Showing 1–50 of 108 results for author: Hamma, A

QTris: a pedagogical board game to teach Quantum Mechanics

Authors: Alessandro Amabile, Maria Bondani, Immacolata De Simone, Michela Nazzaro, Michele Viscardi, Alioscia Hamma

Abstract: In this paper we introduce the new version of QTris, a board game designed to teach and learn Quantum Mechanics within the framework of Quantum Information and Computation. The key idea behind the game is that every game sequence simulates a process on a system of qubits. Thus, QTris can be effectively integrated as a pedagogical tool to teach Quantum Mechanics at high-school level following a two… ▽ More In this paper we introduce the new version of QTris, a board game designed to teach and learn Quantum Mechanics within the framework of Quantum Information and Computation. The key idea behind the game is that every game sequence simulates a process on a system of qubits. Thus, QTris can be effectively integrated as a pedagogical tool to teach Quantum Mechanics at high-school level following a two-state approach. After arguing in support of this latter approach, we describe QTris' basic rules and some of its possible extensions, emphasizing how the game mechanics puts in clear light key quantum concepts such as incompatibility, probabilistic measurement and unitary transformation. Moreover, we report on the results of a QTris-based educational activity which involved about 150 high-school students and provided encouraging preliminary indications that QTris can be a useful pedagogical platform to promote an immediate understanding of some key concepts of Quantum Mechanics. △ Less

Submitted 10 August, 2026; originally announced August 2026.

Comments: 54 pages, 13 figures

Quantum Chinese Remainder Clock

Authors: Ivri Nagar, Alioscia Hamma, Mikel Palmero, Matthew Radzihovsky, Shouzhuo Yang, Seth Lloyd

Abstract: The Chinese remainder theorem is used in metrology for extending the range of quantum clocks/radar/interferometry, where the phase of a signal is known relative to a set of oscillators with different periods. This paper investigates the performance of a quantum-mechanical Chinese remainder clock, consisting of atoms/oscillators with pairwise coprime periods. We provide the optimal initial state an… ▽ More The Chinese remainder theorem is used in metrology for extending the range of quantum clocks/radar/interferometry, where the phase of a signal is known relative to a set of oscillators with different periods. This paper investigates the performance of a quantum-mechanical Chinese remainder clock, consisting of atoms/oscillators with pairwise coprime periods. We provide the optimal initial state and the optimal Heisenberg-limited quantum measurements for measuring time up to the product of the periods. We introduce a novel fault-tolerant post-processing protocol that allows reconstruction of the correct time even in the presence of errors in the remainders. △ Less

Submitted 11 August, 2026; v1 submitted 8 August, 2026; originally announced August 2026.

Comments: 4 pages, 2 figures

Report number: MIT-CTP/6088

Stabilizer entropy is trustworthy for mixed states

Authors: Gianluca Esposito, Michele Viscardi, Alioscia Hamma

Abstract: Quantifying non-stabilizerness in mixed states is provably intractable, as any strict monotone requires superexponential time. We propose a linear Stabilizer Entropy that acts as a proper non-stabilizerness monotone with overwhelming probability when restricted to non-adaptive Clifford channels acting on flat mixed stabilizer states. Analytical and numerical results for Haar-random states, Cliffor… ▽ More Quantifying non-stabilizerness in mixed states is provably intractable, as any strict monotone requires superexponential time. We propose a linear Stabilizer Entropy that acts as a proper non-stabilizerness monotone with overwhelming probability when restricted to non-adaptive Clifford channels acting on flat mixed stabilizer states. Analytical and numerical results for Haar-random states, Clifford orbits, and random matrix product states show that monotonicity violation probabilities decay as $\exp-ηN$. We also prove the validity of Stabilizer Entropy in specific many-body systems undergoing partial measurements, where the amount of resource never increases for each measurement outcome as well as when averaged over outcome probabilities. Given the hardness of strict alternatives, Stabilizer Entropy emerges as a practical and theoretically justified resource measure. △ Less

Submitted 25 July, 2026; v1 submitted 28 June, 2026; originally announced June 2026.

Induced Resource Theories and Harvesting via Quantum Probes

Authors: Ron Nyström, Simone Cepollaro, Nicola Pranzini, Stefano Cusumano, Alioscia Hamma, Esko Keski-Vakkuri

Abstract: We consider scenarios in which a quantum system with a well-defined resource theory is used as a probe to interact with an environment, such as a quantum field, for which a resource-theoretic description is absent or incomplete. We clarify if and how the harvesting of a resource in the probe can tell us about the state of the environment. This is particularly ambiguous when the probe-environment i… ▽ More We consider scenarios in which a quantum system with a well-defined resource theory is used as a probe to interact with an environment, such as a quantum field, for which a resource-theoretic description is absent or incomplete. We clarify if and how the harvesting of a resource in the probe can tell us about the state of the environment. This is particularly ambiguous when the probe-environment interaction is not a free operation, or the concept of such free operations cannot be defined altogether. We propose a framework and precise conditions under which it becomes possible to interpret resource generation on the probe as evidence of resources in the environment, thereby introducing an effective notion of resources for the latter. Our results clarify in which sense resources can be said to be harvested from the environment and provide a systematic way to analyse such processes beyond fully controlled resource-theoretic settings. More generally, this work may provide a step towards a more general understanding of the interplay of different quantum resources. △ Less

Submitted 15 June, 2026; originally announced June 2026.

Comments: 14 pages, 4 figures

A journey through Flatland: What does the antiflatness of a spectrum teach us?

Authors: Barbara Jasser, Daniele Iannotti, Alioscia Hamma

Abstract: We explore the concept of antiflatness to characterize the structural fluctuations within the entanglement spectrum of a quantum state (i.e., the spectrum of its reduced density operator). As a measure of the interplay between entanglement and magic, two fundamental quantum resources, antiflatness provides second-order information about quantum correlations that standard average measures fail to c… ▽ More We explore the concept of antiflatness to characterize the structural fluctuations within the entanglement spectrum of a quantum state (i.e., the spectrum of its reduced density operator). As a measure of the interplay between entanglement and magic, two fundamental quantum resources, antiflatness provides second-order information about quantum correlations that standard average measures fail to capture. Recognizing that standard majorization theory fundamentally orders states by purity and is structurally blind to spectral fluctuations, we introduce a novel partial ordering known as antiflat majorization, based on the Rényi entropy spread. We define Flatness-Preserving Operations (FPOs), establishing new necessary conditions for state convertibility. Furthermore, we unify different measures of antiflatness-such as Capacity of Entanglement, Linear Rényi spread, and Logarithmic antiflatness-using the frameworks of escort distributions and Bregman divergences. We prove that the Capacity of Entanglement can be expressed as a second derivative of the Kullback-Leibler divergence along the escort trajectory, connecting it with the Quantum Fisher Information. Finally, we demonstrate that absolute maximal antiflatness is not achieved by a single universal state, but rather by a continuous Pareto frontier of extremal states with jump spectra, and we analyze the typicality of these spectral fluctuations using Haar, Bures-Hall and t-doped Clifford random state ensembles. △ Less

Submitted 4 June, 2026; v1 submitted 20 May, 2026; originally announced May 2026.

Non-Local Magic Resources for Fermionic Gaussian States

Authors: Daniele Iannotti, Beatrice Magni, Riccardo Cioli, Alioscia Hamma, Xhek Turkeshi

Abstract: Entanglement and magic are fundamental resources that capture the complexity of quantum many-body systems. Non-local magic isolates the irreducible nonstabilizerness intrinsically tied to entanglement. However, evaluating this quantity generally requires a prohibitive minimization over the full Hilbert space, making it computationally inaccessible beyond a few qubits. Here, we overcome this bottle… ▽ More Entanglement and magic are fundamental resources that capture the complexity of quantum many-body systems. Non-local magic isolates the irreducible nonstabilizerness intrinsically tied to entanglement. However, evaluating this quantity generally requires a prohibitive minimization over the full Hilbert space, making it computationally inaccessible beyond a few qubits. Here, we overcome this bottleneck by establishing a closed-form expression for the non-local stabilizer entropies of fermionic Gaussian states over local Gaussian unitaries, which we prove at Rényi index $α=2$ for arbitrary subsystem size, and which can be evaluated in polynomial time directly from the eigenvalues of the reduced Majorana covariance matrix. We apply this framework to characterize fermionic non-local magic across diverse physical regimes: we derive an exact Page-like curve for typical random states, reveal logarithmic scaling at the quantum critical point of the XY model, and establish a quasiparticle picture for magic generation during out-of-equilibrium quantum quenches. Crucially, because our result relies solely on two-point correlation functions, it provides a scalable route for the experimental estimation of fermionic non-local magic in large-scale quantum processors via fermionic shadow tomography. △ Less

Submitted 20 August, 2026; v1 submitted 29 April, 2026; originally announced April 2026.

Comments: 27 pages, 12 figures; Proof addition of the main result

Probes of chaos over the Clifford group and approach to Haar values

Authors: Stefano Cusumano, Gianluca Esposito, Alioscia Hamma

Abstract: Chaotic behavior of quantum systems can be characterized by the adherence of the expectation values of given probes to moments of the Haar distribution. In this work, we analyze the behavior of several probes of chaos using a technique known as Isospectral Twirling [1]. This consists in fixing the spectrum of the Hamiltonian and picking its eigenvectors at random. Here, we study the transition fro… ▽ More Chaotic behavior of quantum systems can be characterized by the adherence of the expectation values of given probes to moments of the Haar distribution. In this work, we analyze the behavior of several probes of chaos using a technique known as Isospectral Twirling [1]. This consists in fixing the spectrum of the Hamiltonian and picking its eigenvectors at random. Here, we study the transition from stabilizer bases to random bases according to the Haar measure by T-doped random quantum circuits. We then compute the average value of the probes over ensembles of random spectra from Random Matrix Theory, the Gaussian Diagonal Ensemble and the Gaussian Unitary Ensemble, associated with non-chaotic and chaotic behavior respectively. We also study the behavior of such probes over the Toric Code Hamiltonian. △ Less

Submitted 16 June, 2026; v1 submitted 31 March, 2026; originally announced March 2026.

Comments: 76 pages

Non-stabilizerness and U(1) symmetry in chaotic many-body quantum systems

Authors: Daniele Iannotti, Angelo Russotto, Barbara Jasser, Jovan Odavić, Alioscia Hamma

Abstract: We present exact, closed-form results for the non-stabilizerness of random pure states subject to a U(1) symmetry constraint. Using stabilizer entropy as our non-stabilizerness monotone, we derive the average and the variance for U(1)-constrained Haar random states. We show that the presence of a conserved charge leads to a substantial suppression of non-stabilizerness (magic) compared to the unco… ▽ More We present exact, closed-form results for the non-stabilizerness of random pure states subject to a U(1) symmetry constraint. Using stabilizer entropy as our non-stabilizerness monotone, we derive the average and the variance for U(1)-constrained Haar random states. We show that the presence of a conserved charge leads to a substantial suppression of non-stabilizerness (magic) compared to the unconstrained case, and identify a qualitative difference between entanglement and magic response. In the thermodynamic limit, stabilizer entropy exhibits a different leading-order scaling close to a vanishing relative charge density, implying that magic is more robust to charge density fluctuations than entanglement entropy. We test our analytical predictions against midspectrum eigenstates of two chaotic many-body systems with conserved U(1) charge: the complex-fermion Sachdev-Ye-Kitaev (cSYK) model and a Heisenberg XXZ chain with next-to-nearest-neighbour couplings and conserved magnetization. We find an excellent agreement for the non-local cSYK model and systematic deviations for the local XXZ chain, highlighting the role of interaction locality. △ Less

Submitted 15 April, 2026; v1 submitted 30 March, 2026; originally announced March 2026.

Comments: 15 figures, 30 pages

Certifying ergotropy under partial information

Authors: Egle Pagliaro, Leonardo Zambrano, Mir Alimuddin, Alioscia Hamma, Antonio Acín, Donato Farina

Abstract: Ergotropy, the maximum work extractable from a quantum system, is a central resource in quantum physics. Computing ergotropy is well established when the system state is fully known, but its estimation under partial information remains an open problem. Here we introduce a general certification framework that lower bounds ergotropy using only the expectation values of a limited set of arbitrary obs… ▽ More Ergotropy, the maximum work extractable from a quantum system, is a central resource in quantum physics. Computing ergotropy is well established when the system state is fully known, but its estimation under partial information remains an open problem. Here we introduce a general certification framework that lower bounds ergotropy using only the expectation values of a limited set of arbitrary observables. The method naturally applies in the finite-statistics regime, yielding confidence-certified bounds that explicitly incorporate shot noise. We benchmark our approach on both synthetic data and experimental measurements from an IBM quantum processor. This establishes a robust and experimentally accessible tool for certifying extractable work in realistic quantum settings. △ Less

Submitted 19 March, 2026; originally announced March 2026.

Comments: 5 + 5 pages, 4 + 1 figures

Magic of discrete lattice gauge theories

Authors: Gianluca Esposito, Simone Cepollaro, Luigi Cappiello, Alioscia Hamma

Abstract: Simulation of quantum field theories and fundamental interactions are one of the most challenging tasks in modern particle physics. Classical computers generally fail to reproduce accurate results when it comes to strongly coupled theories such as QCD. Recent developments in quantum technologies open up the possibility of simulating such physical regimes by using quantum computers. In this paper,… ▽ More Simulation of quantum field theories and fundamental interactions are one of the most challenging tasks in modern particle physics. Classical computers generally fail to reproduce accurate results when it comes to strongly coupled theories such as QCD. Recent developments in quantum technologies open up the possibility of simulating such physical regimes by using quantum computers. In this paper, we study the quantum resource related to the simulability of a quantum theory, i.e. non-stabilizerness for Lattice Gauge Theory (LGT) with discrete symmetry gauge groups. We show that enforcing gauge constraints for $\mathbb{Z}_l$ LGTs has no cost in terms of this resource and discuss the relation between non-abelianity of the gauge group with the average non-stabilizerness of the gauge invariant Hilbert space. △ Less

Submitted 22 January, 2026; originally announced January 2026.

Journal ref: International Journal of Geometric Methods in Modern Physics Vol. 22, No. 06, 2550003 (2025)

Stabilizer Entropy of Subspaces

Authors: Simone Cepollaro, Gianluca Cuffaro, Matthew B. Weiss, Stefano Cusumano, Alioscia Hamma, Seth Lloyd

Abstract: We consider the costs and benefits of embedding the states of one quantum system within those of another. Such embeddings are ubiquitous, e.g., in error correcting codes and in symmetry-constrained systems. In particular we investigate the impact of embeddings in terms of the resource theory of nonstabilizerness (also known as magic) quantified via the stabilizer entropy (SE). We analytically and… ▽ More We consider the costs and benefits of embedding the states of one quantum system within those of another. Such embeddings are ubiquitous, e.g., in error correcting codes and in symmetry-constrained systems. In particular we investigate the impact of embeddings in terms of the resource theory of nonstabilizerness (also known as magic) quantified via the stabilizer entropy (SE). We analytically and numerically study the stabilizer entropy gap or magic gap: the average gap between the SE of a quantum state realized within a subspace of a larger system and the SE of the quantum state considered on its own. We find that while the stabilizer entropy gap is typically positive, requiring the injection of magic, both zero and negative magic gaps are achievable. This suggests that certain choices of embedding subspace provide strong resource advantages over others. We provide formulas for the average nonstabilizerness of a subspace given its corresponding projector and sufficient conditions for realizing zero or negative gaps: in particular, certain classes of stabilizer codes provide paradigmatic examples of the latter. Through numerical optimization, we find subspaces which achieve both minimal and maximal average SE for a variety of dimensions, and compute the magic gap for specific error-correcting codes and symmetry-induced subspaces. Our results suggest that a judicious choice of embedding can lead to greater efficiency in both classical and quantum simulations. △ Less

Submitted 28 December, 2025; originally announced December 2025.

Comments: 40 pages, 15 figures

Experimental demonstration of non-local magic in a superconducting quantum processor

Authors: Halima Giovanna Ahmad, Gianluca Esposito, Viviana Stasino, Jovan Odavic, Carlo Cosenza, Alessandro Sarno, Pasquale Mastrovito, Michele Viscardi, Stefano Cusumano, Francesco Tafuri, Davide Massarotti, Alioscia Hamma

Abstract: Non-local magic is the non-stabilizerness that no local unitary operation can erase. It captures the joint action of entanglement and magic underlying quantum advantage, and it has never been measured on quantum hardware. Here we report its first experimental demonstration, on a superconducting quantum processing unit, through two independent routes: an optimal local-erasure protocol and a direct,… ▽ More Non-local magic is the non-stabilizerness that no local unitary operation can erase. It captures the joint action of entanglement and magic underlying quantum advantage, and it has never been measured on quantum hardware. Here we report its first experimental demonstration, on a superconducting quantum processing unit, through two independent routes: an optimal local-erasure protocol and a direct, state-agnostic measurement of subsystem purity. The two agree with each other and with theory. Exploiting direct access to the device, we construct a noise model with no free parameters that identifies readout error and a depolarizing controlled-Z channel as the dominant mechanisms, and we show that local and non-local magic can be addressed separately, erasing local magic in situ while preserving the non-local part. Non-local magic provides a hardware benchmark beyond standard gate-fidelity protocols and points toward more reliable pre-fault tolerant devices.The same tools underlie a purity-estimation protocol with exponential speedup and the decoding of Hawking radiation in a black-hole toy model. △ Less

Submitted 30 June, 2026; v1 submitted 19 November, 2025; originally announced November 2025.

Van Hove singularities in stabilizer entropy densities

Authors: Daniele Iannotti, Lorenzo Campos Venuti, Alioscia Hamma

Abstract: The probability distribution of a measure of non-stabilizerness, also known as magic, is investigated for Haar-random pure quantum states. Focusing on the stabilizer Rényi entropies, the associated probability density functions (PDFs) are found to display distinct non-analytic features analogous to Van Hove singularities in condensed matter systems. For a single qubit, the stabilizer purity exhibi… ▽ More The probability distribution of a measure of non-stabilizerness, also known as magic, is investigated for Haar-random pure quantum states. Focusing on the stabilizer Rényi entropies, the associated probability density functions (PDFs) are found to display distinct non-analytic features analogous to Van Hove singularities in condensed matter systems. For a single qubit, the stabilizer purity exhibits a logarithmic divergence at a critical value corresponding to a saddle point on the Bloch sphere. This divergence occurs at the $|H\rangle$-magic states, which hence can be identified as states for which the density of non-stabilizerness in the Hilbert space is infinite. An exact expression for the PDF is derived for the case $α= 2$, with analytical predictions confirmed by numerical simulations. The logarithmic divergence disappears for dimensions $d \ge 3$, in agreement with the behavior of ordinary Van Hove singularities on flat manifolds. In addition, it is shown that, for one qubit, the linear stabilizer entropy is directly related to the partial incompatibility of quantum measurements, one of the defining properties of quantum mechanics, at the basis of Stern-Gerlach experiments. △ Less

Submitted 9 February, 2026; v1 submitted 25 October, 2025; originally announced October 2025.

Comments: 21 pages, 10 figures. Added citations and a general proof for the partially incompatible case on n-qubits

Journal ref: J. Phys. A: Math. Theor. 59 075301 (2026)

Non-Clifford Cost of Random Unitaries

Authors: Lorenzo Leone, Salvatore F. E. Oliviero, Alioscia Hamma, Jens Eisert, Lennart Bittel

Abstract: Recent years have enjoyed a strong interest in exploring properties and applications of random quantum circuits. In this work, we explore the ensemble of $t$-doped Clifford circuits on $n$ qubits, consisting of Clifford circuits interspersed with $t$ single-qubit non-Clifford gates. We establish rigorous convergence bounds towards unitary $k$-designs, revealing the intrinsic cost in terms of non-C… ▽ More Recent years have enjoyed a strong interest in exploring properties and applications of random quantum circuits. In this work, we explore the ensemble of $t$-doped Clifford circuits on $n$ qubits, consisting of Clifford circuits interspersed with $t$ single-qubit non-Clifford gates. We establish rigorous convergence bounds towards unitary $k$-designs, revealing the intrinsic cost in terms of non-Clifford resources in various flavors. First, we analyze the $k$-th order frame potential, which quantifies how well the ensemble of doped Clifford circuits is spread within the unitary group. We prove that a quadratic doping level, $t = \tildeΘ(k^2)$, is both necessary and sufficient to approximate the frame potential of the full unitary group. As a consequence, we refine existing upper bounds on the convergence of the ensemble towards state $k$-designs. Second, we derive tight bounds on the convergence of $t$-doped Clifford circuits towards relative-error $k$-designs, showing that $t = \tildeΘ(nk)$ is both necessary and sufficient for the ensemble to form a relative $\varepsilon$-approximate $k$-design. Similarly, $t = \tildeΘ(n)$ is required to generate pseudo-random unitaries. All these results highlight that generating random unitaries is extremely costly in terms of non-Clifford resources, and that such ensembles fundamentally lie beyond the classical simulability barrier. Additionally, we introduce doped-Clifford Weingarten functions to derive analytic expressions for the twirling operator over the ensemble of random doped Clifford circuits, and we establish their asymptotic behavior in relevant regimes. △ Less

Submitted 5 May, 2026; v1 submitted 15 May, 2025; originally announced May 2025.

Journal ref: PRX Quantum 7, 020321 (2026)

Integrability and Chaos via fractal analysis of Spectral Form Factors: Gaussian approximations and exact results

Authors: Lorenzo Campos Venuti, Jovan Odavić, Alioscia Hamma

Abstract: It is well known that the spectral form factor (SFF) of a possibly degenerate many-body Hamiltonian can be identified with a planar random walk taking steps of unequal length. In this paper we push this identification further and propose to study the chaotic content of a Hamiltonian $H$ via its associated random walk seen as a fractal, using the tools of fractal geometry. In particular we conjectu… ▽ More It is well known that the spectral form factor (SFF) of a possibly degenerate many-body Hamiltonian can be identified with a planar random walk taking steps of unequal length. In this paper we push this identification further and propose to study the chaotic content of a Hamiltonian $H$ via its associated random walk seen as a fractal, using the tools of fractal geometry. In particular we conjecture that for chaotic Hamiltonians the Hausdorff dimension of the frontier of the corresponding random walk approaches the universal value $d_F=4/3$ -- the same value obtained when the random walk describes a Wiener process. Our numerical simulations for non-integrable models confirm this expectation while for quasi-free integrable models we obtain a value $d_F = 1$. Additionally, we numerically show that ``Bethe Ansatz walkers'' fall into a category similar to the non-integrable walkers. To motivate this conjecture we consider many-body Hamiltonians with degenerate but rationally independent eigenvalues. We prove that if the degeneracies satisfy certain Lyapunov conditions, the random walk becomes a Wiener process, $d_F=4/3$, and the distribution of the SFF becomes Gaussian. This is the familiar Gaussian approximation for the SFF which we show to be violated at very low temperature. We also compute the moments of the SFF exactly under milder hypotheses thus solving the classical problem of determining the moments of a random walker taking steps of unequal lengths. Finally, we consider quasi-free Fermionic models with possibly degenerate but rationally independent one-particle spectra. We show that in this case the distribution of the SFF becomes log-Normal and also give the exact form of the moments under milder hypotheses. △ Less

Submitted 27 March, 2026; v1 submitted 8 May, 2025; originally announced May 2025.

Journal ref: Phys. Rev. E 113, 044128 (2026)

Non-stabilizerness and violations of CHSH inequalities

Authors: Stefano Cusumano, Lorenzo Campos Venuti, Simone Cepollaro, Immacolata De Simone, Gianluca Esposito, Daniele Iannotti, Barbara Jasser, Jovan Odavi\' c, Michele Viscardi, Alioscia Hamma

Abstract: We study quantitatively the interplay between entanglement and non-stabilizer resources in violating the CHSH inequalities. We show that, while non-stabilizer resources are necessary, they must have a specific structure, namely they need to be both asymmetric and (surprisingly) {\it local}. We employ stabilizer entropy (SE) to quantify the non-stabilizer resources involved and the probability of v… ▽ More We study quantitatively the interplay between entanglement and non-stabilizer resources in violating the CHSH inequalities. We show that, while non-stabilizer resources are necessary, they must have a specific structure, namely they need to be both asymmetric and (surprisingly) {\it local}. We employ stabilizer entropy (SE) to quantify the non-stabilizer resources involved and the probability of violation given the resources. We show how spectral quantities related to the flatness of entanglement spectrum and its relationship with non-local SE affect the CHSH inequality. Finally, we utilize these results - together with tools from representation theory - to construct a systematic way of building ensembles of states with higher probability of violation. △ Less

Submitted 25 June, 2026; v1 submitted 4 April, 2025; originally announced April 2025.

Comments: 24 pages, 6 figures, 2 tables

Interplay of entanglement structures and stabilizer entropy in spin models

Authors: Michele Viscardi, Marcello Dalmonte, Alioscia Hamma, Emanuele Tirrito

Abstract: Understanding the interplay between nonstabilizerness and entanglement is crucial for uncovering the fundamental origins of quantum complexity. Recent studies have proposed entanglement spectral quantities, such as antiflatness of the entanglement spectrum and entanglement capacity, as effective complexity measures, establishing direct connections to stabilizer Rényi entropies. In this work, we sy… ▽ More Understanding the interplay between nonstabilizerness and entanglement is crucial for uncovering the fundamental origins of quantum complexity. Recent studies have proposed entanglement spectral quantities, such as antiflatness of the entanglement spectrum and entanglement capacity, as effective complexity measures, establishing direct connections to stabilizer Rényi entropies. In this work, we systematically investigate quantum complexity across a diverse range of spin models, analyzing how entanglement structure and nonstabilizerness serve as distinctive signatures of quantum phases. By studying entanglement spectra and stabilizer entropy measures, we demonstrate that these quantities consistently differentiate between distinct phases of matter. Specifically, we provide a detailed analysis of spin chains including the XXZ model, the transverse-field XY model, its extension with Dzyaloshinskii-Moriya interactions, as well as the Cluster Ising and Cluster XY models. Our findings reveal that entanglement spectral properties and magic-based measures serve as intertwined, robust indicators of quantum phase transitions, highlighting their significance in characterizing quantum complexity in many-body systems. △ Less

Submitted 9 December, 2025; v1 submitted 11 March, 2025; originally announced March 2025.

Comments: Submission to SciPost, 40 pages, 13 figures

Journal ref: SciPost Phys. Core 9, 012 (2026)

Stabilizer Entropy and entanglement complexity in the Sachdev-Ye-Kitaev model

Authors: Barbara Jasser, Jovan Odavić, Alioscia Hamma

Abstract: The Sachdev-Ye-Kitaev (SYK) model is of paramount importance for the understanding of both strange metals and a microscopic theory of two-dimensional gravity. We study the interplay between Stabilizer Rényi Entropy (SRE) and entanglement entropy in both the ground state and highly excited states of the SYK4+SYK2 model interpolating the highly chaotic four-body interactions model with the integrabl… ▽ More The Sachdev-Ye-Kitaev (SYK) model is of paramount importance for the understanding of both strange metals and a microscopic theory of two-dimensional gravity. We study the interplay between Stabilizer Rényi Entropy (SRE) and entanglement entropy in both the ground state and highly excited states of the SYK4+SYK2 model interpolating the highly chaotic four-body interactions model with the integrable two-body interactions one. The interplay between these quantities is assessed also through universal statistics of the entanglement spectrum and its anti-flatness. We find that SYK4 is indeed characterized by a complex pattern of both entanglement and non-stabilizer resources while SYK2 is non-universal and not complex. We discuss the fragility and robustness of these features depending on the interpolation parameter. △ Less

Submitted 1 October, 2025; v1 submitted 5 February, 2025; originally announced February 2025.

Journal ref: Phys. Rev. B 112, 174204 (2025)

Entanglement and Stabilizer entropies of random bipartite pure quantum states

Authors: Daniele Iannotti, Gianluca Esposito, Lorenzo Campos Venuti, Alioscia Hamma

Abstract: The interplay between non-stabilizerness and entanglement in random states is a very rich arena of study for the understanding of quantum advantage and complexity. In this work, we tackle the problem of such interplay in random pure quantum states. We show that while there is a strong dependence between entanglement and magic, they are, surprisingly, perfectly uncorrelated. We compute the expectat… ▽ More The interplay between non-stabilizerness and entanglement in random states is a very rich arena of study for the understanding of quantum advantage and complexity. In this work, we tackle the problem of such interplay in random pure quantum states. We show that while there is a strong dependence between entanglement and magic, they are, surprisingly, perfectly uncorrelated. We compute the expectation value of non-stabilizerness given the Schmidt spectrum (and thus entanglement). At a first approximation, entanglement determines the average magic on the Schmidt orbit. However, there is a finer structure in the average magic distinguishing different orbits where the flatness of entanglement spectrum is involved. △ Less

Submitted 10 July, 2025; v1 submitted 31 January, 2025; originally announced January 2025.

Comments: 27 pages, 5 figures; v2: added references; v3: added remarks, figures and references; v4: final, accepted on Quantum journal version

Journal ref: Quantum 9, 1797 (2025)

Harvesting stabilizer entropy and non-locality from a quantum field

Authors: S. Cepollaro, S. Cusumano, A. Hamma, G. Lo Giudice, J. Odavic

Abstract: The harvesting of quantum resources from the vacuum state of a quantum field is a central topic in relativistic quantum information. While several proposals for the harvesting of entanglement from the quantum vacuum exist, less attention has been paid to other quantum resources, such as non-stabilizerness, commonly dubbed {\em magic} and quantified by the Stabilizer Rényi Entropy (SRE). In this wo… ▽ More The harvesting of quantum resources from the vacuum state of a quantum field is a central topic in relativistic quantum information. While several proposals for the harvesting of entanglement from the quantum vacuum exist, less attention has been paid to other quantum resources, such as non-stabilizerness, commonly dubbed {\em magic} and quantified by the Stabilizer Rényi Entropy (SRE). In this work, we show how to harvest SRE from the vacuum state of a massless field using accelerated Unruh-DeWitt detectors in Minkowski spacetime. In particular, one can harvest a particular non-local form of SRE that cannot be erased by local unitary operations. This non-local SRE is a fundamental quantity to study the interplay between entanglement and non-stabilizer resources. We conclude our work with an analysis of the CHSH inequalities: when restricting to stabilizer measurements, i.e. Pauli measurements, one cannot extract a violation from the quantum field. △ Less

Submitted 19 November, 2025; v1 submitted 16 December, 2024; originally announced December 2024.

Comments: 18 pages, 7 figures

Journal ref: Phys. Rev. D 112, 105012 (2025)

Stabilizer entropy in non-integrable quantum evolutions

Authors: Jovan Odavić, Michele Viscardi, Alioscia Hamma

Abstract: Entanglement and stabilizer entropy are both involved in the onset of complex behavior in quantum many-body systems. Their interplay is at the root of complexity of simulability, scrambling, thermalization and typicality. In this work, we study the dynamics of entanglement, stabilizer entropy, and the anti-flatness of the entanglement spectrum after a quantum quench in a spin chain. We find that f… ▽ More Entanglement and stabilizer entropy are both involved in the onset of complex behavior in quantum many-body systems. Their interplay is at the root of complexity of simulability, scrambling, thermalization and typicality. In this work, we study the dynamics of entanglement, stabilizer entropy, and the anti-flatness of the entanglement spectrum after a quantum quench in a spin chain. We find that free-fermion theories show a gap in the long-time behavior of these resources compared to their random matrix theory value while non-integrable models saturate it. △ Less

Submitted 25 July, 2025; v1 submitted 13 December, 2024; originally announced December 2024.

Report number: QB10187B

Journal ref: Phys. Rev. B 112, 104301 (2025)

Invested and Potential Magic Resources in Measurement-Based Quantum Computation

Authors: Gongchu Li, Lei Chen, Si-Qi Zhang, Xu-Song Hong, Huaqing Xu, Yuancheng Liu, You Zhou, Geng Chen, Chuan-Feng Li, Alioscia Hamma, Guang-Can Guo

Abstract: Magic states and magic gates are crucial for achieving universal quantum computation, but important questions about how magic resources should be implemented to attain maximal quantum advantage have remained unexplored, especially in the context of measurement-based quantum computation (MQC). This work bridges the gap between MQC and the resource theory of magic by introducing the key concepts of… ▽ More Magic states and magic gates are crucial for achieving universal quantum computation, but important questions about how magic resources should be implemented to attain maximal quantum advantage have remained unexplored, especially in the context of measurement-based quantum computation (MQC). This work bridges the gap between MQC and the resource theory of magic by introducing the key concepts of "invested" and "potential" magic resources. The former quantifies the magic cost associated with MQC, serving as both a resource witness and a feasible upper bound for the practical realization, and is gate-order independent; The latter represents the maximal achievable magic resource in a given graph structure defining MQC. We utilize both concepts to analyze the quantum Fourier transform (QFT) and provide a fresh perspective on the universality of MQC, highlighting the crucial role of non-Pauli measurements in injecting magic. In particular, we theoretically prove that high-dimensional graphs can generate an exponential advantage of MQC compared to classical computing. We demonstrate experimentally our theoretical findings in a high-fidelity four-photon setup, surpassing conventional magic state injection (MSI) methods in both qubit efficiency and resource utilization. Our findings pave the way for future research exploring magic resource optimization and novel distillation schemes within the MQC framework, advancing fault-tolerant universal quantum computation. △ Less

Submitted 16 November, 2025; v1 submitted 4 August, 2024; originally announced August 2024.

Comments: 5 pages, 3 figures

Journal ref: Phys. Rev. Lett. 135, 160203 (2025)

Magic phase transition and non-local complexity in generalized $W$ State

Authors: A. G. Catalano, J. Odavić, G. Torre, A. Hamma, F. Franchini, S. M. Giampaolo

Abstract: We employ the Stabilizer Renyi Entropy (SRE) to characterize a quantum phase transition that has so far eluded any standard description and can thus now be explained in terms of the interplay between its non-stabilizer properties and entanglement. The transition under consideration separates a region with a unique ground state from one with a degenerate ground state manifold spanned by states with… ▽ More We employ the Stabilizer Renyi Entropy (SRE) to characterize a quantum phase transition that has so far eluded any standard description and can thus now be explained in terms of the interplay between its non-stabilizer properties and entanglement. The transition under consideration separates a region with a unique ground state from one with a degenerate ground state manifold spanned by states with finite and opposite (intensive) momenta. We show that SRE has a jump at the crossing points, while the entanglement entropy remains continuous. Moreover, by leveraging on a Clifford circuit mapping, we connect the observed jump in SRE to that occurring between standard and generalized $W$-states with finite momenta. This mapping allows us to quantify the SRE discontinuity analytically. △ Less

Submitted 21 April, 2025; v1 submitted 27 June, 2024; originally announced June 2024.

Comments: 8 pages including supplementary material, 3 figures

Report number: RBI-ThPhys-2024-12

Journal ref: SciPost Phys. Core 8, 078 (2025)

Geometric methods in quantum information and entanglement variational principle

Authors: Daniele Iannotti, Alioscia Hamma

Abstract: Geometrical methods in quantum information are very promising for both providing technical tools and intuition into difficult control or optimization problems. Moreover, they are of fundamental importance in connecting pure geometrical theories, like GR, to quantum mechanics, like in the AdS/CFT correspondence. In this paper, we first make a survey of the most important settings in which geometric… ▽ More Geometrical methods in quantum information are very promising for both providing technical tools and intuition into difficult control or optimization problems. Moreover, they are of fundamental importance in connecting pure geometrical theories, like GR, to quantum mechanics, like in the AdS/CFT correspondence. In this paper, we first make a survey of the most important settings in which geometrical methods have proven useful to quantum information theory. Then, we lay down a general framework for an action principle for quantum resources like entanglement, coherence, and anti-flatness. We discuss the case of a two-qubit system. △ Less

Submitted 19 March, 2024; originally announced March 2024.

Comments: Slighted expanded version with respect to the article that appeared in the International Journal of Geometric Methods in Modern Physics (https://doi.org/10.1142/s0219887824400103)

Journal ref: International Journal of Geometric Methods in Modern Physics (2024)

Gravitational back-reaction is magical

Authors: ChunJun Cao, Gong Cheng, Alioscia Hamma, Lorenzo Leone, William Munizzi, Savatore F. E. Oliviero

Abstract: We study the interplay between magic and entanglement in quantum many-body systems. We show that non-local magic, which is supported by the quantum correlations is lower bounded by the non-flatness of entanglement spectrum and upper bounded by the amount of entanglement in the system. We then argue that a smoothed version of non-local magic bounds the hardness of classical simulations for incompre… ▽ More We study the interplay between magic and entanglement in quantum many-body systems. We show that non-local magic, which is supported by the quantum correlations is lower bounded by the non-flatness of entanglement spectrum and upper bounded by the amount of entanglement in the system. We then argue that a smoothed version of non-local magic bounds the hardness of classical simulations for incompressible states. In conformal field theories, we conjecture that the non-local magic should scale linearly with entanglement entropy but sublinearly when an approximation of the state is allowed. We support the conjectures using both analytical arguments based on unitary distillation and numerical data from an Ising CFT. If the CFT has a holographic dual, then we prove that the non-local magic vanishes if and only if there is no gravitational back-reaction. Furthermore, we show that non-local magic is approximately equal to the rate of change of the minimal surface area in response to the change of cosmic brane tension in the bulk. △ Less

Submitted 2 March, 2026; v1 submitted 11 March, 2024; originally announced March 2024.

Comments: 39 pages, 21 figures; Updated version to match journal submission

Journal ref: PRX Quantum 6, 040375 (2025)

Stabilizer entropy of quantum tetrahedra

Authors: Simone Cepollaro, Goffredo Chirco, Gianluca Cuffaro, Gianluca Esposito, Alioscia Hamma

Abstract: How complex is the structure of quantum geometry? In several approaches, the spacetime atoms are obtained by the SU(2) intertwiner called quantum tetrahedron. The complexity of this construction has a concrete consequence in recent efforts to simulate such models and toward experimental demonstrations of quantum gravity effects. There are, therefore, both a computational and an experimental comple… ▽ More How complex is the structure of quantum geometry? In several approaches, the spacetime atoms are obtained by the SU(2) intertwiner called quantum tetrahedron. The complexity of this construction has a concrete consequence in recent efforts to simulate such models and toward experimental demonstrations of quantum gravity effects. There are, therefore, both a computational and an experimental complexity inherent to this class of models. In this paper, we study this complexity under the lens of stabilizer entropy (SE). We calculate the SE of the gauge-invariant basis states and its average in the SU(2) gauge invariant subspace. We find that the states of definite volume are singled out by the (near) maximal SE and give precise bounds to the verification protocols for experimental demonstrations on available quantum computers. △ Less

Submitted 26 February, 2024; v1 submitted 12 February, 2024; originally announced February 2024.

Comments: 8 pages, 3 figures, 9 pages supplementary material

Journal ref: Phys. Rev. D 109 (2024), 126008

Optimal encoding of two dissipative interacting qubits

Authors: G. Di Bello, G. De Filippis, A. Hamma, C. A. Perroni

Abstract: We investigate a system of two coupled qubits interacting with an Ohmic bath as a physical model for the implementation of one logical qubit. In this model, the interaction with the other qubit represents unitary noise while the Ohmic bath is responsible for finite temperature. In the presence of a one-dimensional decoherence-free subspace (DFS), we show that, while this is not sufficient to prote… ▽ More We investigate a system of two coupled qubits interacting with an Ohmic bath as a physical model for the implementation of one logical qubit. In this model, the interaction with the other qubit represents unitary noise while the Ohmic bath is responsible for finite temperature. In the presence of a one-dimensional decoherence-free subspace (DFS), we show that, while this is not sufficient to protect a qubit from decoherence, it can be exploited to encode one logical qubit with greater performance than the physical one. We show different possible strategies for the optimal encoding of a logical qubit through a numerical analysis based on matrix product states. This method reproduces faithfully the results of perturbative calculations, but it can be extended to cases of crucial interest for physical implementations, e.g., in the case of strong coupling with the bath. As a result, a logical qubit encoded in the subspace which is the direct sum of the antiferromagnetic states in Bell basis, the DFS and the one in the triplet, is the optimally robust one, as it takes advantage of both the anchoring to the DFS and the protection from the antiferromagnetic interaction. These authors contributed equally to this work, and their names are listed in alphabetical order. △ Less

Submitted 18 January, 2024; v1 submitted 9 October, 2023; originally announced October 2023.

Comments: 13 pages, 12 figures

Journal ref: Phys. Rev. B 109, 014304 (2024)

Learning t-doped stabilizer states

Authors: Lorenzo Leone, Salvatore F. E. Oliviero, Alioscia Hamma

Abstract: In this paper, we present a learning algorithm aimed at learning states obtained from computational basis states by Clifford circuits doped with a finite number $t$ of $T$-gates. The algorithm learns an exact tomographic description of $t$-doped stabilizer states in terms of Pauli observables. This is possible because such states are countable and form a discrete set. To tackle the problem, we int… ▽ More In this paper, we present a learning algorithm aimed at learning states obtained from computational basis states by Clifford circuits doped with a finite number $t$ of $T$-gates. The algorithm learns an exact tomographic description of $t$-doped stabilizer states in terms of Pauli observables. This is possible because such states are countable and form a discrete set. To tackle the problem, we introduce a novel algebraic framework for $t$-doped stabilizer states, which extends beyond $T$-gates and includes doping with any kind of local non-Clifford gate. The algorithm requires resources of complexity $\text{poly}(n,2^t)$ and exhibits an exponentially small probability of failure. △ Less

Submitted 21 May, 2024; v1 submitted 24 May, 2023; originally announced May 2023.

Comments: L.L. and S.O. contributed equally to this work

Journal ref: Quantum 8, 1361 (2024)

Logarithmic light cone, slow entanglement growth, and quantum memory

Authors: Yu Zeng, Alioscia Hamma, Yu-Ran Zhang, Qiang Liu, Rengang Li, Heng Fan, Wu-Ming Liu

Abstract: Effective light cones, characterized by Lieb-Robinson bounds, emerge in nonrelativistic local quantum systems. Here, we present several analytical results derived from logarithmic light cones (LLCs). Possible origins of LLCs include the one-dimensional (1D) disordered XXZ model and a phenomenological model of many-body localization (MBL). In the LLC regime, we prove that, for arbitrary spatial dim… ▽ More Effective light cones, characterized by Lieb-Robinson bounds, emerge in nonrelativistic local quantum systems. Here, we present several analytical results derived from logarithmic light cones (LLCs). Possible origins of LLCs include the one-dimensional (1D) disordered XXZ model and a phenomenological model of many-body localization (MBL). In the LLC regime, we prove that, for arbitrary spatial dimensions and any initial pure state, entanglement growth is upper-bounded by logarithmic time with an additional subleading \emph{double-logarithmic} correction -- arising from a real asymptotic solution of the \emph{Lambert W} function -- valid up to the asymptotic time limit. In the context of the 1D disordered XXZ model, this result resolves the ambiguity in distinguishing between logarithmic and power-law fits of entanglement growth in numerical studies; we also propose a falsifiable phenomenological functional form for the entanglement growth that agrees with existing numerical results. We show that information scrambling is logarithmically slow in the LLC regime. Furthermore, we demonstrate that the LLC supports long-lived quantum memories -- quantum codes with macroscopic code distance and lifetimes that scale exponentially with system size -- under unitary time evolution. Our analytical results provide benchmarks for future numerical studies of the MBL regime at large time scales. △ Less

Submitted 27 August, 2025; v1 submitted 14 May, 2023; originally announced May 2023.

Comments: 11 pages, 2 figures. In version 3, the proof of Theorem 1 was refined and we demonstrated that the logarithmic light cone implies that entanglement grows at most logarithmically with time, with an additional double-logarithmic correction. In version 4, we add a section on phenomenological interpretation involving the double-logarithmic correction

Journal ref: Phys. Rev. B 112, 094202 (2025)

Stabilizer entropy dynamics after a quantum quench

Authors: Davide Rattacaso, Lorenzo Leone, Salvatore F. E. Oliviero, Alioscia Hamma

Abstract: Stabilizer entropies (SE) measure deviations from stabilizer resources and as such are a fundamental ingredient for quantum advantage. In particular, the interplay of SE and entanglement is at the root of the complexity of classically simulating quantum many-body systems. In this paper, we study the dynamics of SE in a quantum many-body system away from the equilibrium after a quantum quench in an… ▽ More Stabilizer entropies (SE) measure deviations from stabilizer resources and as such are a fundamental ingredient for quantum advantage. In particular, the interplay of SE and entanglement is at the root of the complexity of classically simulating quantum many-body systems. In this paper, we study the dynamics of SE in a quantum many-body system away from the equilibrium after a quantum quench in an integrable system. We obtain two main results: (i) we show that SE, despite being an L-extensive quantity, equilibrates in a time that scales at most linearly with the subsystem size; and (ii) we show that there is a SE length increasing linearly in time, akin to correlations and entanglement spreading. △ Less

Submitted 5 November, 2023; v1 submitted 26 April, 2023; originally announced April 2023.

Comments: 13 pages, 9 figures

Journal ref: Phys. Rev. A 108, 042407 (2023)

The scrambling power of gravity in black hole radiation

Authors: Xuan-Lin Su, Alioscia Hamma, Antonino Marciano

Abstract: The black hole information paradox remains a profound challenge in theoretical physics. Among the proposed resolutions, the soft-hair approach stands out for its independence from any specific quantum gravity model. In this paper, we investigate how the inclusion of soft degrees of freedom in the unitary evolution of quantum electrodynamics, within a spacetime collapsing into a Reissner-Nordstrom… ▽ More The black hole information paradox remains a profound challenge in theoretical physics. Among the proposed resolutions, the soft-hair approach stands out for its independence from any specific quantum gravity model. In this paper, we investigate how the inclusion of soft degrees of freedom in the unitary evolution of quantum electrodynamics, within a spacetime collapsing into a Reissner-Nordstrom black hole, leads to information scrambling. By evaluating the tripartite mutual information of this unitary evolution, we estimate the degree of information scrambling in the corresponding quantum channel. Our results show that the presence of soft degrees of freedom induces scrambling of information initially encoded in hard degrees of freedom, driven by quantum electrodynamics interactions and the nontrivial transformations arising from the non-uniqueness of the vacuum in the collapsing spacetime. This lays the groundwork for a deeper understanding of the black hole information paradox, particularly the mechanisms behind information scrambling. △ Less

Submitted 15 October, 2025; v1 submitted 24 April, 2023; originally announced April 2023.

Comments: 19 pages, 1 figure

Scrambling Power of Soft Photons

Authors: Xuan-Lin Su, Alioscia Hamma, Antonino Marciano

Abstract: Observable scattering processes entail emission-absorption of soft photons. As these degrees of freedom go undetected, some information is lost. Whether some of this information can be recovered in the observation of the hard photons, depends of the actual pattern of the scrambling of information. We compute the information scrambling of photon scattering by the tripartite mutual information in te… ▽ More Observable scattering processes entail emission-absorption of soft photons. As these degrees of freedom go undetected, some information is lost. Whether some of this information can be recovered in the observation of the hard photons, depends of the actual pattern of the scrambling of information. We compute the information scrambling of photon scattering by the tripartite mutual information in terms of the 2-Renyi entropy, and find a finite amount of scrambling is present. The developed procedure thus sheds novel light on the black hole information loss paradox, showing that scrambling is a byproduct of decoherence achieved by the scattering system in its interaction with the environment, due to the emission-absorption of soft photons in fully unitary processes. △ Less

Submitted 3 May, 2023; v1 submitted 24 April, 2023; originally announced April 2023.

Comments: 23 pages, 4 figures

Quantifying non-stabilizerness through entanglement spectrum flatness

Authors: Emanuele Tirrito, Poetri Sonya Tarabunga, Gugliemo Lami, Titas Chanda, Lorenzo Leone, Salvatore F. E. Oliviero, Marcello Dalmonte, Mario Collura, Alioscia Hamma

Abstract: Non-stabilizerness - also colloquially referred to as magic - is the a resource for advantage in quantum computing and lies in the access to non-Clifford operations. Developing a comprehensive understanding of how non-stabilizerness can be quantified and how it relates other quantum resources is crucial for studying and characterizing the origin of quantum complexity. In this work, we establish a… ▽ More Non-stabilizerness - also colloquially referred to as magic - is the a resource for advantage in quantum computing and lies in the access to non-Clifford operations. Developing a comprehensive understanding of how non-stabilizerness can be quantified and how it relates other quantum resources is crucial for studying and characterizing the origin of quantum complexity. In this work, we establish a direct connection between non-stabilizerness and entanglement spectrum flatness for a pure quantum state. We show that this connection can be exploited to efficiently probe non-stabilizerness even in presence of noise. Our results reveal a direct connection between non-stabilizerness and entanglement response, and define a clear experimental protocol to probe non-stabilizerness in cold atom and solid-state platforms. △ Less

Submitted 8 May, 2024; v1 submitted 3 April, 2023; originally announced April 2023.

Journal ref: Phys. Rev. A 109, L040401 (2024)

Phase transition in Stabilizer Entropy and efficient purity estimation

Authors: Lorenzo Leone, Salvatore F. E. Oliviero, Gianluca Esposito, Alioscia Hamma

Abstract: Stabilizer Entropy (SE) quantifies the spread of a state in the basis of Pauli operators. It is a computationally tractable measure of non-stabilizerness and thus a useful resource for quantum computation. SE can be moved around a quantum system, effectively purifying a subsystem from its complex features. We show that there is a phase transition in the residual subsystem SE as a function of the d… ▽ More Stabilizer Entropy (SE) quantifies the spread of a state in the basis of Pauli operators. It is a computationally tractable measure of non-stabilizerness and thus a useful resource for quantum computation. SE can be moved around a quantum system, effectively purifying a subsystem from its complex features. We show that there is a phase transition in the residual subsystem SE as a function of the density of non-Clifford resources. This phase transition has important operational consequences: it marks the onset of a subsystem purity estimation protocol that requires $poly(n)exp(t)$ many queries to a circuit containing $t$ non-Clifford gates that prepares the state from a stabilizer state. Then, for $t=O(\log_2 n)$, it estimates the purity with polynomial resources and, for highly entangled states, attains an exponential speed-up over the known state-of-the-art algorithms. △ Less

Submitted 4 March, 2024; v1 submitted 15 February, 2023; originally announced February 2023.

Journal ref: Phys. Rev. A 109, 032403 (2024)

Learning efficient decoders for quasi-chaotic quantum scramblers

Authors: Lorenzo Leone, Salvatore F. E. Oliviero, Seth Lloyd, Alioscia Hamma

Abstract: Scrambling of quantum information is an important feature at the root of randomization and benchmarking protocols, the onset of quantum chaos, and black-hole physics. Unscrambling this information is possible given perfect knowledge of the scrambler [arXiv:1710.03363.]. We show that one can retrieve the scrambled information even without any previous knowledge of the scrambler, by a learning algor… ▽ More Scrambling of quantum information is an important feature at the root of randomization and benchmarking protocols, the onset of quantum chaos, and black-hole physics. Unscrambling this information is possible given perfect knowledge of the scrambler [arXiv:1710.03363.]. We show that one can retrieve the scrambled information even without any previous knowledge of the scrambler, by a learning algorithm that allows the building of an efficient decoder. Remarkably, the decoder is classical in the sense that it can be efficiently represented on a classical computer as a Clifford operator. It is striking that a classical decoder can retrieve with fidelity one all the information scrambled by a random unitary that cannot be efficiently simulated on a classical computer, as long as there is no full-fledged quantum chaos. This result shows that one can learn the salient properties of quantum unitaries in a classical form, and sheds a new light on the meaning of quantum chaos. Furthermore, we obtain results concerning the algebraic structure of $t$-doped Clifford circuits, i.e., Clifford circuits containing t non-Clifford gates, their gate complexity, and learnability that are of independent interest. In particular, we show that a $t$-doped Clifford circuit $U_t$ can be decomposed into two Clifford circuits $U_{0},U^{\prime}_0$ that sandwich a local unitary operator $u_t$, i.e., $U_t=U_{0} u_{t}U_{0}^{\prime}$. The local unitary operator $u_t$ contains $t$ non-Clifford gates and acts nontrivially on at most $t$ qubits. As simple corollaries, the gate complexity of the $t$-doped Clifford circuit $U_t$ is $O(n^2+t^3)$, and it admits a efficient process tomography using $\mathrm{poly}(n,2^t)$ resources. △ Less

Submitted 4 March, 2024; v1 submitted 21 December, 2022; originally announced December 2022.

Comments: Corrected the typos and emphasized several results on learning Clifford circuits that were previously overlooked in the previous version

Journal ref: Phys. Rev. A 109, 022429 (2024)

Unscrambling Quantum Information with Clifford decoders

Authors: Salvatore F. E. Oliviero, Lorenzo Leone, Seth Lloyd, Alioscia Hamma

Abstract: Quantum information scrambling is a unitary process that destroys local correlations and spreads information throughout the system, effectively hiding it in nonlocal degrees of freedom. In principle, unscrambling this information is possible with perfect knowledge of the unitary dynamics [B. Yoshida and A. Kitaev, arXiv:1710.03363.]. However, this Letter demonstrates that even without previous kno… ▽ More Quantum information scrambling is a unitary process that destroys local correlations and spreads information throughout the system, effectively hiding it in nonlocal degrees of freedom. In principle, unscrambling this information is possible with perfect knowledge of the unitary dynamics [B. Yoshida and A. Kitaev, arXiv:1710.03363.]. However, this Letter demonstrates that even without previous knowledge of the internal dynamics, information can be efficiently decoded from an unknown scrambler by monitoring the outgoing information of a local subsystem. Surprisingly, we show that scramblers with unknown internal dynamics, which are rapidly mixing but not fully chaotic, can be decoded using Clifford decoders. The essential properties of a scrambling unitary can be efficiently recovered, even if the process is exponentially complex. Specifically, we establish that a unitary operator composed of $t$ non-Clifford gates admits a Clifford decoder up to $t\le n$. △ Less

Submitted 4 March, 2024; v1 submitted 21 December, 2022; originally announced December 2022.

Report number: LA-UR-22-33044

Journal ref: Phys. Rev. Lett. 132, 080402 (2024)

Entanglement complexity of the Rokhsar-Kivelson-sign wavefunctions

Authors: Stefano Piemontese, Tommaso Roscilde, Alioscia Hamma

Abstract: In this paper we study the transitions of entanglement complexity in an exemplary family of states - the Rokhsar-Kivelson-sign wavefunctions - whose degree of entanglement is controlled by a single parameter. This family of states is known to feature a transition between a phase exhibiting volume-law scaling of entanglement entropy and a phase with sub-extensive scaling of entanglement, reminiscen… ▽ More In this paper we study the transitions of entanglement complexity in an exemplary family of states - the Rokhsar-Kivelson-sign wavefunctions - whose degree of entanglement is controlled by a single parameter. This family of states is known to feature a transition between a phase exhibiting volume-law scaling of entanglement entropy and a phase with sub-extensive scaling of entanglement, reminiscent of the many-body-localization transition of disordered quantum Hamiltonians [Physical Review B 92, 214204 (2015)]. We study the singularities of the Rokhsar-Kivelson-sign wavefunctions and their entanglement complexity across the transition using several tools from quantum information theory: fidelity metric; entanglement spectrum statistics; entanglement entropy fluctuations; stabilizer Rényi Entropy; and the performance of a disentangling algorithm. Across the whole volume-law phase the states feature universal entanglement spectrum statistics. Yet a "super-universal" regime appears for small values of the control parameter in which all metrics become independent of the parameter itself; the entanglement entropy as well as the stabilizer Rényi entropy appear to approach their theoretical maximum; the entanglement fluctuations scale to zero as in output states of random universal circuits, and the disentangling algorithm has essentially null efficiency. All these indicators consistently reveal a complex pattern of entanglement. In the sub-volume-law phase, on the other hand, the entanglement spectrum statistics is no longer universal, entanglement fluctuations are larger and exhibiting a non-universal scaling; and the efficiency of the disentangling algorithm becomes finite. Our results, based on model wavefunctions, suggest that a similar combination of entanglement scaling properties and of entanglement complexity features may be found in high-energy Hamiltonian eigenstates. △ Less

Submitted 25 April, 2023; v1 submitted 2 November, 2022; originally announced November 2022.

Comments: Journal version. Added 1 paragraph and 1 image

Journal ref: Phys. Rev. B 107, 134202 (2023)

Complexity of frustration: a new source of non-local non-stabilizerness

Authors: J. Odavić, T. Haug, G. Torre, A. Hamma, F. Franchini, S. M. Giampaolo

Abstract: We advance the characterization of complexity in quantum many-body systems by examining $W$-states embedded in a spin chain. Such states show an amount of non-stabilizerness or "magic" (measured as the Stabilizer Rényi Entropy -SRE-) that grows logarithmic with the number of qubits/spins. We focus on systems whose Hamiltonian admits a classical point with an extensive degeneracy. Near these points… ▽ More We advance the characterization of complexity in quantum many-body systems by examining $W$-states embedded in a spin chain. Such states show an amount of non-stabilizerness or "magic" (measured as the Stabilizer Rényi Entropy -SRE-) that grows logarithmic with the number of qubits/spins. We focus on systems whose Hamiltonian admits a classical point with an extensive degeneracy. Near these points, a Clifford circuit can convert the ground state into a $W$-state, while in the rest of the phase to which the classic point belongs, it is dressed with local quantum correlations. Topological frustrated quantum spin-chains host phases with the desired phenomenology, and we show that their ground state's SRE is the sum of that of the $W$-states plus an extensive local contribution. Our work reveals that $W$-states/frustrated ground states display a non-local degree of complexity that can be harvested as a quantum resource and has no counterpart in GHZ states/non-frustrated systems. △ Less

Submitted 21 August, 2023; v1 submitted 21 September, 2022; originally announced September 2022.

Comments: 8 pages, 3 figures

Report number: RBI-ThPhys-2023-56

Journal ref: SciPost Phys. 15, 131 (2023)

Bounded light cone and robust topological order out of equilibrium

Authors: Yu Zeng, Alioscia Hamma, Yu-Ran Zhang, Jun-Peng Cao, Heng Fan, Wu-Ming Liu

Abstract: The ground state degeneracy of topologically ordered gapped Hamiltonians is the bedrock for self-correcting quantum memories, which are unfortunately not stable away from equilibrium even at zero temperature. This plague precludes practical robust self-correction since stability at zero temperature is a prerequisite for finite-temperature robustness. In this work, we show that the emergence of a b… ▽ More The ground state degeneracy of topologically ordered gapped Hamiltonians is the bedrock for self-correcting quantum memories, which are unfortunately not stable away from equilibrium even at zero temperature. This plague precludes practical robust self-correction since stability at zero temperature is a prerequisite for finite-temperature robustness. In this work, we show that the emergence of a bounded light cone renders the unitary time evolution a quasi-adiabatic continuation that preserves topological order, with the initial ground space retaining its macroscopic distance at all times as a quantum code. We also show how bounded light cones can emerge through suitable perturbations in Kitaev's toric code and honeycomb model. Our results suggest that topological orders and self-correcting quantum memories can be dynamically robust at zero temperature. △ Less

Submitted 28 October, 2024; v1 submitted 29 August, 2022; originally announced August 2022.

Comments: 7+5 pages, 4+3 figures. the Supplemental Material is updated. In version 4, we show the emergence of a bounded light cone in the disordered Kitaev's honeycomb model

Journal ref: Phys. Rev. A 111, 022414 (2025)

Retrieving information from a black hole using quantum machine learning

Authors: Lorenzo Leone, Salvatore F. E. Oliviero, Stefano Piemontese, Sarah True, Alioscia Hamma

Abstract: In a seminal paper[JHEP09(2007)120], Hayden and Preskill showed that information can be retrieved from a black hole that is sufficiently scrambling, assuming that the retriever has perfect control of the emitted Hawking radiation and perfect knowledge of the internal dynamics of the black hole. In this paper, we show that for $t-$doped Clifford black holes - that is, black holes modeled by random… ▽ More In a seminal paper[JHEP09(2007)120], Hayden and Preskill showed that information can be retrieved from a black hole that is sufficiently scrambling, assuming that the retriever has perfect control of the emitted Hawking radiation and perfect knowledge of the internal dynamics of the black hole. In this paper, we show that for $t-$doped Clifford black holes - that is, black holes modeled by random Clifford circuits doped with an amount $t$ of non-Clifford resources - an information retrieval decoder can be learned with fidelity scaling as $\exp(-αt)$ using quantum machine learning while having access only to out-of-time-order correlation functions. We show that the crossover between learnability and non-learnability is driven by the amount of non-stabilizerness present in the black hole and sketch a different approach to quantum complexity. △ Less

Submitted 29 December, 2022; v1 submitted 13 June, 2022; originally announced June 2022.

Journal ref: Phys. Rev. A 106, 062434(2022)

Magic-state resource theory for the ground state of the transverse-field Ising model

Authors: Salvatore F. E. Oliviero, Lorenzo Leone, Alioscia Hamma

Abstract: Ground states of quantum many-body systems are both entangled and possess a kind of quantum complexity as their preparation requires universal resources that go beyond the Clifford group and stabilizer states. These resources - sometimes described as magic - are also the crucial ingredient for quantum advantage. We study the behavior of the stabilizer Rényi entropy in the integrable transverse fie… ▽ More Ground states of quantum many-body systems are both entangled and possess a kind of quantum complexity as their preparation requires universal resources that go beyond the Clifford group and stabilizer states. These resources - sometimes described as magic - are also the crucial ingredient for quantum advantage. We study the behavior of the stabilizer Rényi entropy in the integrable transverse field Ising spin chain. We show that the locality of interactions results in a localized stabilizer Rényi entropy in the gapped phase thus making this quantity computable in terms of local quantities in the gapped phase, while measurements involving $L$ spins are necessary at the critical point to obtain an error scaling with $O(L^{-1})$. △ Less

Submitted 23 October, 2022; v1 submitted 4 May, 2022; originally announced May 2022.

Journal ref: Phys. Rev. A 106, 042426 (2022)

Nonstabilizerness determining the hardness of direct fidelity estimation

Authors: Lorenzo Leone, Salvatore F. E. Oliviero, Alioscia Hamma

Abstract: In this work, we show how the resource theory of nonstabilizerness quantifies the hardness of direct fidelity estimation protocols. In particular, the resources needed for a direct fidelity estimation conducted on generic states, such as Pauli fidelity estimation and shadow fidelity estimation protocols, grow exponentially with the stabilizer Rényi entropy. Remarkably, these protocols are shown to… ▽ More In this work, we show how the resource theory of nonstabilizerness quantifies the hardness of direct fidelity estimation protocols. In particular, the resources needed for a direct fidelity estimation conducted on generic states, such as Pauli fidelity estimation and shadow fidelity estimation protocols, grow exponentially with the stabilizer Rényi entropy. Remarkably, these protocols are shown to be feasible only for those states that are useless to attain any quantum speedup or advantage. This result suggests the impossibility of estimating efficiently fidelity for generic states and, at the same time, leaves the window open to those protocols specialized at directly estimating the fidelity of particular states. We then extend our results to quantum evolutions, showing that the resources needed to certify the quality of the implementation of a given unitary $U$ are governed by the nonstabilizerness in the Choi state associated with $U$, which is shown to possess a profound connection with out-of-time order correlators. △ Less

Submitted 4 March, 2023; v1 submitted 6 April, 2022; originally announced April 2022.

Journal ref: Phys. Rev. A 107, 022429 (2023)

Measuring magic on a quantum processor

Authors: Salvatore F. E. Oliviero, Lorenzo Leone, Alioscia Hamma, Seth Lloyd

Abstract: Magic states are the resource that allows quantum computers to attain an advantage over classical computers. This resource consists in the deviation from a property called stabilizerness which in turn implies that stabilizer circuits can be efficiently simulated on a classical computer. Without magic, no quantum computer can do anything that a classical computer cannot do. Given the importance of… ▽ More Magic states are the resource that allows quantum computers to attain an advantage over classical computers. This resource consists in the deviation from a property called stabilizerness which in turn implies that stabilizer circuits can be efficiently simulated on a classical computer. Without magic, no quantum computer can do anything that a classical computer cannot do. Given the importance of magic for quantum computation, it would be useful to have a method for measuring the amount of magic in a quantum state. In this work, we propose and experimentally demonstrate a protocol for measuring magic based on randomized measurements. Our experiments are carried out on two IBM Quantum Falcon processors. This protocol can provide a characterization of the effectiveness of a quantum hardware in producing states that cannot be effectively simulated on a classical computer. We show how from these measurements one can construct realistic noise models affecting the hardware. △ Less

Submitted 23 December, 2022; v1 submitted 31 March, 2022; originally announced April 2022.

Comments: Salvatore F.E. Oliviero and Lorenzo Leone contributed equally to this paper

Journal ref: npj Quantum Inf 8, 148 (2022)

Transitions in Entanglement Complexity in Random Circuits

Authors: Sarah True, Alioscia Hamma

Abstract: Entanglement is the defining characteristic of quantum mechanics. Bipartite entanglement is characterized by the von Neumann entropy. Entanglement is not just described by a number, however; it is also characterized by its level of complexity. The complexity of entanglement is at the root of the onset of quantum chaos, universal distribution of entanglement spectrum statistics, hardness of a disen… ▽ More Entanglement is the defining characteristic of quantum mechanics. Bipartite entanglement is characterized by the von Neumann entropy. Entanglement is not just described by a number, however; it is also characterized by its level of complexity. The complexity of entanglement is at the root of the onset of quantum chaos, universal distribution of entanglement spectrum statistics, hardness of a disentangling algorithm and of the quantum machine learning of an unknown random circuit, and universal temporal entanglement fluctuations. In this paper, we numerically show how a crossover from a simple pattern of entanglement to a universal, complex pattern can be driven by doping a random Clifford circuit with $T$ gates. This work shows that quantum complexity and complex entanglement stem from the conjunction of entanglement and non-stabilizer resources, also known as magic. △ Less

Submitted 20 September, 2022; v1 submitted 5 February, 2022; originally announced February 2022.

Journal ref: Quantum 6, 818 (2022)

Local Convertibility in quantum spin systems

Authors: Luigi Amico, Vladimir Korepin, Alioscia Hamma, Salvatore Marco Giampaolo, Fabio Franchini

Abstract: Local Convertibility refers to the possibility of transforming a given state into a target one, just by means of LOCC with respect to a given bipartition of the system and it is possible if and only if all the Renyi-entropies of the initial state are smaller than those of the target state. We apply this concept to adiabatic evolutions and ask whether they can be rendered through LOCC in the sense… ▽ More Local Convertibility refers to the possibility of transforming a given state into a target one, just by means of LOCC with respect to a given bipartition of the system and it is possible if and only if all the Renyi-entropies of the initial state are smaller than those of the target state. We apply this concept to adiabatic evolutions and ask whether they can be rendered through LOCC in the sense above. We argue that a lack of differential local convertibility (dLC) signals a higher computational power of the system's quantum phase, which is also usually connected with the existence of long-range entanglement, topological order, or edge-states. Remarkably, dLC can detect these global properties already by considering small subsystems. Moreover, we connect dLC to spontaneous symmetry breaking by arguing that states with finite order parameters must be the most classical ones and thus be locally convertible. △ Less

Submitted 12 August, 2022; v1 submitted 25 January, 2022; originally announced January 2022.

Comments: Chapter for the Springer Volume "Entanglement in Spin Chains - Theory and Quantum Technology Applications", 33 pages, many figures, typos corrected. arXiv admin note: text overlap with arXiv:1604.06403, arXiv:1306.6685

Report number: RBI-ThPhys-2022-1

Entanglement of random hypergraph states

Authors: You Zhou, Alioscia Hamma

Abstract: Random quantum states and operations are of fundamental and practical interests. In this work, we investigate the entanglement properties of random hypergraph states, which generalize the notion of graph states by applying generalized controlled-phase gates on an initial reference product state. In particular, we study the two ensembles generated by random Controlled-Z(CZ) and Controlled-Controlle… ▽ More Random quantum states and operations are of fundamental and practical interests. In this work, we investigate the entanglement properties of random hypergraph states, which generalize the notion of graph states by applying generalized controlled-phase gates on an initial reference product state. In particular, we study the two ensembles generated by random Controlled-Z(CZ) and Controlled-Controlled-Z(CCZ) gates, respectively. By applying tensor network representation and combinational counting, we analytically show that the average subsystem purity and entanglement entropy for the two ensembles feature the same volume law, but greatly differ in typicality, namely the purity fluctuation is small and universal for the CCZ ensemble while it is large for the CZ ensemble. We discuss the implications of these results for the onset of entanglement complexity and quantum chaos. △ Less

Submitted 1 August, 2022; v1 submitted 14 October, 2021; originally announced October 2021.

Comments: 14+8 pages,9 figures,published version

Journal ref: Phys. Rev. A 106, 012410 (2022)

Towards a Geometrization of Quantum Complexity and Chaos

Authors: Davide Rattacaso, Patrizia Vitale, Alioscia Hamma

Abstract: In this paper, we show how the restriction of the Quantum Geometric Tensor to manifolds of states that can be generated through local interactions provides a new tool to understand the consequences of locality in physics. After a review of a first result in this context, consisting in a geometric out-of-equilibrium extension of the quantum phase transitions, we argue the opportunity and the useful… ▽ More In this paper, we show how the restriction of the Quantum Geometric Tensor to manifolds of states that can be generated through local interactions provides a new tool to understand the consequences of locality in physics. After a review of a first result in this context, consisting in a geometric out-of-equilibrium extension of the quantum phase transitions, we argue the opportunity and the usefulness to exploit the Quantum Geometric Tensor to geometrize quantum chaos and complexity. △ Less

Submitted 14 July, 2021; originally announced July 2021.

Comments: Conference paper

Journal ref: In: Nielsen F., Barbaresco F. (eds) Geometric Science of Information. GSI 2021. Lecture Notes in Computer Science, vol 12829. Springer. (2021)

Stabilizer Rényi entropy

Authors: Lorenzo Leone, Salvatore F. E. Oliviero, Alioscia Hamma

Abstract: We introduce a novel measure for the quantum property of nonstabilizerness - commonly known as "magic" - by considering the Rényi entropy of the probability distribution associated to a pure quantum state given by the square of the expectation value of Pauli strings in that state. We show that this is a good measure of nonstabilizerness from the point of view of resource theory and show bounds wit… ▽ More We introduce a novel measure for the quantum property of nonstabilizerness - commonly known as "magic" - by considering the Rényi entropy of the probability distribution associated to a pure quantum state given by the square of the expectation value of Pauli strings in that state. We show that this is a good measure of nonstabilizerness from the point of view of resource theory and show bounds with other known measures. The stabilizer Rényi entropy has the advantage of being easily computable because it does not need a minimization procedure. We present a protocol for an experimental measurement by randomized measurements. We show that the nonstabilizerness is intimately connected to out-of-time-order correlation functions and that maximal levels of nonstabilizerness are necessary for quantum chaos. △ Less

Submitted 11 March, 2022; v1 submitted 23 June, 2021; originally announced June 2021.

Journal ref: Phys. Rev. Lett. 128, 050402 (2022)

Stability of topological purity under random local unitaries

Authors: Salvatore F. E. Oliviero, Lorenzo Leone, You Zhou, Alioscia Hamma

Abstract: In this work, we provide an analytical proof of the robustness of topological entanglement under a model of random local perturbations. We define a notion of average topological subsystem purity and show that, in the context of quantum double models, this quantity does detect topological order and is robust under the action of a random quantum circuit of shallow depth. In this work, we provide an analytical proof of the robustness of topological entanglement under a model of random local perturbations. We define a notion of average topological subsystem purity and show that, in the context of quantum double models, this quantity does detect topological order and is robust under the action of a random quantum circuit of shallow depth. △ Less

Submitted 24 June, 2021; v1 submitted 8 June, 2021; originally announced June 2021.

Comments: Added new references

Journal ref: SciPost Phys. 12, 096 (2022)

Transitions in entanglement complexity in random quantum circuits by measurements

Authors: Salvatore F. E. Oliviero, Lorenzo Leone, Alioscia Hamma

Abstract: Random Clifford circuits doped with non Clifford gates exhibit transitions to universal entanglement spectrum statistics[1] and quantum chaotic behavior. In [2] we proved that the injection of $O(n)$ non Clifford gates into a $n$-qubit Clifford circuit drives the transition towards the universal value of the purity fluctuations. In this paper, we show that doping a Clifford circuit with $O(n)$ sin… ▽ More Random Clifford circuits doped with non Clifford gates exhibit transitions to universal entanglement spectrum statistics[1] and quantum chaotic behavior. In [2] we proved that the injection of $O(n)$ non Clifford gates into a $n$-qubit Clifford circuit drives the transition towards the universal value of the purity fluctuations. In this paper, we show that doping a Clifford circuit with $O(n)$ single qubit non Clifford measurements is both necessary and sufficient to drive the transition to universal fluctuations of the purity. △ Less

Submitted 4 October, 2021; v1 submitted 12 March, 2021; originally announced March 2021.

Journal ref: Physics Letters A, 127721(2021)