Today’s Noisy Intermediate-Scale Quantum (NISQ) computers have scientific value, but quantum machines with broad practical value must be protected against noise using quantum error correction and fault-tolerant protocols. Recent studies of quantum error correction on actual hardware are opening a new era of quantum information processing. Error-corrected computers capable of performing one million quantum operations or more may be realized soon, raising a compelling question for the quantum community: What are the potential uses of these megaquop machines?

A fundamental challenge in quantum information science is certifying that an n-qubit state$\rho$prepared in the lab closely matches a target state$\vert \psi\rangle$. Previous approaches to this problem often require deep quantum circuits, exponentially many single-qubit measurements, or are limited to specific state families. In this work, we introduce a new method that leverages a connection between state certification and the mixing time of a random walk, allowing almost all n-qubit target states, including those with exponential circuit complexity, to be certified with only$\mathrm{O}(n^{2})$single-qubit measurements. Our protocol is broadly compatible with various experimental platforms and has applications in benchmarking quantum systems, optimizing quantum circuits, and efficiently learning and verifying representations of quantum states—such as neural networks and tensor networks—using only single-qubit measurements. Moreover, these verified representations enable the efficient prediction of highly non-local properties of$\rho$that would otherwise require an exponential number of measurements.

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.

We reconsider the black hole firewall puzzle, emphasizing that quantum error-correction, computational complexity, and pseudorandomness are crucial concepts for understanding the black hole interior. We assume that the Hawking radiation emitted by an old black hole is pseudorandom, meaning that it cannot be distinguished from a perfectly thermal state by any efficient quantum computation acting on the radiation alone. We then infer the existence of a subspace of the radiation system which we interpret as an encoding of the black hole interior. This encoded interior is entangled with the late outgoing Hawking quanta emitted by the old black hole, and is inaccessible to computationally bounded observers who are outside the black hole. Specifically, efficient operations acting on the radiation, those with quantum computational complexity polynomial in the entropy of the remaining black hole, commute with a complete set of logical operators acting on the encoded interior, up to corrections which are exponentially small in the entropy. Thus, under our pseudorandomness assumption, the black hole interior is well protected from exterior observers as long as the remaining black hole is macroscopic. On the other hand, if the radiation is not pseudorandom, an exterior observer may be able to create a firewall by applying a polynomial-time quantum computation to the radiation.

This paper prove the security of the Bennett-Brassard (BB84) quantum key distribution protocol in the case where the source and detector are under the limited control of an adversary. This proof applies when both the source and the detector have small basis-dependent flaws, as is typical in practical implementations of the protocol. The estimation of the key generation rate in some special cases: sources that emit weak coherent states, detectors with basis-dependent efficiency, and misaligned sources and detectors.