arXiv subjects

Quynh T. Nguyen

Publications and source records attributed to Quynh T. Nguyen.

18 recordsLinked to original sources

arXiv2026

Classical Adversarial Fault-Tolerance and PCPs

We show how to compile an arbitrary classical circuit into a fault-tolerant circuit, which performs the desired computation even when an almost-linear number of bits are adversarially chosen and corrupted in each timestep. Using a variant of this fault-tolerance scheme that only detects (rather than corrects) corruptions, we give a new construction of probabilistically checkable proofs (PCPs) for NP with polylogarithmic query complexity. This PCP construction from fault-tolerance presents a promising candidate for quantization by the work of Anshu, Breuckmann, and Nguyen (STOC'24), who provided a roadmap for constructing quantum PCPs via fault-tolerance.

cs.CC↗

arXiv2025

Poincaré Duality and Multiplicative Structures on Quantum Codes

Quantum LDPC codes have attracted intense interest due to their advantageous properties for realizing efficient fault-tolerant quantum computing. In particular, sheaf codes represent a novel framework that encompasses all well-known good qLDPC codes with profound underlying mathematics. In this work, we generalize Poincaré duality from manifolds to both classical and quantum codes defined via sheaf theory on $t$-dimensional cell complexes. Viewing important code properties including the encoding rate, code distance, local testability soundness, and efficient decoders as parameters of the underlying (co)chain complexes, we rigorously prove a duality relationship between the $i$-th chain and the $(t-i)$-th cochain of sheaf codes. We further build multiplicative structures such as cup and cap products on sheaved chain complexes, inspired by the standard notions of multiplicative structures and Poincaré duality on manifolds. This immediately leads to an explicit isomorphism between (co)homology groups of sheaf codes via a cap product. As an application, we obtain transversal disjoint logical $\mathrm{C}Z$ gates with $k_{\mathrm{C}Z}=Θ(n)$ on families of good qLDPC and almost-good quantum locally testable codes. Moreover, we provide multiple new methods to construct transversal circuits composed of $\mathrm{C}\mathrm{C}Z$ gates as well as for higher order controlled-$Z$ that are provably logical operations on the code space. We conjecture that they generate nontrivial logical actions, pointing towards fault-tolerant non-Clifford gates on nearly optimal qLDPC sheaf codes. Mathematically, our results are built on establishing the equivalence between sheaf cohomology in the derived-functor sense, Čech cohomology, and the cohomology of sheaf codes, thereby introducing new mathematical tools into quantum coding theory.

quant-ph↗

arXiv2025

On the complexity of unique quantum witnesses and quantum approximate counting

We study the long-standing open question on the power of unique witnesses in quantum protocols, which asks if $\textsf{UniqueQMA}$, a variant of $\textsf{QMA}$ whose accepting witness space is 1-dimensional, contains $\mathsf{QMA}$ under quantum reductions. This work rules out any black-box reduction from $\mathsf{QMA}$ to $\mathsf{UniqueQMA}$ by showing a quantum oracle separation between $\mathsf{BQP}^\mathsf{UniqueQMA}$ and $\mathsf{QMA}$. This provides a contrast to the classical case, where the Valiant-Vazirani theorem shows a black-box randomized reduction from $\mathsf{UniqueNP}$ to $\mathsf{NP}$, and suggests the need for studying the structure of the ground space of local Hamiltonians in distilling a potential unique witness. Via similar techniques, we show, relative to a quantum oracle, that $\mathsf{QMA}^\mathsf{QMA}$ cannot decide quantum approximate counting, ruling out a quantum analogue of Stockmeyer's algorithm in the black-box setting. We then ask a natural question; what structural properties of the local Hamiltonian problem can we exploit? We introduce a physically motivated candidate by showing that the ground energy of local Hamiltonians that satisfy a computational variant of the eigenstate thermalization hypothesis (ETH) can be estimated through a $\mathsf{UniqueQMA}$ protocol. Our protocol can be viewed as a quantum expander test in a low energy subspace of the Hamiltonian and verifies a unique entangled state across two copies of the subspace. This allows us to conclude that if $\mathsf{UniqueQMA}$ is not equivalent to $\mathsf{QMA}$, then $\mathsf{QMA}$-hard Hamiltonians must violate ETH under adversarial perturbations. This also serves as evidence that chaotic local Hamiltonians, such as the SYK model may be computationally simpler than general local Hamiltonians.

quant-ph↗

arXiv2025

Quantum fault tolerance with constant-space and logarithmic-time overheads

In a model of fault-tolerant quantum computation with quick and noiseless polyloglog-time auxiliary classical computation, we construct a fault tolerance protocol with constant-space and $\widetilde{O}(\log N)$-time overhead, where $\widetilde{O}(\cdot)$ hides sub-polylog factors. Our construction utilizes constant-rate quantum locally testable codes (qLTC), new fault-tolerant gadgets on qLTCs and qLDPC codes, and a new analysis framework. In particular, 1) we develop a new simple and self-contained construction of magic state distillation for qubits using qudit quantum Reed-Solomon codes with $(\log \frac{1}{\varepsilon})^γ$ spacetime overhead, where $γ\rightarrow 0$. 2) We prove that the recent family of almost-good qLTCs of Dinur-Lin-Vidick admit parallel single-shot decoders against adversarial errors of weight scaling with the code distance. 3) We develop logical state preparation and logical gate procedures with $\widetilde{O}(1)$-spacetime overhead on qLTCs. 4) To combine these ingredients, we introduce a new framework of fault tolerance analysis called the weight enumerator formalism. The framework permits easy formal composition of fault-tolerant gadgets, so we expect it to be of independent interest. Our work gives the lowest spacetime overhead to date, which, for the first time, matches that of classical fault tolerance up to sub-polylog factors. We conjecture this is optimal up to sub-polylog factors.

quant-ph↗

arXiv2025

Composable Quantum Fault-Tolerance

Proving threshold theorems for fault-tolerant quantum computation is a burdensome endeavor with many moving parts that come together in relatively formulaic but lengthy ways. It is difficult and rare to combine elements from multiple papers into a single formal threshold proof, due to the use of different measures of fault-tolerance. In this work, we introduce composable fault-tolerance, a framework that decouples the probabilistic analysis of the noise distribution from the combinatorial analysis of circuit correctness, and enables threshold proofs to compose independently analyzed gadgets easily and rigorously. Within this framework, we provide a library of standard and commonly used gadgets such as memory and logic implemented by constant-depth circuits for quantum low-density parity check codes and distillation. As sample applications, we explicitly write down a threshold proof for computation with surface code and re-derive the constant space-overhead fault-tolerant scheme of Gottesman using gadgets from this library. We expect that future fault-tolerance proofs may focus on the analysis of novel techniques while leaving the standard components to the composable fault-tolerance framework, with the formal proof following the intuitive ``napkin math'' exactly.

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

Learning quantum Gibbs states locally and efficiently

Learning the Hamiltonian underlying a quantum many-body system in thermal equilibrium is a fundamental task in quantum learning theory and experimental sciences. To learn the Gibbs state of local Hamiltonians at any inverse temperature $β$, the state-of-the-art provable algorithms fall short of the optimal sample and computational complexity, in sharp contrast with the locality and simplicity in the classical cases. In this work, we present a learning algorithm that learns each local term of a $n$-qubit $D$-dimensional Hamiltonian to an additive error $ε$ with sample complexity $\tilde{O}\left(\frac{e^{\mathrm{poly}(β)}}{β^2ε^2}\right)\log(n)$. The protocol uses parallelizable local quantum measurements that act within bounded regions of the lattice and near-linear-time classical post-processing. Thus, our complexity is near optimal with respect to $n,ε$ and is polynomially tight with respect to $β$. We also give a learning algorithm for Hamiltonians with bounded interaction degree with sample and time complexities of similar scaling on $n$ but worse on $β, ε$. At the heart of our algorithm is the interplay between locality, the Kubo-Martin-Schwinger condition, and the operator Fourier transform at arbitrary temperatures.

quant-ph↗

arXiv2024

Good binary quantum codes with transversal CCZ gate

We give an asymptotically good family of quantum CSS codes on qubits with a transversal CCZ gate, meaning that the parallel logical CCZ on all logical qubits is performed by parallel physical CCZs on (a subset of) physical qubits. The construction is based on the observation that any classical code satisfying a multiplication property can be used to construct a quantum CSS code with transversal (qudit) CCZ. To obtain a constant-rate and linear-distance family, we then instantiate this construction with a classical good family of algebraic-geometry codes on a non-binary, but constant-sized, alphabet. Finally, we use a technique from the arithmetic secret sharing literature to reduce the alphabet to binary. As a corollary, the constructed code family provides a magic state distillation scheme with constant space overhead.

quant-ph↗

arXiv2024

Circuit-to-Hamiltonian from tensor networks and fault tolerance

We define a map from an arbitrary quantum circuit to a local Hamiltonian whose ground state encodes the quantum computation. All previous maps relied on the Feynman-Kitaev construction, which introduces an ancillary `clock register' to track the computational steps. Our construction, on the other hand, relies on injective tensor networks with associated parent Hamiltonians, avoiding the introduction of a clock register. This comes at the cost of the ground state containing only a noisy version of the quantum computation, with independent stochastic noise. We can remedy this - making our construction robust - by using quantum fault tolerance. In addition to the stochastic noise, we show that any state with energy density exponentially small in the circuit depth encodes a noisy version of the quantum computation with adversarial noise. We also show that any `combinatorial state' with energy density polynomially small in depth encodes the quantum computation with adversarial noise. This serves as evidence that any state with energy density polynomially small in depth has a similar property. As applications, we give a new proof of the QMA-completeness of the local Hamiltonian problem (with logarithmic locality) and show that contracting injective tensor networks to additive error is BQP-hard. We also discuss the implication of our construction to the quantum PCP conjecture, combining with an observation that QMA verification can be done in logarithmic depth.

quant-ph↗

arXiv2024

Theory for Equivariant Quantum Neural Networks

Quantum neural network architectures that have little-to-no inductive biases are known to face trainability and generalization issues. Inspired by a similar problem, recent breakthroughs in machine learning address this challenge by creating models encoding the symmetries of the learning task. This is materialized through the usage of equivariant neural networks whose action commutes with that of the symmetry. In this work, we import these ideas to the quantum realm by presenting a comprehensive theoretical framework to design equivariant quantum neural networks (EQNN) for essentially any relevant symmetry group. We develop multiple methods to construct equivariant layers for EQNNs and analyze their advantages and drawbacks. Our methods can find unitary or general equivariant quantum channels efficiently even when the symmetry group is exponentially large or continuous. As a special implementation, we show how standard quantum convolutional neural networks (QCNN) can be generalized to group-equivariant QCNNs where both the convolution and pooling layers are equivariant to the symmetry group. We then numerically demonstrate the effectiveness of a SU(2)-equivariant QCNN over symmetry-agnostic QCNN on a classification task of phases of matter in the bond-alternating Heisenberg model. Our framework can be readily applied to virtually all areas of quantum machine learning. Lastly, we discuss about how symmetry-informed models such as EQNNs provide hopes to alleviate central challenges such as barren plateaus, poor local minima, and sample complexity.

quant-ph↗

arXiv2024

Theoretical Guarantees for Permutation-Equivariant Quantum Neural Networks

Despite the great promise of quantum machine learning models, there are several challenges one must overcome before unlocking their full potential. For instance, models based on quantum neural networks (QNNs) can suffer from excessive local minima and barren plateaus in their training landscapes. Recently, the nascent field of geometric quantum machine learning (GQML) has emerged as a potential solution to some of those issues. The key insight of GQML is that one should design architectures, such as equivariant QNNs, encoding the symmetries of the problem at hand. Here, we focus on problems with permutation symmetry (i.e., the group of symmetry $S_n$), and show how to build $S_n$-equivariant QNNs. We provide an analytical study of their performance, proving that they do not suffer from barren plateaus, quickly reach overparametrization, and generalize well from small amounts of data. To verify our results, we perform numerical simulations for a graph state classification task. Our work provides the first theoretical guarantees for equivariant QNNs, thus indicating the extreme power and potential of GQML.

quant-ph↗

arXiv2023

The mixed Schur transform: efficient quantum circuit and applications

The Schur transform, which block-diagonalizes the tensor representation $U^{\otimes n}$ of the unitary group $\mathbf{U}_d$ on $n$ qudits, is an important primitive in quantum information and theoretical physics. We give a generalization of its quantum circuit implementation due to Bacon, Chuang, and Harrow (SODA 2007) to the case of mixed tensor $U^{\otimes n} \otimes \bar{U}^{\otimes m}$, where $\bar{U}$ is the dual representation. This representation is the symmetry of unitary-equivariant channels, which find various applications in quantum majority vote, multiport-based teleportation, asymmetric state cloning, black-box unitary transformations, etc. The "mixed" Schur transform contains several natural extensions of the representation theory used in the Schur transform, in which the main ingredient is a duality between the mixed tensor representations and the walled Brauer algebra. Another element is an efficient implementation of a "dual" Clebsch-Gordan transform for $\bar{U}$. The overall circuit has complexity $\widetilde{O} ((n+m)d^4)$. Finally, we show how the mixed Schur transform enables efficient implementation of unitary-equivariant channels in various settings and discuss other potential applications, including an extension of permutational quantum computing that includes partial transposes.

quant-ph↗

arXiv2023

Improving the speed of variational quantum algorithms for quantum error correction

We consider the problem of devising a suitable Quantum Error Correction (QEC) procedures for a generic quantum noise acting on a quantum circuit. In general, there is no analytic universal procedure to obtain the encoding and correction unitary gates, and the problem is even harder if the noise is unknown and has to be reconstructed. The existing procedures rely on Variational Quantum Algorithms (VQAs) and are very difficult to train since the size of the gradient of the cost function decays exponentially with the number of qubits. We address this problem using a cost function based on the Quantum Wasserstein distance of order 1 ($QW_1$). At variance with other quantum distances typically adopted in quantum information processing, $QW_1$ lacks the unitary invariance property which makes it a suitable tool to avoid to get trapped in local minima. Focusing on a simple noise model for which an exact QEC solution is known and can be used as a theoretical benchmark, we run a series of numerical tests that show how, guiding the VQA search through the $QW_1$, can indeed significantly increase both the probability of a successful training and the fidelity of the recovered state, with respect to the results one obtains when using conventional approaches.

quant-ph↗

arXiv2023

Fluctuation-driven, topology-stabilized order in a correlated nodal semimetal

The interplay between strong electron correlation and band topology is at the forefront of condensed matter research. As a direct consequence of correlation, magnetism enriches topological phases and also has promising functional applications. However, the influence of topology on magnetism remains unclear, and the main research effort has been limited to ground state magnetic orders. Here we report a novel order above the magnetic transition temperature in magnetic Weyl semimetal (WSM) CeAlGe. Such order shows a number of anomalies in electrical and thermal transport, and neutron scattering measurements. We attribute this order to the coupling of Weyl fermions and magnetic fluctuations originating from a three-dimensional Seiberg-Witten monopole, which qualitatively agrees well with the observations. Our work reveals a prominent role topology may play in tailoring electron correlation beyond ground state ordering, and offers a new avenue to investigate emergent electronic properties in magnetic topological materials.

cond-mat.str-el↗

arXiv2023

Representation Theory for Geometric Quantum Machine Learning

Recent advances in classical machine learning have shown that creating models with inductive biases encoding the symmetries of a problem can greatly improve performance. Importation of these ideas, combined with an existing rich body of work at the nexus of quantum theory and symmetry, has given rise to the field of Geometric Quantum Machine Learning (GQML). Following the success of its classical counterpart, it is reasonable to expect that GQML will play a crucial role in developing problem-specific and quantum-aware models capable of achieving a computational advantage. Despite the simplicity of the main idea of GQML -- create architectures respecting the symmetries of the data -- its practical implementation requires a significant amount of knowledge of group representation theory. We present an introduction to representation theory tools from the optics of quantum learning, driven by key examples involving discrete and continuous groups. These examples are sewn together by an exposition outlining the formal capture of GQML symmetries via "label invariance under the action of a group representation", a brief (but rigorous) tour through finite and compact Lie group representation theory, a reexamination of ubiquitous tools like Haar integration and twirling, and an overview of some successful strategies for detecting symmetries.

quant-ph↗

arXiv2022

Block-encoding dense and full-rank kernels using hierarchical matrices: applications in quantum numerical linear algebra

Many quantum algorithms for numerical linear algebra assume black-box access to a block-encoding of the matrix of interest, which is a strong assumption when the matrix is not sparse. Kernel matrices, which arise from discretizing a kernel function $k(x,x')$, have a variety of applications in mathematics and engineering. They are generally dense and full-rank. Classically, the celebrated fast multipole method performs matrix multiplication on kernel matrices of dimension $N$ in time almost linear in $N$ by using the linear algebraic framework of hierarchical matrices. In light of this success, we propose a block-encoding scheme of the hierarchical matrix structure on a quantum computer. When applied to many physical kernel matrices, our method can improve the runtime of solving quantum linear systems of dimension $N$ to $O(κ\operatorname{polylog}(\frac{N}{\varepsilon}))$, where $κ$ and $\varepsilon$ are the condition number and error bound of the matrix operation. This runtime is near-optimal and, in terms of $N$, exponentially improves over prior quantum linear systems algorithms in the case of dense and full-rank kernel matrices. We discuss possible applications of our methodology in solving integral equations and accelerating computations in N-body problems.

quant-ph↗

arXiv2022

Quantum algorithms for group convolution, cross-correlation, and equivariant transformations

Group convolutions and cross-correlations, which are equivariant to the actions of group elements, are commonly used in mathematics to analyze or take advantage of symmetries inherent in a given problem setting. Here, we provide efficient quantum algorithms for performing linear group convolutions and cross-correlations on data stored as quantum states. Runtimes for our algorithms are logarithmic in the dimension of the group thus offering an exponential speedup compared to classical algorithms when input data is provided as a quantum state and linear operations are well conditioned. Motivated by the rich literature on quantum algorithms for solving algebraic problems, our theoretical framework opens a path for quantizing many algorithms in machine learning and numerical methods that employ group operations.

quant-ph↗

arXiv2021

Direct prediction of phonon density of states with Euclidean neural networks

Machine learning has demonstrated great power in materials design, discovery, and property prediction. However, despite the success of machine learning in predicting discrete properties, challenges remain for continuous property prediction. The challenge is aggravated in crystalline solids due to crystallographic symmetry considerations and data scarcity. Here we demonstrate the direct prediction of phonon density of states using only atomic species and positions as input. We apply Euclidean neural networks, which by construction are equivariant to 3D rotations, translations, and inversion and thereby capture full crystal symmetry, and achieve high-quality prediction using a small training set of $\sim 10^{3}$ examples with over 64 atom types. Our predictive model reproduces key features of experimental data and even generalizes to materials with unseen elements,and is naturally suited to efficiently predict alloy systems without additional computational cost. We demonstrate the potential of our network by predicting a broad number of high phononic specific heat capacity materials. Our work indicates an efficient approach to explore materials' phonon structure, and can further enable rapid screening for high-performance thermal storage materials and phonon-mediated superconductors.

physics.comp-ph↗