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Jens Eisert

Publications and source records attributed to Jens Eisert.

At least 55 records · Page 3Linked to original sources
arXiv2026

No Universal Purification in Quantum Mechanics

Many central tasks in fundamental physics and quantum information processing are possible only insofar as mixed quantum states can be made purer. In this work, we prove that the linearity and positivity of quantum mechanics impose general restrictions on quantum purification, unveiling a new fundamental principle of quantum information processing. We first establish that no quantum operation can transform a finite number of copies of an unknown quantum state or channel into an exactly pure output that depends non-trivially on the input, thereby ruling out an important form of universal purification in both static and dynamical settings. Building on this, we show that, upon relaxing the requirement of exact purity, one can establish quantitative sample-complexity lower bounds for approximate purification that hold for arbitrary physically allowed strategies, whose scaling matches the performance of purification-related tasks across several different areas of quantum information processing. Moreover, this lower bound leads to a generalized standard quantum limit for learning arbitrary functions of a quantum state, greatly extending earlier results based on quantum Fisher information and revealing a deep connection between purification and quantum learning. Extending this principle to other important settings, we establish, for the first time, an exponential sample-complexity lower bound for approximate pure dilation state preparation and a no-go theorem for approximate bosonic Gaussian state purification with passive Gaussian operations, establishing much more stringent limitations under practical operational constraints.

quant-ph↗
arXiv2026

Approximability limits for bounded-degree max-LINSAT and implications for decoded quantum interferometry

For general max-k-XORSAT with $k \geq 3$, no polynomial-time algorithm can do substantially better than random guessing on worst-case instances unless $\mathsf{P} = \mathsf{NP}$: approximating beyond the random-assignment value of $1/2$ is $\mathsf{NP}$-hard. The picture changes when each variable appears in at most $D$ constraints. In that bounded-degree setting, polynomial-time algorithms can provably beat the random baseline by an additive amount of order $1/\sqrt{D}$. For Boolean instances, this scaling is known to be optimal: the matching hardness result is due to Trevisan, while the corresponding algorithmic guarantee was established by Barak et al. Whether the same holds over general finite fields, and what it implies for quantum algorithms, has not been established. We make this connection explicit and extend the hardness to max-E$k$-LINSAT$(q,r)$ with bounded degree $D$ and over arbitrary finite fields $\mathbb{F}_q$, proving that it is $\mathsf{NP}$-hard to exceed $r/q + \mathcal{O}_{q,r}(1/\sqrt{D})$. These results provide the complexity-theoretic benchmark for the bounded-degree instances targeted by decoded quantum interferometry (DQI), QAOA, and classical heuristics. Any quantum advantage on bounded-degree instances is therefore confined to the constant prefactor. We further show that in the context of DQI and on $(k,D)$-regular instances, this prefactor is sensitive to the nature of the decoder: DQI with classical decoders faces an information-theoretic $1/\sqrt{D \log D}$ barrier that prevents it from matching the hardness scaling, while DQI with quantum decoders is compatible with the $1/\sqrt{D}$ scaling -- identifying quantum decoding as the key ingredient for matching the complexity-theoretic scaling with DQI.

quant-ph↗
arXiv2026

Computational regimes in matrix-product-state-based quantum trajectory simulations

Efficient simulation of open quantum systems is central to modeling noisy quantum hardware and many-body dynamics. In trajectory-based tensor network methods, cost is often associated with trajectory-level quantities such as entanglement growth or bond dimension. However, the total cost of a fixed-accuracy simulation also depends on statistical sampling, and the interplay between per-trajectory complexity and sampling effort remains poorly understood. Here we introduce a cost-resolved framework for matrix product state (MPS)-based quantum trajectory simulations that decomposes total cost into memory per trajectory, runtime per trajectory, and sampling effort. We show that physically equivalent stochastic unravelings of the same Lindblad dynamics do not necessarily reduce total cost, but instead redistribute cost between trajectory complexity and statistical convergence. This trade-off is quantified by two dimensionless inflation factors: a bond dimension inflation $α$ and a sampling inflation $κ$, which together determine the preferred unraveling under hardware-dependent memory and parallelism constraints. We provide a practical protocol for extracting $(α,κ)$ from modest pilot simulations and demonstrate it using benchmarks across multiple noise channels. The resulting decision maps show that the computationally favorable unraveling can change with noise strength, time-step resolution, system size, and available parallelism. These results establish unraveling choice as a hardware-aware simulation design problem rather than an intrinsic optimization of trajectory entanglement alone.

quant-ph↗
arXiv2026

Detecting entanglement from few partial transpose moments and their decay via weight enumerators

The $p_3$-PPT criterion is an experimentally viable relaxation of the well-known positive partial transposition (PPT) criterion for the certification of quantum entanglement. Recently, it has been generalized to various families of entanglement criteria based on the PT moments $p_k=$Tr$[(ρ^Γ)^k]$, where $ρ^Γ$ denotes the partially transposed density matrix of a quantum state $ρ$. While most of these generalizations are strictly more powerful than the $p_3$-PPT criterion, their $m$-th level versions usually rely on the availability of $p_k$ for all moment orders $k\le m$. Here, we show that one can alternatively compare any three PT moments of orders $k p_k^xp_m^{1-x} $ must be entangled, where $x=(m-l)/(m-k)$. Using the example of locally depolarized GHZ states, we identify the most promising versions of these three-moment criteria and compare their performance with a broad range of entanglement criteria. In the case of globally depolarized stabilizer states, we prove that having access to $p_k$ for $k \le 5$ is sufficient to reproduce the full PPT criterion. More generally, we show that the Stieltjes-$m$ criterion is as powerful as the PPT criterion whenever $ρ^Γ$ has no more than $(m+1)/2$ distinct eigenvalues. Finally, we introduce a notion of quantum weight enumerators that capture the decay of $p_k$ under local white noise for arbitrary quantum states and illustrate this concept for an AME state. Our results contribute to the growing body of literature on higher-moment PPT relaxations and modern applications of weight enumerators in quantum error correction and information theory.

quant-ph↗
arXiv2026

Experimental measurement and a physical interpretation of quantum shadow enumerators

Throughout its history, the theory of quantum error correction has heavily benefited from translating classical concepts into the quantum setting. In particular, classical notions of weight enumerators, which relate to the performance of an error-correcting code, and MacWilliams' identity, which helps to compute enumerators, have been generalized to the quantum case. In this work, we establish a distinct relationship between the theoretical machinery of quantum weight enumerators and a seemingly unrelated physics experiment: we prove that Rains' quantum shadow enumerators - a powerful mathematical tool - arise as probabilities of observing fixed numbers of triplets in a Bell sampling experiment. This insight allows us to develop here a rigorous framework for the direct measurement of quantum weight enumerators, thus enabling experimental and theoretical studies of the entanglement structure of any quantum error-correcting code or state under investigation. On top of that, we derive concrete sample complexity bounds and physically-motivated robustness guarantees against unavoidable experimental imperfections. Finally, we experimentally demonstrate the possibility of directly measuring weight enumerators on a trapped-ion quantum computer. Our experimental findings are in good agreement with theoretical predictions and illuminate how entanglement theory and quantum error correction can cross-fertilize each other once Bell sampling experiments are combined with the theoretical machinery of quantum weight enumerators.

quant-ph↗
arXiv2026

The abelian state hidden subgroup problem: Learning stabilizer groups and beyond

Identifying the symmetry properties of quantum states is a central theme in quantum information theory and quantum many-body physics. In this work, we investigate quantum learning problems in which the goal is to identify a hidden symmetry of an unknown quantum state. Building on the recent formulation of the state hidden subgroup problem (StateHSP), we focus on abelian groups and develop an efficient quantum algorithm that learns any hidden symmetry subgroup using a generalized form of Fourier sampling. We showcase the versatility of the approach in three concrete applications: These are learning (i) qubit and qudit stabilizer groups, (ii) cuts along which a state is unentangled, and (iii) hidden translation symmetries. Through these applications, we reveal that well-known quantum learning primitives, such as Bell sampling and Bell difference sampling, are, in fact, special cases of Fourier sampling. Our results highlight the broad potential of the StateHSP framework for symmetry-based quantum learning tasks and provide protocols that are easier to implement on near-term quantum devices.

quant-ph↗
arXiv2026

How hard is it to verify a classical shadow?

Classical shadows are succinct classical representations of quantum states which allow one to encode a set of properties P of a quantum state rho, while only requiring measurements on logarithmically many copies of rho in the size of P. In this work, we initiate the study of verification of classical shadows, denoted classical shadow validity (CSV), from the perspective of computational complexity, which asks: Given a classical shadow S, how hard is it to verify that S predicts the measurement statistics of a quantum state? We first show that even for the elegantly simple classical shadow protocol of [Huang, Kueng, Preskill, Nature Physics 2020] utilizing local Clifford measurements, CSV is QMA-complete. This hardness continues to hold for the high-dimensional extension of said protocol due to [Mao, Yi, and Zhu, PRL 2025]. In contrast, we show that for the HKP and MYZ protocols utilizing global Clifford measurements, CSV can be "dequantized" for low-Frobenius norm observables, i.e., solved in randomized poly-time with standard sampling assumptions. Among other results, we also show that CSV for exponentially many observables is complete for a quantum generalization of the second level of the polynomial hierarchy, yielding the first natural complete problem for such a class.

quant-ph↗
arXiv2026

Mind the gaps: The fraught road to quantum advantage

Quantum computing is advancing rapidly, yet substantial gaps separate today's noisy intermediate-scale quantum (NISQ) devices from tomorrow's fault-tolerant application-scale quantum (FASQ) machines. We identify four related hurdles along the road ahead: (i) from error mitigation to active error detection and correction, (ii) from rudimentary error correction to scalable fault tolerance, (iii) from early heuristics to mature, verifiable algorithms, and (iv) from exploratory simulators to credible advantage in quantum simulation. Targeting these transitions will accelerate progress toward broadly useful quantum computing.

quant-ph↗
arXiv2026

Strategy optimization for quantum conference key agreement in asymmetric star networks

The distribution of entangled states is a core task for quantum networks facilitating quantum communication, and the use of multipartite entangled states comes with its own set of considerations. In this work, we analyze a quantum conference agreement protocol based on GHZ states in a network with a central station to which multiple clients are connected. Using comprehensive numerical simulations, we investigate how minor variations in the scenario-such as the number of parties, the number of memories, and asymmetric distances from the central station-can drastically influence the performance of the protocol. In particular, we demonstrate that it is crucial to adjust the strategy by optimizing cutoff times. From a broader perspective, we argue that numerical simulations are an indispensable tool for protocol design for devising realistic schemes for quantum communication.

quant-ph↗
arXiv2026

Non-Clifford Cost of Random Unitaries

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.

quant-ph↗
arXiv2026

Stability of digital and analog quantum simulations under noise

Quantum simulation is a central application of near-term quantum devices, pursued in both analog and digital architectures. A key challenge for both paradigms is the effect of imperfections and noise on predictive power. In this work, we present a rigorous and physically transparent comparison of the stability of digital and analog quantum simulators under a variety of perturbative noise models. We provide rigorous worst- and average-case error bounds for noisy quantum simulation of local observables. We find that the two paradigms show comparable scaling in the worst case, while exhibiting different forms of enhanced error cancellation on average. We further analyze Gaussian and Brownian noise processes, deriving concentration bounds that capture typical deviations beyond worst-case guarantees. These results provide a unified framework for quantifying the robustness of noisy quantum simulations and identify regimes where digital methods have intrinsic advantages and when we can see similar behavior.

quant-ph↗
arXiv2026

Optimal algorithmic complexity of inference in quantum kernel methods

Quantum kernel methods are among the leading candidates for achieving quantum advantage in supervised learning. A key bottleneck is the cost of inference: evaluating a trained model on new data requires estimating a weighted sum $\sum_{i=1}^N α_i k(x,x_i)$ of $N$ kernel values to additive precision $\varepsilon$, where $α$ is the vector of trained coefficients. The standard approach estimates each term independently via sampling, yielding a query complexity of $O(N\lVertα\rVert_2^2/\varepsilon^2)$. In this work, we identify two independent axes for improvement: (1) How individual kernel values are estimated (sampling versus quantum amplitude estimation), and (2) how the sum is approximated (term-by-term versus via a single observable), and systematically analyze all combinations thereof. The query-optimal combination, encoding the full inference sum as the expectation value of a single observable and applying quantum amplitude estimation, achieves a query complexity of $O(\lVertα\rVert_1/\varepsilon)$, removing the dependence on $N$ from the query count and yielding a quadratic improvement in both $\lVertα\rVert_1$ and $\varepsilon$. We prove a matching lower bound of $Ω(\lVertα\rVert_1/\varepsilon)$, establishing query-optimality of our approach up to logarithmic factors. Beyond query complexity, we also analyze how these improvements translate into gate costs and show that the query-optimal strategy is not always optimal in practice from the perspective of gate complexity. Our results provide both a query-optimal algorithm and a practically optimal choice of strategy depending on hardware capabilities, along with a complete landscape of intermediate methods to guide practitioners. All algorithms require only amplitude estimation as a subroutine and are thus natural candidates for early-fault-tolerant implementations.

quant-ph↗
arXiv2026

Using deep learning to construct stochastic local search SAT solvers with performance bounds

The Boolean Satisfiability problem (SAT), as the prototypical $\mathsf{NP}$-complete problem, is crucial in both theoretical computer science and practical applications. To address this problem, stochastic local search (SLS) algorithms, which iteratively and randomly update candidate assignments, present an important and theoretically well-studied class of solvers. Recent theoretical advancements have identified conditions under which SLS solvers efficiently solve SAT instances, provided they have access to suitable ``oracles'', i.e., instance-specific distribution samples. We propose leveraging machine learning models, particularly graph neural networks (GNN), as oracles to enhance the performance of SLS solvers. Our approach, evaluated on random and pseudo-industrial SAT instances, demonstrates a significant performance improvement regarding step counts and solved instances. Our work bridges theoretical results and practical applications, highlighting the potential of purpose-trained SAT solvers with performance guarantees.

cs.AI↗
arXiv2026

Holonomic quantum computation: a scalable adiabatic architecture

Holonomic quantum computation exploits the geometric evolution of eigenspaces of a degenerate Hamiltonian to implement unitary evolution of computational states. In this work we introduce a framework for performing scalable quantum computation in atom experiments through a universal set of fully holonomic adiabatic gates. Through a detailed differential geometric analysis, we elucidate the geometric nature of these gates and their inherent robustness against classical control errors and other noise sources. The concepts that we introduce here are expected to be widely applicable to the understanding and design of error robustness in generic holonomic protocols. To underscore the practical feasibility of our approach, we contextualize our gate design within recent advancements in Rydberg-based quantum computing and simulation.

quant-ph↗
arXiv2026

Continuum field theory of matchgate tensor network ensembles

Tensor networks provide discrete representations of quantum many-body systems, yet their precise connection to continuum field theories remains relatively poorly understood. Invoking a notion of typicality, we develop a continuum description for random ensembles of two-dimensional fermionic matchgate tensor networks with spatially fluctuating parameters. As a diagnostic of the resulting universal physics, we analyze disorder-averaged moments of fermionic two-point functions, both in flat geometry and on a hyperbolic disk, where curvature reshapes their long-distance structure. We show that disorder drives universal long-distance behavior governed by a nonlinear sigma-model of symmetry class D with a topological term, placing random matchgate networks in direct correspondence with the thermal quantum Hall problem. The resulting phase structure includes localized phases, quantum Hall criticality, and a robust thermal metal with diffusive correlations and spontaneous replica-symmetry breaking. Weak non-Gaussian deformations reduce the symmetry to discrete permutations, generate a mass for the Goldstone modes, and suppress long-range correlations. In this way, typicality offers a route from ensembles of discrete tensor networks to continuum quantum field theories.

cond-mat.dis-nn↗
arXiv2026

Computational relative entropy

Our capacity to process information depends on the computational power at our disposal. Information theory captures our ability to distinguish states or communicate messages when it is unconstrained with unrivaled beauty and elegance. For computationally bounded observers the situation is quite different -- they can, for example, be fooled to believe that distributions are more random than they actually are. In our work, we build a new foundation for a computational quantum information theory that captures the essence of complexity-constrained information theory while retaining the look and feel of the unbounded asymptotic theory. As our fundamental quantity, we define the computational relative entropy as the optimal error exponent in asymmetric hypothesis testing when restricted to polynomially many copies and quantum gates, defined in a mathematically rigorous way. Building on this foundation, we prove a computational analogue of Stein's lemma, establish computational versions of fundamental inequalities like Pinsker's bound, and demonstrate a computational smoothing property showing that computationally indistinguishable states yield equivalent information measures. We derive a computational entropy that operationally characterizes optimal compression rates for quantum states under computational limitations and show that our quantities apply to computational entanglement theory, proving a computational version of the Rains bound. Our framework reveals striking separations between computational and unbounded information measures, including quantum-classical gaps that arise from cryptographic assumptions, demonstrating that computational constraints fundamentally alter the information-theoretic landscape and open new research directions at the intersection of quantum information, complexity theory, and cryptography.

quant-ph↗
arXiv2026

High-threshold decoding of non-Pauli codes for 2D universality

Topological codes have many desirable properties that allow fault-tolerant quantum computation with relatively low overhead. A core challenge for these codes, however, is to achieve a low-overhead universal gate set with limited connectivity. In this work, we explore a non-Pauli stabilizer code that can be used to complete a universal gate set on topological toric and surface codes in strictly two dimensions. Fault-tolerant syndrome extraction for the non-Pauli code requires mid-circuit $X$ corrections, a key difference to conventional Pauli codes. We construct and benchmark a just-in-time (JIT) matching decoder to reliably decide these corrections. Under a phenomenological error model with equally likely physical and measurement errors, we find a high threshold of $\approx 2.5\,\%$, close to the $\approx 2.9\,\%$ of a decoder with access to the full syndrome history. We also perform a finite-size scaling analysis to estimate how the logical error rate scales below threshold and verify an exponential suppression in both physical error rate and in the system size. A second global decoding step for $Z$ errors is required and the non-Clifford gates in the circuit reduce the threshold from $\approx 2.9\,\%$ to $\approx 1.8\,\%$ with a naive decoder. We show how $Z$ decoding can be improved using knowledge of the $X$ corrections, pushing the threshold to $\approx 2.2\,\%$. Our results suggest non-Clifford logic in 2D codes could perform comparably to 2D quantum memory. Our formalism for efficient benchmarking and decoding directly generalizes to a broader family of CSS codes whose $X$ stabilizers are twisted by diagonal Clifford operators, and spacetime versions thereof, defined by CSS-like circuits enriched by $CCZ$, $CS$, and $T$ gates.

quant-ph↗
arXiv2026

Unifying communication paradigms in measurement-based delegated quantum computing

Delegated quantum computing (DQC) allows clients with low quantum capabilities to outsource computations to a server hosting a quantum computer. This process is often envisioned within the measurement-based quantum computing framework, as it naturally facilitates blindness of inputs and computation. Hence, the overall process of setting up and conducting the computation encompasses a sequence of three stages: preparing the qubits, entangling the qubits to obtain the resource state, and measuring the qubits to run the computation. There are two primary approaches to distributing these stages between the client and the server that impose different constraints on cryptographic techniques and experimental implementations. In the prepare-and-send setting, the client prepares the qubits and sends them to the server, while in the receive-and-measure setting, the client receives the qubits from the server and measures them. Although these settings have been extensively studied independently, their interrelation and whether setting-dependent theoretical constraints are inevitable remain unclear. By implementing the key components of most DQC protocols in the respective missing setting, we provide a method to build prospective protocols in both settings simultaneously and to translate existing protocols from one setting into the other.

quant-ph↗