It is shown that every one-dimensional Hamiltonian with short-range interactions admits a quantum Gibbs sampler [CKG23] with a system-size independent spectral gap at all finite temperatures. Consequently, their Gibbs states can be prepared in polylogarithmic depth, and satisfy exponential clustering of correlations, generalizing [Ara69].

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 constant inverse temperature, 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 Hamiltonian on any bounded-degree graph to a constant additive error with the optimal sample complexity $\mathcal{O}(\log n)$. The protocol uses parallelizable local quantum measurements that act within bounded neighborhoods of the graph and near-linear-time classical post-processing. We also give a learning algorithm for lattice Hamiltonians with near-optimal scaling on the learning precision and the inverse temperature. At the heart of our algorithm is the interplay between locality, the Kubo-MartinSchwinger condition, and the operator Fourier transform at arbitrary temperatures.

Random reversible and quantum circuits form random walks on the alternating group Alt(2n) and unitary group SU(2n), respectively, with each random gate as one step of the walk. Existing bounds on the spectral gap for the t-th moment of these random walks have inverse-polynomial dependence in both n and t. We prove that the gap for random reversible circuits is Ω(n−3) for all t≥1, and the gap for random quantum circuits is Ω(n−3) for t ≤ Θ(2n/2).Importantly, these gaps are independent of t in the respective regimes. We can further improve both gaps to n−1/polylog(n, t) for t ≤ 2Θ(n), which is tight up to polylog factors in n and t. Our spectral gap results have a number of consequences:1)Random reversible circuits with $\mathcal{O}\left( {{n^4}t} \right)$ gates form multiplicative-error t-wise independent (even) permutations for all t ≥ 1; for t ≤ Θ(2n/6.1), we show that $\tilde {\mathcal{O}}\left( {{n^2}t} \right)$ gates suffice.2)Random quantum circuits with $\mathcal{O}\left( {{n^4}t} \right)$ gates form multiplicative-error unitary t-designs for t ≤Θ(2n/2); for t ≤ Θ(22n/5), we show that $\tilde {\mathcal{O}}\left( {{n^2}t} \right)$ gates suffice.3)The robust quantum circuit complexity of random quantum circuits grows linearly for an exponentially long time, proving the robust Brown–Susskind conjecture [1], [2]. We also show an analogous result for random reversible circuits.Our spectral gap bounds are proven by reducing random quantum circuits to a more structured walk: a modification of the "PFC ensemble" from [3] together with an expander on the alternating group due to Kassabov [4], for which we give an efficient implementation using reversible circuits. In our reduction, we approximate the structured walk with local random circuits without losing the gap, which uses tools from the study of frustration-free Hamiltonians.

A quantum system coupled to a bath at some fixed, finite temperature converges to its Gibbs state. This thermalization process defines a natural, physically-motivated model of quantum computation. However, whether quantum computational advantage can be achieved within this realistic physical setup has remained open, due to the challenge of finding systems that thermalize quickly, but are classically intractable. Here we consider sampling from the measurement outcome distribution of quantum Gibbs states at constant temperatures, and prove that this task demonstrates quantum computational advantage. We design a family of commuting local Hamiltonians (parent Hamiltonians of shallow quantum circuits) and prove that they rapidly converge to their Gibbs states under the standard physical model of thermalization (as a continuous-time quantum Markov chain). On the other hand, we show that no polynomial time classical algorithm can sample from the measurement outcome distribution by reducing to the classical hardness of sampling from noiseless shallow quantum circuits. The key step in the reduction is constructing a fault-tolerance scheme for shallow IQP circuits against input noise.

A unitary k-design is an ensemble of unitaries that matches the first$k$moments of the Haar measure. In this work, we provide two efficient constructions of k-designs on n-qubits using new random matrix theory techniques. Our first construction is based on exponentiating sums of random i.i.d. Hermitian matrices and uses O(k2n2)-many gates. In the spirit of central limit theorems, we show that this random sum approximates the Gaussian Unitary Ensemble (GUE). We then show that the product of just two exponentiated GUE matrices is already approximately Haar random. Our second construction is based on products of exponentiated sums of random permutations and uses Õ($k$poly ($n$)) many gates. The$k$dependence is optimal (up to polylogarithmic factors) and is inherited from the efficiency of existing k-wise independent permutations. Furthermore, replacing random permutations with quantum-secure pseudorandom permutations (PRPs), we also obtain a pseudorandom unitary (PRU) ensemble that is secure under nonadaptive queries. A central feature of both proofs is a new connection between the polynomial method in quantum query complexity and the large-dimension ($N$) expansion in random matrix theory. In particular, the first construction uses the polynomial method to control high moments of certain random matrix ensembles without requiring delicate Weingarten calculations. In doing so, we define and solve a moment problem on the unit circle, asking whether a finite number of equally weighted points can reproduce a given set of moments. In our second construction, the key step is to exhibit an orthonormal basis for irreducible representations of the partition algebra that has a low-degree large-$N$expansion. This allows us to show that the distinguishing probability is a low-degree rational polynomial of the dimension$N$.

Finding ground states of quantum many-body systems is known to be hard for both classical and quantum computers. As a result, when Nature cools a quantum system in a low-temperature thermal bath, the ground state cannot always be found efficiently. Instead, Nature finds a local minimum of the energy. In this work, we study the problem of finding local minima in quantum systems under thermal perturbations. While local minima are much easier to find than ground states, we show that finding a local minimum is computationally hard for classical computers, even when the task is to output a single-qubit observable at any local minimum. In contrast, we prove that a quantum computer can always find a local minimum efficiently using a thermal gradient descent algorithm that mimics the cooling process in Nature. To establish the classical hardness of finding local minima, we consider a family of two-dimensional Hamiltonians such that any problem solvable by polynomial-time quantum algorithms can be reduced to finding local minima of these Hamiltonians. Therefore, cooling systems to local minima is universal for quantum computation, and, assuming quantum computation is more powerful than classical computation, finding local minima is classically hard and quantumly easy.

Duckiepond is an education and research development environment that includes software systems, educational materials, and of a fleet of autonomous surface vehicles Duckieboat. Duckieboats are designed to be easily reproducible with parts from a 3D printer and other commercially available parts, with flexible software that leverages several open source packages. The Duckiepond environment is modeled after Duckietown and AI Driving Olympics environments: Duckieboats rely only on one monocular camera, IMU, and GPS, and perform all ML processing using onboard embedded computers. Duckiepond coordinates commonly used middlewares (ROS and MOOS) and containerized software packages in Docker, making it easy to deploy. The combination of learning-based methods together with classic methods enables important maritime missions: track and trail, navigation, and coordinate among Duckieboats to avoid collisions. Duckieboats have been operating in a man-made lake, reservoir and river environments. All software, hardware, and educational materials are openly available (https://robotx-nctu.github.io/duckiepond), with the goal of supporting research and education communities across related domains.