Rouzé, Cambyse, Daniel Stilck França, and Álvaro M. Alhambra. "Efficient thermalization and universal quantum computing with quantum Gibbs samplers." arXiv preprint. DOI: 10.48550/arXiv.2403.12691.
ABSTRACT: The preparation of thermal states of matter is a crucial task in quantum simulation. In this work, we prove that a recently introduced, efficiently implementable dissipative evolution thermalizes to the Gibbs state in time scaling polynomially with system size at high enough temperatures, and for any Hamiltonian that satisfies a Lieb-Robinson bound, such as local Hamiltonians on a lattice. Furthermore, we show the efficient adiabatic preparation of the associated purifications or ``thermofield double'' states. To the best of our knowledge, these are the first results rigorously establishing the efficient preparation of high-temperature Gibbs states and their purifications. In the low-temperature regime, we show that implementing this family of dissipative evolutions for inverse temperatures logarithmic in the system's size is polynomially equivalent to standard quantum computation. On a technical level, for high temperatures, our proof makes use of the mapping of the generator of the evolution into a Hamiltonian, and then analysing it as perturbation of the Hamiltonian corresponding to infinite temperature. For low temperature, we instead perform a perturbation at zero temperature of the Laplace transform of the energy observable at fixed runtime, and resort to circuit-to-Hamiltonian mappings akin to the proof of universality of quantum adiabatic computing. Taken together, our results show that a family of quasi-local dissipative evolutions efficiently prepares a large class of quantum many-body states of interest, and has the potential to mirror the success of classical Monte Carlo methods for quantum many-body systems.
Mele, Antonio Anna, Armando Angrisani, Soumik Ghosh, Sumeet Khatri, Jens Eisert, Daniel Stilck França, and Yihui Quek. "Noise-induced shallow circuits and absence of barren plateaus." arXiv preprint. DOI: 10.48550/arXiv.2403.13927.
ABSTRACT: Motivated by realistic hardware considerations of the pre-fault-tolerant era, we comprehensively study the impact of uncorrected noise on quantum circuits. We first show that any noise `truncates' most quantum circuits to effectively logarithmic depth, in the task of computing Pauli expectation values. We then prove that quantum circuits under any non-unital noise exhibit lack of barren plateaus for cost functions composed of local observables. But, by leveraging the effective shallowness, we also design a classical algorithm to estimate Pauli expectation values within inverse-polynomial additive error with high probability over the ensemble. Its runtime is independent of circuit depth and it operates in polynomial time in the number of qubits for one-dimensional architectures and quasi-polynomial time for higher-dimensional ones. Taken together, our results showcase that, unless we carefully engineer the circuits to take advantage of the noise, it is unlikely that noisy quantum circuits are preferable over shallow quantum circuits for algorithms that output Pauli expectation value estimates, like many variational quantum machine learning proposals. Moreover, we anticipate that our work could provide valuable insights into the fundamental open question about the complexity of sampling from (possibly non-unital) noisy random circuits.
Julià-Farré, S., Argüello-Luengo, J., Henriet, L., & Dauphin, A. (2024). "Quantized Thouless pumps protected by interactions in dimerized Rydberg tweezer arrays". arXiv preprint. DOI: 10.48550/arXiv.2402.09311.
ABSTRACT: We study Thouless pumps, i.e., adiabatic topological transport, in an interacting spin chain described by the dimerized XXZ Hamiltonian. In the noninteracting case, quantized Thouless pumps can only occur when a topological singularity is encircled adiabatically. In contrast, here we show that, in the presence of interactions, such topological transport can even persist for exotic paths in which the system gets arbitrarily close to the singularity. We illustrate the robustness of these exotic Thouless pumps through the behavior of the noninteracting singularity, which for sufficiently strong interactions splits into two singularities separated by a spontaneous antiferromagnetic insulator. We perform a numerical benchmark of these phenomena by means of tensor network simulations of ground-state physics and real-time adiabatic dynamics. Finally, we propose an experimental protocol with Floquet-driven Rydberg tweezer arrays.
Chevallier, C., Vovrosh, J., de Hond, J., Dagrada, M., Dauphin, A., & Elfving, V. E. (2024), "Variational protocols for emulating digital gates using analog control with always-on interactions". arXiv preprint. DOI: 10.48550/arXiv.2402.07653.
ABSTRACT: We design variational pulse sequences tailored for neutral atom quantum simulators and show that we can engineer layers of single-qubit and multi-qubit gates. As an application, we discuss how the proposed method can be used to perform refocusing algorithms, SWAP networks, and ultimately quantum chemistry simulations. While the theoretical protocol we develop still has experimental limitations, it paves the way, with some further optimisation, for the use of analog quantum processors for variational quantum algorithms, including those not previously considered compatible with analog mode.
Julià-Farré, S., Vovrosh, J., & Dauphin, A. (2024). "Amorphous quantum magnets in a two-dimensional Rydberg atom array". arXiv preprint. DOI: 10.48550/arXiv.2402.02852.
ABSTRACT: Amorphous solids, i.e., systems which feature well-defined short-range properties but lack long-range order, constitute an important research topic in condensed matter. While their microscopic structure is known to differ from their crystalline counterpart, there are still many open questions concerning the emergent collective behavior in amorphous materials. This is particularly the case in the quantum regime, where the numerical simulations are extremely challenging. In this article, we instead propose to explore amorphous quantum magnets with an analog quantum simulator. To this end, we first present an algorithm to generate amorphous quantum magnets, suitable for Rydberg simulators of the Ising model. Subsequently, we use semiclassical approaches to get a preliminary insight of the physics of the model. In particular, we calculate mean-field phase diagrams, and use the linear-spin-wave theory to study localization properties and dynamical structure factors of the excitations. Finally, we outline an experimental proposal based on Rydberg atoms in programmable tweezer arrays, thus opening the road towards the study of amorphous quantum magnets in regimes difficult to simulate classically.
Villoria, A., Basold, H., & Laarman, A. (2023). "Enriching diagrams with algebraic operations". arXiv preprint. DOI: 10.48550/arXiv.2310.11288.
ABSTRACT: In this paper, we extend diagrammatic reasoning in monoidal categories with algebraic operations and equations. We achieve this by considering monoidal categories that are enriched in the category of Eilenberg-Moore algebras for a monad. Under the condition that this monad is monoidal and affine, we construct an adjunction between symmetric monoidal categories and symmetric monoidal categories enriched over algebras for the monad. This allows us to devise an extension, and its semantics, of the ZX-calculus with probabilistic choices by freely enriching over convex algebras, which are the algebras of the finite distribution monad. We show how this construction can be used for diagrammatic reasoning of noise in quantum systems.
Skolik, A., Cattelan, M., Yarkoni, S., Bäck, T., & Dunjko, V. "Equivariant quantum circuits for learning on weighted graphs". npj Quantum Inf 9, 47 (2023). DOI: 10.1038/s41534-023-00710-y.
ABSTRACT: Variational quantum algorithms are the leading candidate for advantage on near-term quantum hardware. When training a parametrized quantum circuit in this setting to solve a specific problem, the choice of ansatz is one of the most important factors that determines the trainability and performance of the algorithm. In quantum machine learning (QML), however, the literature on ansatzes that are motivated by the training data structure is scarce. In this work, we introduce an ansatz for learning tasks on weighted graphs that respects an important graph symmetry, namely equivariance under node permutations. We evaluate the performance of this ansatz on a complex learning task, namely neural combinatorial optimization, where a machine learning model is used to learn a heuristic for a combinatorial optimization problem. We analytically and numerically study the performance of our model, and our results strengthen the notion that symmetry-preserving ansatzes are a key to success in QML.
Gil-Fuster, E., Eisert, J., & Dunjko, V. (2023). On the expressivity of embedding quantum kernels. Machine Learning: Science and Technology. DOI: 10.1088/2632-2153/ad2f51.
ABSTRACT: One of the most natural connections between quantum and classical machine learning has been established in the context of kernel methods. Kernel methods rely on kernels, which are inner products of feature vectors living in large feature spaces. Quantum kernels are typically evaluated by explicitly constructing quantum feature states and then taking their inner product, here called embedding quantum kernels. Since classical kernels are usually evaluated without using the feature vectors explicitly, we wonder how expressive embedding quantum kernels are. In this work, we raise the fundamental question: can all quantum kernels be expressed as the inner product of quantum feature states? Our first result is positive: Invoking computational universality, we find that for any kernel function there always exists a corresponding quantum feature map and an embedding quantum kernel. The more operational reading of the question is concerned with efficient constructions, however. In a second part, we formalize the question of universality of efficient embedding quantum kernels. For shift-invariant kernels, we use the technique of random Fourier features to show that they are universal within the broad class of all kernels which allow a variant of efficient Fourier sampling. We then extend this result to a new class of so-called composition kernels, which we show also contains projected quantum kernels introduced in recent works. After proving the universality of embedding quantum kernels for both shift-invariant and composition kernels, we identify the directions towards new, more exotic, and unexplored quantum kernel families, for which it still remains open whether they correspond to efficient embedding quantum kernels.
Koridon, E., Fraxanet, J., Dauphin, A., Visscher, L., O'Brien, T. E., and Polla, S., “A hybrid quantum algorithm to detect conical intersections”, Quantum 8, 1259 (2024). DOI: 10.22331/q-2024-02-20-1259.
ABSTRACT: Conical intersections are topologically protected crossings between the potential energy surfaces of a molecular Hamiltonian, known to play an important role in chemical processes such as photoisomerization and non-radiative relaxation. They are characterized by a non-zero Berry phase, which is a topological invariant defined on a closed path in atomic coordinate space, taking the value π when the path encircles the intersection manifold. In this work, we show that for real molecular Hamiltonians, the Berry phase can be obtained by tracing a local optimum of a variational ansatz along the chosen path and estimating the overlap between the initial and final state with a control-free Hadamard test. Moreover, by discretizing the path into N points, we can use N single Newton-Raphson steps to update our state non-variationally. Finally, since the Berry phase can only take two discrete values (0 or π), our procedure succeeds even for a cumulative error bounded by a constant; this allows us to bound the total sampling cost and to readily verify the success of the procedure. We demonstrate numerically the application of our algorithm on small toy models of the formaldimine molecule (H2C=NH).
Pérez-Salinas, A., Draškić, R., Tura, J., and Dunjko, V., “Shallow quantum circuits for deeper problems”, Phys. Rev. A 108, 062423. DOI: 10.1103/PhysRevA.108.062423.
ABSTRACT: State-of-the-art quantum computers can only reliably execute circuits with limited qubit numbers and computational depth. This severely reduces the scope of algorithms that can be run. While numerous techniques have been invented to exploit few-qubit devices, corresponding schemes for depth-limited computations are less explored. This work investigates to what extent we can mimic the performance of a deeper quantum computation by repeatedly using a shallower device. We propose a method for this purpose, inspired by Feynman simulation, where a given circuit is chopped into two pieces. The first piece is executed and measured early on, and the second piece is run based on the previous outcome. This method is inefficient if applied in a straightforward manner due to the high number of possible outcomes. To mitigate this issue, we propose a shallow variational circuit, whose purpose is to maintain the complexity of the method within predefined tolerable limits, and provide an optimization method to find such a circuit. The composition of these components of the methods is called “reduce&chop.” As we discuss, this approach works for certain cases of interest. We believe this work may stimulate new research towards exploiting the potential of shallow quantum computers.
