arXiv subjects

N. L. Diaz

Publications and source records attributed to N. L. Diaz.

17 recordsLinked to original sources

arXiv2026

No-go theorems for sublinear-depth group designs

Constructing ensembles of circuits which efficiently approximate the Haar measure over various groups is a long-standing and fundamental problem in quantum information theory. Recently it was shown that one can obtain approximate designs over the unitary group with depths scaling logarithmically in the number of qubits, but that no sublinear-depth approximate designs exist over the orthogonal group. Here we derive, for any group $G$ possessing an invariant state $G^{\otimes k} \lvertΨ\rangle= \lvertΨ\rangle$, a lower bound on the diamond distance between the $k$\textsuperscript{th} moment operator of any ensemble of elements of $G$, and that of the Haar measure over $G$. We then use this bound to prove that for many groups of interest, no subset of $G$ consisting of sublinear-depth one-dimensional circuits with local gates can form an approximate $k$-design over $G$. More generally, on a $D$-dimensional lattice, our results imply that such group designs require depths scaling at least as $n^{1/D}$. Moreover, for most of the groups we consider we find that such ensembles can, with high probability, be distinguished from $k$-designs by a single shot of a constant-depth measurement. Among other examples, we show that there is a constant separation between (a) the maximum depth and gate count for which no circuit can approximate even the second moment of random matchgate circuits, and (b) the depth and gate count required to implement the matchgate Haar distribution exactly. We furthermore rule out the existence of sublinear-depth $8$-designs over the Clifford group. Finally, we relax the assumption of working with local gates, and prove the impossibility of obtaining approximate designs over $G$ using any circuit comprised of a sublinear number of gates generated by Pauli strings.

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arXiv2026

Unifying spacetime approaches to quantum mechanics

Recent efforts to formulate quantum mechanics in a way that treats space and time on a more equal footing have led to a large variety of spacetime-oriented approaches. In this work we present a detailed study of spacetime states, the objects that play the role of quantum states in the recently introduced framework of spacetime quantum mechanics, and show that the main proposals in the literature are different manifestations of the same underlying object. Path integrals, quantum states over time, pseudo-density matrices, the Page and Wootters mechanism, superdensity operators, and timelike-entanglement proposals all arise from spacetime states through particular evaluations, reduced information, linear maps, or quantum channels. This unification provides explicit mathematical representations of these formalisms, reveals relations among them, and clarifies the spacetime information each one captures. We also study the broader relevance of the spacetime-state point of view for Leggett-Garg inequalities, OTOCs, temporal tensor networks, fermionic systems, relativistic QFTs, quantum reference frames, and classical physics, together with additional insights and perspectives revealed by the common unifying framework.

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arXiv2026

From quantum time to manifestly covariant QFT: On the need for a quantum-action-based quantization

In quantum time (QT) schemes, time is promoted to a degree of freedom, allowing Lorentz covariance to be made explicit for single particles. We ask whether this can be lifted to QFT, so that Lorentz covariance becomes manifest at the Hilbert-space level, rather than being hidden as in the standard canonical formulation. We address this question by proposing a second-quantized approach in which the elementary particle is the QT particle itself, leading naturally to the notion of spacetime field algebras and of quantum action. We show, however, that a naive many-body construction runs into inconsistencies. To pinpoint their origin we introduce a classical counterpart of the second-quantized formalism, spacetime classical mechanics (SCM), and prove a no-go theorem: Dirac quantization of SCM collapses back to standard QFT and therefore hides covariance. We circumvent this problem by presenting a quantum-action-based quantization that yields a spacetime version of quantum mechanics (SQM), making covariance manifest for (interacting) QFTs. Finally, we show that this resolution is tied to a genuine spacetime generalization of the notion of quantum state, required by causality and closely connected to recent ``states over time'' proposals and, in dS/CFT-motivated settings, to microscopic notions of timelike entanglement and emergent time.

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arXiv2026

The commutant of fermionic Gaussian unitaries

In this work, we characterize the $t$-th order commutants of fermionic Gaussian unitaries and of their particle-preserving subgroup acting on $n$ fermionic modes. These commutants govern Haar averages over the corresponding groups and therefore play a central role in fermionic randomized protocols, invariant theory, and resource quantification. Using Howe dualities, we show that the particle-preserving commutant is generated by generalized copy-hopping operators, while that for general Gaussian commutant is generated by generalized quadratic Majorana bilinears together with parity. We then derive closed formulas for the dimensions of both commutants as functions of $t$ and $n$, and develop constructive Gelfand--Tsetlin procedures to obtain explicit orthonormal bases, with detailed low-$t$ examples. Our framework also clarifies the structure of replicated fermionic states and connects naturally to measures of fermionic correlations, generalized Plücker-type constraints, and the stabilizer entropy of fermionic Gaussian states. These results provide a unified algebraic description of higher-order invariants for fermionic Gaussian dynamics.

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arXiv2026

Analyzing the free states of one quantum resource theory as resource states of another

In the context of quantum resource theories (QRTs), free states are defined as those which can be obtained at no cost under a certain restricted set of conditions. However, when taking a free state from one QRT and evaluating it through the optics of another QRT, it might well turn out that the state is now extremely resourceful. Such realization has recently prompted numerous works characterizing states across several QRTs. In this work we contribute to this body of knowledge by analyzing the resourcefulness in free states for--and across witnesses of--the QRTs of multipartite entanglement, fermionic non-Gaussianity, imaginarity, realness, spin coherence, Clifford non-stabilizerness, $S_n$-equivariance and non-uniform entanglement. We provide rigorous theoretical results as well as present numerical studies that showcase the rich and complex behavior that arises in this type of cross-examination.

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arXiv2026

Spacetime quantum mechanics for bosonic and fermionic systems

We provide a Hilbert space approach to quantum mechanics where space and time are treated on an equal footing. Our approach replaces the standard dependence on an external classical time parameter with a spacetime-symmetric algebraic structure, thereby unifying the axioms that traditionally distinguish the treatment of spacelike and timelike separations. Standard quantum evolution can be recovered from timelike correlators, defined by means of a quantum action operator, a quantum version of the action of classical mechanics. The corresponding map also provides a novel perspective on the path integral formulation, which, in the case of fermions, yields an alternative to the use of Grassmann variables. In addition, the formalism can be interpreted in terms of generalized quantum states, codifying both the conventional information of a quantum system at a given time and its evolution. We show that these states are solutions to a quantum principle of stationary action grounded in timelike correlations and pseudo-entropies

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arXiv2026

Group Fourier filtering of quantum resources in quantum phase space

Recently, it has been shown that group Fourier analysis of quantum states, i.e., decomposing them into the irreducible representations (irreps) of a symmetry group, enables new ways to characterize their resourcefulness. Given that quantum phase spaces (QPSs) provide an alternative description of quantum systems, and thus of the group's representation, one may wonder how such harmonic analysis changes. In this work we show that for general compact Lie-group quantum resource theories (QRTs), the entire family of Stratonovich-Weyl quantum phase space representations-characterized by the Cahill-Glauber parameter $s$-has a clear resource-theoretic and signal-processing meaning. Specifically, changing $s$ implements a group Fourier filter that can be continuously tuned to favor low-dimensional irreps where free states have most of their support ($s=-1$), leave the spectrum unchanged ($s=0$), or highlight resourceful, high-dimensional irreps ($s=1$). As such, distinct QPSs constitute veritable group Fourier filters for resources. Moreover, we show that the norms of the QRT's free state Fourier components completely characterize all QPSs. Finally, we uncover an $s$-duality relating the phase space spectra of free states and typical (Haar-random) highly resourceful states through a shift in $s$. Overall, our results provide a new interpretation of QPSs and promote them to a signal-processing framework for diagnosing, filtering, and visualizing quantum resources.

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arXiv2025

Parallel-in-time quantum simulation via Page and Wootters quantum time

In the past few decades, researchers have created a veritable zoo of quantum algorithms by drawing inspiration from classical computing, information theory, and even from physical phenomena. Here we present quantum algorithms for parallel-in-time simulations that are inspired by the Page and Wootters formalism. In this framework, and thus in our algorithms, the classical time-variable of quantum mechanics is promoted to the quantum realm by introducing a Hilbert space of ``clock'' qubits which are then entangled with the ``system'' qubits. We show that our algorithms can compute temporal properties over $N$ different times of many-body systems by only using $\log(N)$ clock qubits. As such, we achieve an exponential trade-off between time and spatial complexities. In addition, we rigorously prove that the entanglement created between the system qubits and the clock qubits has operational meaning, as it encodes valuable information about the system's dynamics. We also provide a circuit depth estimation of all the protocols, showing a running time advantage in computation times over traditional sequential-in-time algorithms. In particular, for the case when the dynamics are determined by the Aubry--Andre model, we present a hybrid method for which our algorithms have a depth that only scales as $\mathcal{O}(\log(N)n)$. As a by-product, we can relate the previous schemes to the problem of equilibration of an isolated quantum system, thus indicating that our framework enables a new dimension for studying dynamical properties of many-body systems.

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arXiv2025

Does provable absence of barren plateaus imply classical simulability?

A large amount of effort has recently been put into understanding the barren plateau phenomenon. In this perspective article, we face the increasingly loud elephant in the room and ask a question that has been hinted at by many but not explicitly addressed: Can the structure that allows one to avoid barren plateaus also be leveraged to efficiently simulate the loss classically? We collect evidence-on a case-by-case basis-that many commonly used models whose loss landscapes avoid barren plateaus can also admit classical simulation, provided that one can collect some classical data from quantum devices during an initial data acquisition phase. This follows from the observation that barren plateaus result from a curse of dimensionality, and that current approaches for solving them end up encoding the problem into some small, classically simulable, subspaces. Thus, while stressing that quantum computers can be essential for collecting data, our analysis sheds doubt on the information processing capabilities of many parametrized quantum circuits with provably barren plateau-free landscapes. We end by discussing the (many) caveats in our arguments including the limitations of average case arguments, the role of smart initializations, models that fall outside our assumptions, the potential for provably superpolynomial advantages and the possibility that, once larger devices become available, parametrized quantum circuits could heuristically outperform our analytic expectations.

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arXiv2025

A unified approach to quantum resource theories and a new class of free operations

In quantum resource theories (QRTs) certain quantum states and operations are deemed more valuable than others. While the determination of the ``free'' elements is usually guided by the constraints of some experimental setup, this can make it difficult to study similarities and differences between QRTs. In this work, we argue that QRTs follow from the choice of a preferred algebraic structure $\mathcal{E}$ to be preserved, thus setting the free operations as the automorphisms of $\mathcal{E}$. We illustrate our finding by determining $\mathcal{E}$ for the QRTs of entanglement, Clifford stabilizerness, purity, imaginarity, fermionic Gaussianity, reference frames, thermodynamics and coherence; showing instances where $\mathcal{E}$ is a Lie algebra, group, ring, or even a simple set. This unified understanding allows us to generalize the concept of stochastic local operations and classical communication (SLOCC) to identify novel resource non-increasing operations for Lie-algebra based QRTs, thus finding a new solution to an open problem in the literature. We showcase the sanity of our new set of operations by rigorously proving that they map free states to free states, as well as determine more general situations where these transformations strictly do not increase the resource of a state.

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arXiv2025

Optimal Haar random fermionic linear optics circuits

Sampling unitary Fermionic Linear Optics (FLO), or matchgate circuits, has become a fundamental tool in quantum information. Such capability enables a large number of applications ranging from randomized benchmarking of continuous gate sets, to fermionic classical shadows. In this work, we introduce optimal algorithms to sample over the non-particle-preserving (active) and particle-preserving (passive) FLO Haar measures. In particular, we provide appropriate distributions for the gates of $n$-qubit parametrized circuits which produce random active and passive FLO. In contrast to previous approaches, which either incur classical $\mathcal{O}(n^3)$ compilation costs or have suboptimal depths, our methods directly output circuits which simultaneously achieve an optimal down-to-the-constant-factor $Θ(n)$ depth and $Θ(n^2)$ gate count; with only a $Θ(n^2)$ classical overhead. Finally, we also provide quantum circuits to sample Clifford FLO with an optimal $Θ(n^2)$ gate count.

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arXiv2025

Path Integrals from Spacetime Quantum Actions

The possibility of extending the canonical formulation of quantum mechanics (QM) to a space-time symmetric form has recently attracted wide interest. In this context, a recent proposal has shown that a spacetime symmetric many-body extension of the Page and Wootters mechanism naturally leads to the so-called Quantum Action (QA) operator, a quantum version of the action of classical mechanics. In this work, we focus on connecting the QA with the well-established Feynman's Path Integral (PI). In particular, we present a novel formalism which allows one to identify the "sum over histories" with a quantum trace, where the role of the classical action is replaced by the corresponding QA. The trace is defined in the extended Hilbert space resulting from assigning a conventional Hilbert space to each time slice and then taking their tensor product. The formalism opens the way to the application of quantum computation protocols to the evaluation of PIs and general correlation functions, and reveals that different representations of the PI arise from distinct choices of basis in the evaluation of the same trace expression. The Hilbert space embedding of the PIs also discloses a new approach to their continuum time limit. Finally, we discuss how the ensuing canonical-like version of QM inherits many properties from the PI formulation, thus allowing an explicitly covariant treatment of spacetime symmetries.

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arXiv2024

Spacetime quantum and classical mechanics with dynamical foliation

The conventional phase space of classical physics treats space and time differently, and this difference carries over to field theories and quantum mechanics (QM). In this paper, the phase space is enhanced through two main extensions. First, we promote the time choice of the Legendre transform to a dynamical variable. Second, we extend the Poisson brackets of matter fields to a spacetime symmetric form. The ensuing "spacetime phase space" is employed to obtain an explicitly covariant version of Hamilton equations for relativistic field theories. A canonical-like quantization of the formalism is then presented in which the fields satisfy spacetime commutation relations and the foliation is quantum. In this approach, the classical action is also promoted to an operator and retains explicit covariance through its non-separability in the matter-foliation partition. The problem of establishing a correspondence between the new noncausal framework (where fields at different times are independent) and conventional QM is solved through a generalization of spacelike correlators to spacetime. In this generalization, the Hamiltonian is replaced by the action, and conventional particles by off-shell particles. When the foliation is quantized, the previous map is recovered by conditioning on foliation eigenstates, in analogy with the Page and Wootters mechanism. We also provide an interpretation of the correspondence in which the causal structure of a given theory emerges from the quantum correlations between the system and an environment. This idea holds for general quantum systems and allows one to generalize the density matrix to an operator containing the information of correlators both in space and time.

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arXiv2023

Showcasing a Barren Plateau Theory Beyond the Dynamical Lie Algebra

Barren plateaus have emerged as a pivotal challenge for variational quantum computing. Our understanding of this phenomenon underwent a transformative shift with the recent introduction of a Lie algebraic theory capable of explaining most sources of barren plateaus. However, this theory requires either initial states or observables that lie in the circuit's Lie algebra. Focusing on parametrized matchgate circuits, in this work we are able to go beyond this assumption and provide an exact formula for the loss function variance that is valid for arbitrary input states and measurements. Our results reveal that new phenomena emerge when the Lie algebra constraint is relaxed. For instance, we find that the variance does not necessarily vanish inversely with the Lie algebra's dimension. Instead, this measure of expressiveness is replaced by a generalized expressiveness quantity: The dimension of the Lie group modules. By characterizing the operators in these modules as products of Majorana operators, we can introduce a precise notion of generalized globality and show that measuring generalized-global operators leads to barren plateaus. Our work also provides operational meaning to the generalized entanglement as we connect it with known fermionic entanglement measures, and show that it satisfies a monogamy relation. Finally, while parameterized matchgate circuits are not efficiently simulable in general, our results suggest that the structure allowing for trainability may also lead to classical simulability.

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arXiv2021

Spacetime Quantum Actions

We propose a formulation of quantum mechanics in an extended Fock space in which a tensor product structure is applied to time. Subspaces of histories consistent with the dynamics of a particular theory are defined by a direct quantum generalization of the corresponding classical action. The diagonalization of such quantum actions enables us to recover the predictions of conventional quantum mechanics and reveals an extended unitary equivalence between all physical theories. Quantum correlations and coherent effects across time and between distinct theories acquire a rigorous meaning, which is encoded in the rich temporal structure of physical states. Connections with modern relativistic schemes and the path integral formulation also emerge.

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arXiv2019

History state formalism for scalar particles

We present a covariant quantum formalism for scalar particles based on an enlarged Hilbert space. The particular physical theory can be introduced through a timeless Wheeler DeWitt-like equation, whose projection onto four-dimensional coordinates leads to the Klein Gordon equation. The standard quantum mechanical product in the enlarged space, which is invariant and positive definite, implies the usual Klein Gordon product when applied to its eigenstates. Moreover, the standard three-dimensional invariant measure emerges naturally from the flat measure in four dimensions when mass eigenstates are considered, allowing a rigorous identification between definite mass history states and the standard Wigner representation. Connections with the free propagator of scalar field theory and localized states are subsequently derived. The formalism also allows the superposition of different theories and remains valid in the presence of a fixed external field, revealing special orthogonality relations. Other details such as extended identities for the current density, the quantization of parameterized theories and the nonrelativistic limit, with its connection to the Page and Wootters formalism, are discussed. A related consistent second quantization formulation is also introduced.

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arXiv2019

History state formalism for Dirac's theory

We propose a history state formalism for a Dirac particle. By introducing a reference quantum clock system it is first shown that Dirac's equation can be derived by enforcing a timeless Wheeler-DeWitt-like equation for a global state. The Hilbert space of the whole system constitutes a unitary representation of the Lorentz group with respect to a properly defined invariant product, and the proper normalization of global states directly ensures standard Dirac's norm. Moreover, by introducing a second quantum clock, the previous invariant product emerges naturally from a generalized continuity equation. The invariant parameter $τ$ associated with this second clock labels history states for different particles, yielding an observable evolution in the case of an hypothetical superposition of different masses. Analytical expressions for both space-time density and electron-time entanglement are provided for two particular families of electron's states, the former including Pryce localized particles.

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