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András Gilyén

Publications and source records attributed to András Gilyén.

At least 19 recordsLinked to original sources
arXiv2026

Tight bounds for hybrid quantum-classical query algorithms

We study hybrid quantum-classical algorithms in the query model. The computation consists of quantum subroutines that make at most $q$ oracle queries; between subroutines, all qubits are measured and discarded. We prove matching upper and lower bounds for several problems in this model: (i) (Unbiased) Phase estimation up to a precision $ε$ requires $Θ(\frac{1}{q ε^2})$ queries; (ii) (Unbiased) Amplitude estimation up to a precision $ε$ requires $Θ(\frac{p(1-p)}{q ε^2})$ queries; (iii) Search among $N$ items requires $Θ(\frac{N}{q})$ queries; (iv) Two level AND-OR tree (AND of $m$ ORs, with $n$ inputs to each OR) requires $Θ(\frac{nm}{q})$ queries. All of these bounds are optimal up to a constant factor, for all $q$ from 1, corresponding to the classical complexity, to $Q(f)$, the unrestricted quantum query complexity of the respective problem. We also derive a hybrid lower bound for distinguishing two distributions specified by a state-preparation unitary; this bound is tight up to a logarithmic factor. Besides specific lower bounds, an important contribution is developing methods for proving lower bounds on hybrid quantum algorithms (which have been very ad-hoc up to now). The first two bounds follow from a common framework: we analyze probability distributions over measurement transcripts and bound a progress measure that captures their distinguishability. The AND-OR bound requires a more delicate argument, combining two progress measures that track the information available to the algorithm classically and in quantum form, respectively.

quant-ph↗
arXiv2026

The complete phase diagram of detailed-balanced Gibbs samplers for the mean-field Heisenberg model

We study the spectral gap of recently discovered families of KMS-detailed-balanced Gibbs-sampler Lindbladians for the mean-field Heisenberg model on $n$ spins. We show that the spectral gaps of these natural families of Lindbladians admit the same phase diagram: the gap is constant above the critical temperature (i.e., in the disordered phase), scales as $1/\sqrt{n}$ at the critical temperature, and as $1/n$ below the critical temperature (i.e., in the ordered phase). Our result builds on and extends the recent work of Basso et al.~(2026) which characterized this phase diagram for the GNS-detailed-balanced Davies generator, but only partially resolved the gap scaling at the critical temperature.

quant-ph↗
arXiv2026

Solving Sparse SDPs in Sublinear Time: A Classical Algorithm Inspired by the Quantum OR Lemma

We give the first sublinear-time classical solvers for sparse semidefinite programs in the bounded-radius regime, without low-rank assumptions or Frobenius norm dependence on the constraint matrices. For constant precision and bounded primal and dual radii, prior quantum algorithms of Brandão et al. (2019) and van Apeldoorn and Gilyén (2019) achieved $\widetilde{O}(\sqrt{n}+\sqrt{m})$ dependence on matrix dimension $n$ and constraint number $m$. Compared with the $\widetilde{O}(mn)$ runtime of existing classical methods, this suggests a quartic quantum speedup when $m \approx n$. Beyond a usual Grover speedup, this separation relies on the Quantum OR lemma, whose sample-reuse mechanism decouples the cost of Gibbs-state preparation from constraint search. We show that this reuse mechanism is classically realizable for sparse SDPs. Our main technical contribution is a classical procedure for simultaneously estimating many expectation values with respect to a sparse Hamiltonian's Gibbs state. This combines randomized Lánczos filtering with an efficient sampling-based estimator. We also introduce a stochastic online-learning framework for SDP solving, substantially improving accuracy-dependence over standard oracle-based MMWU approaches. Let $s$ denote the the input matrix sparsity and $γ:=Rr/\varepsilon$ capture dependence on the primal $(R)$ and dual $(r)$ radii as well as target accuracy $(\varepsilon)$. When $γ^2\leq\min\{m,n/s\}$, our solver runs in time $\widetilde{O}\left(nsγ^{4.5}+msγ^2\right)$. For $γ=O(1)$, this is $\widetilde{O}\left((n+m)s\right)$ and sublinear in the $O(mns)$ input size. Similar to the quantum algorithms, this matches known lower bounds with respect to $m$ and $n$, up to logarithmic factors. This implies that, with respect to dimensions $m$ and $n$, there is no super-quadratic quantum advantage for generic sparse SDP solving.

quant-ph↗
arXiv2026

Efficiently estimating quantum thermal properties from exponentially fewer samples

We present an efficient protocol for estimating $M$ observables to precision $\varepsilon$ from only $\mathcal{O}(\log (M)/\varepsilon^2)$ copies of a Gibbs state $ρ\propto\exp(-βH)$, given access to its Hamiltonian $H$. The protocol uses single-copy, nonadaptive measurements and a total Hamiltonian simulation time of $\widetilde{\mathcal{O}}(βM/\varepsilon^2)$; we show that the sample complexity is optimal in a black-box setting. The key idea of our protocol is a new interpretation of quantum Gibbs samplers as detailed-balanced measurement channels: measurements that preserve the Gibbs state when outcomes are marginalized, and thus enable Gibbs samples to be reused for measuring subsequent observables. Consequently, shadow tomography of thermal states admits a general efficient algorithm when controlled Hamiltonian simulation is available, substantially boosting the data acquisition rate in quantum thermal simulation workflows.

quant-ph↗
arXiv2026

Methods for Reducing Ancilla-Overhead in Block Encodings

Block encodings are a fundamental primitive in quantum algorithms, but can often have large ancilla overhead. In this work, we introduce novel techniques for reducing this overhead in two distinct ways. In Part I, we prove the existence of a "space-time tradeoff" by deriving an algorithm that, for any block encoding, approximately uncomputes all but one of its ancilla (freeing up those ancillae for reuse in later parts of a quantum algorithm). In Part II, we evaluate the minimum number of ancillae required to perform coherent multiplication of block encodings and introduce a "space-accuracy tradeoff". Specfically, we prove that logarithmic ancillae is optimal for exact multiplication of block encodings, but show that (in certain block encoding regimes) approximate multiplication of block encodings can be achieved to high-precision with just one ancilla.

quant-ph↗
arXiv2026

A streamlined quantum algorithm for topological data analysis with exponentially fewer qubits

Topological invariants of a dataset, such as the number of holes that survive from one length scale to another (persistent Betti numbers) can be used to analyze and classify data in machine learning applications. We present an improved quantum algorithm for computing persistent Betti numbers, and provide an end-to-end complexity analysis. Our approach provides large polynomial time improvements, and an exponential space saving, over existing quantum algorithms. Subject to gap dependencies, our algorithm obtains an almost quintic speedup in the number of datapoints over previously known rigorous classical algorithms for computing the persistent Betti numbers to constant additive error - the salient task for applications. However, we also introduce a quantum-inspired classical power method with provable scaling only quadratically worse than the quantum algorithm. This gives a provable classical algorithm with scaling comparable to existing classical heuristics. We discuss whether quantum algorithms can achieve an exponential speedup for tasks of practical interest, as claimed previously. We conclude that there is currently no evidence for this being the case.

quant-ph↗
arXiv2026

Quantum generalizations of Glauber and Metropolis dynamics

Classical Markov Chain Monte Carlo methods have been essential for simulating statistical physical systems and have proven well applicable to other systems with many degrees of freedom. Motivated by the statistical physics origins, Chen, Kastoryano, and Gilyén [CKG23] proposed a continuous-time quantum thermodynamic analogue to Glauber dynamics that is (i) exactly detailed balanced, (ii) efficiently implementable, and (iii) quasi-local for geometrically local systems. Physically, their construction resembles the dissipative dynamics arising from weak system-bath interaction. In this work, we give an efficiently implementable discrete-time counterpart to any continuous-time quantum Gibbs sampler. Our construction preserves the desirable features (i)-(iii) while does not decrease the spectral gap. Also, we give an alternative highly coherent quantum generalization of detailed balanced dynamics that resembles another physically derived master equation, and propose a smooth interpolation between this and earlier constructions. Moreover, we show how to make earlier Metropolis-style Gibbs samplers (which estimate energies both before and after jumps) exactly detailed balanced. We study generic properties of all constructions, including the uniqueness of the fixed point, the (quasi-)locality of the resulting operators. Finally, we prove that the spectral gap of our new highly coherent Gibbs sampler is constant at high temperatures, thereby it mixes fast. We hope that our systematic approach to quantum Glauber and Metropolis dynamics will lead to widespread applications in various domains.

quant-ph↗
arXiv2025

An efficient and exact noncommutative quantum Gibbs sampler

Preparing thermal and ground states is an essential quantum algorithmic task for quantum simulation. In this work, we construct the first efficiently implementable and exactly detailed-balanced Lindbladian for Gibbs states of arbitrary noncommutative Hamiltonians. Our construction can also be regarded as a continuous-time quantum analog of the Metropolis-Hastings algorithm. To prepare the quantum Gibbs state, our algorithm invokes Hamiltonian simulation for a time proportional to the mixing time and the inverse temperature $β$, up to polylogarithmic factors. Moreover, the gate complexity reduces significantly for lattice Hamiltonians as the corresponding Lindblad operators are (quasi-) local (with radius $\simβ$) and only depend on local Hamiltonian patches. Meanwhile, purifying our Lindbladians yields a temperature-dependent family of frustration-free "parent Hamiltonians", prescribing an adiabatic path for the canonical purified Gibbs state (i.e., the Thermal Field Double state). These favorable features suggest that our construction serves as a quantum algorithmic counterpart to classical Markov chain Monte Carlo sampling.

quant-ph↗
arXiv2025

Quantum algorithms: A survey of applications and end-to-end complexities

The anticipated applications of quantum computers span across science and industry, ranging from quantum chemistry and many-body physics to optimization, finance, and machine learning. Proposed quantum solutions in these areas typically combine multiple quantum algorithmic primitives into an overall quantum algorithm, which must then incorporate the methods of quantum error correction and fault tolerance to be implemented correctly on quantum hardware. As such, it can be difficult to assess how much a particular application benefits from quantum computing, as the various approaches are often sensitive to intricate technical details about the underlying primitives and their complexities. Here we present a survey of several potential application areas of quantum algorithms and their underlying algorithmic primitives, carefully considering technical caveats and subtleties. We outline the challenges and opportunities in each area in an "end-to-end" fashion by clearly defining the problem being solved alongside the input-output model, instantiating all "oracles," and spelling out all hidden costs. We also compare quantum solutions against state-of-the-art classical methods and complexity-theoretic limitations to evaluate possible quantum speedups. The survey is written in a modular, wiki-like fashion to facilitate navigation of the content. Each primitive and application area is discussed in a standalone section, with its own bibliography of references and embedded hyperlinks that direct to other relevant sections. This structure mirrors that of complex quantum algorithms that involve several layers of abstraction, and it enables rapid evaluation of how end-to-end complexities are impacted when subroutines are altered.

quant-ph↗
arXiv2025

Quantum state preparation without coherent arithmetic

We introduce a versatile method for preparing a quantum state whose amplitudes are given by some known function. Unlike existing approaches, our method does not require handcrafted reversible arithmetic circuits, or quantum table reads, to encode the function values. Instead, we use a template quantum eigenvalue transformation circuit to convert a low cost block encoding of the sine function into the desired function. Our method uses only 4 ancilla qubits (3 if the approximating polynomial has definite parity), providing order-of-magnitude qubit count reductions compared to state-of-the-art approaches, while using a similar number of gates if the function can be well represented by a polynomial or Fourier approximation. Like black-box methods, the complexity of our approach depends on the 'L2-norm filling-fraction' of the function. We demonstrate the algorithmic utility of our method, including preparing Gaussian and Kaiser window states.

quant-ph↗
arXiv2025

A little bit of self-correction

We investigate the emergence of stable subspaces in the low-temperature quantum thermal dynamics of finite spin chains. Our analysis reveals the existence of effective decoherence-free qudit subspaces, persisting for timescales exponential in $β$. Surprisingly, the appearance of metastable subspaces is not directly related to the entanglement structure of the ground state(s). Rather, they arise from symmetry relations in low-lying excited states. Despite their stability within a 'phase', practical realization of stable qubits is hindered by susceptibility to symmetry-breaking perturbations. This work highlights that there can be non-trivial quantum behavior in the thermal dynamics of noncommuting many body models, and opens the door to more extensive studies of self-correction in such systems.

quant-ph↗
arXiv2025

A distillation-teleportation protocol for fault-tolerant QRAM

We present a protocol for fault-tolerantly implementing the logical quantum random access memory (QRAM) operation, given access to a specialized, noisy QRAM device. For coherently accessing classical memories of size $2^n$, our protocol consumes only $\mathrm{poly}(n)$ fault-tolerant quantum resources (logical gates, logical qubits, quantum error correction cycles, etc.), avoiding the need to perform active error correction on all $Ω(2^n)$ components of the QRAM device. This is the first rigorous conceptual demonstration that a specialized, noisy QRAM device could be useful for implementing a fault-tolerant quantum algorithm. In fact, the fidelity of the device can be as low as $1/\mathrm{poly}(n)$. The protocol queries the noisy QRAM device $\mathrm{poly}(n)$ times to prepare a sequence of $n$-qubit QRAM resource states, which are moved to a general-purpose $\mathrm{poly}(n)$-size processor to be encoded into a QEC code, distilled, and fault-tolerantly teleported into the computation. To aid this protocol, we develop a new gate-efficient streaming version of quantum purity amplification that matches the optimal sample complexity in a wide range of parameters and is therefore of independent interest. The exponential reduction in fault-tolerant quantum resources comes at the expense of an exponential quantity of purely classical complexity: each of the $n$ iterations of the protocol requires adaptively updating the $2^n$-size classical dataset and providing the noisy QRAM device with access to the updated dataset at the next iteration. While our protocol demonstrates that QRAM is more compatible with fault-tolerant quantum computation than previously thought, the need for significant classical computational complexity exposes potentially fundamental limitations to realizing a truly $\mathrm{poly}(n)$-cost fault-tolerant QRAM.

quant-ph↗
arXiv2025

Myths around quantum computation before full fault tolerance: What no-go theorems rule out and what they don't

In this perspective article, we revisit and critically evaluate prevailing viewpoints on the capabilities and limitations of near-term quantum computing and its potential transition toward fully fault-tolerant quantum computing. We examine theoretical no-go results and their implications, addressing misconceptions about the practicality of quantum error mitigation techniques and variational quantum algorithms. By emphasizing the nuances of error scaling, circuit depth, and algorithmic feasibility, we highlight viable near-term applications and synergies between error mitigation and early fault-tolerant architectures. Our discussion explores strategies for addressing current challenges, such as barren plateaus in variational circuits and the integration of quantum error mitigation and quantum error correction techniques. We aim to underscore the importance of continued innovation in hardware and algorithmic design to bridge the gap between theoretical potential and practical utility, paving the way for meaningful quantum advantage in the era of late noisy intermediate scale and early fault-tolerant quantum devices.

quant-ph↗
arXiv2024

A Quantum Speed-Up for Approximating the Top Eigenvectors of a Matrix

Finding a good approximation of the top eigenvector of a given $d\times d$ matrix $A$ is a basic and important computational problem, with many applications. We give two different quantum algorithms that, given query access to the entries of a Hermitian matrix $A$ and assuming a constant eigenvalue gap, output a classical description of a good approximation of the top eigenvector: one algorithm with time complexity $\mathcal{\tilde{O}}(d^{1.75})$ and one with time complexity $d^{1.5+o(1)}$ (the first algorithm has a slightly better dependence on the $\ell_2$-error of the approximating vector than the second, and uses different techniques of independent interest). Both of our quantum algorithms provide a polynomial speed-up over the best-possible classical algorithm, which needs $Ω(d^2)$ queries to entries of $A$, and hence $Ω(d^2)$ time. We extend this to a quantum algorithm that outputs a classical description of the subspace spanned by the top-$q$ eigenvectors in time $qd^{1.5+o(1)}$. We also prove a nearly-optimal lower bound of $\tildeΩ(d^{1.5})$ on the quantum query complexity of approximating the top eigenvector. Our quantum algorithms run a version of the classical power method that is robust to certain benign kinds of errors, where we implement each matrix-vector multiplication with small and well-behaved error on a quantum computer, in different ways for the two algorithms. Our first algorithm estimates the matrix-vector product one entry at a time, using a new "Gaussian phase estimation" procedure. Our second algorithm uses block-encoding techniques to compute the matrix-vector product as a quantum state, from which we obtain a classical description by a new time-efficient unbiased pure-state tomography procedure.

quant-ph↗
arXiv2023

On variants of multivariate quantum signal processing and their characterizations

Quantum signal processing (QSP) is a highly successful algorithmic primitive in quantum computing which leads to conceptually simple and efficient quantum algorithms using the block-encoding framework of quantum linear algebra. Multivariate variants of quantum signal processing (MQSP) could be a valuable tool in extending earlier results via implementing multivariate (matrix) polynomials. However, MQSP remains much less understood than its single-variate version lacking a clear characterization of "achievable" multivariate polynomials. We show that Haah's characterization of general univariate QSP can be extended to homogeneous bivariate (commuting) quantum signal processing. We also show a similar result for an alternative inhomogeneous variant when the degree in one of the variables is at most 1, but construct a counterexample where both variables have degree 2, which in turn refutes an earlier characterization proposed / conjectured by Rossi and Chuang for a related restricted class of MQSP. Finally, we describe homogeneous multivariate (non-commuting) QSP variants that break away from the earlier two-dimensional treatment limited by its reliance on Jordan-like decompositions, and might ultimately lead to the development of novel quantum algorithms.

quant-ph↗
arXiv2023

Quantum Thermal State Preparation

Preparing ground states and thermal states is essential for simulating quantum systems on quantum computers. Despite the hope for practical quantum advantage in quantum simulation, popular state preparation approaches have been challenged. Monte Carlo-style quantum Gibbs samplers have emerged as an alternative, but prior proposals have been unsatisfactory due to technical obstacles rooted in energy-time uncertainty. We introduce simple continuous-time quantum Gibbs samplers that overcome these obstacles by efficiently simulating Nature-inspired quantum master equations (Lindbladians). In addition, we construct the first provably accurate and efficient algorithm for preparing certain purified Gibbs states (called thermal field double states in high-energy physics) of rapidly thermalizing systems; this algorithm also benefits from a quantum walk speedup. Our algorithms' costs have a provable dependence on temperature, accuracy, and the mixing time (or spectral gap) of the relevant Lindbladian. We complete the first rigorous proof of finite-time thermalization for physically derived Lindbladians by developing a general analytic framework for nonasymptotic secular approximation and approximate detailed balance. Given the success of classical Markov chain Monte Carlo (MCMC) algorithms and the ubiquity of thermodynamics, we anticipate that quantum Gibbs sampling will become indispensable in quantum computing.

quant-ph↗
arXiv2023

Limitations of the Macaulay matrix approach for using the HHL algorithm to solve multivariate polynomial systems

Recently Chen and Gao~\cite{ChenGao2017} proposed a new quantum algorithm for Boolean polynomial system solving, motivated by the cryptanalysis of some post-quantum cryptosystems. The key idea of their approach is to apply a Quantum Linear System (QLS) algorithm to a Macaulay linear system over $\mathbb{C}$, which is derived from the Boolean polynomial system. The efficiency of their algorithm depends on the condition number of the Macaulay matrix. In this paper, we give a strong lower bound on the condition number as a function of the Hamming weight of the Boolean solution, and show that in many (if not all) cases a Grover-based exhaustive search algorithm outperforms their algorithm. Then, we improve upon Chen and Gao's algorithm by introducing the Boolean Macaulay linear system over $\mathbb{C}$ by reducing the original Macaulay linear system. This improved algorithm could potentially significantly outperform the brute-force algorithm, when the Hamming weight of the solution is logarithmic in the number of Boolean variables. Furthermore, we provide a simple and more elementary proof of correctness for our improved algorithm using a reduction employing the Valiant-Vazirani affine hashing method, and also extend the result to polynomial systems over $\mathbb{F}_q$ improving on subsequent work by Chen, Gao and Yuan \cite{ChenGao2018}. We also suggest a new approach for extracting the solution of the Boolean polynomial system via a generalization of the quantum coupon collector problem \cite{arunachalam2020QuantumCouponCollector}.

quant-ph↗
arXiv2023

Sampling-based sublinear low-rank matrix arithmetic framework for dequantizing quantum machine learning

We present an algorithmic framework for quantum-inspired classical algorithms on close-to-low-rank matrices, generalizing the series of results started by Tang's breakthrough quantum-inspired algorithm for recommendation systems [STOC'19]. Motivated by quantum linear algebra algorithms and the quantum singular value transformation (SVT) framework of Gilyén, Su, Low, and Wiebe [STOC'19], we develop classical algorithms for SVT that run in time independent of input dimension, under suitable quantum-inspired sampling assumptions. Our results give compelling evidence that in the corresponding QRAM data structure input model, quantum SVT does not yield exponential quantum speedups. Since the quantum SVT framework generalizes essentially all known techniques for quantum linear algebra, our results, combined with sampling lemmas from previous work, suffice to generalize all recent results about dequantizing quantum machine learning algorithms. In particular, our classical SVT framework recovers and often improves the dequantization results on recommendation systems, principal component analysis, supervised clustering, support vector machines, low-rank regression, and semidefinite program solving. We also give additional dequantization results on low-rank Hamiltonian simulation and discriminant analysis. Our improvements come from identifying the key feature of the quantum-inspired input model that is at the core of all prior quantum-inspired results: $\ell^2$-norm sampling can approximate matrix products in time independent of their dimension. We reduce all our main results to this fact, making our exposition concise, self-contained, and intuitive.

cs.DS↗