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Louis Golowich

Publications and source records attributed to Louis Golowich.

18 recordsLinked to original sources
arXiv2026

Fault-Tolerant Quantum Computation with Adversarial Errors

We prove a fault-tolerance theorem for quantum computation against adversarial noise. For every quantum circuit on $\bar{N}$ logical qudits of depth $\bar{T}$, we construct a fault-tolerant circuit on $N=\text{poly}(\bar{N})$ physical qudits of depth $\bar{T}\cdot\bar{N}^{o(1)}$, which is robust against an adversary who may arbitrarily choose and corrupt an almost-linear number $N^{1-o(1)}$ of physical qudits at each time step. This robustness significantly improves upon prior fault-tolerance theorems, which assumed corruptions were either local and stochastic, or else only act on a polynomially vanishing fraction of qudits. Our fault-tolerance scheme addresses a key bottleneck towards constructing quantum PCPs via the circuit-to-Hamiltonian mapping of Anshu, Breuckmann, and Nguyen (STOC'24). More fundamentally, our result demonstrates that fault-tolerant quantum computation remains possible under noise models that are global, worst-case, and non-Markovian over the full duration of the computation, directly countering concerns that correlated noise could fundamentally undermine quantum fault tolerance. Our construction is based on a new family of subsystem product codes we develop, which have large dimension and distance along with low-weight parity-checks, and which support transversal non-Clifford gates. We show how to perform single-shot fault-tolerant error correction on these codes using a Floquet-like procedure based on the local testability of classical tensor codes. We then obtain a universal fault-tolerance scheme using repeated code switching in a hypercubic qudit architecture. Finally, we recursively compose our scheme with itself to reduce an initially exponential qudit dimension down to a constant.

quant-ph↗
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

Constant-Overhead Addressable Gates via Single-Shot Code Switching

It is a major challenge to perform addressable and parallel logical operations on constant-rate quantum LDPC (qLDPC) codes. Indeed, the overhead of targeting specific logical qubits represents a crucial bottleneck in many quantum fault-tolerance schemes. We introduce fault-tolerant protocols for performing various addressable as well as parallel logical operations with constant space-time overhead, on a family of constant-rate and polynomial-distance qLDPC codes. Specifically, we construct gadgets for a large class of permutations of logical qubits. We apply these logical permutations to construct gadgets for applying a targeted Hadamard (or $CNOT$) gate on any chosen logical qubit (pair). We also construct gadgets for preparing logical code states, and for applying Hadamard gates on all logical qubits in a codeblock. All of our gadgets use constant quantum space-time overhead along with polynomially bounded classical computation. Prior protocols for such operations required larger overhead, or else relied on codes with certain symmetries that lack known asymptotic constructions. Our codes are given by tensor products of classical codes constructed from lossless expander graphs. Our core technical contribution is a constant-overhead code-switching procedure between 2- and 3-dimensional product codes, which generalizes Bombin's dimensional jump (arXiv:1412.5079). We prove that all of our gadgets exhibit a constant threshold under locally stochastic noise. Along the way, we develop a small-set flip decoder for high-dimensional product codes from lossless expanders. Our techniques yield additional interesting consequences, such as single-shot state preparation of 2-dimensional product codes with constant space-time overhead. We also propose a method for performing parallel non-Clifford gates by extending our techniques to codes supporting transversal application of such gates.

quant-ph↗
arXiv2025

Near-Asymptotically-Good Quantum Codes with Transversal CCZ Gates and Sublinear-Weight Parity-Checks

It is a major challenge to construct good quantum codes supporting fault-tolerant (e.g. transversal) non-Clifford gates with low-weight parity-check measurements. In this paper, we construct the first known quantum codes with linear dimension and distance supporting transversal non-Clifford gates that have sublinear locality (i.e. parity-check weight). Specifically, we construct codes with transversal $CCZ$ gates that have dimension and distance $Θ(N)$ and locality $O(\sqrt{N})$, where $N$ denotes the block length. We furthermore design an efficient decoding algorithm for these codes. The alphabet size of these codes is $q=Θ(\sqrt{N})$, but it can be reduced to a constant (e.g. $q=2$) while incurring a polylogarithmic loss in other parameters. We also show how to decrease the locality to $O(N^{1/3})$, albeit with a larger alphabet size and slightly lower distance. We construct these codes as products of classical codes with appropriate algebraic structure. While our quantum codes are subsystem codes with non-commuting gauge operators, we show they nevertheless permit error correction from noisy syndrome measurements. As byproducts, we prove multiple technical results of independent interest. In particular, our efficient decoder can be viewed as a new multivariate generalization of Prony's method for reconstructing a function from partial access to its Fourier transform. Meanwhile, our distance analysis involves new connections to the classical study of maximally recoverable codes. Our results on product codes also resolve a conjecture of Bravyi & Hastings (2014) in the large-alphabet regime, by providing a new construction of quantum codes with dimension and distance $Θ(N)$ and locality $N^ε$ for arbitrary $ε>0$.

quant-ph↗
arXiv2024

Decoding Quasi-Cyclic Quantum LDPC Codes

Quantum low-density parity-check (qLDPC) codes are an important component in the quest for quantum fault tolerance. Dramatic recent progress on qLDPC codes has led to constructions which are asymptotically good, and which admit linear-time decoders to correct errors affecting a constant fraction of codeword qubits. These constructions, while theoretically explicit, rely on inner codes with strong properties only shown to exist by probabilistic arguments, resulting in lengths that are too large to be practically relevant. In practice, the surface/toric codes, which are the product of two repetition codes, are still often the qLDPC codes of choice. A previous construction based on the lifted product of an expander-based classical LDPC code with a repetition code (Panteleev & Kalachev, 2020) achieved a near-linear distance (of $Ω(N/\log N)$ where $N$ is the number of codeword qubits), and avoids the need for such intractable inner codes. Our main result is an efficient decoding algorithm for these codes that corrects $Θ(N/\log N)$ adversarial errors. En route, we give such an algorithm for the hypergraph product version these codes, which have weaker $Θ(\sqrt{N})$ distance (but are simpler). Our decoding algorithms leverage the fact that the codes we consider are quasi-cyclic, meaning that they respect a cyclic group symmetry. Since the repetition code is not based on expanders, previous approaches to decoding expander-based qLDPC codes, which typically worked by greedily flipping code bits to reduce some potential function, do not apply in our setting. Instead, we reduce our decoding problem (in a black-box manner) to that of decoding classical expander-based LDPC codes under noisy parity-check syndromes. For completeness, we also include a treatment of such classical noisy-syndrome decoding that is sufficient for our application to the quantum setting.

quant-ph↗
arXiv2024

Quantum LDPC Codes of Almost Linear Distance via Homological Products

We present new constructions of quantum codes of linear or close-to-linear distance and dimension with low-weight stabilizers. Only a few constructions of such codes were previously known, and were primarily based on a specific operation from homological algebra, namely the balanced product. In contrast, our constructions are based on a more basic and widely used product, namely the homological product (i.e. the tensor product of chain complexes). Our results help address the natural question: When do homological products preserve good code distance? Our first main result constructs asymptotically good $[[N,Θ(N),Θ(N)]]$ quantum codes with small polynomial stabilizer weight from homological products of codes with a property called product-expansion. This notion was recently introduced and used to bound the distance of balanced product quantum codes; we apply it instead to homological products. For every $ε>0$, our second main result constructs close-to-linear distance $[[N,N^{1-ε},N^{1-ε}]]$ (subsystem) quantum LDPC codes with constant stabilizer weight from iterated homological products of a constant-sized quantum locally testable code. The key insight here is that by using subsystem codes (but still with constant-weight stabilizers), we can circumvent a particular obstruction that limited the distance of many prior product code constructions to at most $\tilde{O}(\sqrt{N})$.

quant-ph↗
arXiv2024

Quantum LDPC Codes with Transversal Non-Clifford Gates via Products of Algebraic Codes

For every integer $r\geq 2$ and every $ε>0$, we construct an explicit infinite family of quantum LDPC codes supporting a transversal $C^{r-1}Z$ gate with length $N$, dimension $K\geq N^{1-ε}$, distance $D\geq N^{1/r}/\operatorname{poly}(\log N)$, and stabilizer weight $w\leq\operatorname{poly}(\log N)$. The previous state of the art construction (in most parameter regimes) was the $r$-dimensional color code, which has only constant dimension $K=O(1)$, and otherwise has the same parameters up to polylogarithmic factors. Our construction provides the first known codes with low-weight stabilizers that are capable of magic state distillation with arbitrarily small yield parameter $γ=\log(N/K)/\log(D)>0$. A classical analogue of transversal $C^{r-1}Z$ gates is given by the multiplication property, which requires component-wise products of classical codewords to belong to another similar code. As a byproduct of our techniques, we also obtain a new construction of classical locally testable codes with such a multiplication property. We construct our codes as products of chain complexes associated to classical LDPC codes, which in turn we obtain by imposing local Reed-Solomon codes on a specific spectral expander that we construct. We prove that our codes support the desired transversal $C^{r-1}Z$ gates by using the multiplication property to combine local circuits based on the topological structure.

quant-ph↗
arXiv2024

Asymptotically Good Quantum Codes with Transversal Non-Clifford Gates

We construct quantum codes that support transversal $CCZ$ gates over qudits of arbitrary prime power dimension $q$ (including $q=2$) such that the code dimension and distance grow linearly in the block length. The only previously known construction with such linear dimension and distance required a growing alphabet size $q$ (Krishna & Tillich, 2019). Our codes imply protocols for magic state distillation with overhead exponent $γ=\log(n/k)/\log(d)\rightarrow 0$ as the block length $n\rightarrow\infty$, where $k$ and $d$ denote the code dimension and distance respectively. It was previously an open question to obtain such a protocol with a contant alphabet size $q$. We construct our codes by combining two modular components, namely, (i) a transformation from classical codes satisfying certain properties to quantum codes supporting transversal $CCZ$ gates, and (ii) a concatenation scheme for reducing the alphabet size of codes supporting transversal $CCZ$ gates. For this scheme we introduce a quantum analogue of multiplication-friendly codes, which provide a way to express multiplication over a field in terms of a subfield. We obtain our asymptotically good construction by instantiating (i) with algebraic-geometric codes, and applying a constant number of iterations of (ii). We also give an alternative construction with nearly asymptotically good parameters ($k,d=n/2^{O(\log^*n)}$) by instantiating (i) with Reed-Solomon codes and then performing a superconstant number of iterations of (ii).

quant-ph↗
arXiv2024

New Explicit Constant-Degree Lossless Expanders

We present a new explicit construction of onesided bipartite lossless expanders of constant degree, with arbitrary constant ratio between the sizes of the two vertex sets. Our construction is simpler to state and analyze than the only prior construction of Capalbo, Reingold, Vadhan, and Wigderson (2002), and achieves improvements in some parameters. We construct our lossless expanders by imposing the structure of a constant-sized lossless expander "gadget" within the neighborhoods of a large bipartite spectral expander; similar constructions were previously used to obtain the weaker notion of unique-neighbor expansion. Our analysis simply consists of elementary counting arguments and an application of the expander mixing lemma.

math.CO↗
arXiv2023

From Grassmannian to Simplicial High-Dimensional Expanders

In this paper, we present a new construction of simplicial complexes of subpolynomial degree with arbitrarily good local spectral expansion. Previously, the only known high-dimensional expanders (HDXs) with arbitrarily good expansion and less than polynomial degree were based on one of two constructions, namely Ramanujan complexes and coset complexes. In contrast, our construction is a Cayley complex over the group $\mathbb{F}_2^k$, with Cayley generating set given by a Grassmannian HDX. Our construction is in part motivated by a coding-theoretic interpretation of Grassmannian HDXs that we present, which provides a formal connection between Grassmannian HDXs, simplicial HDXs, and LDPC codes. We apply this interpretation to prove a general characterization of the 1-homology groups over $\mathbb{F}_2$ of Cayley simplicial complexes over $\mathbb{F}_2^k$. Using this result, we construct simplicial complexes on $N$ vertices with arbitrarily good local expansion for which the dimension of the 1-homology group grows as $Ω(\log^2N)$. No prior constructions in the literature have been shown to achieve as large a 1-homology group.

math.CO↗
arXiv2023

NLTS Hamiltonians and Strongly-Explicit SoS Lower Bounds from Low-Rate Quantum LDPC Codes

Recent constructions of the first asymptotically good quantum LDPC (qLDPC) codes led to two breakthroughs in complexity theory: the NLTS (No Low-Energy Trivial States) theorem (Anshu, Breuckmann, and Nirkhe, STOC'23), and explicit lower bounds against a linear number of levels of the Sum-of-Squares (SoS) hierarchy (Hopkins and Lin, FOCS'22). In this work, we obtain improvements to both of these results using qLDPC codes of low rate: - Whereas Anshu et al. only obtained NLTS Hamiltonians from qLDPC codes of linear dimension, we show the stronger result that qLDPC codes of arbitrarily small positive dimension yield NLTS Hamiltonians. - The SoS lower bounds of Hopkins and Lin are only weakly explicit because they require running Gaussian elimination to find a nontrivial codeword, which takes polynomial time. We resolve this shortcoming by introducing a new method of planting a strongly explicit nontrivial codeword in linear-distance qLDPC codes, which in turn yields strongly explicit SoS lower bounds. Our "planted" qLDPC codes may be of independent interest, as they provide a new way of ensuring a qLDPC code has positive dimension without resorting to parity check counting, and therefore provide more flexibility in the code construction.

quant-ph↗
arXiv2023

Quantum Locally Recoverable Codes

Classical locally recoverable codes, which permit highly efficient recovery from localized errors as well as global recovery from larger errors, provide some of the most useful codes for distributed data storage in practice. In this paper, we initiate the study of quantum locally recoverable codes (qLRCs). In the long term, like their classical counterparts, such qLRCs may be used for large-scale quantum data storage. Our results also have concrete implications for quantum LDPC codes, which are applicable to near-term quantum error-correction. After defining quantum local recoverability, we provide an explicit construction of qLRCs based on the classical LRCs of Tamo and Barg (2014), which we show have (1) a close-to-optimal rate-distance tradeoff (i.e. near the Singleton bound), (2) an efficient decoder, and (3) permit good spatial locality in a physical implementation. Although the analysis is significantly more involved than in the classical case, we obtain close-to-optimal parameters by introducing a "folded" version of our quantum Tamo-Barg (qTB) codes, which we then analyze using a combination of algebraic techniques. We furthermore present and analyze two additional constructions using more basic techniques, namely random qLRCs, and qLRCs from AEL distance amplification. Each of these constructions has some advantages, but neither achieves all 3 properties of our folded qTB codes described above. We complement these constructions with Singleton-like bounds that show our qLRC constructions achieve close-to-optimal parameters. We also apply these results to obtain Singleton-like bounds for qLDPC codes, which to the best of our knowledge are novel. We then show that even the weakest form of a stronger locality property called local correctability, which permits more robust local recovery and is achieved by certain classical codes, is impossible quantumly.

quant-ph↗
arXiv2023

A New Berry-Esseen Theorem for Expander Walks

We prove that the sum of $t$ boolean-valued random variables sampled by a random walk on a regular expander converges in total variation distance to a discrete normal distribution at a rate of $O(λ/t^{1/2-o(1)})$, where $λ$ is the second largest eigenvalue of the random walk matrix in absolute value. To the best of our knowledge, among known Berry-Esseen bounds for Markov chains, our result is the first to show convergence in total variation distance, and is also the first to incorporate a linear dependence on expansion $λ$. In contrast, prior Markov chain Berry-Esseen bounds showed a convergence rate of $O(1/\sqrt{t})$ in weaker metrics such as Kolmogorov distance. Our result also improves upon prior work in the pseudorandomness literature, which showed that the total variation distance is $O(λ)$ when the approximating distribution is taken to be a binomial distribution. We achieve the faster $O(λ/t^{1/2-o(1)})$ convergence rate by generalizing the binomial distribution to discrete normals of arbitrary variance. We specifically construct discrete normals using a random walk on an appropriate 2-state Markov chain. Our bound can therefore be viewed as a regularity lemma that reduces the study of arbitrary expanders to a small class of particularly simple expanders.

math.PR↗
arXiv2022

Approaching the Quantum Singleton Bound with Approximate Error Correction

It is well known that no quantum error correcting code of rate $R$ can correct adversarial errors on more than a $(1-R)/4$ fraction of symbols. But what if we only require our codes to *approximately* recover the message? We construct efficiently-decodable approximate quantum codes against adversarial error rates approaching the quantum Singleton bound of $(1-R)/2$, for any constant rate $R$. Moreover, the size of the alphabet is a constant independent of the message length and the recovery error is exponentially small in the message length. Central to our construction is a notion of quantum list decoding and an implementation involving folded quantum Reed-Solomon codes.

quant-ph↗
arXiv2021

Improved Product-Based High-Dimensional Expanders

High-dimensional expanders generalize the notion of expander graphs to higher-dimensional simplicial complexes. In contrast to expander graphs, only a handful of high-dimensional expander constructions have been proposed, and no elementary combinatorial construction with near-optimal expansion is known. In this paper, we introduce an improved combinatorial high-dimensional expander construction, by modifying a previous construction of Liu, Mohanty, and Yang (ITCS 2020), which is based on a high-dimensional variant of a tensor product. Our construction achieves a spectral gap of $Ω(\frac{1}{k^2})$ for random walks on the $k$-dimensional faces, which is only quadratically worse than the optimal bound of $Θ(\frac{1}{k})$. Previous combinatorial constructions, including that of Liu, Mohanty, and Yang, only achieved a spectral gap that is exponentially small in $k$. We also present reasoning that suggests our construction is optimal among similar product-based constructions.

cs.DM↗
arXiv2017

New Classes of Set-Sequential Trees

A graph is called set-sequential if its vertices can be labeled with distinct nonzero vectors in $\mathbb{F}_2^n$ such that when each edge is labeled with the sum$\pmod{2}$ of its vertices, every nonzero vector in $\mathbb{F}_2^n$ is the label for either a single vertex or a single edge. We resolve certain cases of a conjecture of Balister, Gyori, and Schelp in order to show many new classes of trees to be set-sequential. We show that all caterpillars $T$ of diameter $k$ such that $k \leq 18$ or $|V(T)| \geq 2^{k-1}$ are set-sequential, where $T$ has only odd-degree vertices and $|T| = 2^{n-1}$ for some positive integer $n$. We also present a new method of recursively constructing set-sequential trees.

math.CO↗
arXiv2017

Maximum Size of a Family of Pairwise Graph-Different Permutations

Two permutations of the vertices of a graph $G$ are called $G$-different if there exists an index $i$ such that $i$-th entry of the two permutations form an edge in $G$. We bound or determine the maximum size of a family of pairwise $G$-different permutations for various graphs $G$. We show that for all balanced bipartite graphs $G$ of order $n$ with minimum degree $n/2 - o(n)$, the maximum number of pairwise $G$-different permutations of the vertices of $G$ is $2^{(1-o(1))n}$. We also present examples of bipartite graphs $G$ with maximum degree $O(\log n)$ that have this property. We explore the problem of bounding the maximum size of a family of pairwise graph-different permutations when an unlimited number of disjoint vertices is added to a given graph. We determine this exact value for the graph of 2 disjoint edges, and present some asymptotic bounds relating to this value for graphs consisting of the union of $n/2$ disjoint edges.

math.CO↗