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A fault-tolerant approach to reliable quantum memory is essential for scalable quantum computing, as physical qubits are susceptible to noise. Quantum error correction (QEC) must be continuously performed to prolong the memory lifetime. In QEC, error syndromes are generated rapidly, often within the execution time of a few quantum gates, requiring decoders to process this error data with equal speed. A typical QEC cycle involves multiple rounds of syndrome measurements, causing potential error locations to scale rapidly with the code size and the number of measurement rounds. However, no such decoders currently exist for general quantum low-density parity-check codes. In this paper, we propose a fault-tolerant belief propagation (FTBP) decoder that utilizes a space-time Tanner graph across multiple rounds of syndrome extraction with mixed-alphabet error variables. To enhance FTBP, we introduce a technique of probabilistic error consolidation to mitigate degeneracy effects and short cycles. Additionally, we propose an adaptive sliding window procedure that captures long error events across window boundaries and adjusts the decoding in real time. Our simulations demonstrate high error thresholds of 0.4%-0.87% and strong error-floor performance for topological code families, including rotated toric, toric color, and twisted XZZX toric codes., Comment: 18 pages, 22 figures

We present a theoretical investigation into the dynamics of a viscous gravity current subjected to spatially-finite lubrication (i.e., a `slippery patch'). The work is motivated by grounded ice sheets flowing across patches of basal meltwater which reduce the ice-bed frictional coupling, causing perturbations enhancing ice motion, with implications for increased ice flux into the ocean and sea level rise. The flow is characterized by transitions between shear- and extension-dominated dynamics, which necessitates boundary-layer solutions at the transition points. We develop a depth-integrated analytical model of Newtonian flow which concisely reveals fundamental relationships between ice sheet geometry (thickness, surface slope, and slippery patch length) and the magnitude and spatial extent of resulting horizontal deviatoric stresses. This reduced-order analytical model shows good quantitative agreement with numerical simulations using 2-D Newtonian Stokes equations, which are further extended to the case of a non-Newtonian flow. From the reduced-order model, we rationalize that the slippery patch-induced stress perturbations are exponentially-decaying functions of distance upstream away from the patch onset. We also show that the amplitude of the perturbation scales linearly with the surface slope and patch length while the decay lengthscale scales linearly with ice thickness. These fundamental relationships have implications for the response of the Greenland Ice Sheet to the inland expansion of basal meltwater presence over the coming warming decades.

Deep learning frameworks have become powerful tools for approaching scientific problems such as turbulent flow, which has wide-ranging applications. In practice, however, existing scientific machine learning approaches have difficulty fitting complex, multi-scale dynamical systems to very high precision, as required in scientific contexts. We propose using the novel multistage neural network approach with a spectrum-informed initialization to learn the residue from the previous stage, utilizing the spectral biases associated with neural networks to capture high frequency features in the residue, and successfully tackle the spectral bias of neural networks. This approach allows the neural network to fit target functions to double floating-point machine precision $O(10^{-16})$., Comment: 8 pages, 3 figures, ICML 2024 workshop (AI for Science: Scaling in AI for Scientific Discovery)

Given that quantum error correction processes are unreliable, an efficient error syndrome extraction circuit should use fewer ancillary qubits, quantum gates, and measurements, while maintaining low circuit depth, to minimizing the circuit area, roughly defined as the product of circuit depth and the number of physical qubits. We propose to design parallel flagged syndrome extraction with shared flag qubits for quantum stabilizer codes. Versatile parallelization techniques are employed to minimize the required circuit area, thereby improving the error threshold and overall performance. Specifically, all the measurement outcomes in multiple rounds of syndrome extraction are integrated into a lookup table decoder, allowing us to parallelize multiple stabilizer measurements with shared flag qubits. We present flag-sharing and fully parallel schemes for the [[17,1,5]] and [[19,1,5]] Calderbank-Shor-Steane (CSS) codes. This methodology extends to the [[5,1,3]] non-CSS code, achieving the minimum known circuit area. Numerical simulations have demonstrated improved pseudothresholds for these codes by up to an order of magnitude compared to previous schemes in the literature., Comment: 19 pages, 22 figures

Determining the energy gap in a quantum many-body system is critical to understanding its behavior and is important in quantum chemistry and condensed matter physics. The challenge of determining the energy gap requires identifying both the excited and ground states of a system. In this work, we consider preparing the prior distribution and circuits for the eigenspectrum of time-independent Hamiltonians, which can benefit both classical and quantum algorithms for solving eigenvalue problems. The proposed algorithm unfolds in three strategic steps: Hamiltonian transformation, parameter representation, and classical clustering. These steps are underpinned by two key insights: the use of quantum circuits to approximate the ground state of transformed Hamiltonians and the analysis of parameter representation to distinguish between eigenvectors. The algorithm is showcased through applications to the 1D Heisenberg system and the LiH molecular system, highlighting its potential for both near-term quantum devices and fault-tolerant quantum devices. The paper also explores the scalability of the method and its performance across various settings, setting the stage for more resource-efficient quantum computations that are both accurate and fast. The findings presented here mark a new insight into hybrid algorithms, offering a pathway to overcoming current computational challenges.

Gaussian quantum information processing with continuous-variable (CV) quantum information carriers holds significant promise for applications in quantum communication and quantum internet. However, applying Gaussian state distillation and quantum error correction (QEC) faces limitations imposed by no-go results concerning local Gaussian unitary operations and classical communications. This paper introduces a Gaussian QEC protocol that relies solely on local Gaussian resources. A pivotal component of our approach is CV gate teleportation using entangled Gaussian states, which facilitates the implementation of the partial transpose operation on a quantum channel. Consequently, we can efficiently construct a two-mode noise-polarized channel from two noisy Gaussian channels. Furthermore, this QEC protocol naturally extends to a nonlocal Gaussian state distillation protocol., Comment: Lemma 3 and Lemma 4 are incorrect

We discuss the emerging advances and opportunities at the intersection of machine learning (ML) and climate physics, highlighting the use of ML techniques, including supervised, unsupervised, and equation discovery, to accelerate climate knowledge discoveries and simulations. We delineate two distinct yet complementary aspects: (1) ML for climate physics and (2) ML for climate simulations. While physics-free ML-based models, such as ML-based weather forecasting, have demonstrated success when data is abundant and stationary, the physics knowledge and interpretability of ML models become crucial in the small-data/non-stationary regime to ensure generalizability. Given the absence of observations, the long-term future climate falls into the small-data regime. Therefore, ML for climate physics holds a critical role in addressing the challenges of ML for climate simulations. We emphasize the need for collaboration among climate physics, ML theory, and numerical analysis to achieve reliable ML-based models for climate applications.

Molecular ground state simulation is a promising application of quantum computing. Nevertheless, this question has been shown as a QMA-complete problem, indicating that its complexity increases with the size of the molecule. To address this challenge, we focus on reducing the computation cost of molecular ground state simulation. In this study, we present a mathematical framework named Subspace Restriction Scheme (SRS), based on the Qubit Efficiency Encoding (QEE) method. Within this framework, we introduce and test a novel subspace, Multiplicity Hund Subspace (MH), which is generated by Hund's rule and selected based on molecular multiplicity. Our testing data and mathematical proofs demonstrate that MH subspace significantly reduces qubit usage compared to the classic Multiplicity Subspace and Particle Conservation Subspace (PC). For example, the ground state simulation of 10 electrons in 50 orbitals only requires 35 qubits, compared to the 44 qubits by PC and 100 qubits by the Jordan-Wigner transform. Furthermore, leveraging the reduced computation cost, we examine SRS for seldom tested larger molecules such as $\mathrm{CH}_{4}$ and $\mathrm{H_{2}O_{2}}$ without frozen any orbital and find that the MH subspace has less ground-state energy difference in the tested 15 molecules compared to PC subspace.

This article explores the application of coding techniques for fault-tolerant quantum computation and extends their usage to fault-tolerant quantum communication. We review repeater-based quantum networks, emphasizing the roles of coding theory and fault-tolerant quantum operations, particularly in the context of quantum teleportation. We highlight that fault-tolerant implementation of the Bell measurement enables reliable quantum communication without requiring a universal set of quantum gates. Finally, we discuss various quantum code candidates for achieving higher transmission rates., Comment: 7 pages, 5 figures

In this paper, we explore the application of semidefinite programming to the realm of quantum codes, specifically focusing on codeword stabilized (CWS) codes with entanglement assistance. Notably, we utilize the isotropic subgroup of the CWS group and the set of word operators of a CWS-type quantum code to derive an upper bound on the minimum distance. Furthermore, this characterization can be incorporated into the associated distance enumerators, enabling us to construct semidefinite constraints that lead to SDP bounds on the minimum distance or size of CWS-type quantum codes. We illustrate several instances where SDP bounds outperform LP bounds, and there are even cases where LP fails to yield meaningful results, while SDP consistently provides tighter and relevant bounds. Finally, we also provide interpretations of the Shor-Laflamme weight enumerators and shadow enumerators for codeword stabilized codes, enhancing our understanding of quantum codes., Comment: 15 pages, 2 tables

Quantum stabilizer codes often face the challenge of syndrome errors due to error-prone measurements. To address this issue, multiple rounds of syndrome extraction are typically employed to obtain reliable error syndromes. In this paper, we consider phenomenological decoding problems, where data qubit errors may occur between two syndrome extractions, and each syndrome measurement can be faulty. To handle these diverse error sources, we define a generalized check matrix over mixed quaternary and binary alphabets to characterize their error syndromes. This generalized check matrix leads to the creation of a Tanner graph comprising quaternary and binary variable nodes, which facilitates the development of belief propagation (BP) decoding algorithms to tackle phenomenological errors. Importantly, our BP decoders are applicable to general sparse quantum codes. Through simulations of quantum memory protected by rotated toric codes, we demonstrates an error threshold of 3.3% in the phenomenological noise model. Additionally, we propose a method to construct effective redundant stabilizer checks for single-shot error correction. Simulations show that BP decoding performs exceptionally well, even when the syndrome error rate greatly exceeds the data error rate., Comment: 14 pages, 9 figures, 1 table

Liquid-water oceans likely underlie the ice shells of Europa and Enceladus, but ocean properties are challenging to measure due to the overlying ice. Here, we consider gravity-driven flow of the ice shells of icy satellites and relate this to ocean freeze and melt rates. We employ a first-principles approach applicable to conductive ice shells in a Cartesian geometry. We derive a scaling law under which ocean freeze/melt rates can be estimated from shell-thickness measurements. Under a steady-state assumption, ocean freeze/melt rates can be inferred from measurements of ice thickness, given a basal viscosity. Depending on a characteristic thickness scale and basal viscosity, characteristic freeze/melt rates range from around O(10$^{-1}$) to O(10$^{-5}$) mm/year. Our scaling is validated with ice-penetrating radar measurements of ice thickness and modelled snow accumulation for Roosevelt Island, Antarctica. Our model, coupled with observations of shell thickness, could help estimate the magnitudes of ocean freeze/melt rates on icy satellites., Comment: 9 pages, 4 figures. Accepted for publication in MNRAS

Based on a presentation of $\mathcal{C}_n$ and the help of [GAP], we construct the character table of the Clifford group $\mathcal{C}_n$ for $n=1,2,3$. As an application, we can efficiently decompose the (higher power of) tensor product of the matrix representation in those cases. Our results recover some known results in [HWW, WF] and reveal some new phenomena. We prove that when $n \geq 3$, (1) the trivial character is the only linear character for $\mathcal{C}_n$ and hence $\mathcal{C}_n$ equals to its commutator subgroup, (2) the $n$-qubit Pauli group $\mathcal{P}_n$ is the only proper non-trivial normal subgroup of $\mathcal{C}_n$, (3) the matrix representation $\mathcal{M}_{2^n}$ is a faithful representation for $\mathcal{C}_n$. As a byproduct, we give a presentation of the finite symplectic group $Sp(2n,2)$ in terms of generators and relations., Comment: 15 pages; comments and suggestions are welcome. Two conjectures in last version are proved and one is disproved; references updated

One of the primary objectives in the field of quantum state learning is to develop algorithms that are time-efficient for learning states generated from quantum circuits. Earlier investigations have demonstrated time-efficient algorithms for states generated from Clifford circuits with at most $\log(n)$ non-Clifford gates. However, these algorithms necessitate multi-copy measurements, posing implementation challenges in the near term due to the requisite quantum memory. On the contrary, using solely single-qubit measurements in the computational basis is insufficient in learning even the output distribution of a Clifford circuit with one additional $T$ gate under reasonable post-quantum cryptographic assumptions. In this work, we introduce an efficient quantum algorithm that employs only nonadaptive single-copy measurement to learn states produced by Clifford circuits with a maximum of $O(\log n)$ non-Clifford gates, filling a gap between the previous positive and negative results., Comment: 8 pages

Inspired by presentations by the authors during a workshop organized at the Princeton Center for Theoretical Science (PCTS) in January 2022, we present a perspective on some of the outstanding questions related to the "physics of the ground beneath our feet." These identified challenges are intrinsically shared with the field of Soft Matter but also have unique aspects when the natural environment is studied., Comment: Perspective Paper, 30 pages, 15 figures

Deep learning techniques are increasingly applied to scientific problems, where the precision of networks is crucial. Despite being deemed as universal function approximators, neural networks, in practice, struggle to reduce the prediction errors below $O(10^{-5})$ even with large network size and extended training iterations. To address this issue, we developed the multi-stage neural networks that divides the training process into different stages, with each stage using a new network that is optimized to fit the residue from the previous stage. Across successive stages, the residue magnitudes decreases substantially and follows an inverse power-law relationship with the residue frequencies. The multi-stage neural networks effectively mitigate the spectral biases associated with regular neural networks, enabling them to capture the high frequency feature of target functions. We demonstrate that the prediction error from the multi-stage training for both regression problems and physics-informed neural networks can nearly reach the machine-precision $O(10^{-16})$ of double-floating point within a finite number of iterations. Such levels of accuracy are rarely attainable using single neural networks alone., Comment: 38 pages, 17 pages

In this paper, we investigate the use of quaternary reliability statistics for ordered statistics decoding (OSD) of quantum codes. OSD can be used to improve the performance of belief propagation (BP) decoding when it fails to correct the error syndrome. We propose an approach that leverages quaternary reliability information and the hard-decision history output by BP to perform reliability sorting for OSD. This approach improves upon previous methods that separately treat X and Z errors, by preserving the X/Z correlations during the sorting step. Our simulations show that the refined BP with scalar messages and the proposed OSD outperforms previous BP-OSD combinations. We achieve thresholds of roughly 17.5%--17.7% for toric, surface, and XZZX codes, and 15.42% for hexagonal planar color codes., Comment: 5 pages, 3 figures, 2 tables (thresholds improved)

Factorization Machine (FM) is the most commonly used model to build a recommendation system since it can incorporate side information to improve performance. However, producing item suggestions for a given user with a trained FM is time-consuming. It requires a run-time of $O((N_m \log N_m)^2)$, where $N_m$ is the number of items in the dataset. To address this problem, we propose a quadratic unconstrained binary optimization (QUBO) scheme to combine with FM and apply quantum annealing (QA) computation. Compared to classical methods, this hybrid algorithm provides a faster than quadratic speedup in finding good user suggestions. We then demonstrate the aforementioned computational advantage on current NISQ hardware by experimenting with a real example on a D-Wave annealer., Comment: 9 pages, 3 figures

We address the maximum size of binary codes and binary constant weight codes with few distances. Previous works established a number of bounds for these quantities as well as the exact values for a range of small code lengths. As our main results, we determine the exact size of maximal binary codes with two distances for all lengths $n\ge 6$ as well as the exact size of maximal binary constant weight codes with 2,3, and 4 distances for several values of the weight and for all but small lengths., Comment: Main text 23 pp. and Appendix 17pp

To perform achieve fault-tolerant quantum computation, one can use flagged syndrome extraction with fewer ancilla qubits. However, it suffers from long circuit depth if one stabilizer is measured at a time. Previously, Reichardt showed that it is possible to measure multiple stabilizers with at most one shared flag qubit for certain small quantum codes. In this paper, we propose a procedure for general Calderbank-Shor-Steane codes of distance three so that multiple $Z$-stabilizers ($X$-stabilizers) can be fault-tolerantly measured in parallel with one shared flag qubit. We simulate the memory and computation pseudo-thresholds for various code schemes. In particular, our parallel scheme based on Shor's nine-qubit code performs better than known seven- and nine-qubit schemes in the literature.

The syndrome decoding problem is known to be NP-hard. We use the quantum approximate optimization algorithm (QAOA) to solve the syndrome decoding problem with elegantly-designed generator- and check-based cost Hamiltonians for classical and quantum codes. Simulations of the level-4 check-based QAOA decoding of the [7,4,3] Hamming code, as well as the level-4 generator-based QAOA decoding of the [[5,1,3]] quantum code, demonstrate decoding performances that match the maximum likelihood decoding. In addition, we show that a combinatorial optimization problem with additional redundant clauses may be more suitable for QAOA, while the number of qubits remains the same. Furthermore, we show that the QAOA decoding of a quantum code is inherently degenerate. That is, degenerate errors of comparable weight will be returned by QAOA with comparable probability. This is supported by simulations of the generator-based QAOA decoding of the [[9,1,3]] Shor code.

We study the maximum cardinality problem of a set of few distances in the Hamming and Johnson spaces. We formulate semidefinite programs for this problem and extend the 2011 works by Barg-Musin and Musin-Nozaki. As our main result, we find new parameters for which the maximum size of two- and three-distance sets is known exactly., Comment: Renew the whole article to add more results and authors

We describe explicitly the chamber structure of the movable cone for a general complete intersection Calabi--Yau threefold in a non-split $(n + 4)$-dimensional $\mathbb{P}^{n}$-ruled Fano manifold of index $n + 1$ and Picard number two. Moreover, all birational minimal models of such Calabi--Yau threefolds are found whose number is finite., Comment: 27 pages. The final version is accepted for publication in J. Geom. Phys

We study the upper bounds for $A(n,d)$, the maximum size of codewords with length $n$ and Hamming distance at least $d$. Schrijver studied the Terwilliger algebra of the Hamming scheme and proposed a semidefinite program to bound $A(n, d)$. We derive more sophisticated matrix inequalities based on a split Terwilliger algebra to improve Schrijver's semidefinite programming bounds on $A(n, d)$. In particular, we improve the semidefinite programming bounds on $A(18,4)$ to $6551$., Comment: 15 pages

Topological quantum codes are favored because they allow qubit layouts that are suitable for practical implementation. An $N$-qubit topological code can be decoded by minimum-weight perfect matching (MWPM) with complexity $O(\text{poly}(N))$ if it is of CSS-type. Recently it is shown that various quantum codes, including non-CSS codes, can be decoded by an adapted belief propagation with memory effects (denoted MBP) with complexity almost linear in $N$. In this paper, we show that various two-dimensional topological codes, CSS or non-CSS, regardless of the layout, can be decoded by MBP, including color codes and twisted XZZX codes. We will comprehensively compare these codes in terms of code efficiency and decoding performance, assuming perfect error syndromes.

Whether there exist finite time blow-up solutions for the 2-D Boussinesq and the 3-D Euler equations are of fundamental importance to the field of fluid mechanics. We develop a new numerical framework, employing physics-informed neural networks (PINNs), that discover, for the first time, a smooth self-similar blow-up profile for both equations. The solution itself could form the basis of a future computer-assisted proof of blow-up for both equations. In addition, we demonstrate PINNs could be successfully applied to find unstable self-similar solutions to fluid equations by constructing the first example of an unstable self-similar solution to the C\'ordoba-C\'ordoba-Fontelos equation. We show that our numerical framework is both robust and adaptable to various other equations., Comment: Main paper: 6 pages, 3 figures. Supplementary material: 15 pages, 12 figures. To appear in Physical Review Letters

While quantum weight enumerators establish some of the best upper bounds on the minimum distance of quantum error-correcting codes, these bounds are not optimized to quantify the performance of quantum codes under the effect of arbitrary quantum channels that describe bespoke noise models. Herein, for any Kraus decomposition of any given quantum channel, we introduce corresponding quantum weight enumerators that naturally generalize the Shor-Laflamme quantum weight enumerators. We establish an indirect linear relationship between these generalized quantum weight enumerators by introducing an auxiliary exact weight enumerator that completely quantifies the quantum code's projector, and is independent of the underlying noise process. By additionally working within the framework of approximate quantum error correction, we establish a general framework for constructing a linear program that is infeasible whenever approximate quantum error correcting codes with corresponding parameters do not exist. Our linear programming framework allows us to establish the non-existence of certain quantum codes that approximately correct amplitude damping errors, and obtain non-trivial upper bounds on the maximum dimension of a broad family of permutation-invariant quantum codes., Comment: 16 pages. arXiv admin note: substantial text overlap with arXiv:2001.03976

The purpose of the article is to explain a new method to establish the existence of an exceptional collection of length three for a fake projective plane M with non-trivial automorphism group, related to a conjecture of Galkin-Katzarkov-Mellit-Shinder in 2015. Our method shows that 30 fake projective planes support such a sequence, most of which are new. In particular, this provides many new H-phantom categories., Comment: 39 pages. To appear on International Mathematics Research Notices