arXiv subjects

Yihui Quek

Publications and source records attributed to Yihui Quek.

At least 19 recordsLinked to original sources

arXiv2026

Post-Quantum Cryptography from Quantum Stabilizer Decoding

Post-quantum cryptography currently rests on a small number of hardness assumptions, posing significant risks should any one of them be compromised. This vulnerability motivates the search for new and cryptographically versatile assumptions that make a convincing case for quantum hardness. In this work, we argue that decoding random quantum stabilizer codes -- a quantum analog of the well-studied LPN problem -- is an excellent candidate. This task occupies a unique middle ground: it is inherently native to quantum computation, yet admits an equivalent formulation with purely classical input and output, as recently shown by Khesin et al. (STOC '26). We prove that the average-case hardness of quantum stabilizer decoding implies the core primitives of classical Cryptomania, including public-key encryption (PKE) and oblivious transfer (OT), as well as one-way functions. Our constructions are moreover practical: our PKE scheme achieves essentially the same efficiency as state-of-the-art LPN-based PKE, and our OT is round-optimal. We also provide substantial evidence that stabilizer decoding does not reduce to LPN, suggesting that the former problem constitutes a genuinely new post-quantum assumption. Our primary technical contributions are twofold. First, we give a reduction from random quantum stabilizer decoding to an average-case problem closely resembling LPN, but which is equipped with additional symplectic algebraic structure. While this structure is essential to the quantum nature of the problem, it raises significant barriers to cryptographic security reductions. Second, we develop a new suit of scrambling techniques for such structured linear spaces, and use them to produce rigorous security proofs for all of our constructions.

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arXiv2025

Design boosters: from constant-time quantum chaos to $\infty$-designs and beyond

We study a counterintuitive property of 'conditioning' on the result of measuring a subsystem of a quantum state: such conditioning can boost design quality, at the cost of increased system size. We work in the setting of deep thermalization from many-body physics: starting from a bipartite state on a global system $(A,B)$ drawn from a $k$-design, we measure system $B$ in the computational basis, keep the outcome and examine the state that remains in system $A$, approximating the overall ensemble (the 'projected ensemble') by a $k'$-design. We ask: how does the design quality change due to this procedure, or how does $k'$ compare to $k$? We give the first rigorous example of unitary dynamics generating a state such that, projection at very early (constant) times can boost design randomness. These dynamics are those of quantum chaos, modeled by the evolution of a Hamiltonian drawn from the Gaussian Unitary Ensemble (GUE). We show that, even though a state generated by such dynamics at constant time only forms a $k=\mathcal{O}(1)$ design, the projected ensemble is Haar-random (or a $k'=\infty$ design) in the thermodynamic limit (i.e. when $N_B=\infty$). This phenomenon persists even with weaker and more physically realistic assumptions; our results can be appropriately applied to non-GUE Hamiltonians that nevertheless show likely chaotic signatures in their eigenbases. Moreover, we show that with no assumption on how the global state was generated, a $k$-design experiences a degradation in design quality to $k' = \lfloor k/2 \rfloor$. This improves upon best prior results on the deep thermalization of designs. Together, our contributions argue for design boosting as a result of chaos and showcase a novel mechanism to generate good designs.

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arXiv2025

Hamiltonian Decoded Quantum Interferometry

We introduce Hamiltonian Decoded Quantum Interferometry (HDQI), a quantum algorithm that utilizes coherent Bell measurements and the symplectic representation of the Pauli group to reduce Gibbs sampling and Hamiltonian optimization to classical decoding. For a signed Pauli Hamiltonian $H$ and any degree-$\ell$ polynomial ${P}$, HDQI prepares a purification of the density matrix $\rho_{P}(H) \propto {P}^2(H)$ by solving a combination of two tasks: decoding $\ell$ errors on a classical code defined by $H$, and preparing a pilot state that encodes the anti-commutation structure of $H$. Choosing $P(x)$ to approximate $\exp(-\beta x/2)$ yields Gibbs states at inverse temperature $\beta$; other choices prepare approximate ground states, microcanonical ensembles, and other spectral filters. For local Hamiltonians, the corresponding decoding problem is that of LDPC codes. Preparing the pilot state is always efficient for commuting Hamiltonians, but highly non-trivial for non-commuting Hamiltonians. Nevertheless, we prove that this state admits an efficient matrix product state representation for Hamiltonians whose anti-commutation graph decomposes into connected components of logarithmic size. We show that HDQI efficiently prepares Gibbs states at arbitrary temperatures for a class of physically motivated commuting Hamiltonians -- including the toric code and Haah's cubic code -- but we also develop a matching efficient classical algorithm for this task. For a non-commuting semiclassical spin glass and commuting stabilizer Hamiltonians with quantum defects, HDQI prepares Gibbs states up to a constant inverse-temperature threshold using polynomial quantum resources and quasi-polynomial classical pre-processing. These results position HDQI as a versatile algorithmic primitive and the first extension of Regev's reduction to non-abelian groups.

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arXiv2025

Quantum advantage from random geometrically-two-local Hamiltonian dynamics

Classical hardness-of-sampling results are largely established for random quantum circuits, whereas analog simulators natively realize time evolutions under geometrically local Hamiltonians. Does a typical such Hamiltonian already yield classically-intractable dynamics? We answer this question in the affirmative for the ensemble of geometrically-2-local Hamiltonians with Gaussian coefficients, evolved for constant time. This naturally leads to a quantum advantage scheme with clear prospects for experimental realization, necessitating only course-grained control. We give strong evidence of hardness for this physically-relevant ensemble. We develop the first worst-to-average-case reduction for approximating output probabilities of (time-independent) geometrically-2-local Hamiltonian evolutions. Our reduction proceeds by nonstandard means: while we also leverage polynomial interpolation, unlike previous works, we reduce directly to an evaluator for the exact distribution over Hamiltonians, from which we are trying to prove that sampling is hard. Previous works instead sampled from various perturbations of the true distribution, introducing additional constraints meant to keep the perturbation, measured in total variation distance, under control. We dispense with this step. Our reduction consists in a robust multivariate polynomial interpolation, reduced to sequential robust univariate interpolations via the symmetries of the Gaussian. We circumvent the fact that random Hamiltonians lack a hiding symmetry, a key property in previous proofs. We also contribute an algorithmic version of Berlekamp-Welch to deal with errored evaluations, solving an open problem from the RCS literature. We expect the machinery we develop to find use in average-case Hamiltonian complexity, filling in a gap in this literature which has thus far focussed on worst-case hardness results.

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arXiv2025

Information-Computation Gaps in Quantum Learning via Low-Degree Likelihood

In a variety of physically relevant settings for learning from quantum data, designing protocols that can computationally efficiently extract information remains largely an art, and there are important cases where we believe this to be impossible, that is, where there is an information-computation gap. While there is a large array of tools in the classical literature for giving evidence for average-case hardness of statistical inference problems, the corresponding tools in the quantum literature are far more limited. One such framework in the classical literature, the low-degree method, makes predictions about hardness of inference problems based on the failure of estimators given by low-degree polynomials. In this work, we extend this framework to the quantum setting. We establish a general connection between state designs and low-degree hardness. We use this to obtain the first information-computation gaps for learning Gibbs states of random, sparse, non-local Hamiltonians. We also use it to prove hardness for learning random shallow quantum circuit states in a challenging model where states can be measured in adaptively chosen bases. To our knowledge, the ability to model adaptivity within the low-degree framework was open even in classical settings. In addition, we also obtain a low-degree hardness result for quantum error mitigation against strategies with single-qubit measurements. We define a new quantum generalization of the planted biclique problem and identify the threshold at which this problem becomes computationally hard for protocols that perform local measurements. Interestingly, the complexity landscape for this problem shifts when going from local measurements to more entangled single-copy measurements. We show average-case hardness for the "standard" variant of Learning Stabilizers with Noise and for agnostically learning product states.

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arXiv2024

The Learning Stabilizers with Noise problem

Random classical codes have good error correcting properties, and yet they are notoriously hard to decode in practice. Despite many decades of extensive study, the fastest known algorithms still run in exponential time. The Learning Parity with Noise (LPN) problem, which can be seen as the task of decoding a random linear code in the presence of noise, has thus emerged as a prominent hardness assumption with numerous applications in both cryptography and learning theory. Is there a natural quantum analog of the LPN problem? In this work, we introduce the Learning Stabilizers with Noise (LSN) problem, the task of decoding a random stabilizer code in the presence of local depolarizing noise. We give both polynomial-time and exponential-time quantum algorithms for solving LSN in various depolarizing noise regimes, ranging from extremely low noise, to low constant noise rates, and even higher noise rates up to a threshold. Next, we provide concrete evidence that LSN is hard. First, we show that LSN includes LPN as a special case, which suggests that it is at least as hard as its classical counterpart. Second, we prove a worst-case to average-case reduction for variants of LSN. We then ask: what is the computational complexity of solving LSN? Because the task features quantum inputs, its complexity cannot be characterized by traditional complexity classes. Instead, we show that the LSN problem lies in a recently introduced (distributional and oracle) unitary synthesis class. Finally, we identify several applications of our LSN assumption, ranging from the construction of quantum bit commitment schemes to the computational limitations of learning from quantum data.

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arXiv2024

Simulating quantum chaos without chaos

Quantum chaos is a quantum many-body phenomenon that is associated with a number of intricate properties, such as level repulsion in energy spectra or distinct scalings of out-of-time ordered correlation functions. In this work, we introduce a novel class of "pseudochaotic" quantum Hamiltonians that fundamentally challenges the conventional understanding of quantum chaos and its relationship to computational complexity. Our ensemble is computationally indistinguishable from the Gaussian unitary ensemble (GUE) of strongly-interacting Hamiltonians, widely considered to be a quintessential model for quantum chaos. Surprisingly, despite this effective indistinguishability, our Hamiltonians lack all conventional signatures of chaos: it exhibits Poissonian level statistics, low operator complexity, and weak scrambling properties. This stark contrast between efficient computational indistinguishability and traditional chaos indicators calls into question fundamental assumptions about the nature of quantum chaos. We, furthermore, give an efficient quantum algorithm to simulate Hamiltonians from our ensemble, even though simulating Hamiltonians from the true GUE is known to require exponential time. Our work establishes fundamental limitations on Hamiltonian learning and testing protocols and derives stronger bounds on entanglement and magic state distillation. These results reveal a surprising separation between computational and information-theoretic perspectives on quantum chaos, opening new avenues for research at the intersection of quantum chaos, computational complexity, and quantum information. Above all, it challenges conventional notions of what it fundamentally means to actually observe complex quantum systems.

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arXiv2024

Noise-induced shallow circuits and absence of barren plateaus

Motivated by realistic hardware considerations of the pre-fault-tolerant era, we comprehensively study the impact of uncorrected noise on quantum circuits. We first show that in the task of estimating observable expectation values any noise truncates most quantum circuits to effectively logarithmic depth. We then prove that quantum circuits under any non-unital noise do not exhibit barren plateaus for cost functions composed of local observables. However, by using the effective shallowness, we also design an efficient classical algorithm to estimate observable expectation values within any constant additive accuracy, with high probability over the choice of the circuit, in any circuit architecture. Taken together, our results establish that, unless we carefully engineer quantum circuits to take advantage of the noise, noisy quantum circuits are unlikely to offer an advantage over shallow ones for algorithms that output observable expectation value estimates, such as many variational quantum machine learning proposals.

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arXiv2023

Learning quantum states and unitaries of bounded gate complexity

While quantum state tomography is notoriously hard, most states hold little interest to practically-minded tomographers. Given that states and unitaries appearing in Nature are of bounded gate complexity, it is natural to ask if efficient learning becomes possible. In this work, we prove that to learn a state generated by a quantum circuit with $G$ two-qubit gates to a small trace distance, a sample complexity scaling linearly in $G$ is necessary and sufficient. We also prove that the optimal query complexity to learn a unitary generated by $G$ gates to a small average-case error scales linearly in $G$. While sample-efficient learning can be achieved, we show that under reasonable cryptographic conjectures, the computational complexity for learning states and unitaries of gate complexity $G$ must scale exponentially in $G$. We illustrate how these results establish fundamental limitations on the expressivity of quantum machine learning models and provide new perspectives on no-free-lunch theorems in unitary learning. Together, our results answer how the complexity of learning quantum states and unitaries relate to the complexity of creating these states and unitaries.

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arXiv2023

Pseudomagic Quantum States

Notions of nonstabilizerness, or "magic", quantify how non-classical quantum states are in a precise sense: states exhibiting low nonstabilizerness preclude quantum advantage. We introduce 'pseudomagic' ensembles of quantum states that, despite low nonstabilizerness, are computationally indistinguishable from those with high nonstabilizerness. Previously, such computational indistinguishability has been studied with respect to entanglement, introducing the concept of pseudoentanglement. However, we demonstrate that pseudomagic neither follows from pseudoentanglement nor implies it. In terms of applications, the study of pseudomagic offers fresh insights into the theory of quantum scrambling: it uncovers states that, even though they originate from non-scrambling unitaries, remain indistinguishable from scrambled states to any physical observer. Additional applications include new lower bounds on state synthesis problems, property testing protocols, and implications for quantum cryptography. Our work is driven by the observation that only quantities measurable by a computationally bounded observer - intrinsically limited by finite-time computational constraints - hold physical significance. Ultimately, our findings suggest that nonstabilizerness is a 'hide-able' characteristic of quantum states: some states are much more magical than is apparent to a computationally bounded observer.

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arXiv2022

Learning quantum processes without input control

We introduce a general statistical learning theory for processes that take as input a classical random variable and output a quantum state. Our setting is motivated by the practical situation in which one desires to learn a quantum process governed by classical parameters that are out of one's control. This framework is applicable, for example, to the study of astronomical phenomena, disordered systems and biological processes not controlled by the observer. We provide an algorithm for learning with high probability in this setting with a finite amount of samples, even if the concept class is infinite. To do this, we review and adapt existing algorithms for shadow tomography and hypothesis selection, and combine their guarantees with the uniform convergence on the data of the loss functions of interest. As a by-product we obtain sufficient conditions for performing shadow tomography of classical-quantum states with a number of copies which depends on the dimension of the quantum register, but not on the dimension of the classical one. We give concrete examples of processes that can be learned in this manner, based on quantum circuits or physically motivated classes, such as systems governed by Hamiltonians with random perturbations or data-dependent phase-shifts.

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arXiv2022

Exponentially tighter bounds on limitations of quantum error mitigation

Quantum error mitigation has been proposed as a means to combat unwanted and unavoidable errors in near-term quantum computing without the heavy resource overheads required by fault tolerant schemes. Recently, error mitigation has been successfully applied to reduce noise in near-term applications. In this work, however, we identify strong limitations to the degree to which quantum noise can be effectively `undone' for larger system sizes. Our framework rigorously captures large classes of error mitigation schemes in use today. By relating error mitigation to a statistical inference problem, we show that even at shallow circuit depths comparable to the current experiments, a superpolynomial number of samples is needed in the worst case to estimate the expectation values of noiseless observables, the principal task of error mitigation. Notably, our construction implies that scrambling due to noise can kick in at exponentially smaller depths than previously thought. They also impact other near-term applications, constraining kernel estimation in quantum machine learning, causing an earlier emergence of noise-induced barren plateaus in variational quantum algorithms and ruling out exponential quantum speed-ups in estimating expectation values in the presence of noise or preparing the ground state of a Hamiltonian.

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arXiv2022

A single $T$-gate makes distribution learning hard

The task of learning a probability distribution from samples is ubiquitous across the natural sciences. The output distributions of local quantum circuits form a particularly interesting class of distributions, of key importance both to quantum advantage proposals and a variety of quantum machine learning algorithms. In this work, we provide an extensive characterization of the learnability of the output distributions of local quantum circuits. Our first result yields insight into the relationship between the efficient learnability and the efficient simulatability of these distributions. Specifically, we prove that the density modelling problem associated with Clifford circuits can be efficiently solved, while for depth $d=n^{\Omega(1)}$ circuits the injection of a single $T$-gate into the circuit renders this problem hard. This result shows that efficient simulatability does not imply efficient learnability. Our second set of results provides insight into the potential and limitations of quantum generative modelling algorithms. We first show that the generative modelling problem associated with depth $d=n^{\Omega(1)}$ local quantum circuits is hard for any learning algorithm, classical or quantum. As a consequence, one cannot use a quantum algorithm to gain a practical advantage for this task. We then show that, for a wide variety of the most practically relevant learning algorithms -- including hybrid-quantum classical algorithms -- even the generative modelling problem associated with depth $d=\omega(\log(n))$ Clifford circuits is hard. This result places limitations on the applicability of near-term hybrid quantum-classical generative modelling algorithms.

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arXiv2022

Multivariate trace estimation in constant quantum depth

There is a folkloric belief that a depth-$\Theta(m)$ quantum circuit is needed to estimate the trace of the product of $m$ density matrices (i.e., a multivariate trace), a subroutine crucial to applications in condensed matter and quantum information science. We prove that this belief is overly conservative by constructing a constant quantum-depth circuit for the task, inspired by the method of Shor error correction. Furthermore, our circuit demands only local gates in a two dimensional circuit -- we show how to implement it in a highly parallelized way on an architecture similar to that of Google's Sycamore processor. With these features, our algorithm brings the central task of multivariate trace estimation closer to the capabilities of near-term quantum processors. We instantiate the latter application with a theorem on estimating nonlinear functions of quantum states with "well-behaved" polynomial approximations.

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arXiv2022

Quantum algorithm for Petz recovery channels and pretty good measurements

The Petz recovery channel plays an important role in quantum information science as an operation that approximately reverses the effect of a quantum channel. The pretty good measurement is a special case of the Petz recovery channel, and it allows for near-optimal state discrimination. A hurdle to the experimental realization of these vaunted theoretical tools is the lack of a systematic and efficient method to implement them. This paper sets out to rectify this lack: using the recently developed tools of quantum singular value transformation and oblivious amplitude amplification, we provide a quantum algorithm to implement the Petz recovery channel when given the ability to perform the channel that one wishes to reverse. Moreover, we prove that, in some sense, our quantum algorithm's usage of the channel implementation cannot be improved by more than a quadratic factor. Our quantum algorithm also provides a procedure to perform pretty good measurements when given multiple copies of the states that one is trying to distinguish.

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arXiv2021

Learnability of the output distributions of local quantum circuits

There is currently a large interest in understanding the potential advantages quantum devices can offer for probabilistic modelling. In this work we investigate, within two different oracle models, the probably approximately correct (PAC) learnability of quantum circuit Born machines, i.e., the output distributions of local quantum circuits. We first show a negative result, namely, that the output distributions of super-logarithmic depth Clifford circuits are not sample-efficiently learnable in the statistical query model, i.e., when given query access to empirical expectation values of bounded functions over the sample space. This immediately implies the hardness, for both quantum and classical algorithms, of learning from statistical queries the output distributions of local quantum circuits using any gate set which includes the Clifford group. As many practical generative modelling algorithms use statistical queries -- including those for training quantum circuit Born machines -- our result is broadly applicable and strongly limits the possibility of a meaningful quantum advantage for learning the output distributions of local quantum circuits. As a positive result, we show that in a more powerful oracle model, namely when directly given access to samples, the output distributions of local Clifford circuits are computationally efficiently PAC learnable by a classical learner. Our results are equally applicable to the problems of learning an algorithm for generating samples from the target distribution (generative modelling) and learning an algorithm for evaluating its probabilities (density modelling). They provide the first rigorous insights into the learnability of output distributions of local quantum circuits from the probabilistic modelling perspective.

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arXiv2021

Fast algorithm for quantum polar decomposition, pretty-good measurements, and the Procrustes problem

The polar decomposition of a matrix is a key element in the quantum linear algebra toolbox. We show that the problem of quantum polar decomposition, recently studied in Lloyd et al. [LBP+20], has a simple and concise implementation via the quantum singular value transform (QSVT). We focus on the applications to pretty-good measurements, a close-to-optimal measurement to distinguish quantum states, and the quantum Procrustes problem, the task of learning an optimal unitary mapping between given `input' and `output' quantum states. By transforming the state-preparation unitaries into a block-encoding, a pre-requisite for QSVT, we develop algorithms for these problems whose gate complexity exhibits a polynomial advantage in the size and condition number of the input compared to alternative approaches for the same problem settings [LBP+20, GLMQW20]. For these applications of the polar decomposition, we also obtain an exponential speedup in precision compared to [LBP+20], as the block-encodings remove the need for the costly density matrix exponentiation step. We contribute a rigorous analysis of the approach of [LBP+20].

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arXiv2021

Private learning implies quantum stability

Learning an unknown $n$-qubit quantum state $ρ$ is a fundamental challenge in quantum computing. Information-theoretically, it is known that tomography requires exponential in $n$ many copies of $ρ$ to estimate it up to trace distance. Motivated by computational learning theory, Aaronson et al. introduced many (weaker) learning models: the PAC model of learning states (Proceedings of Royal Society A'07), shadow tomography (STOC'18) for learning "shadows" of a state, a model that also requires learners to be differentially private (STOC'19) and the online model of learning states (NeurIPS'18). In these models it was shown that an unknown state can be learned "approximately" using linear-in-$n$ many copies of rho. But is there any relationship between these models? In this paper we prove a sequence of (information-theoretic) implications from differentially-private PAC learning, to communication complexity, to online learning and then to quantum stability. Our main result generalizes the recent work of Bun, Livni and Moran (Journal of the ACM'21) who showed that finite Littlestone dimension (of Boolean-valued concept classes) implies PAC learnability in the (approximate) differentially private (DP) setting. We first consider their work in the real-valued setting and further extend their techniques to the setting of learning quantum states. Key to our results is our generic quantum online learner, Robust Standard Optimal Algorithm (RSOA), which is robust to adversarial imprecision. We then show information-theoretic implications between DP learning quantum states in the PAC model, learnability of quantum states in the one-way communication model, online learning of quantum states, quantum stability (which is our conceptual contribution), various combinatorial parameters and give further applications to gentle shadow tomography and noisy quantum state learning.

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