Papers

We study a variant of Schumacher compression. The task asks for the minimal memory cost for compressing many copies of a density matrix with known spectrum and unknown eigenbasis, without preserving its purification. For fixed dimension and distinct positive eigenvalues, we obtain the cost through its additive constant. Achievability uses a generalization of Werner's cloning map to $\mathrm{GL}(d,\mathbb C)$ irreducible representations, with finite trace-distance bounds controlled by highest-weight differences and row gaps. The matching converse follows from a quantitative form of Koashi–Imoto incompressibility for irreducible group orbits under a spectral-gap assumption. We also identify the relation between this quantum minimum description length and universal lossless coding overhead, and its connection to free entropy is explained in a companion letter. We provide a Lean certificate for our proofs.

In free probability, Voiculescu's free entropy counts matrix microstates that approximate an operator. It is a non-commutative analog of the Shannon entropy, distinct from any quantum entropy of a density operator, but it has had no physical meaning in quantum theory. We define a finite-dimensional physical free entropy at finite resolution and show that it directly characterizes, up to an explicit constant, the minimal quantum memory needed to program many copies of a state with known eigenvalues and unknown eigenbasis. This task is a variant of Schumacher compression, and we call its optimal cost the quantum minimum description length. We briefly discuss how free entropy differs from quantum entropies and its potential applications in quantum physics.

Since the introduction of the complexity class QMA as a quantum-verifier analogue of NP (Kitaev, 1997), many have wondered whether quantum proofs are necessary or classical proofs suffice - that is, whether QMA = QCMA or QCMA != QMA (Aharonov and Naveh, 2002; Aaronson and Kuperberg, CCC '07). This longstanding question was recently answered by works of Bostanci, Haferkamp, Nirkhe, and Zhandry (STOC '26) and Bostanci, Huang, and Vaikuntanathan (FOCS '26), which showed that quantum proofs are more powerful than classical ones in the classical-oracle setting. However, it remains unclear what exactly makes quantum proofs more powerful than classical ones. In the information-theoretic setting, a family of quantum states is not classicalizable if and only if it is unclonable. Indeed, these recent works also explicitly highlight the unclonability of their quantum proofs as a key mechanism behind their separations, and their arguments crucially rely on this property. This raises the question of whether unclonability is necessary for quantum proofs to be more powerful than classical ones. In this work, we show that even clonable quantum proofs can be more powerful than classical ones relative to a classical oracle by constructing a classical oracle O such that QCMA^O != ClonableQMA^O. This resolves the open question of Nehoran and Zhandry (ITCS '24), who established the analogous separation relative to a quantum oracle. We also show a classical-oracle separation between BQP/clonableqpoly and BQP/poly, and give applications of our results to quantum cryptography.

We consider the computational task of anti-symmetrization of an unknown mixed quantum state: Given iid copies of a rank-$r$ qudit state $\rho=\sum_{i=1}^r \lambda_i |\varphi_i\rangle\!\langle\varphi_i|$, prepare a copy of the state proportional to $\Pi_{\mathrm{anti}}^r (|\varphi_1\rangle\otimes \ldots\otimes|\varphi_r\rangle)$, with $\Pi_{\mathrm{anti}}^r$ the anti-symmetric subspace projector. We first give a simple algorithm for anti-symmetrization that uses $\widetilde{O}(r / \lambda_{\min})$ copies of $\rho$, where $\lambda_{\min}$ is the minimum non-zero eigenvalue of $\rho$. This algorithm works even in the streaming setting with small working memory, and it uses only simple quantum operations like controlled-SWAPs and constantly many layers of single-qubit gates. As an application of this streaming anti-symmetrization procedure, we give a reduction from gapped mixed-state one-way state generators to pure-state one-way state generators. Then, using tools from Schur-Weyl duality, we give and analyze a second (non-streaming) anti-symmetrization procedure that uses $O(r / \lambda_{\min})$ copies of $\rho$, which then serves as a building block in new algorithms for weak Schur sampling and unitary Schur sampling, outperforming existing approaches in some regimes of number of copies $n$ and local dimension $d$, as well as in an algorithm for optimal purity amplification for qudit states.

Partial distinguishability affects multiphoton interference, which is central to large-scale photonic experiments such as Boson Sampling and to applications of multiphoton states across quantum technologies. Characterizing the internal degrees of freedom responsible for this behavior is challenging: in many scenarios they cannot be accessed directly and even their dimension or physical nature may be unknown a priori. Here we introduce Gram-matrix tomography for independently prepared photons with pure or nearly pure internal states. For pure states, the Gram matrix contains all information required to predict the statistics of experiments that are insensitive to the internal degrees of freedom. We reconstruct this matrix, up to physically irrelevant phase choices, from a quadratic number of measurable Bargmann invariants (quantities encoding overlap magnitudes and collective phases). We access these quantities using two complementary approaches: targeted Fourier interferometers and two- and three-body photon-number correlations measured in a single random interferometer. We derive explicit error bounds that control the total-variation distance between predicted and actual outcome distributions for any subsequent experiment that is insensitive to the internal states. These guarantees account for finite sampling and deviations from purity. Crucially, the total sample complexity of the protocol is independent of the internal Hilbert-space dimension and polynomial under mild assumptions on overlaps of internal states. Our protocol uses standard interferometric measurements and turns them into a robust and operationally grounded characterization of partially distinguishable photons.

A major practical obstacle to Regev's factoring algorithm is its large space usage. In particular, the algorithm was believed to be incompatible with "qubit recycling," which streams the control qubits in phase estimation rather than storing them all simultaneously. We show how to fix this streaming problem. We build on the construction of Ragavan and Vaikuntanathan, which used Fibonacci accumulators to reduce the qubit count from $O(n^{1.5})$ to $11.32n$. We make two key improvements. First, we use superposition masking to prevent unwanted information from entering the accumulators, removing the need for uncomputation. Second, we show how to perform phase estimation using Fibonacci powers of a unitary operation rather than powers of two. This avoids various encoding conversions required by the construction of Ragavan and Vaikuntanathan. After these improvements, the phase-estimation qubits are used only once in the circuit, allowing them to be streamed. Our improvements reduce the space usage of Regev's factoring algorithm to $n+M(n)+O(\sqrt{n})$, where $M(n)$ is the space required for an in-place multiplication. For example, using the in-place multiplier of Luo et al. (2026), Regev's algorithm can factor an $n$-bit integer using $4n+O(\sqrt{n})$ qubits. Although our construction still falls short of the $n/2+\epsilon n$ qubits sufficient for leading implementations of Shor's algorithm, it is the lowest-space instantiation of Regev's algorithm to date and suggests that Regev's algorithm is still far from fully optimized.

The Eastin-Knill theorem rules out a universal set of transversal logical gates on any code that detects single-qubit errors. The theorem constrains one partition of the qubits at a time. We show that depth-one layers of few-qubit gates, each respecting its own partition, can together generate a universal logical gate set on an error-correcting code. Each logical gate generator is then a single depth-one layer followed by error correction of the same stabilizer code, with no magic-state distillation, code switching, gauge fixing, lattice surgery, or other gadgets. We show that such codes can be found by applying an entangling circuit to a lower-distance "seed" code admitting a depth-one non-Clifford gate and an appropriate "kernel" code. We construct 47 codes with depth-one generating sets that tolerate at least one faulty gate per layer, including a [[37,1,7]] and a [[59,1,9]] code, both of check weight at most eight.

How can random order change the role of quantum memory in streaming? Later classical input can restore the usefulness of a quantum state consumed by earlier queries. We call this replenishment. We construct an artificial problem based on Hidden Matching, with repeated coordinate data and online matching requests. It admits a one-pass quantum algorithm using polylogarithmic space in uniformly random order, but unconditionally requires polynomial space both classically in random order and quantumly when all updates precede the requests. To prove the quantum lower bound, we strengthen the consumability bounds of Gilboa, Jain, and McClean for Multiple Hidden Matching. Without prior entanglement, any quantum encoding supporting $r$ independent matching requests requires $\Omega(r)$ qubits for $r\le N^{1/2-\delta}$ and every fixed $\delta\in(0,1/2)$, even with simultaneous revelation and arbitrary joint decoding. For sequential requests, the linear bound extends through $r=\Theta(\sqrt N)$. We also adapt Kallaugher's triangle-counting algorithm to uniformly random streams in which every edge is repeated equally often. Rebuilding the quantum sketch and resampling the classical estimator improve its expected-space bound in suitable parameter regimes. Finally, we extend the robust Noisy Gap Cycle framework of Assadi and Sundaresan to quantum streaming. A quantum communication lower bound for Block Hidden XOR yields an $\Omega(n)$ space lower bound in random edge order for large enough cycles, with consequences for several graph problems. Thus random order can enable replenishment of small quantum representations, while substantial space requirements persist for other tasks.

The no-cloning theorem is a cornerstone result in quantum mechanics that forbids copying of general quantum information. More quantitatively, even given $N$ copies of an unknown state, any channel producing an $(N+1)$-copy state incurs a nonzero trace-distance error on some inputs. Turning this around, one can ask about the sample complexity of approximate $N\rightarrow N+1$ cloning: for a desired small error $\epsilon$, what $N$ suffices? For arbitrary pure states on a Hilbert space $H$, the answer is $N\simeq \dim H/\epsilon$ - astronomically large for most $H$ of interest. What if the input is promised to lie in a structured family? We examine a range of families fundamental to many-body physics and quantum information: $n$-mode fermionic Gaussian and Slater states, $n$-mode bosonic Gaussian states, and qudit phase states. For fermions, we construct optimal cloning channels, reducing the complexity from exponential to polynomial in $n$. For bosonic Gaussian states, we construct an explicit cloner that certifies polynomial complexity, without any constraint on the energy of the state; this channel, however, is not optimal in general. The fermionic and bosonic results follow from a single representation-theoretic framework, which generalizes previous work by Werner and by Chiribella and Yang; it also yields the cloners' unitary implementation. Furthermore, we explain why for many families the cloning complexity is linear in the dimension of the family manifold, via a controlled saddle-point evaluation of the frame potential at high $N$. The same calculation accounts for the analogous scaling of tomography complexity. The above framework does not cover phase states; for these we give an independent construction of a channel with optimal cloning fidelity. Our work complements recent results on cloning stabilizer states.

Measurement incompatibility enables quantum advantages in communication and state discrimination. It is also necessary for Bell nonlocality, which is used in device-independent protocols for quantum key distribution and private randomness generation. Classical communication can influence the receiver's outcome and must therefore be accounted for when certifying measurement incompatibility. We introduce union maximal leakage, which quantifies the information needed for a common classical explanation of all observed probabilities and is computed from them by a linear program. If a message of at most $k$ bits precedes the receiver's independent measurement choice, compatible measurements for each message give union maximal leakage at most $k$. This bound holds with any input-independent nonsignalling shared resource. In quantum theory, a violation also implies that the initially shared state is entangled. Conversely, union maximal leakage determines the minimum classical communication needed to reproduce the observations with compatible measurements and arbitrary nonsignalling assistance. For a fixed shared quantum state and receiver measurements, a violation is possible exactly when, for some message value, the receiver's measurements are Bell-nonlocal against two-outcome sender measurements on that state. For binary receiver outcomes, we show that a violation also certifies randomness private from the sender, even when the receiver's measurement depends on the sender's message and the sender retains quantum side information. We quantify this privacy through an explicit conditional entropy bound and establish composably secure randomness extraction in sequential protocols with device memory.

Random circuit sampling experiments, the leading demonstrations of quantum computational advantage, are tested with the linear cross-entropy benchmark (XEB), which scores outcomes by their ideal probabilities. We study what a high XEB score certifies about an untrusted quantum device. In the quantum query model, we show that the honest sampler's ideal score acts as a Tsirelson bound for efficient devices and certified randomness. First, exceeding the ideal score by a constant requires $\Omega(N^{1/3})$ queries for the Haar and Fourier oracle ensembles with $N$ outcomes, and this bound is tight; this proves an earlier conjecture for Haar-random states and extends it to Fourier sampling. Second, for Fourier sampling and for Haar-random states accessed through the canonical state-preparation oracle, the $n$-bit output of a device that makes polynomially many queries and scores within $o(1)$ of the ideal score has $n-O(\log n)$ bits of smooth min-entropy, even against an adversary who is entangled with the device and later learns the oracle; this is nearly optimal. As a corollary, sampling the collision distribution, which is proportional to the square of the ideal distribution, has query complexity $\Theta(N^{1/3})$, and a coherent version of the same algorithm approximately prepares quantum Hadamard products of states with uniform amplitude magnitudes. Both lower bounds rest on progress measures that bound the excess over the ideal score and increase only slightly with each query. For Fourier sampling the measure is the norm of the coherent amplitudes for removing pairs in a purification of the random function, combining the polynomial method with the compressed oracle technique; for Haar-random states it is the square root of the average level in an expansion in Dirichlet orthogonal polynomials, controlled by an exact identity for the effect of one observation.

Learning the Hamiltonian of a many-body system from its dynamics is a central task in quantum science, yet the algorithms with the strongest provable guarantees assume some level of quantum control–fast, arbitrary single-qubit gates interleaved with time evolution, and measurements in arbitrary bases–that is beyond the capabilities of near-term analog quantum simulators. Motivated by analog atom- and ion-based platforms, we study Hamiltonian learning under minimal access models. Uniform state preparation and measurements: We first consider the setting where in every experiment, one can rotate each qubit to the same state, perform short-time evolution, and measure every qubit in the same basis. Surprisingly, we show that for generic 2-local Hamiltonians on any interaction graph, all of the parameters can be reconstructed from such experiments. Computational basis state preparation and measurements: We then consider a similarly constrained setting, but where state preparation and measurement are restricted to the computational basis. For nearest-neighbor Hamiltonians with only Pauli $X/Z$ interactions, a class which captures contemporary Rydberg atom platforms, we show that over 1D and 2D rectangular lattices, all of the parameters can be reconstructed from such experiments up to unavoidable gauges. Our protocols introduce new techniques for solving structured polynomial systems over an extensive number of parameters. Taken together, our results suggest that one can learn a great deal from the dynamics of quantum many-body systems even under the most stringent experimental constraints.

We consider the following data compression problem. Given a string $x \in \{0,1\}^m$ of Hamming weight at most $n$, compress it into a shorter string $y \in \{0,1\}^s$ so that any bit $x_i$ of $x$ can be retrieved without any error using at most $t$ quantum queries to the standard oracle encoding of $y$. If queries are allowed to be adaptive we show how optimal compression up to a logarithmic factor can be achieved. If the queries are required to be made non-adaptively, we show schemes whose space is optimal in its dependence on $m$ except for a logarithmic factor, and is at most quadratically worse when compared to the optimum in its dependence on $n$.

A major open problem in quantum complexity is the question of unitary synthesis—namely, is it possible to efficiently implement any unitary, given the ability to evaluate any classical function in superposition? Inspired by a recent work by Brakerski and Yuen (CRYPTO 2026), we propose an “average-case” unitary synthesis problem, which asks whether it is possible to efficiently implement a Haar random unitary given access to a random Boolean function. We then demonstrate a superpolynomial query lower bound, establishing that unitary synthesis in this model is impossible. Along with our main result, we settle in the negative a question Brakerski and Yuen pose, of whether scalable pseudorandom unitaries (PRUs) can be implemented in the ROM-PRU model. In particular, we show that in the ROM-PRU model, no $\Omega(N^{1+\gamma})$ unitary design on $n$ qubits, where $N=2^n$, can be efficiently achieved for any $\gamma > 0$. On the other hand, we construct an $O(N)$-design in $\text{poly}(n)$ queries in the same model, surpassing the previously best known results which constructed $O(\sqrt{N})$-designs.

The Clifford hierarchy plays a central role in fault-tolerant quantum computation, but its higher levels remain only partially understood. Recent work on quantum higher order Fourier analysis characterizes membership in the Clifford hierarchy using quantum derivatives, but does not provide an explicit description of the individual levels. In this work, we address this problem by showing that the collection of quantum derivatives of a unitary operator forms a non-abelian \(1\)-cocycle, and conversely that every such \(1\)-cocycle arises from a unitary operator. This yields a recursive cohomological characterization of the Clifford hierarchy. We then specialize this framework to the third level and decompose the cocycle data into symplectic, affine, and phase components, each admitting a cohomological interpretation. As applications, we give a new proof that every two-qudit gate in the third level of the Clifford hierarchy is semi-Clifford, and prove for the first time that every three-qudit gate in the third level is semi-Clifford.

Preparing stabilizer-code states with shallow circuits is an important primitive for near-term quantum error correction in superconducting circuits, where logical-state initialization requires four-body stabilizer correlations from native one- and two-body controls. We introduce a digital-analog method for the preparation of surface-code ground-state manifolds. The method exploits a structural property of the adiabatic gauge potential, mapping stabilizer-local counterdiabatic terms to digital-analog blocks on the superconducting layout using fixed-angle two-qubit XY gates, single-qubit rotations, and analog exchange-interaction evolutions. The dressed blocks generate higher-order operators used as the variational ansatz to determine the counterdiabatic terms within the Sels-Polkovnikov adiabatic-gauge-potential framework. We benchmark checkerboard plaquette-stabilizer grids up to 4 x 4; the square-grid instances correspond to rotated surface codes, while the rectangular cases probe the scaling of the method. For all grids, the proposed method generates the dominant four-body counterdiabatic basis and, at short evolution times, substantially improves ground-state preparation compared with bare adiabatic evolution. We further generalize the construction to n x m stabilizer lattices and show that digital-analog synthesis can reduce the entangling depth by an order of magnitude. Together, these results establish a direct connection between stabilizer geometry, the structure of the counterdiabatic gauge potential, and digital-analog control for measurement-free preparation of stabilizer-code states on near-term superconducting architectures.

We develop an operational approach to strong-to-weak spontaneous symmetry breaking (sw-SSB) of a non-Abelian symmetry group G. We formulate it as the decoding task of distinguishing two strongly symmetric states that differ by a pair of inserted irreps, after both are corrupted by strongly symmetric decoherence. For finite G, this task is dual to distinguishing logical states of the quantum double D(G) under pure-charge noise. We derive a hierarchy of statistical mechanics models describing optimal decoding from the fusion outcomes measured on successively larger nested regions. As for maximum-likelihood decoders of Abelian codes the measured irreps act as quenched disorder obeying a Nishimori-like condition. Strikingly, because local irreps do not determine how they fuse globally, the strong $G\times G$ symmetry is realized only nonlocally, and irreps trivially charged under Z(G) (the center of G) can hide inside noise-generated ones. As a result, the strong symmetry spontaneously breaks down to Z(G), times the weak symmetry. The only transition at nonzero error rate is then an ordering of the center, whose threshold cannot decrease as the hierarchy is ascended. This conclusion holds at any finite level of the hierarchy for non-nilpotent groups. We confirm these predictions with tensor-network calculations for $S_3$ and SU(2). At the first level, the $S_3$ model, whose center is trivial, shows no transition, and the fidelity correlations of all irreps provably decay to non-zero values. The SU(2) model exhibits a $Z_2$ transition consistent with the Nishimori-Ising universality class, where the fidelity correlations of spin-$1/2$ irreps become long-range. The threshold rises from the first to the second level of the hierarchy by $16\%$. Our results provide a systematic framework for studying non-Abelian sw-SSB and for constructing decoders for non Abelian topological codes.

The Hohenberg-Mermin-Wagner theorem motivates the expectation that continuous symmetries remain unbroken in one-dimensional quantum ground states. Here, contrary to this expectation, we demonstrate the spontaneous breaking of U(1) spin-rotation symmetry and the emergence of an interacting quantum critical point in lattice spin-$1/2$ and spin-$1$ chains. Using subsystem Binder scaling, we locate the critical point and determine the correlation-length critical exponent as $\nu = 0.60 \pm 0.02$, distinct from the Gaussian value $0.5$ but consistent with the perturbative renormalization-group prediction. We further analyze the finite-size scaling of the lowest neutral excitation gap and obtain the dynamical critical exponent $z=1.986 \pm 0.005$. The result $z < 2$ is incompatible with microscopic realizations of this critical theory by frustration-free models, implying a mechanism for continuous symmetry breaking in one-dimensional quantum ground states distinct from frustration freeness.

Provable guarantees for preparing quantum Gibbs states, such as rapid mixing or gapped coherent preparation paths, are known only for restricted Hamiltonian families and usually must be re-derived for each new family. Building on homomorphic polynomial transduction, a generalization of decoded quantum interferometry (DQI) and Hamiltonian DQI (HDQI), we show how relative decoding transfers such a guarantee from a source Hamiltonian $H_A$ to a target $H_B$. Writing each Hamiltonian as a sum of $m$ Pauli terms, the subsets of terms whose product is proportional to the identity, its relations, form a binary linear code, $K_A$ for the source and $K_B$ for the target, with the Pauli-label matrix as parity-check matrix as in DQI. Starting from the canonical thermofield double (TFD) of $H_A$, a Bell transform, a reversible label map, and a coherent decoder for the quotient code $K_B/K_A$ prepare an approximate TFD of $H_B$, and hence its Gibbs state. Because this decoder only resolves target relations absent from the source, the reachable inverse temperature is set by the relative distance $d_{\rm rel}$, the fewest terms in any such relation. It can far exceed the ordinary distance $d_{\rm ord}$, the fewest terms in any target relation, which bounds the uniform exact decoding radius of DQI and HDQI. We show that linear $d_{\rm rel}$ with efficient decoding at linear radius certifies a constant inverse temperature. As an example, starting from a nearest-neighbor spin chain whose TFD is known to be preparable at every finite temperature, a sparse classical parity-check matrix yields bounded-degree, geometrically nonlocal, noncommuting targets with $d_{\rm ord}=3$ and $d_{\rm rel}=\Theta(m)$. To our knowledge, this gives a new class of efficiently preparable TFDs.

Time reversal $\Theta$ squares to $-1$ on a half-integer spin, so every state is orthogonal to its time reverse. This sign has been read only from quantum objects. The equilibrium state loses it, but the classical record keeps it. For a detector that resolves handedness, the probabilities that one stretch of record is followed by the time reverse of another measure the cohomology class of time reversal, $\Theta^2=\pm1$. Their matrix carries this sign in its signature, a topological integer that survives equal emission and absorption losses. A programmed emitter on quantum processors reads it in about $10^3$ shots.

A quantum channel is called incompatibility-breaking if it always destroys the incompatibility of quantum measurements. On the other extreme are incompatibility-preserving channels, which never destroy incompatibility. This work studies both types of channels through the mathematical framework of zonotopes and zonoids, an approach that provides a complete characterization of compatibility for dichotomic measurements and is particularly powerful in the qubit setting. Using the language of zonoids, we present necessary and sufficient conditions for when a qubit unital channel is incompatibility-breaking, which are also sufficient for nonunital channels. As one of our main results, we demonstrate the superactivation phenomenon of two incompatibility-breaking qubit channels that are no longer incompatibility-breaking when used in parallel. Translating this result into the task of quantum steering, it says that two copies of a two-qubit Werner state can be steered at noise threshold $1/2$, even though this is impossible using just one copy. For the problem of incompatibility preservation, we prove that a quantum channel with matching input and output dimensions will preserve general incompatibility if and only if it is a unitary transformation. When the output dimension is larger, this result no longer holds, and we offer a conjectured form of a general incompatibility-preserving channel.

Building on our earlier program-evolution workflow guided by large language models (LLMs), we study weight-five bivariate bicycle (BB) and perturbed bivariate bicycle (PBB) codes. The resulting catalogue contains 1,142 distinct code proposals, including 1,081 nonbaseline proposals attributable to LLM-generated programs. Across the catalogue, we certify connected Calderbank–Shor–Steane (CSS) realizations [[96,4,10]], [[140,6,10]], and [[180,4,14]]. A post-search comparison certifies seven imported Lin–Pryadko archive constructions. For leading parameter triples also represented in that archive, we provide exact distance evidence, explicit bivariate presentations, and verified component reductions. A basis-independent connectivity analysis identifies 409 of the 1,142 catalogue entries as disconnected and shows that 73.1% of the classes with exact distance certificates contain repeated connected components. Algebraic analysis organizes the connected CSS classes into order-3, order-7, and order-15 cyclotomic-kernel strata. The strongest exact connected PBB parameter point is [[216,4,10]], attained by two distinct component classes. Among the 936 distinct CSS proposals from the LLM-guided campaign with a recorded positive distance, 816 (87.18%) are certified at $d\geq5$. For comparison, three random-search controls each sample 6,444 CSS code proposals uniformly without replacement, using the same per-lattice and encoded-dimension sample counts as the LLM-guided campaign. In these controls, 4,672–4,785 proposals (72.50–74.26%) meet the same criterion. The LLM-guided campaign has the higher certified yield, while the random controls cover more connected classes. Together, these results extend LLM-guided discovery to a more constrained code family and provide a reproducible structural and exact-distance account of its strongest candidates.

GPU-based optimization faces two communication bottlenecks: coordinating frequent requests and delivering dense models that exceed device memory. We present a discrete simulated bifurcation (dSB) architecture that addresses both through NVIDIA DOCA GPUNetIO. For resident models, a persistent service receives field updates, executes each solve within one GPU thread block, and returns the result. Exact integer coupling sums, GPU work queues, and batched transmission keep the receive–solve–reply path on the device without a dedicated CPU data-path core. In comparisons with socket-based servers using the same solver, the largest latency gains occur under concurrent load. As the offered load increases from 400 to 800 thousand requests per second, median round-trip latency rises by only 6%. At the highest tested load, median and 99th-percentile latencies are 189 and 218~$\mu$s, compared with 288 and 609~$\mu$s for the tuned persistent CPU proxy across repeated runs. For models larger than device memory, a streaming solver retains dynamical state on the GPU and reuses incoming coupling tiles across replicas. It evaluates ten-million-variable dense binary matrices at approximately 307~Gb/s, consuming a 12.5-TB logical matrix through a 64-MiB packet buffer. Ground-state recovery on planted instances and agreement with reference executions verify the computation. Together, the two modes scale dSB to concurrent requests and dense models beyond GPU memory.

Global condition-number dependence can substantially overestimate the difficulty of preparing normalized quantum linear-system solutions. Many existing beyond-conditioning approaches are favorable when globally ill-conditioned directions have limited relevance to the target solution. Here we study a harder regime for time-evolution quantum Krylov linear solving (QKS), where a vanishing eigenvalue remains solution-relevant and must be retained. We identify a QKS regime in which the solution-relevant spectrum is compressed into a compact reduced space where the ill-conditioning is concentrated along a soft direction aligned with the solution, so that normalization removes the divergent amplification along that direction and leaves only a bounded transverse response. Consequently, the normalized-state complexity need not inherit the divergence of the global condition number. Our contributions are threefold: (1) We show that the solution-relevant information admits a compact and stable QKS representation, with bounded subspace dimension, evolution time, and reconstruction overhead as the relevant eigenvalue vanishes. (2) We show that normalized-state difficulty is governed by whether inverse amplification changes the solution direction, rather than by the small eigenvalue alone; even an arbitrarily ill-conditioned reduced system can remain directionally stable when the amplification is solution-aligned. (3) We prove that this stability persists under finite projected-system errors and transfers to the actual QKS output, yielding an end-to-end complexity bound without inverse dependence on the retained small eigenvalue. An explicit separation family further shows that a full inverse-amplification criterion can diverge while the corresponding QKS quantities remain bounded.

We present a discrete analogue of geometric quantization over finite fields which, when applied to certain classical error-correcting codes, yields quantum error-correcting codes. This provides a physical interpretation for a previously known quantum-classical correspondence in coding theory.

Estimating the logical error rate (LER) of a quantum error-correcting (QEC) code by Monte Carlo sampling takes $O(p^{-\lceil d/2 \rceil}/\varepsilon^2)$ samples at physical error rate $p$, code distance $d$ and relative error $\varepsilon$, which becomes intractable at large distances and low error rates, and especially for concatenated codes, whose distance grows as $d^{\ell}$ with the number of levels $\ell$. The fail fast method reduces this cost by sampling the failures of each weight separately, but it only works for unconcatenated codes, and does not model the soft information that the levels of a concatenated code pass to each other. We propose a predictor that takes the stabilizers, the minimum-weight logical operators and, for odd $d$, the logical operators of weight $d + 1$ of a code, and computes, without sampling, the leading monomial $c\,p^{o}$ of the LER when expressed as a polynomial in $p$, at every level of concatenation, in time polynomial in the code parameters and linear in $\ell$ for fixed budgets. We prove that our prediction is exact for any unconcatenated stabilizer code, for every logical observable and arbitrary per-qubit error rates, whenever no budget is exhausted. Furthermore, for concatenated codes, we construct a recursive approach that computes the leading monomial exactly in exponential time, and our predictor approximates this approach. Our predictions match exact enumeration at level 1 on every code we evaluate, and up to three levels for the iceberg code. Using our predictor, we evaluate and compare $37$ code configurations under four noise biases in $48$ core-hours, where Monte Carlo sampling would need more than $10^{12}$ shots at $p = 10^{-3}$ for $113$ of these $148$ cases.

Information causality (IC) was introduced as a physical principle that imposes constraints on the non-local correlations between two parties (Alice and Bob) in a Bell scenario. Specifically, it states that the information that Bob can gain about an $n$-bit database held by Alice is bounded by the information she communicates to him. To date, the question whether IC for any $n \in \mathbb{N}$ singles out the set of quantum correlations remains open, meaning that it is unknown whether for any number of inputs $m$ and outputs $d$ of the two parties the set of correlations that satisfy IC exactly coincides with the quantum correlations. In this work, we answer this question in the negative. Firstly, we show that for any finite database length $n \geq 2$, there is a bipartite set of correlations that satisfies IC for all compatible protocols and that exhibits post-quantum correlations in the Bell scenario with $n+1$ inputs and $2$ outcomes per party. This result relies on constructing an entropic proof of IC for GPTs that may be of independent interest. We further show that in the Bell scenario with $n+1$ inputs and $2$ outcomes per party, our constructed sets of correlations lead to neither a superset nor a subset of the quantum correlations. Secondly, we construct a set of correlations in the Bell scenario with $3$ inputs and $2$ outputs per party that satisfies IC (for all database sizes $n \in \mathbb{N}$) and that includes post-quantum points. Consequently, all known formulations of information causality fail to fully characterize the bipartite set of quantum correlations, already in the $3$-input, $2$-outcome Bell scenario.

Increasing capacity can induce a machine learning model to generalize worse until it perfectly interpolates the training data, and better once it moves beyond interpolation. This inversion, known as double descent, is well understood for independent features. For correlated features, its shape is governed by the spectrum of the feature covariance, but existing analyses typically yield only implicit equations. We first close this gap for classical linear regression, analytically characterizing the entire risk curve through two quantities, namely the condition number of the feature population covariance and a spectral measure of correlation strength. Then, we apply this theory on quantum models, focusing on quantum extreme learning machines (QELMs). We show that three tunable design knobs, i.e., encoding strength, input reuploads, and measurement shots, each regularize the QELM output covariance, and derive conditions under which the interpolation threshold remains stable. Finite measurement shot noise can be leveraged to guarantee this stability. Numerical experiments confirm each prediction, yielding concrete design principles for QELMs that generalize well beyond interpolation.

We introduce contraction tree boundary matrix product states (CT-bMPS), a scalable approximate contraction method for general closed tensor networks. CT-bMPS represents the boundary tensor and its complementary environment at each edge of a contraction tree as matrix product states. This representation allows local compression based on a reduced transition matrix to incorporate environment information along arbitrary contraction trees, while keeping the time and memory costs polynomial in the network size and bond dimension. We benchmark the method on a variety of models and applications, including Ising partition functions, random tensor networks, kicked Ising dynamics, and quantum error correction decoding. These benchmarks demonstrate improved accuracy and computational efficiency over existing general-purpose approximate contraction methods, together with the ability to handle larger and more complex networks. Optimized contraction trees and environment updates substantially improve accuracy without increasing the bond dimension. These results establish CT-bMPS as a general and scalable approximation framework with flexible contraction orders and environment-based compression for a wide range of applications.

Mid-circuit measurements with feedforward reduce the optimal worst-case quantum depth for exact $n$-qubit unitary synthesis from $\Theta(n)$ to $O(1)$, using arbitrary one- and two-qubit gates and unrestricted ancillary space. Measurements and resets without feedforward remain subject to the circuit causal-cone bound. Our deterministic construction uses single-qubit measurements, $O(1)$ rounds of parity feedforward, and $O(n4^n\log(n+2))$ total qubits, with all ancillas starting and ending in $|0\rangle$. Classical processing is not counted in quantum depth. We teleport only the matrix error of an approximation by Nehoran and Yuen. An exactly prepared ancillary state encodes this error, whose action is added back by coherent interference. At operator-norm error $2^{-n}$, one step of amplitude amplification suffices. Coherent fanout circuits achieve the same constant-depth bound, while circuits using only one- and two-qubit gates attain depth $\Theta(n)$ with $O(4^n\log^2(n+2))$ gates. Generic targets require $\Omega(4^n/(h+1))$ qubits at depth $h$, even under unrestricted classical control. Our depth-width trade-offs meet this bound within a factor $O(n^2)$ for $1\le h\le2^n$.