Sparse quantum state preparation with improved Toffoli cost

Quantum 10, 2208 (2026). https://doi.org/10.22331/q-2026-09-10-2208 The preparation of quantum states is one of the most fundamental tasks in quantum computing, and a key primitive in many quantum algorithms. Of particular interest to areas such as quantum simulation and linear-system solvers are sparse quantum states, which contain only a small number $s$ of non-zero computational basis states compared to a generic state. In this work, we present an approach that prepares $s$-sparse states on $n$ qubits, reducing the number of Toffoli gates required compared to prior art. We work in the established framework of first preparing a dense state on a $\lceil{\log(s)}\rceil$-qubit sub-register, and then mapping this state to the target state via an isometry, with the latter step dominating the cost of the full algorithm. The speed-up is achieved by designing an efficient algorithm for finding and implementing the isometry. The worst-case Toffoli cost of our isometry circuit, which may be viewed as a batched version of an approach by Malvetti et al., is essentially $2s$ for sufficiently large values of $n$, yielding roughly a $\log(s)/2$ improvement factor over the state-of-the-art. In numerical benchmarks on randomly chosen states, the cost is closer to $s$. With the improved isometry circuit, we examine the dense-state preparation step and present ways to optimize the joint cost of both steps, particularly in the case of target states with purely real coefficients, by outsourcing some sub-tasks from the dense-state preparation to the isometry.

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Streaming Belief Propagation on Mixed-Alphabet Tanner Graphs for Practical Quantum Memory

Quantum 10, 2207 (2026). https://doi.org/10.22331/q-2026-09-10-2207 Reliable quantum memory under circuit-level noise requires decoders that can process syndrome information continuously and at a rate comparable to its generation. In practical quantum error correction (QEC), repeated syndrome measurements cause the number of potential error locations to grow rapidly with both code size and time. In this paper, we propose a streaming mixed-alphabet belief propagation (SM-BP) decoder. We construct a space-time Tanner graph across multiple rounds of syndrome extraction with mixed-alphabet error variables, preserving correlations arising from multi-qubit faults. Additionally, we propose an adaptive sliding window procedure that captures long error events across window boundaries and adjusts the decoding in real time. To enhance SM-BP, we introduce a technique of probabilistic error consolidation to mitigate degeneracy effects and short cycles. Our simulations demonstrate high error thresholds of 0.4% to 0.87% and strong error-floor performance for topological code families, including rotated toric, toric color, and twisted XZZX toric codes. These results show that SM-BP provides a practical decoding framework for continuous QEC under circuit-level noise.

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Solving the Nonlinear Vlasov Equation on a Quantum Computer

Quantum 10, 2206 (2026). https://doi.org/10.22331/q-2026-09-10-2206 The practical applicability of a recent Carleman-linearization-based quantum algorithm for solving ordinary differential equations (ODEs) with quadratic nonlinearities is investigated for the nonlinear electrostatic Vlasov equation with Krook-type collision operators. The equation is discretized on a (1+1)-dimensional phase-space grid and mapped onto the input of the quantum algorithm. Upper bounds for the query and gate complexities are derived in the limit of large grid sizes and found to be polynomially larger than the time complexity of the corresponding classical algorithms, primarily due to the dimension, sparsity, and norm of the Carleman-linearized evolution matrix. The convergence criteria are shown to impose severe restrictions on physically relevant plasma applications, requiring dissipation levels far exceeding those provided by the Krook operator.

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Breaking the Orthogonality Barrier in Quantum LDPC Codes

Quantum 10, 2205 (2026). https://doi.org/10.22331/q-2026-09-09-2205 Classical low-density parity-check (LDPC) codes are a widely deployed and well-established technology, forming the backbone of modern communication and storage systems. It is well known that, in this classical setting, increasing the girth of the Tanner graph while maintaining regular degree distributions leads simultaneously to good belief-propagation (BP) decoding performance and large minimum distance. In the quantum setting, however, this principle does not directly apply because quantum LDPC codes must satisfy additional orthogonality constraints between their parity-check matrices. When one enforces both orthogonality and regularity in a straightforward manner, the girth is typically reduced and the minimum distance becomes structurally upper bounded. In this work, we overcome this limitation by using permutation matrices with controlled commutativity and by restricting the orthogonality constraints to only the active part of the construction, while preserving regular check-matrix structures. This design circumvents conventional structural distance limitations induced by parent-matrix orthogonality, and enables the construction of quantum LDPC codes with large girth while avoiding latent low-weight logical operators. As a concrete demonstration, we construct a girth-8, (3,12)-regular $[[9216,4612, \leq 48]]$ quantum LDPC code and show that, under BP decoding combined with a low-complexity post-processing algorithm, it achieves a frame error rate as low as $10^{-8}$ on the depolarizing channel with error probability $4 \%$.

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Faster Quantum Simulation Of Markovian Open Quantum Systems Via Randomisation

Quantum 10, 2204 (2026). https://doi.org/10.22331/q-2026-09-03-2204 When simulating the dynamics of open quantum systems with quantum computers, it is essential to accurately approximate the system's behaviour while preserving the physicality of its evolution. Traditionally, for Markovian open quantum systems, this has been achieved using first and second-order Trotter-Suzuki product formulas or probabilistic algorithms. In this work, we introduce novel non-probabilistic algorithms for simulating Markovian open quantum systems using randomisation. Our methods, including first and second-order randomised Trotter-Suzuki formulas and the QDRIFT channel, not only maintain the physicality of the system's evolution but also enhance the scalability and precision of quantum simulations. We derive error bounds and step count limits for these techniques, bypassing the need for the mixing lemma typically employed in Hamiltonian simulation proofs. Furthermore, we implement these randomised algorithms using Classical Sampling (CS), demonstrating their gate complexity advantages over deterministic TS product formulas. This work systematically extends powerful randomisation techniques from Hamiltonian simulation to the general setting of Markovian open quantum systems, highlighting their potential to enable faster and more accurate simulations.

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Quantum Max d-Cut via qudit swap operators

Quantum 10, 2203 (2026). https://doi.org/10.22331/q-2026-09-03-2203 Quantum Max Cut (QMC) problem for systems of qubits is an example of a 2-local Hamiltonian problem, and a prominent paradigm in computational complexity theory. This paper investigates the algebraic structure of a higher-dimensional analog of the QMC problem for systems of qudits. The Quantum Max $d$-Cut ($d$-QMC) problem asks for the largest eigenvalue of a Hamiltonian on a graph with $n$ vertices whose edges correspond to swap operators acting on $(\mathbb C^d)^{\otimes n}$. The algebra generated by the swap operators is identified as a quotient of a free algebra modulo symmetric group relations and a single additional relation of degree $d$. This presentation leads to a tailored hierarchy of semidefinite programs, leveraging noncommutative polynomial optimization (NPO) methods, that converges to the solution of the $d$-QMC problem. For a large class of complete bipartite graphs, exact solutions for the $d$-QMC problem are derived using the representation theory of symmetric groups and Littlewood-Richardson coefficients. Lastly, the paper addresses a refined $d$-QMC problem focused on finding the largest eigenvalue within each isotypic component (irreducible block) of the graph Hamiltonian. It is shown that the spectrum of the star graph Hamiltonian distinguishes between isotypic components of the $3$-QMC problem. For general $d$, low-degree relations for separating isotypic components are presented, enabling adaptation of the global NPO hierarchy to efficiently compute the largest eigenvalue in each isotypic component.

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Floquetifying stabiliser codes with distance-preserving rewrites

Quantum 10, 2202 (2026). https://doi.org/10.22331/q-2026-09-03-2202 Stabiliser codes with large weight measurements can be challenging to implement fault-tolerantly. To overcome this, we propose a Floquetification procedure which, given a stabiliser code, synthesises a novel Floquet code that only uses single- and two-qubit operations. Moreover, this procedure preserves the distance and number of logicals of the original code. The new Floquet code requires additional physical qubits. This overhead is linear in the weight of the largest measurement of the original code. Our method is based on the ZX calculus, a graphical language for representing and rewriting quantum circuits. However, a problem arises with the use of ZX in the context of rewriting error-correcting codes: ZX rewrites generally do not preserve code distance. Tackling this issue, we define the notion of distance-preserving rewrite that enables the transformation of error-correcting codes without changing their distance. These distance-preserving rewrites are used to decompose arbitrary weight stabiliser measurements into quantum circuits with single- and two-qubit operations. As we only use distance-preserving rewrites, we are guaranteed that a single error in the resulting circuit creates at most a single error on the data qubits. These decompositions enable us to generalise the Floquetification procedure of Townsend-Teague et al [83] to arbitrary stabiliser codes, provably preserving the distance and number of logicals of the original code.

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Reducing Spatial and Temporal Dimensionality in the Multidimensional Caldeira-Leggett Model

Quantum 10, 2201 (2026). https://doi.org/10.22331/q-2026-09-02-2201 Focusing on the real-time dynamics of the reduced density matrix of the multidimensional Caldeira-Leggett model, several techniques are adopted in this paper to reduce the spatial and temporal dimensionality, combined into an efficient algorithm. From a spatial perspective, an equivalent formulation of the Dyson series is presented. With the aid of a low-rank approximation, the spatial dimensionality of open quantum system simulations is halved. From a temporal perspective, the frozen Gaussian approximation is used to approximate both the evolution operator and the interaction operator in the multidimensional Caldeira-Leggett model. This reduces the high-dimensional integrals to one- and two-dimensional integrals independent of the truncation level of the Dyson series. Through these techniques, we design an efficient algorithm whose validity is verified through several numerical experiments, including a two-dimensional double slit simulation.

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Quantum Simulation of Nuclear Dynamics in First Quantization

Quantum 10, 2200 (2026). https://doi.org/10.22331/q-2026-09-02-2200 The study of real time dynamics of nuclear systems is of great importance to provide theoretical predictions of cross sections relevant for both terrestrial experiments as well as applications in astrophysics. First principles simulations of these dynamical processes is however hindered by an exponential cost in classical resources and the possibility of performing scalable simulations using quantum computers is currently an active field of research. In this work we provide the first complete characterization of the resource requirements for studying nuclear dynamics with the full Leading Order (LO) pionless EFT Hamiltonian in first quantization employing simulation strategies using both product formulas as well as Quantum Signal Processing. In particular, we show that time evolution of such an Hamiltonian can be performed with polynomial resources in the number of particles, and logarithmic resources in the number of single-particle basis states. This result provides an exponential improvement compared with previous work on the same Hamiltonian model in second quantization. We find that interesting simulations for low energy nuclear scattering could be achievable with tens of millions of T gates and few hundred logical qubits suggesting that the study of simple nuclear reactions could be amenable for early fault tolerant quantum platforms.

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Symmetry-enriched topological order in tensor networks: Defects, gauging and anyon condensation

Quantum 10, 2199 (2026). https://doi.org/10.22331/q-2026-09-01-2199 We study symmetry-enriched topological order in two-dimensional tensor network states by using graded matrix product operator algebras to represent symmetry-induced domain walls. A close connection to the theory of graded unitary fusion categories is established. Tensor network representations of the topological defect superselection sectors are constructed for all domain walls. The emergent symmetry-enriched topological order is extracted from these representations, including the symmetry action on the underlying anyons. Dual phase transitions, induced by gauging a global symmetry, and condensation of a bosonic subtheory, are analyzed and the relationship between topological orders on either side of the transition is derived. Several examples are worked through explicitly.

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A brief history of quantum vs classical computational advantage

Quantum 10, 2198 (2026). https://doi.org/10.22331/q-2026-09-01-2198 In this review article we summarize all experiments claiming quantum computational advantage to date. Our review highlights challenges, loopholes, and refutations appearing in subsequent work to provide a complete picture of the current statuses of these experiments. In addition, we also discuss theoretical computational advantage in example problems such as approximate optimization and recommendation systems. Finally, we review recent experiments in quantum error correction --- the biggest frontier to reach experimental quantum advantage in Shor's algorithm.

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Non-Gaussian Noise Magnetometry Using Local Spin Qubits

Quantum 10, 2197 (2026). https://doi.org/10.22331/q-2026-08-27-2197 Atomic scale qubits, as may be realized in nitrogen vacancy (NV) centers in diamond, offer the opportunity to study magnetic field noise with nanometer scale spatial resolution. Using these spin qubits, one can learn a great deal about the magnetic-field noise correlations, and correspondingly the collective-mode spectra, in quantum materials and devices. However, to date these tools have been essentially restricted to studying Gaussian noise processes – equivalent to linear-response. In this work we will show how to extend these techniques beyond the Gaussian regime and show how to unambiguously measure higher-order magnetic noise cumulants in a local, spatially resolved way. We unveil two protocols for doing this; the first uses a single spin-qubit and different dynamical decoupling sequences to extract non-Markovian and non-Gaussian spin-echo noise. The second protocol uses two-qubit coincidence measurements to study spatially non-local cumulants in the magnetic noise. We then demonstrate the utility of these protocols by considering a model of a bath of non-interacting two-level systems, as well as a model involving spatially correlated magnetic fluctuations near a second-order Ising phase transition. In both cases, we highlight how this technique can be used to measure in a real many-body system how fluctuation dynamics converge towards the central limit theorem as a function of effective bath size. We then conclude by discussing some promising applications and extensions of this method.

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On the relation between perspective-neutral, algebraic, and effective quantum reference frames

Quantum 10, 2196 (2026). https://doi.org/10.22331/q-2026-08-20-2196 The framework of internal quantum reference frames (QRFs) constitutes a universal toolset for dealing with symmetries in quantum theory and has led to new revelations in quantum gravity, gauge theories and foundational physics. Multiple approaches have emerged, sometimes differing in scope and the way symmetries are implemented, raising the question as to their relation. Here, we investigate the relation between three approaches to QRFs for gauge symmetries, namely the $effective$ semiclassical, $algebraic$, and $perspective-neutral$ (PN) approaches. Rather than constructing Hilbert spaces, as the PN approach, the effective approach is based on a quantum phase space parametrized by expectation values and fluctuations, while the emphasis of the algebraic approach is on the state space of complex linear functionals on a kinematical algebra. Nevertheless, external frame information is treated as gauge in all three formalisms, manifested in constraints on states and algebra. We show that these three approaches are, in fact, equivalent for ideal QRFs, distinguished by sharp orientations, which is the previous setting of the first two approaches. Our demonstration pertains to single constraints, including relativistic ones, and encompasses QRF changes. In particular, the QRF transformations of the PN framework agree semiclassically with those of the older effective approach, by which it was inspired. As a physical application, we explore the QRF covariance of uncertainties and fluctuations, which turn out to be frame dependent. This is particularly well-suited for the effective and algebraic approaches, for which these quantities form a natural basis. Finally, we pave the way towards extending these two approaches to non-ideal QRFs by studying the projection and gauge-fixing operations of the Page-Wootters formalism, built into the PN framework, on algebraic states.

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Simple logical quantum computation with concatenated symplectic double codes

Quantum 10, 2195 (2026). https://doi.org/10.22331/q-2026-08-19-2195 There have been significant recent advances in constructing theoretical and practical quantum error correcting codes that function well as quantum memories; however, performing fault-tolerant logical gates on these codes is less studied, and the protocols that do exist often require significant complexity. Building off the symplectic double construction, we investigate concatenated symplectic double codes, which have a rich set of logical gates implementable using only physical single-qubit gates and qubit relabeling. Combined with injected fold-transversal gates, the full Clifford group on a single codeblock is achieved through a functionally simple circuit. We perform circuit-level simulations of state preparation and quantum error correction on these codes and show that they have promising performance at near state-of-the-art physical error rates. As such, we argue that concatenated symplectic double codes are strong contenders as the underlying computational code on medium- to large-scale quantum computers.

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Quantifying mixed-state entanglement via partial transpose and realignment moments

Quantum 10, 2194 (2026). https://doi.org/10.22331/q-2026-08-19-2194 Entanglement plays a crucial role in quantum information science and many-body physics, yet quantifying it in mixed quantum many-body systems has remained a notoriously difficult problem. Here, we introduce families of quantitative entanglement witnesses, constructed from partial transpose and realignment moments, which provide rigorous bounds on entanglement monotones as well as entanglement dimensionality. Our witnesses can be efficiently measured using SWAP tests or variants of Bell measurements, thus making them directly implementable on current hardware. Leveraging our witnesses, we present several novel results on entanglement properties of mixed states, both in quantum information and many-body physics. We develop efficient algorithms to test whether mixed states with bounded entropy have low or high entanglement, which previously was only possible for pure states. We also provide an efficient algorithm to test the Schmidt rank using only two-copy measurements, and the operator Schmidt rank using four-copy measurements. Further, our witnesses robustly certify the quantum circuit depth in the presence of noise, as well as the Schmidt rank of mixed states. Finally, we show that the entanglement phase diagram of Haar random states, quantified by the partial transpose negativity, can be fully established solely by computing our witness, a result that also applies to any state $4$-design. Our witnesses can also be efficiently computed for matrix product states, thus enabling the characterization of entanglement in extensive many-body systems. Finally, we make progress on the entanglement required for quantum cryptography, establishing rigorous limits on pseudoentanglement and pseudorandom density matrices with bounded entropy. Our work opens new avenues for quantifying entanglement in large and noisy quantum systems.

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Quantum phase estimation with optimal confidence interval using three control qubits

Quantum 10, 2193 (2026). https://doi.org/10.22331/q-2026-08-19-2193 Quantum phase estimation is an important routine in many quantum algorithms, particularly for estimating the ground state energy in quantum chemistry simulations. This estimation involves applying powers of a unitary to the ground state, controlled by an auxiliary state prepared on a control register. In many applications the goal is to provide a confidence interval for the phase estimate, and optimal performance is provided by a discrete prolate spheroidal sequence. We show how to prepare the corresponding state in a far more efficient way than prior work. We find that a matrix product state representation with a bond dimension of 4 is sufficient to give a highly accurate approximation for all dimensions tested, up to $2^{24}$. This matrix product state can be efficiently prepared using a sequence of simple three-qubit operations. When the dimension is a power of 2, the phase estimation can be performed with only three qubits for the control register, making it suitable for early-generation fault-tolerant quantum computers with a limited number of logical qubits. 

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Measurement incompatibility in Bayesian multiparameter quantum estimation

Quantum 10, 2192 (2026). https://doi.org/10.22331/q-2026-08-13-2192 We present a comprehensive and pedagogical formulation of Bayesian multiparameter quantum estimation. Within this framework, we analyse the role of measurement incompatibility and establish its quantitative effect on attainable precision. We achieve this by deriving upper bounds based on the pretty good measurement – a notion from hypothesis testing – combined with the evaluation of the Nagaoka-Hayashi lower bound. In general, we prove that, as in the many-copy regime of local estimation theory, incompatibility can at most double the minimum loss relative to the idealised scenario in which individually optimal measurements are assumed jointly implementable. Therefore, in practical situations, the latter may provide a sufficient and computationally efficient benchmark without solving the full optimisation problem. Our results, which we illustrate through applications of discrete phase imaging, phase and dephasing estimation, and qubit sensing, provide analytical and numerical tools for assessing ultimate precision limits and the role of measurement incompatibility in Bayesian multiparameter quantum metrology, including an open-source package for all the bounds discussed here.

quantum-journal.org

Catalytic $z$-rotations in constant $T$-depth

Quantum 10, 2191 (2026). https://doi.org/10.22331/q-2026-08-13-2191 We show that the $T$-depth of any single-qubit $z$-rotation can be reduced to $3$ if a certain catalyst state is available. To achieve an $\epsilon$-approximation, it suffices to have a catalyst state of size polynomial in $\log(1/\epsilon)$. This implies that $\mathsf{QNC}^0_f/\mathsf{qpoly}$ admits a finite universal gate set consisting of Clifford+$T$. In particular, there are catalytic constant $T$-depth circuits that approximate multi-qubit Toffoli, adder, and quantum Fourier transform arbitrarily well. We also show that the catalyst state can be prepared in time polynomial in $\log (1/\epsilon)$.

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Robust topological quantum state transfer with long-range interactions in Rydberg arrays

Quantum 10, 2190 (2026). https://doi.org/10.22331/q-2026-08-13-2190 We develop a theoretical framework for fast, robust and high-fidelity topological quantum state transfer in one-dimensional systems with long-range couplings, motivated by chains of Rydberg atoms with dipole–dipole interactions. Such long-range interactions naturally give rise to extended Su–Schrieffer–Heeger and Rice–Mele models supporting topologically protected edge states. We show that these edge states enable high-fidelity edge-to-edge excitation transfer using both time-independent protocols, based on coherent edge state dynamics, and time-dependent protocols, based on adiabatic modulation of system parameters. Long-range couplings play a central role by enhancing the relevant energy gaps, leading to a substantial improvement in transfer efficiency compared to nearest neighbour models. The resulting transfer is robust against positional disorder, reflecting its topological origin and highlighting the potential of long-range interacting platforms for reliable quantum state transfer.

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Efficient Graph State Generation in Linear Optics

Quantum 10, 2189 (2026). https://doi.org/10.22331/q-2026-08-13-2189 Graph states are central resources for quantum information processing, supporting applications in computation, communication, and error correction. In photonic systems, they are typically assembled from smaller entangled states using probabilistic fusion gates, which demand many photons and suffer from low success rates. We present an optimized scheme for directly generating caterpillar graph states (CGSs)—essential resource states for constructing high-dimensional lattice graph states—using only single-photon sources, linear optics, and heralded measurements. Based on the linear quantum graph (LQG) picture, our method produces CGSs efficiently. For CGSs of length $l\ge 3$, it requires $l-2$ fewer photons and achieves a success rate $2^{l-2}$ times higher than fusion-based approaches. These results demonstrate that the LQG picture provides a powerful and flexible route to generating complex photonic graph states for efficient quantum information processing.

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Unconditional Quantum Advantage for Sampling with Shallow Circuits

Quantum 10, 2188 (2026). https://doi.org/10.22331/q-2026-08-12-2188 Recent work by Bravyi, Gosset, and Koenig showed that there exists a search problem that a constant-depth quantum circuit can solve, but that any constant-depth classical circuit with bounded fan-in cannot. They also pose the question: Can we achieve a similar proof of separation for an input-independent sampling task? In this paper, we show that the answer to this question is yes when the number of random input bits given to the classical circuit is bounded. We introduce a distribution $D_{n}$ over $\\{0,1\\}^n$ and construct a constant-depth uniform quantum circuit family $\\{C_n\\}_n$ such that $C_n$ samples from a distribution close to $D_{n}$ in total variation distance. For any $\delta \lt 1$ we also prove, unconditionally, that any classical circuit with bounded fan-in gates that takes as input $kn + n^\delta$ i.i.d. Bernouli random variables with entropy $1/k$ and produces output close to $D_{n}$ in total variation distance has depth $\Omega(\log \log n)$. This gives an unconditional proof that constant-depth quantum circuits can sample from distributions that can't be reproduced by constant-depth bounded fan-in classical circuits, even up to additive error. We also show a similar separation between constant-depth quantum circuits with advice and classical circuits with bounded fan-in and fan-out, but access to an unbounded number of i.i.d random inputs. The distribution $D_n$ and classical circuit lower bounds are inspired by work of Viola, in which he shows a different (but related) distribution cannot be sampled from approximately by constant-depth bounded fan-in classical circuits.

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Quantum Resource Comparison for Two Leading Surface Code Lattice Surgery Approaches

Quantum 10, 2187 (2026). https://doi.org/10.22331/q-2026-08-10-2187 Hamiltonian simulation is one of the most promising candidates for the demonstration of quantum advantage within the next ten years, and several studies have proposed end-to-end resource estimates for executing such algorithms on fault-tolerant quantum processors. Usually, these resource estimates are based upon the assumption that quantum error correction is implemented using the surface code, and that the best surface code compilation scheme involves serializing input circuits by eliminating all Clifford gates. This transformation is thought to make best use of the native multi-body measurement (lattice surgery) instruction set available to surface codes. Some work, however, has suggested that direct compilation from Clifford+T to lattice surgery operations may be beneficial for circuits that have high degrees of logical parallelism. In this study, we analyze the resource costs for implementing Hamiltonian simulation using example approaches from each of these leading surface code compilation families. The Hamiltonians whose dynamics we consider are those of the transverse-field Ising model in several geometries, the Kitaev honeycomb model, and the $\mathrm{\alpha-RuCl_3}$ complex under a time-varying magnetic field. We show, among other things, that the optimal scheme depends on whether Hamiltonian simulation is implemented using the quantum signal processing or Trotter-Suzuki algorithms, with Trotterization benefiting by orders of magnitude from direct Clifford+T compilation for these applications. Our results suggest that surface code quantum computers should not have a one-size-fits-all compilation scheme, but that smart compilers should predict the optimal scheme based upon high-level quantities from logical circuits such as average circuit density, numbers of logical qubits, and T fraction.

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Disorder-Induced Entanglement Phase Transitions in Non-Hermitian Systems with Skin Effects

Quantum 10, 2186 (2026). https://doi.org/10.22331/q-2026-08-07-2186 Non-Hermitian dynamics is ubiquitous in various physical systems. While recent study shows that such a dynamics leads to an area-law scaling of the entanglement entropy due to the non-Hermitian skin effects, it remains unclear how disorder changes the behavior of the entanglement entropy in a non-Hermitian system with skin effects. Here we study the dynamics of a many-body state of free fermions in the paradigmatic Hatano-Nelson model with open boundaries, and find that the area-law behavior of the entanglement entropy in the pristine Hatano-Nelson model develops into a logarithmic scaling for small disorder strength. As we further increase the disorder strength, the system reenters an area-law regime through an entanglement phase transition. At the critical point, the entanglement entropy exhibits a universal algebraic scaling. We further demonstrate the absence of a conformal invariance in the log-law regime by examining the subsystem entanglement entropy, the connected correlation function and the mutual information. Finally, we show the existence of disorder induced entanglement phase transitions in the Hatano-Nelson model with periodic boundaries.

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Promise of Graph Sparsification and Decomposition for Noise Reduction in QAOA: Analysis for Trapped-Ion Compilations

Quantum 10, 2185 (2026). https://doi.org/10.22331/q-2026-08-07-2185 We develop new approximate compilation schemes that significantly reduce the expense of compiling the Quantum Approximate Optimization Algorithm (QAOA) for solving the Max-Cut problem. Our main focus is on compilation with trapped-ion simulators using Pauli-$X$ operations and all-to-all Ising Hamiltonian $H_\text{Ising}$ evolution generated by Molmer-Sorensen or optical dipole force interactions, though some of our results also apply to standard gate-based compilations. Our results are based on principles of graph sparsification and decomposition; the former reduces the number of edges in a graph while maintaining its cut structure, while the latter breaks a weighted graph into a small number of unweighted graphs. Though these techniques have been used as heuristics in various hybrid quantum algorithms, there have been no guarantees on their performance, to the best of our knowledge. This work provides the first provable guarantees using sparsification and decomposition to improve quantum noise resilience and reduce quantum circuit complexity. For quantum hardware that uses edge-by-edge QAOA compilations, sparsification leads to a direct reduction in circuit complexity. For trapped-ion quantum simulators implementing all-to-all $H_{Ising}$ pulses, we show that for a $(1-\epsilon)$ factor loss in the Max-Cut approximation ($\epsilon \gt 0)$, our compilations improve the (worst-case) number of $H_{Ising}$ pulses from $O(n^2)$ to $O(n\log(n/\epsilon))$ and the (worst-case) number of Pauli-$X$ bit flips from $O(n^2)$ to $O\left(\frac{n\log(n/\epsilon)}{\epsilon^2}\right)$ for $n$-node graphs. This is an asymptotic improvement for any constant $\epsilon \gt 0$. We demonstrate that significant improvements to the approximation ratio are obtained using decomposition in simulated trapped-ion experiments with dephasing noise. We further present a generic argument showing that sparsification results in an exponentially improved circuit fidelity lower bound in digital computing schemes based on one- and two-qubit gates, which are relevant to a wide variety of hardwares such as superconducting qubits and certain neutral atom or trapped ion setups, and more sophisticated noise models. We anticipate these approximate compilation techniques will be useful tools in a variety of future quantum computing experiments.

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Detection of a Rényi Index Dependent Transition in Entanglement Entropy Scaling

Quantum 10, 2184 (2026). https://doi.org/10.22331/q-2026-08-07-2184 The scaling of entanglement with subsystem size encodes key information about phases and criticality, but the von Neumann entropy is costly to access in experiments and simulations, often requiring full state tomography. The second Rényi entropy is readily measured using two-copy protocols and is often used as a proxy for the von Neumann entanglement entropy, where it is assumed to track its asymptotic scaling. Sugino and Korepiny (Int. J. Mod. Phys. B 32, 1850306 (2018)) revealed that in the ground state of some highly constrained spin models, the scaling of the von Neumann and Rényi entropies can differ, varying from power law to logarithmic scaling as a function of the Rényi index. Here, we construct a number-conserving many-body state that demonstrates a Rényi-index-dependent change in the leading entanglement scaling, generalizing previous results to the case of interacting fermions. We introduce a symmetry-aware lower bound on the von Neumann entropy built from charge-resolved Rényi entropies that can provide a protocol for diagnosing anomalous entanglement scaling from experimentally accessible data.

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Entanglement growth in the dark intervals of a locally monitored free-fermion chain

Quantum 10, 2183 (2026). https://doi.org/10.22331/q-2026-08-03-2183 We consider a free fermionic chain with monitoring of the particle density on a single site of the chain and study the entanglement dynamics of quantum jump trajectories. We show that the entanglement entropy grows in time towards a stationary state which display volume law scaling of the entropy, in stark contrast with both the unitary dynamics after a local quench and the no-click limit corresponding to full post-selection. We explain the extensive entanglement growth as a consequence of the peculiar distribution of quantum jumps in time, which display superpoissonian waiting time distribution characterised by a bunching of quantum jumps followed by long dark intervals where no-clicks are detected, akin to the distribution of fluorescence light in a driven atom. We show that the presence of dark intervals is the key feature to explain the effect and that by increasing the number of sites which are monitored the volume law scaling gives away to the Zeno effect and its associated area law.

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On the quantum computational complexity of classical linear dynamics with geometrically local interactions: Dequantization and universality

Quantum 10, 2182 (2026). https://doi.org/10.22331/q-2026-08-03-2182 The simulation of large-scale classical systems in exponentially small space on quantum computers has gained attention. The prior work demonstrated that a quantum algorithm offers an exponential speedup over any classical algorithm in simulating classical dynamics with long-range interactions. However, many real-world classical systems, such as those arising from partial differential equations, exhibit only local interactions. The question remains whether quantum algorithms can still provide exponential speedup under this condition. In this work, we thoroughly characterize the computational complexity of simulating such geometrically local systems on quantum computers. First, we dequantize the quantum algorithm for simulating short-time (polynomial-time) dynamics of such systems. This implies that the problem of simulating this dynamics does not yield any exponential quantum advantage. Second, we show that simulating short-time dynamics is at least as hard as polynomial-time and linear-space probabilistic classical computation. Third, we show that the computational complexity of simulating long-time (exponential-time) dynamics is captured by exponential-time and polynomial-space quantum computation. This suggests a super-polynomial time advantage when restricting the computation to polynomial-space, or an exponential space advantage otherwise. This work offers new insights into the complexity of classical dynamics governed by partial differential equations, providing a pathway for achieving quantum advantage in practical problems.

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Entanglement-assisted Quasi-cyclic Quantum Low-density Parity-check Codes over Qubits

Quantum 10, 2181 (2026). https://doi.org/10.22331/q-2026-07-31-2181 We construct several families of entanglement-assisted quasi-cyclic quantum LDPC (EA-QC-QLDPC) codes via structured tilings of permutation matrices. The entanglement-unassisted portion of the joint Tanner graph of the proposed EA-QC-QLDPC code derived from two distinct classical QC-LDPC codes is free of 4-cycles. Notably, one of the proposed families constructed from two distinct classical codes requires only a ${single}$ shared Bell pair between the quantum transmitter and receiver, highlighting its resource efficiency. We also analytically determine the exact code rates for some of the proposed constructions. Furthermore, two of the proposed families of EA-QC-QLDPC codes are derived from a single classical code whose Tanner graphs possess girth greater than six, further enhancing their error-correcting performance. We also propose an encoding scheme with improved complexity by exploiting the proposed code structure. The performance of the proposed codes is assessed under both random and burst error models under the depolarizing and Markovian noise actions. Simulation results reveal nearly one order of improvement in error-correction performance with the quaternary block-layered normalized min-sum (QBLNMS) decoder compared to the layered binary sum-product decoder over both depolarizing and Markovian channels. Using the QBLNMS decoder over a quaternary alphabet, we demonstrate that correlated Pauli errors can be effectively handled within the decoding framework. Furthermore, under the QBLNMS decoding, the proposed codes achieve ${significant}$ performance improvements compared to prior works and can effectively handle both random and burst errors. The code constructions are scalable across various coding rates and quantum payloads, crucial for practical quantum communication and computing systems.

quantum-journal.org

Quantifiers and witnesses for the nonclassicality of measurements and of states

Quantum 10, 2180 (2026). https://doi.org/10.22331/q-2026-07-30-2180 In recent work [Phys. Rev. X 16, 021050], we proposed a unified notion of nonclassicality that applies to arbitrary processes in quantum theory, including individual quantum states, measurements, and sets thereof. This notion is derived from the principle of generalized noncontextuality, but in a novel manner that applies to individual processes rather than full experiments or theories. In the present work, we develop semidefinite-programming-based certificates and witnesses for the nonclassicality of states, sources, measurements, and sets thereof. These theory-dependent methods complement theory-independent approaches based on noncontextuality inequalities. We demonstrate the framework through a variety of explicit examples.

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Stabilizer Ranks, Barnes Wall Lattices and Magic Monotones

Quantum 10, 2179 (2026). https://doi.org/10.22331/q-2026-07-29-2179 In 2024, Kliuchnikov and Schönnenbeck showed a connection between the Barnes Wall lattices, stabilizer states and Clifford operations. In this work, we study their results and relate them to the problem of lower bounding stabilizer ranks. We show the first quantitative lower bound on stabilizer fidelity as a function of stabilizer ranks, which reproduces the linear-by-log lower bound for $\chi_{\delta}({|{H}\rangle^{ \otimes n}})$, i.e, on the approximate stabilizer rank of $|H\rangle^{\otimes n}$. In fact, we show that the lower bound holds even when the fidelity between the approximation and ${|H\rangle}^{\otimes n}$ is exponentially small, which is currently the best lower bound in this regime. Next, we define a new magic monotone for pure states, the Barnes Wall norm, and its corresponding approximate variant. We upper bound these monotones by the $CS$-count of state preparation, and also by the stabilizer ranks. In particular, the upper bound given by the $CS$-count is tight, in the sense that we exhibit states that achieve the bound. Apart from these results, we give a Fidelity Amplification algorithm, which provides a trade-off between approximation error and the stabilizer rank. As a corollary, it gives us a way to compose approximate stabilizer decompositions into approximate decompositions of their tensor products. Finally, we provide an alternate, elementary proof of the existence and density of product states with maximal stabilizer ranks, which was first proven by Lovitz and Steffan (2022), where they used results from algebraic geometry.

quantum-journal.org