we rigorously study the applicability of BP to many-body quantum systems. For a state represented as a PEPS satisfying a ``loop-decay" condition, we prove that BP supplemented by cluster corrections approximates local observables with exponentially small relative error, and we give explicit formulas expressing local expectation values as BP predictions dressed by connected clusters intersecting the observable region. This representation establishes a direct link between cluster corrections and physical correlation functions. As a result, we show that ``loop-decay" \emph{necessarily implies} exponential decay of connected correlations, yielding sharp, rigorous criteria for when BP can and cannot succeed, and ruling out its validity at critical points. Numerical simulations of the two- and three-dimensional transverse field Ising model at zero and finite temperature confirm our analytical predictions, demonstrating quantitative accuracy deep in gapped phases and systematic failure near criticality.

quant-ph↗
arXiv2026

Quantum-Inspired Algorithm for Classical Spin Hamiltonians Based on Matrix Product Operators

We propose a tensor-network (TN) approach for solving classical optimization problems that is inspired by spectral filtering and sampling on quantum states. We first shift and scale an Ising Hamiltonian of the cost function so that all eigenvalues become non-negative and the ground states correspond to the the largest eigenvalues, which are then amplified by power iteration. We represent the transformed Hamiltonian as a matrix product operator (MPO) and form an immense power of this object via truncated MPO-MPO contractions, embedding the resulting operator into a matrix product state for sampling in the computational basis. In contrast to the density-matrix renormalization group, our approach provides a straightforward route to systematic improvement by increasing the bond dimension and is better at avoiding local minima. We also study the performance of this power method in the context of a higher-order Ising Hamiltonian on a heavy-hexagonal lattice, making a comparison with simulated annealing. These results highlight the potential of quantum-inspired algorithms for solving optimization problems and provide a baseline for assessing and developing quantum algorithms.

quant-ph↗
arXiv2025

Simulating dynamics of the two-dimensional transverse-field Ising model: a comparative study of large-scale classical numerics

The quantum dynamics of many-qubit systems is an outstanding problem that has recently driven significant advances in both numerical methods and programmable quantum processing units. In this work, we employ a comprehensive toolbox of state-of-the-art numerical approaches to classically simulate the dynamics of the two-dimensional transverse field Ising model. Our methods include three different tensor network techniques -- matrix product states, tree-tensor networks, and two-dimensional tensor-networks under the belief propagation approximation -- as well as time-dependent variational Monte Carlo with Neural Quantum States. We focus on two paradigmatic dynamical protocols: (i) quantum annealing through a critical point and (ii) post-quench dynamics. Our extensive results show the quantitative predictions of various state-of-the-art numerical methods providing a benchmark for future numerical investigations and experimental studies with the aim to push the limitations on classical and QPUs. In particular, our work connects classical simulability to different regimes associated with quantum dynamics in Rydberg arrays - namely, quasi-adiabatic dynamics, the Kibble-Zurek mechanism, and quantum quenches.

quant-ph↗
arXiv2025

Probabilistic Computing Optimization of Complex Spin-Glass Topologies

Spin glass systems as lattices of disordered magnets with random interactions have important implications within the theory of magnetization and applications to a wide-range of hard combinatorial optimization problems. Nevertheless, despite sustained efforts, algorithms that attain both high accuracy and efficiency remain elusive. Due to their topologies being low-$k$-partite such systems are well suited to a probabilistic computing (PC) approach using probabilistic bits (P-bits). Here we present complex spin glass topologies solved on a simulated PC realization of an Ising machine. First, we considered a number of three dimensional Edwards-Anderson cubic spin-glasses randomly generated as well as found in the literature as a benchmark. Second, biclique topologies were identified as a likely candidate for a comparative advantage compared to other state-of-the-art techniques, with a range of sizes simulated. We find that the number of iterations necessary to find solutions of a given quality has constant scaling with system size past a saturation point if one assumes perfect parallelization of the hardware. Therefore a PC architecture can trade the computational depth of other methods for parallelized width by connecting a number of P-bits that scales linearly in system size. This constant scaling is shown to persist across a number of solution qualities, up to a certain limit beyond which resource constraints limited further investigation. The saturation point varies between topologies and qualities and becomes exponentially hard in the limit of finding the ground truth. Furthermore we demonstrate that our PC architecture can solve spin-glass topologies to the same quality as the most advanced quantum annealer in minutes, making modest assumptions about their implementation on hardware.

cond-mat.dis-nn↗
arXiv2025

A Quantum Information Perspective on Many-Body Dispersive Forces

Despite its ubiquity, the quantum many-body properties of dispersion remain poorly understood. Here, we investigate the entanglement distribution in assemblies of quantum Drude oscillators, minimal models for dispersion-bound systems. We establish an analytic relationship between entanglement and correlation energy and show how entanglement monogamy determines whether many-body corrections to the pair potential are attractive, repulsive, or zero. These findings, demonstrated in trimers and extended lattices, apply in more general chemical environments where dispersion coexists with other cohesive forces.

quant-ph↗