Papers

A fundamental task in quantum information is to approximate a pure quantum state in terms of sparse states or, for a bipartite system, states of bounded Schmidt rank. The optimal deterministic approximation in each case is straightforward, and maximizes the fidelity: keep the largest entries or singular values. On the other hand, random mixtures of sparse states can achieve quadratically improved trace distances, and yield nontrivial bounds on other distance measures like the robustness. In this work, we give efficient algorithms for finding mixtures of sparse states that optimally approximate a given pure state in either trace distance or robustness. These algorithms also yield descriptions of efficiently samplable ensembles of sparse, or less-entangled, states that correspond to these optimal mixed approximations. This can be used for the truncation step of algorithms for matrix product states, improving their accuracy while using no extra memory and no asymptotic increase in runtime, and we demonstrate this improvement numerically. Our proofs rely on tools from convex optimization and zero-sum games, as well as rigorous gurarantees for computing and sampling from maximum-entropy distributions. In particular, a key step in our construction is conditional Poisson sampling, which is the well-studied problem of drawing fixed-size subsets from a population with known inclusion probabilities. We give improved algorithms for carrying out this sampling, which may be of independent interest.

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

The complexity of matrix multiplication is a central topic in computer science. While the focus has traditionally been on exact algorithms, a long line of literature also considers randomized algorithms, which return an approximate solution in faster time. In this work, we adopt a unifying perspective that frames these randomized algorithms in terms of mean estimation. Using it, we first give refined analyses of classical algorithms based on random walks by Cohen-Lewis (`99), and based on sketching by Sarlós (`06) and Drineas-Kannan-Mahoney (`06). We then propose an improvement on Cohen-Lewis that yields a single classical algorithm that is faster than all the other approaches, if we assume no use of (exact) fast matrix multiplication as a subroutine. Second, we demonstrate a quantum speedup on top of these algorithms by using the recent quantum multivariate mean estimation algorithm by Cornelissen-Hamoudi-Jerbi (`22).

Quantum theory is in principle compatible with scenarios where physical processes occur in an indefinite order, potentially yielding advantages in a broad range of information processing tasks. However, advantages in communication, the most basic form of information processing, have so far remained controversial and hard to prove. Here we provide a framework for assessing the role of causal order in communication, by comparing different causal structures under the constraint that the allowed operations must not generate signaling from signaling-incapable devices. Using this framework, we establish a clear-cut advantage of indefinite causal order, and, at the same time, we identify a series of fundamental limits to the communication power of causal structures in quantum mechanics. Notably, we find that a special form of indefinite causal order, obtained by coherently controlling the order of two processes, enhances the transmission of classical messages in a one-shot scenario, but no quantum operation with indefinite order can offer advantages over shared entanglement when asymptotically many uses of the same communication device are employed. Overall, our results unveil non-trivial relations between communication, causal order, entanglement, and no-signaling quantum processes.

We prove that recognizing the phase of matter of an unknown quantum state is quantum computationally hard. More specifically, we show that the quantum computational time of any phase recognition algorithm must grow exponentially in the range of correlations $\xi$ of the unknown state. This exponential growth renders the problem practically infeasible for even moderate correlation ranges, and leads to super-polynomial quantum computational time in the system size $n$ whenever $\xi = \omega(\log n)$. Our results apply to a substantial portion of all known phases of matter, including symmetry-breaking phases and symmetry-protected topological phases for any discrete on-site symmetry group in any spatial dimension. To establish this hardness, we extend the study of pseudorandom unitaries (PRUs) to quantum systems with symmetries. We prove that symmetric PRUs exist under standard cryptographic conjectures, and can be constructed in extremely low circuit depths. We also establish hardness for systems with translation invariance and purely classical phases of matter. A key technical limitation is that the locality of the parent Hamiltonians of the states we consider is linear in $\xi$; the complexity of phase recognition for Hamiltonians with constant locality remains an important open question.

In this work, we study the asymptotic behavior of protocols that localize entanglement in large multi-qubit states onto a subset of qubits by measuring the remaining qubits. We use the maximal average n-tangle that can be generated on a fixed subsystem by measuring its complement – either with local or global measurements – as our key figure of merit. These quantities are known respectively as the localizable entanglement (LE) and the entanglement of assistance (EA). We build upon the work of [arXiv:2411.04080] that proposed a polynomial-time test, based on the EA, for whether it is possible to transform certain graph states into others using local measurements. We show, using properties of the EA, that this test is effective and useful in large systems for a wide range of sizes of the measured subsystem. In particular, we use this test to demonstrate the surprising result that general local unitaries and global measurements will typically not provide an advantage over the more experimentally feasible local Clifford unitaries and local Pauli measurements in transforming large linear cluster states into GHZ states. Finally, we derive concentration inequalities for the LE and EA over Haar-random states which indicate that the localized entanglement structure has a striking dependence on the locality of the measurement. In deriving these concentration inequalities, we develop several technical tools that may be of independent interest.

Hamiltonian learning protocols are becoming valuable tools for benchmarking and verifying quantum computers and simulators, yet few rigorous protocols address explicitly time-dependent dynamics. We give a scalable protocol for learning the time-dependent coefficients of a locally interacting $n$-qubit system on a graph $\mathsf{G}$ of effective dimension $D$. The unknown generator contains geometrically local Hamiltonian terms and weak onsite Pauli-dephasing terms with time-dependent, nonnegative rates. Its coefficient functions and their derivatives admit a certified simultaneous $C^1$ approximation in a known finite-dimensional function class that supports stable interpolation. The only experimental primitives are preparation of product Pauli eigenstates, evolution for chosen times, and product-Pauli measurements. Under an explicit weak-dissipation condition, the Lieb–Robinson estimates proved here, and a certified effective approximation/interpolation degree $m_{\rm eff}$, the protocol reconstructs every coefficient uniformly on a constant interval with error $\epsilon$ and success probability $1-\delta$ using $\widetilde{\mathcal{O}}\!\left(\epsilon^{-2}\operatorname{poly}(m_{\rm eff})\log(n\delta^{-1})\right)$ samples and polynomial classical processing. The method estimates derivatives of finite-time channel overlaps by stable interpolation, uses process shadows to learn the required local transfer matrices, and constructs sparse diagonally dominant systems through localized semidefinite programs.

Characterizing quantum many-body systems is a fundamental problem across physics, chemistry, and materials science. While significant progress has been made, many existing Hamiltonian learning protocols demand digital quantum control over the entire system, creating a disconnect from many real-world settings that provide access only through small, local probes. Motivated by this, we introduce and formalize the problem of quantum probe tomography, where one seeks to learn the parameters of a many-body Hamiltonian using a single local probe access to a small subsystem of a many-body thermal state undergoing time evolution. We address the identifiability problem of determining which Hamiltonians can be distinguished from probe data through a new combination of tools from algebraic geometry and smoothed analysis. Using this approach, we prove that generic Hamiltonians in various physically natural families are identifiable up to simple, unavoidable structural symmetries. Building on these insights, we design the first efficient end-to-end algorithm for probe tomography that learns Hamiltonian parameters to accuracy $\varepsilon$, with query complexity scaling polynomially in $1/\varepsilon$ and classical post-processing time scaling polylogarithmically in $1/\varepsilon$. In particular, we demonstrate that translation- and rotation-invariant nearest-neighbor Hamiltonians on square lattices in one, two, and three dimensions can be efficiently reconstructed from single-site probes of the Gibbs state, up to inversion symmetry about the probed site. Our results demonstrate that robust Hamiltonian learning remains achievable even under severely constrained experimental access.

Glassiness – a phenomenon in physics characterized by a rough free-energy landscape – implies hardness for stable classical algorithms. For example, it can obstruct constant-time Langevin dynamics and message-passing in random $k$-SAT and max-cut instances. We provide an analogous framework for average-case quantum complexity showing that a natural family of quantum algorithms (e.g., Lindbladian evolution) fails for natural Hamiltonian ensembles (e.g., random 3-local Hamiltonians). Specifically, we prove that the standard notion of quantum glassiness based on replica symmetry breaking obstructs stable quantum algorithms for Gibbs sampling, which we define by a Lipschitz temperature dependence in quantum Wasserstein complexity. Our proof relies on showing that such algorithms fail to capture a structural phase transition in the Gibbs state, where glassiness causes the Gibbs state to decompose into clusters extensively separated in quantum Wasserstein distance. This yields average-case lower bounds for constant-time local Lindbladian evolution and shallow variational circuits. Unlike mixing time lower bounds, our results hold even when dynamics are initialized from the maximally mixed state. We apply these lower bounds to non-commuting, non-stoquastic Hamiltonians by showing a glass transition via the replica trick. We find that the ensemble of all 3-local Pauli strings with independent Gaussian coefficients is average-case hard, while providing analytical evidence that the general $p$-local Pauli ensemble is non-glassy for sufficiently large constant $p$, in contrast to its classical (Ising $p$-spin, always glassy) and fermionic (SYK, never glassy) counterparts.

We present a generic compiler that converts any $\mathsf{MIP}^{*}$ protocol into a succinct interactive argument where the communication and the verifier are classical, and where post-quantum soundness relies on the post-quantum sub-exponential hardness of the Learning with Errors ($\mathsf{LWE}$) problem. Prior to this work, such a compiler for $\mathsf{MIP}^{*}$ was given by Kalai, Lombardi, Vaikuntanathan and Yang (STOC 2022), but the post-quantum soundness of this compiler is still under investigation. More generally, our compiler can be applied to any $\mathsf{QIP}$ protocol which is sound only against semi-malicious provers that follow the prescribed protocol, but with possibly malicious initial state. Our compiler consists of two steps. We first show that if a language $\mathcal{L}$ has a $\mathsf{QIP}$ with semi-malicious soundness, where the prover runs in time $T$, then $\mathcal{L} \in \mathsf{QMATIME}(T)$. Then we construct a succinct classical argument for any such language, where the communication complexity grows polylogarithmically with $T$, under the post-quantum sub-exponential hardness of $\mathsf{LWE}$. Note: After this work was finished, an independent and concurrent work (Baroni et al. 2025) resolved the question of quantum soundness of the KLVY compiler.

Community detection is a foundational problem in data science. Its natural extension to hypergraphs captures higher-order correlations beyond pairwise interactions. In this work, we develop a quantum algorithm for hypergraph community detection that achieves a quartic quantum speedup over the best known classical algorithm, along with superpolynomial savings in space. Our algorithm is based on the Kikuchi method, which we extend beyond previously considered problems such as Tensor PCA and $p$XORSAT to a broad family of generalized stochastic block models. To demonstrate (near) optimality of this method, we prove matching lower bounds (up to logarithmic factors) in the low-degree framework, showing that the algorithm saturates a smooth statistical-computational tradeoff. The quantum speedup arises from a quantized version of the Kikuchi method and is based on the efficient preparation of a guiding state correlated with the underlying community structure. Our work suggests that prior quantum speedups using the Kikuchi method are sufficiently robust to encompass a broader set of problems than previously believed; we conjecture that a quantity known as marginal order characterizes the existence of these quantum speedups.

We study coherence distillation under time-translation-invariant operations: given many copies of a quantum state containing coherence in the energy eigenbasis, the aim is to produce a purer coherent state while respecting the time-translation symmetry. This symmetry ensures that the output remains synchronized with the input and that the process can be realized by energy-conserving unitaries coupling the system to a reservoir initially in an energy eigenstate, thereby modeling thermal operations supplemented by a work reservoir or battery. For qubit systems, we determine the optimal asymptotic fidelity and show that it is governed by the purity of coherence, a measure of asymmetry derived from the right logarithmic derivative (RLD) Fisher information. In particular, we find that the lowest achievable infidelity (one minus fidelity) scales as $1/N$ times the reciprocal of the purity of coherence of each input qubit, where $N$ is the number of copies, giving this quantity a clear operational meaning. We additionally study many other interesting aspects of the coherence distillation problem for qubits, including computing higher-order corrections to the lowest achievable infidelity up to $O(1/N^3)$, and expressing the optimal channel as a boundary value problem that can be solved numerically.

Fractional quantum Hall states are promising platforms for topological quantum computation due to their capacity to encode quantum information in topologically degenerate ground states and in the fusion space of non-abelian anyons. We investigate how the information encoded in two paradigmatic states, the Laughlin and Moore-Read states, is affected by density decoherence – coupling of local charge density to non-thermal noise. We identify a critical filling factor $\nu_c$, above which the quantum information remains fully recoverable for arbitrarily strong decoherence. The $\nu=1/3$ Laughlin state and $\nu = 1/2$ Moore-Read state both lie within this range. Below $\nu_c$ both classes of states undergo a decoherence induced Berezinskii-Kosterlitz-Thousless (BKT) transition into a critical decohered phase. For Laughlin states, information encoded in the topological ground state manifold degrades continuously with decoherence strength inside this critical phase, vanishing only in the limit of infinite decoherence strength. On the other hand, quantum information encoded in the fusion space of non-abelian anyons of the Moore-Read states remains fully recoverable for arbitrary strong decoherence even beyond the BKT transition. These results lend further support to the promise of non-Abelian FQH states as platforms for topological quantum computation and raises the question of how errors in such states can be corrected.

In this article, we study a nonlocal game with two questions and three answers per player, which was first considered by Feige in 1991, and show that there is quantum advantage in this game. We prove that the game is a robust self-test for the $3$-dimensional maximally entangled state. Furthermore, we show that the game can be seen as the "or" of two games that each do not have quantum advantage. Lastly, we investigate the behavior of the game with respect to parallel repetition in the classical, quantum and non-signalling case and obtain perfect parallel repetition of the non-signalling value if Feige's game is repeated an even amount of times.

Two circuits are considered to be equivalent under noise if the effect of faults on one circuit is no worse than the effect of faults on the other circuit. We call this relationship fault equivalence. Fault equivalence offers a way to transform circuits while provably preserving their fault-tolerant properties, enabling a framework for fault-tolerant circuit synthesis and optimisation that is correct by construction. The ZX calculus offers a diagrammatic way to represent and reason about quantum circuits and is a useful tool for manipulating circuits while preserving fault equivalence. For this, the usual set of ZX rewrites has to be restricted to not only preserve the underlying linear map represented by the diagram, but also fault equivalence. In this work, we provide a set of ZX rewrites that are sound and complete for fault equivalence of Clifford ZX diagrams. This means that any equivalence that can be derived using the proposed rules is certain to be correct, and any correct equivalence can be derived using only these rules. For this, we utilise diagrammatic constructions called fault gadgets to reason about arbitrary, possibly correlated Pauli faults in ZX diagrams. Fault gadgets allow us to separate the diagram into a fault-free part, which captures the noise-free behaviour of a diagram, and a noisy part that enumerates the effects of all possible faults. Using this, we provide a unique normal form for ZX diagrams under noise and show that any diagram can be brought into this normal form using our proposed rule set. Leveraging the constructive nature of our proof of completeness, we implemented a software library to check whether two ZX diagrams with given noise models are fault-equivalent.

The instantaneous quantum polynomial (IQP) quantum circuit Born machine (QCBM) has been proposed as a promising quantum generative model over bitstrings. Recent works have shown that the training of IQP-QCBM is classically tractable w.r.t. the so-called Gaussian kernel maximum mean discrepancy (MMD) loss function, while maintaining the potential of a quantum advantage for sampling itself. Nonetheless, the model has a number of aspects where improvements would be important for more general utility: (1) the basic model is known to be not universal - i.e. it is not capable of representing arbitrary distributions, and it was not known whether it is possible to achieve universality by adding hidden (ancillary) qubits; (2) a fixed Gaussian kernel used in the MMD loss can cause training issues, e.g., vanishing gradients. In this paper, we resolve the first question and make decisive strides on the second. We prove that for an $n$-qubit IQP generator, adding $n + 1$ hidden qubits makes the model universal. For the latter, we propose a kernel-adaptive training method, where the kernel is adversarially trained. We show that in the kernel-adaptive method, the convergence of the MMD value implies weak convergence in distribution of the generator. We also analytically analyze the limitations of the MMD-based training method. Finally, we verify the performance benefits on the dataset crafted to spotlight improvements by the suggested method. The results show that kernel-adaptive training outperforms a fixed Gaussian kernel in total variation distance, and the gap increases with the dataset dimensionality. These modifications and analyses shed light on the limits and potential of these new quantum generative methods, which could offer the first truly scalable insights in the comparative capacities of classical versus quantum models, even without access to scalable quantum computers.

We perform sensitive nuclear magnetic resonance (NMR) with spin ensembles which are polarized by nitrogen vacancy centers (NV centers) in diamond at room-temperature. With a near shot-noise-limited photoluminescence detection and a highly uniform magnetic field, we resolve sharp NMR features arising from multiple spin clusters. In particular, we investigate the coupling between nuclear spins and NV centers in the neutral and negatively charged states. Further, we perform high precision NMR and coherent control of families of carbon 13 nuclear spin ensembles in the $m_s$=0 level of the NV ground state. Applying an off-axis magnetic field reveals the various sites associated with the otherwise degenerate couplings of the carbon 13 sites around the NV electronic spin providing access to all the hyperfine tensor components. Last, we observe spectroscopic signatures of pairs of nuclear spins coupled to the same NV center. These results are relevant for ensemble measurements of dynamical polarization that currently rely on expensive nuclear magnetic resonance systems as well as for recently proposed nuclear spin gyroscopes.

The assumed hardness of the Shortest Vector Problem in high-dimensional lattices is one of the cornerstones of post-quantum cryptography. The fastest known heuristic attacks on SVP are via so-called sieving methods. While these still take exponential time in the dimension $d$, they are significantly faster than non-heuristic approaches and their heuristic assumptions are verified by extensive experiments. $k$-Tuple sieving is an iterative method where each iteration takes as input a large number of lattice vectors of a certain norm, and produces an equal number of lattice vectors of slightly smaller norm, by taking sums and differences of $k$ of the input vectors. Iterating these ”sieving steps” sufficiently many times produces a short lattice vector. The fastest attacks (both classical and quantum) are for $k=2$, but taking larger $k$ reduces the amount of memory required for the attack. In this paper we improve the quantum time complexity of 3-tuple sieving from $2^{0.3098 d}$ to $2^{0.2846 d}$, using a two-level amplitude amplification aided by a preprocessing step that associates the given lattice vectors with nearby ”center points” to focus the search on the neighborhoods of these center points. Our algorithm uses $2^{0.1887d}$ classical bits and QCRAM bits, and $2^{o(d)}$ qubits. This is the fastest known quantum algorithm for SVP when total memory is limited to $2^{0.1887d}$.

We study the complexity of two closely related learning problems, one quantum and one classical. In the quantum setting, we consider agnostic tomography for the natural class of product mixed states. Given $N$ copies of an $n$-qubit state $\rho$, the goal is to output a nearly optimal product mixed state approximation in trace distance. While recent work has focused on pure-state ansatz (e.g., product or stabilizer states), no polynomial-time guarantees were previously known for mixed-state ansatz. In the classical setting, we study robust learning of binary product distributions: given samples from an unknown distribution on ${0,1}^n$, the goal is to output a nearly optimal product approximation. Our main contributions are as follows. (1) We give a semi-agnostic tomography algorithm for product mixed states with polynomial sample and computational complexity achieving error $O(\mathrm{opt}\log(1/\mathrm{opt}))$, where $\mathrm{opt}$ is the trace distance to the best product approximation. This is the first efficient algorithm with any nontrivial agnostic guarantee for mixed-state ansatz, using only single-qubit, single-copy measurements. We also prove a Quantum Statistical Query lower bound showing near-optimality, and an unconditional lower bound demonstrating that adaptivity is necessary under single-qubit measurements. (2) We give a semi-agnostic algorithm for robustly learning binary product distributions with matching guarantees and establish a Statistical Query lower bound, essentially resolving the efficient robust learnability of this class and improving on prior work since Diakonikolas et al. (2016).

The first-principles formulation of quantum mechanics relevant for quantum chemistry and trapped quantum gases involves particles in the continuous space $\mathbb R^d$. We present a unified framework and modular algorithm for learning external potentials $V$ with free-fermion models in the continuum. Compared to the lattice-based approaches, the continuum presents new mathematical challenges: the state space is infinite-dimensional and the Hamiltonian contains the Laplacian, which is unbounded in the continuum and produces an unbounded speed of information propagation. We address these through novel optimization methods and information-propagation bounds in combination with a priori regularity assumptions on the external potential. The resulting algorithm provides a unified and robust approach to learn parametric interactions (e.g., Coulomb potentials or periodic potentials) and general smooth functions. Our results lay the foundation for a scalable and generalizable toolkit to learn Hamiltonians in continuous space.

We present a platform-agnostic modular architecture that addresses the increasingly fragmented landscape of quantum computing benchmarking by decoupling problem generation, circuit execution, and results analysis into independent, interoperable components. Supporting over 20 benchmark variants ranging from simple algorithmic tests like Bernstein-Vazirani to complex Hamiltonian simulation with observable calculations, the system integrates with multiple circuit generation APIs (Qiskit, CUDA-Q, Cirq) and enables diverse workflows. We validate the architecture through successful integration with Sandia's $\textit{pyGSTi}$ for advanced circuit analysis and CUDA-Q for multi-GPU HPC simulations. Extensibility of the system is demonstrated by implementing dynamic circuit variants of existing benchmarks and a new quantum reinforcement learning benchmark, which become readily available across multiple execution and analysis modes. Our primary contribution is identifying and formalizing modular interfaces that enable interoperability between incompatible benchmarking frameworks, demonstrating that standardized interfaces reduce ecosystem fragmentation while preserving optimization flexibility. This architecture has been developed as a key enhancement to the continually evolving QED-C Application-Oriented Performance Benchmarks for Quantum Computing suite.

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

Given a positive integer k, it is natural to ask for a formula for the distance between a given density matrix (i.e., mixed quantum state) and the set of density matrices of rank at most k. This problem has already been solved when "distance" is measured in the trace or Frobenius norm. We solve it for all other unitary similarity invariant norms. We also present some consequences of our formula. For example, in the trace and Frobenius norms, the density matrix that is farthest from the set of low-rank density matrices is the maximally-mixed state, but this is not true in many other unitary similarity invariant norms.

Flow models are a cornerstone of modern machine learning. They are generative models that progressively transform probability distributions according to learned dynamics. Specifically, they learn a continuous-time Markov process that efficiently maps samples from a simple source distribution into samples from a complex target distribution. We show that these models are naturally related to the Schrödinger equation, for an unusual Hamiltonian on continuous variables. Moreover, we prove that the dynamics generated by this Hamiltonian can be efficiently simulated on a quantum computer. Together, these results give a quantum algorithm for preparing coherent encodings (a.k.a., qsamples) for a vast family of probability distributions–namely, those expressible by flow models–by reducing the task to an existing classical learning problem, plus Hamiltonian simulation. For statistical problems defined by flow models, such as mean estimation and property testing, this enables the use of quantum algorithms tailored to qsamples, which may offer advantages over classical algorithms based only on samples from a flow model. More broadly, these results reveal a close connection between state-of-the-art machine learning models, such as flow matching and diffusion models, and one of the main expected capabilities of quantum computers: simulating quantum dynamics.

We show that all Clifford circuits under interspersed depolarizing noise lose memory of their input exponentially quickly, even when given access to a constant supply of fresh qubits in arbitrary states. This is somewhat surprising given the result of Aharonov et al. [STOC1997] which gives a fault-tolerant protocol for general quantum circuits using a supply of fresh qubits. Our result shows that such a protocol is impossible using only Clifford gates demonstrating that non-Clifford gates are fundamentally required to store information for long periods of time.

Random unitaries sampled from the Haar measure serve as fundamental models for generic quantum many-body dynamics. Under standard cryptographic assumptions, recent works have constructed polynomial-size quantum circuits that are computationally indistinguishable from Haar-random unitaries, establishing the concept of pseudorandom unitaries (PRUs). While PRUs have found broad implications in many-body physics, they fail to capture the energy conservation that governs physical systems. In this work, we investigate the computational complexity of generating PRUs that conserve energy under a fixed and known Hamiltonian $H$. We provide an efficient construction of energy-conserving PRUs when $H$ is local and commuting with random coefficients. Conversely, we prove that for certain translationally invariant one-dimensional $H$, there exists an efficient quantum algorithm that can distinguish truly random energy-conserving unitaries from any polynomial-size quantum circuit. This establishes that energy-conserving PRUs cannot exist for these Hamiltonians. Furthermore, we prove that determining whether energy-conserving PRUs exist for a given family of one-dimensional local Hamiltonians is an undecidable problem. Our results reveal an unexpected computational barrier that fundamentally separates the generation of generic random unitaries from those obeying the basic physical constraint of energy conservation.

Estimating the state of an open quantum system monitored over time requires incorporating information from past measurements (filtering) and, for improved accuracy, also from future measurements (smoothing). While classical smoothing is well understood within a Bayesian framework, its quantum generalization has been challenging, leading to distinct and seemingly incompatible approaches. In this work, we demonstrate that quantum state smoothing hinges on a uniquely quantum feature: the fundamental dependence of retrodiction on prior correlations. We introduce auxiliary systems into the prior belief to capture correlations formed during preparation and evolution and develop a comprehensive framework for quantum state smoothing based on extended Bayesian retrodiction. This framework identifies all previous approaches as different choices of the extended prior, and naturally extends it to other choices that have not been considered before. We also give an information-theoretic characterization of the choices of prior, in terms of the average entropy of the smoothed states. Our results establish quantum state smoothing as a fundamentally retrodictive process just like classical smoothing, with proper quantum features clearly identified.

Recent advances in quantum Gibbs sampling leave open the central question of rapid mixing near and below phase transitions. This challenge is especially relevant for code Hamiltonians whose Gibbs states capture phenomena such as the thermal stability of quantum topological order. In this work, we formulate a new Markov chain, Code Swendsen-Wang dynamics, which uses global updates to prepare the Gibbs states of arbitrary code Hamiltonians. We establish Code Swendsen-Wang dynamics as the right generalization of Swendsen-Wang dynamics for the Ising model to quantum and classical code Hamiltonians: it mixes rapidly for all previously known code Hamiltonians with efficient Gibbs samplers, resolves the central open case of the 4D toric code, and meets fundamental barriers exactly at first-order phase transitions.

The nature of randomness and complexity growth in systems governed by unitary dynamics is a fundamental question in quantum many-body physics. This problem has motivated the study of models such as local random circuits and their convergence to Haar-random unitaries in the long-time limit. However, these models do not correspond to any family of physical time-independent Hamiltonians. In this work, we address this gap by studying the indistinguishability of time-independent Hamiltonian dynamics from truly random unitaries. On one hand, we establish a no-go result showing that for any ensemble of constant-local Hamiltonians and any evolution times, the resulting time-evolution unitary can be efficiently distinguished from Haar-random and fails to form a $2$-design or a pseudorandom unitary (PRU). On the other hand, we prove that this limitation can be overcome by increasing the locality slightly: there exist ensembles of random polylog-local Hamiltonians in one-dimension such that under constant evolution time, the resulting time-evolution unitary is indistinguishable from Haar-random, i.e. it forms both a unitary $k$-design and a PRU. Moreover, these Hamiltonians can be efficiently simulated under standard cryptographic assumptions.

Pebble games, an abstraction from classical reversible computing, have found use in the design of quantum circuits for inherently sequential tasks. Gidney showed that allowing Hadamard basis measurements during pebble games can dramatically improve costs – an extension termed "spooky pebble games" because the measurements leave temporary phase errors called ghosts. Separately, previous work by Blocki et al. studied the benefits of parallelism in pebble games. In this work we define and study parallel spooky pebble games, showing that parallelism and spookiness can yield impressive gains when used together. First, we show by construction that a line graph of length $\ell$ can be pebbled in depth $2\ell$ (exactly optimal) using space $\leq 2.47\log \ell$. Then, to explore pebbling schemes using even less space, we use a highly optimized $A^*$ search implemented in Julia to find the lowest-depth parallel spooky pebbling possible for a range of concrete line graph lengths $\ell$ given a constant number of pebbles $s$. We then show that these techniques can significantly reduce the cost of the arithmetic in Regev's factoring algorithm. For example, we find that 4096-bit integers $N$ can be factored in multiplication depth 193, which outperforms the 680 required of previous variants of Regev and the 444 reported by Ekerå and Gärtner for Shor's algorithm. While the space required for Shor's algorithm is considerably less than any variant of Regev's algorithm including ours, and thus Shor likely remains the best candidate for the first quantum factorization of large integers, our results show that implementations of Regev's algorithm are far from fully optimized, and Regev's algorithm may have practical importance in the future. We also believe our pebbling techniques are applicable in quantum cryptanalysis beyond integer factorization, and in quantum circuit compilation more broadly.