arXiv subjects

Jens Eisert

Publications and source records attributed to Jens Eisert.

At least 19 recordsLinked to original sources

arXiv2026

Universal quantum coding

Entanglement distillation from noisy bipartite states and reliable quantum communication over noisy channels are fundamental tasks in quantum information theory. Yet existing coding schemes typically require prior knowledge of the underlying state or channel, or assume independent or weakly correlated quantum information sources. Here, using Schur-Weyl duality, we establish a universal quantum error-correction principle that overcomes these limitations. We show that the irreducible permutation spaces arising from the local Schur-Weyl decomposition of multiple copies of an unknown bipartite mixed state form maximally entangled quantum code spaces whose error structure is governed entirely by representation theory. Correcting these symmetry-resolved errors yields protocols for entanglement distillation and quantum communication that achieve coherent-information rates with exponentially vanishing error and optimal second-order corrections, even in the case of almost i.i.d. resources. Our approach also provides sharp finite-copy guarantees that are worst-case optimal at every finite block length, including in the sample-efficient regime, where the number of copies scales only polynomially with the number of qubits. Together, these results establish Schur-Weyl duality as a general mechanism for universal, robust, and optimal quantum coding, unifying entanglement distillation, quantum error correction, and quantum communication within a common representation-theoretic framework.

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arXiv2026

Infinite magic and the emergence of the continuum

Which features of a quantum state are needed for a relativistic quantum field theory to emerge from a lattice system? We address this question by establishing a quantitative relationship between the type classification of hyperfinite von Neumann algebras and the quantum-information-theoretic resource of nonstabilizerness, also called magic. Our results establish two complementary parts of this relationship. First, we prove that any stabilizer system on a lattice defines a canonical GNS Hilbert space in which local von Neumann algebras do not contain type III factors in their central decomposition. As an explicit example, we prove that for stabilizer systems with short-range order, the local algebras are isomorphic to tensor products of hyperfinite factors of type I or type II. This demonstrates that the absence of magic is accompanied by the absence of type III factors in the local algebras, therefore prohibiting the appearance of a quantum field theory. Second, for the hyperfinite factors of Araki, Woods and Powers, we prove a quantitative equivalence: The limiting factor is of type III if and only if the locally optimized min-relative entropy of magic diverges. This divergence is necessary to generate the continuous spectrum associated to a type III factor. Thus, in this setting, the emergence of a type III factor requires not merely the presence of magic, but an infinite amount of it. These results provide a rigorous setting supporting the proposal that infinite magic is a necessary structural ingredient for the emergence of a relativistic continuum from lattice systems.

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arXiv2026

Nonadditivity in fermionic quantum Shannon theory

We study collective state preparation and channel encoding in fermionic systems under parity superselection and composition governed by the canonical anticommutation relations. For a two-mode Gaussian state $τ$, we obtain the exact entanglement of formation, $E_{\mathrm F}^{\mathrm{phys}}(τ^{\otimes m})=\lceil m/2\rceil\log2$ for every $m\geq1$. Two copies have the same formation value as one, violating additivity and strong superadditivity, while regularization halves the single-copy value. For an associated Gaussian channel, we determine the minimum output entropy and Holevo information at every blocklength and prove nonadditivity of both. Together, these results give exact answers to the fermionic counterparts of the four additivity questions associated with Shor and Hastings. One physical use carries no classical information, whereas two uses transmit one bit perfectly. The unassisted classical and zero-error classical capacities both equal one half bit per mode. A sharp finite-block bound gives a strong converse: the decoding success probability vanishes at every rate above capacity. These optimizations cover all physical pure-state decompositions and channel inputs, with Gaussian ensembles and classical codes attaining the optima. The formation and minimum output entropy formulas hold for every Rényi order $α\in(0,\infty]$. We also determine the unassisted quantum and zero-error quantum capacities, both equal to one half logical qubit per mode, and prove a strong converse: entanglement fidelity vanishes at every rate above capacity. For a noisy Gaussian family, we determine the one- and two-copy formation values and prove strict nonadditivity at full rank. For $τ$, Bell violation under local parity-preserving measurements first appears at three copies.

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arXiv2026

The magic cost of fermionic randomness

Sampling-based quantum advantage raises a basic resource question: how much non-Gaussianity is needed to reproduce the random-matrix statistics often associated with quantum chaos? Passive fermionic linear optics preserves Gaussian structure, and even Haar-random mode mixing fails to anticoncentrate for Slater inputs. We show that overcoming this obstruction requires an extensive input resource. For any pure half-filled input state, the one-body defect $ν$, defined as the summed occupation variance of the natural orbitals, must scale linearly with system size. For products of bounded-size blocks on $4N$ modes, this condition is also sufficient, with normalized collision moment asymptotic to $2N/ν$, compared with the Porter-Thomas value 2. For four-mode products, the full Porter-Thomas moment hierarchy is recovered precisely when the defect density tends to its maximum. Assuming average-case hardness, approximate classical sampling remains hard near maximal density, with an error tolerance that increases with the density. Two-copy number fluctuations measure $ν$ directly, and the same quantity lower-bounds the combined cost of non-Gaussian resource modes and parity-preserving local gates in deterministic exact preparation. Fermionic randomness therefore carries an extensive magic cost, with a quantitative link between sampling statistics, preparation resources, and an experimentally accessible witness.

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arXiv2026

Measurement-induced non-commutativity in adaptive fermionic linear optics

Fermionic linear optics (FLO) with Gaussian resources is known to be efficiently classically simulable. We show that this changes for fermions with internal degrees of freedom once the circuit is equipped with mid-circuit number monitoring and classical feedforward. In our architecture, the measurement record routes the selected blocks into a fixed-order Bell-fusion pairing geometry. On the level of the classical description, the permutation sum then no longer collapses to a single determinant or Pfaffian. Each accepted branch expands as a signed sum of path-ordered products of typically non-commuting dressed blocks, and branch amplitudes are matrix elements of the resulting non-commutative trace polynomials. Numerically, we observe Porter-Thomas statistics for the conditional branch distribution and rapid growth of the minimal order-respecting matrix product operator bond dimension. Under reasonable complexity assumptions, these results identify mid-circuit measurement-induced non-commutativity as a route to sampling hardness for noninteracting fermions, with the non-Gaussianity confined to a fixed preparation-and-readout interface rather than to the coherent FLO evolution.

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arXiv2026

Nearest-neighbour gates are all you need: High-rate quantum low-density parity-check codes on a planar grid

High-performance quantum low-density parity-check codes promise substantial reductions in the overhead of fault-tolerant quantum computation, but most constructions require long-range connectivity or qubit shuttling, both of which are difficult to realise in superconducting architectures. Here we introduce a family of quantum low-density parity-check codes that, for the first time, combines planar open-boundary layouts, finite-size advantages over surface codes, and syndrome extraction using only nearest-neighbour gates on a planar grid of qubits with either square or hexagonal connectivity. The resulting circuits achieve optimal constant-depth stabiliser measurement, independent of code size, and naturally remove leakage from the system by exchanging the roles of check and data qubits at each syndrome extraction round. We find finite-size instances such as a [[323,14,15]] code, whose code-efficiency ratio is nearly an order of magnitude larger than that of rotated surface-code patches. At around 40 circuit qubits per logical qubit, the best directional tile-code layouts reduce the per-logical per-round logical error rate by a factor of 100 relative to rotated surface-code memories. These results show that the advantages of quantum low-density parity-check codes can survive compilation into strictly planar nearest-neighbour circuits, bringing low-overhead fault-tolerant memories closer to near-term hardware.

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arXiv2026

A provable quantum advantage for approximate optimization via decoded quantum interferometry

Decoded quantum interferometry (DQI) is a novel paradigm for tackling approximate optimization problems on quantum computers. This framework comes with strong performance guarantees and exploits a well-established duality between optimization and coding theory. A central question, however, is whether DQI can actually provably outperform all polynomial-time classical algorithms. In this work, we establish such an advantage in an oracle setting: we consider an optimization task called folded optimal polynomial intersection (folded OPI), where the acceptance sets are chosen randomly and accessed through membership oracles. We establish a strict gap between the approximation ratio achievable by any polynomial-time classical algorithm and the approximation ratio achieved by the DQI algorithm. Our proof builds on Jordan et al.'s DQI framework for approximate optimization and extends the classical lower-bound method underlying Yamakawa and Zhandry's exact-search oracle separation to approximation. Building on recent developments by Sun and Wootters, Horinaga and Yamakawa, and Jo, we further show that a modified version of the DQI algorithm achieves a strictly higher score guarantee on typical sampled folded OPI instances. As a concrete example, for code rate $0.3$, DQI achieves expected scores of approximately $0.85$. In contrast, exceeding the classical threshold of $0.65$ by any fixed amount with constant probability on sampled instances requires super-polynomially many classical membership queries.

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arXiv2026

An operational continuum limit of quantum combs

Quantum combs are powerful conceptual tools for capturing multi-time processes in quantum information theory, constituting the most general quantum mechanical process. But, despite their causal nature, they lack a meaningful physical connection to time -- and are, by and large, arguably incompatible with it without extra structure. The subclass of quantum combs which assumes an underlying process is described by the so-called process tensor framework, which has been successfully used to study and characterise non-Markovian open quantum systems. But, although process tensors are motivated by an underlying dynamics, it is not a priori clear how to connect them to a continuous process tensor object mathematically -- leaving an uncomfortable conceptual gap. In this work, we take a decisive step toward remedying this situation. We introduce a fully continuous process tensor framework by showing how the discrete multi-partite Choi matrix becomes a vector in bosonic Fock space, which is intrinsically and rigorously defined in the continuum. With this equipped, we lay out the core structural elements of this framework and its properties. This translation allows for an information-theoretic treatment of multi-time correlations in the continuum via the analysis of their continuous matrix product state representatives. Our work closes a gap in the quantum information literature, and opens up the opportunity for the application of many-body physics insights to our understanding of quantum stochastic processes in the continuum.

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arXiv2026

Efficient certification of time-reversal symmetry requires entanglement

Time-reversal symmetry is a fundamental principle of physics describing the invariance of physical laws under reversal of the direction of time. We formulate a Bell-inequality-like test of this antiunitary symmetry using only forward access and trusted quantum operations: entanglement converts temporal input--output relations into measurable spatial exchange symmetry. For $n$-qubit unitary dynamics, we prove that reliably distinguishing the time-reversal-symmetric circular ensembles from Haar-random dynamics requires $Ω(\min\{2^{n/2},2^{n-e}\})$ queries for any classically adaptive protocol. Here, $e=\min\{e_{\mathrm s},e_{\mathrm m}\}$ with $e_{\mathrm s}$ and $e_{\mathrm m}$ representing the probe and measurement logarithmic entanglement negativities, respectively. Maximally entangled probes and SWAP measurements reduce this cost to a constant number of queries. Furthermore, we develop a time-reversal symmetry test for arbitrary fixed, compatible probes and measurements, relate its query complexity to their logarithmic negativities, and match the lower-bound scaling in the high-entanglement regime by optimizing the probe and measurement. Our results establish a quantitative connection between entanglement and time-reversal symmetry, bridging two central concepts in quantum information science and fundamental physics.

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arXiv2026

Approximation theorems for fermionic Gaussian states

A recurring principle in many-body physics is that sufficiently distributed interactions or correlations lead to an effectively mean-field description. For fermionic Gaussian states, we establish a quantitative version of this principle directly at the level of Majorana covariance matrices. The main observation is that the admissibility condition for fermionic covariance matrices imposes a quantitative monogamy constraint on two-point correlations: a fixed region has only a bounded covariance budget to distribute among many disjoint regions. We use this principle in three directions. First, for quadratic fermionic Hamiltonians on finite simple graphs of high degree, we obtain product-state approximations to both the ground-state energy and the finite-temperature free energy, with error of order $D^{-1/2}$ for $D$-regular graphs. Second, we prove a finite fermionic Gaussian de Finetti theorem: a $k$-copy Gaussian state that is $n$-exchangeable under graded fermionic copy permutations is $O(k/n)$-close in trace norm to the product of its one-copy marginal. In particular, infinite exchangeability within the Gaussian class implies exact product structure. Third, for spatial fermionic Gaussian states with exponentially clustering mutual information, we prove that the conditional mutual information across a separating buffer decays as $O(r^{-1})$, and hence, by recoverability, the state admits an $O(r^{-1/2})$ trace-norm approximation by a recovered state. These results show that product and de Finetti approximations and recoverability estimates for fermionic Gaussian systems can all be derived directly from covariance-level structure.

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arXiv2026

Parallel algebraic surgery for qLDPC codes with constant shuttling depth

Reconfigurable atom arrays provide a promising platform for high-rate quantum low-density parity-check codes, but their computational advantage depends on whether logical operations can be implemented without incurring prohibitive shuttling and space overheads. We introduce algebraic surgery for lifted product codes, co-designing the logical measurements, auxiliary codes, and atom shuttling to enable hardware-efficient Pauli-product measurements. These protocols provide flexible logical addressability subject to the cyclic-symmetry constraint, enabling a broad class of parallel logical measurements, including canonical logical operators with heavily overlapping physical supports and joint measurements within or between code blocks even with different lift sizes, while achieving constant shuttling depth on neutral-atom processors under explicit layout assumptions. In a representative hardware-model benchmark, a cycle supporting 33 parallel canonical logical measurements is about $2\times$ faster than the compared single-observable graph-based surgery schedules, with competitive qubit overhead. These shuttling and circuit-level simulations confirm that the algebraic structure of quantum low-density parity-check codes can be used not only to reduce space overhead, but also to simplify physical implementation and expand the set of efficiently accessible logical operations on neutral-atom processors.

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arXiv2026

Testing quantum Gaussianity with constant sample complexity

Efficiently testing whether a quantum state possesses a given structure is both a fundamental and a practical task in quantum information. Among the most important structured families are Gaussian states, which underpin quantum optics and many-body physics while defining paradigmatic regimes of efficient classical simulation. Despite the central role of Gaussian states, optimal tests of Gaussianity have remained unknown. Here we give optimal and robust algorithms for testing fermionic and bosonic Gaussian states, with sample complexity independent of system size, particle number, and energy. Remarkably, the protocols implement experimentally feasible measurements associated with known exact symmetry characterizations of Gaussianity, using Bell sampling on two copies for fermions and passive interferometry with photon counting on three copies for bosons. We further obtain tolerant testers requiring joint measurements on a larger, but still constant, number of copies. At the heart of our analysis is a unified approach---which we call the gradient-flow method---that applies across all structured families considered here and underlies our optimal testers: representation theory turns approximate satisfaction of the defining symmetry into quantitative control of the gradient of the rejection probability, allowing a continuous descent from the input state to the target family whose length bounds their distance.

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arXiv2026

Ultra-high-distance quantum memories from amplified qLDPC codes

A large code distance does not by itself guarantee a well-protected quantum memory, as faults during syndrome extraction can propagate into correlated data errors. Certifying circuit distance becomes demanding as circuits grow. In this work, we introduce distance amplifiers, a modular approach to increasing both code distance and certified circuit-level protection in quantum low-density parity-check codes. The construction tensors a base code with an amplifier, adapting their syndrome-extraction circuits to the amplified code. To certify the resulting memory, we develop a fault-response certificate that yields rigorous lower bounds on circuit distance. Suitable gate ordering and flag qubits control correlated errors, allowing us to establish explicit conditions under which the amplified circuit distance equals the product of the constituent circuit distances. We prove this multiplicative relation for three types of amplifiers: the flagged $[[4,2,2]]$, hypergraph-product, and rotated surface codes. The certificate extends recursively, so the circuit distance multiplies exactly at every amplification step. For example, four successive applications of the flagged $[[4,2,2]]$ amplifier to a $[[18,4,4]]$ seed yield a $[[13320,64,64]]$ code with the circuit distance $d_{\mathrm{circ}} \geq 48$ and check weights $w \leq 14$. We further show how individual logical qubits can be tracked explicitly through amplification. By enabling high-distance quantum memories to be built and certified from smaller building blocks, our framework offers a systematic route toward scalable fault-tolerant quantum computing.

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arXiv2026

Exact Operator Complexity Measures from Random Matchgate Unitaries

Matchgate circuits describe classically tractable dynamics in a physically relevant setting while nonetheless exhibiting features characteristic of complex quantum systems, such as high amounts of magic. This invites the broader question: to what extent are typical complexity measures saturated under trivial dynamics? Here, we address this by studying the behaviour of different Heisenberg-picture complexity measures in random matchgate ensembles. Our main technical contribution is a tractable and generalisible free-fermionic unitary Weingarten calculus formulated entirely in the Majorana basis. This method is tailored to computing moments of operators, and simplifies to a combinatorics problem which depends on the boundary conditions prescribed by a given operator functional. Using this framework, we derive closed-form expressions for the typical local-operator entanglement, operator stabiliser entropy, and higher-order out-of-time-ordered correlators. For extensive initial Majorana strings, both operator entanglement and stabiliser entropy obey volume laws, even though the underlying circuits remain classically simulable. Out-of-time-ordered correlators can likewise become exponentially small, showing that their late-time saturation value alone does not witness computational hardness and constituting a classically tractable example of emergent free independence. Finally, we relate these separations to operator resource theories, showing how operator entanglement beyond the Gaussian baseline lower-bounds the number of non-Gaussian gates in doped matchgate circuits. Our results present a practical toolbox for computing exact averages over random matchgates, while clarifying the distinction between quantum resources versus simulation complexity, and their dependence on the representation being exploited.

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arXiv2026

Dilation theorem for continuum quantum stochastic processes

Connecting mathematical formalism in open quantum systems to its underlying physics necessitates the notion of a dilation, a way to bridge stochastic dynamics with deterministic Schrödinger evolution on a larger space. Although dilations are well known in the literature for states, channels, and quantum combs, there is a substantial gap when it comes to the fully general setting of non-Markovian dynamics on a continuous interval. In an accompanying work, we introduce a continuous process tensor (cPT) framework wherein these processes are represented by vectors in multi-species bosonic Fock spaces of $L^2$ functions. This accommodates all multi-time statistics of a quantum system on an interval. Here, we present a corresponding dilation theorem and show that all cPTs can be approximated arbitrarily well by an explicit closed description with system and environment coupled by a bounded Hamiltonian. As well as its physical appeal and demonstrating completeness of the framework, we show that such regular representations actually underpin the formalism, defining a dense subspace of all cPTs that is easy to handle. This leads to a continuum-suitable version of Choi duality, and permits the rigorous treatment of continuum controls.

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arXiv2026

The power of oracle access: Optimal sample and query complexity of the abelian state hidden subgroup problem

In the quest to identify further quantum algorithms exhibiting superpolynomial speed-ups, a recurring theme is that the complexity of a problem is largely shaped by the input access model. Here, we study this phenomenon for the state hidden subgroup problem (StateHSP), a quantum generalization of the hidden subgroup problem in which the goal is to identify the symmetries of an unknown quantum state. For finite abelian groups, existing Fourier-sampling algorithms use $O(\log(|G|)/ε)$ copies of the state, but whether this scaling is optimal has remained open. We settle the complexity of the abelian StateHSP in both the previously studied sample model and a new query model, which is a stronger and operationally natural generalization that provides access to the state-preparation unitary and its inverse. In the query model, we give a time-efficient quantum algorithm using $O(\log(|G/H|)/\sqrtε)$ forward and inverse queries, and prove a matching $Ω(\log(|G/H|)/\sqrtε)$ lower bound which holds even in the stronger conjugate-query and controlled-query settings. By contrast, we show that in the sample model, $Θ(\log(|G/H|)/ε)$ copies are both sufficient and information-theoretically necessary, even if one allows for arbitrary collective measurements. Thus, the quadratic improvement in $ε$ genuinely arises from coherent access to the preparation circuit. As applications, we obtain faster algorithms for learning stabilizer groups, locating unentanglement, and identifying hidden translation symmetries.

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arXiv2026

Energy-independent tomography of Gaussian states

The exploration of tomography of bosonic Gaussian states is presumably as old as quantum optics, but only recently, their precise and rigorous study have been moving into the focus of attention, motivated by technological developments. In this work, we present an efficient and experimentally feasible Gaussian state tomography algorithm with provable recovery trace-distance guarantees, whose sample complexity depends only on the number of modes, and - remarkably - is independent of the state's photon number or energy, up to doubly logarithmic factors. Our algorithm yields a doubly-exponential improvement over existing methods, and it employs operations that are readily accessible in experimental settings: the preparation of an auxiliary squeezed vacuum, passive Gaussian unitaries, and homodyne detection. At its core lies an adaptive strategy that systematically reduces the total squeezing of the system, enabling efficient tomography. Quite surprisingly, this proves that estimating a Gaussian state in trace distance is generally more efficient than directly estimating its covariance matrix. Our algorithm is particularly well-suited for applications in quantum metrology and sensing, where highly squeezed - and hence high-energy - states are commonly employed. As a further contribution, we establish improved sample complexity bounds for standard heterodyne tomography, equipping this widely used protocol with rigorous trace-norm guarantees.

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arXiv2026

Efficient learning of Clifford disentanglers and typical $t$-doped unitaries with exponentially more $T$ gates

Highly entangled and highly non-stabilizer quantum states need not be hard to learn. We give efficient algorithms for testing and recovering hidden tensor-product structure in unknown pure state vectors of the form $\lvertψ\rangle = U_C \bigotimes_i \lvertψ_i\rangle$, where $U_C$ is an arbitrary unknown Clifford unitary. Although the Clifford can thoroughly scramble the visible product structure, we prove that the Bell distribution retains a characteristic family of quadratic symmetries. By simultaneously block-diagonalising these symmetries, our algorithms linearize the problem and manage to recover both a disentangling Clifford and the hidden partitions with polynomial sample and computational complexity. This may be viewed as an extension of the abelian StateHSP paradigm in which classical post-processing exposes genuinely quadratic structure. Applied to Choi states, the method yields efficient proper learning algorithms for typical $t$-doped Clifford unitaries in regimes containing exponentially more $T$ gates than previously accessible: the required condition fails only for an exponentially small fraction of circuits when $t\sim n$, and continues to hold for a constant fraction even when $t=2n$. Our framework also provides tools for compressing structured many-body Hamiltonians and suggests benchmarking protocols for encoded logical product states in the early fault-tolerant regime.

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