Computer Science

The derandomization of MA, the probabilistic version of NP, is a long standing open question. In this work, we connect this problem to a variant of another major problem: the quantum PCP conjecture. Our connection goes through the surprising quantum characterization of MA by Bravyi and Terhal. They proved the MA-completeness of the problem of deciding whether the groundenergy of a uniform stoquastic local Hamiltonian is zero or inverse polynomial. We show that the gapped version of this problem, i.e. deciding if a given uniform stoquastic local Hamiltonian is frustration-free or has energy at least some constant ϵ, is in NP. Thus, if there exists a gap-amplification procedure for uniform stoquastic Local Hamiltonians (in analogy to the gap amplification procedure for constraint satisfaction problems in the original PCP theorem), then MA = NP (and vice versa). Furthermore, if this gap amplification procedure exhibits some additional (natural) properties, then P = RP. We feel this work opens up a rich set of new directions to explore, which might lead to progress on both quantum PCP and derandomization. Joint work with Dorit Aharonov.

This will be an introductory talk to self-testing as an approach to certifying quantum systems. The theory of self-testing has developed significantly in recent years. It has found applications to quantum cryptography, to characterizing the complexity of multiprover interactive proofs with entangled provers, as well as connections to foundational questions about quantum correlation sets and entanglement. In this talk, I will discuss the model, the assumptions, some common techniques, applications, and open questions.

In this talk, we consider a variant of entangled non-local games where the players are allowed to share infinitely many copies of noisy EPR states. We provide an upper bound on the copies of noisy EPR states for the players to approximate the values of games to an arbitrary precision if the games are binary. The arguments are built on the recent framework about the decidability of the non-interactive simulation of joint distributions with significant extension. A series new techniques on the Fourier analysis on random operator spaces are introduced including a quantum invariance principle and a hypercontractive inequality for random operators. These novel tools are interesting on their own right and may have further applications in quantum information theory and quantum complexity theory.

The AdS/CFT correspondence is central to efforts to reconcile gravity and quantum mechanics. It posits a duality between a quantum gravity theory and a quantum mechanical theory, embodied in a map known as the "dictionary" which is a homomorphism between the theories. This dictionary map is not well understood and has only been computed on special, structured instances.  In this talk we introduce cryptographic ideas to the study of AdS/CFT, and provide evidence that either the dictionary must be exponentially hard to compute, or else the quantum Extended Church-Turing thesis must be false in quantum gravity.  The basic argument is that Susskind's "wormhole growth paradox" requires the dictionary to map a quantity which is hard to compute -- essentially the circuit complexity of the dual quantum state -- to something which is easy to compute in the quantum gravity theory. Therefore the dictionary itself must be hard to compute. Our argument requires creating a custom quantum pseudorandomness construction inspired by block ciphers such as the AES and DES cryptosystems.  No background in quantum gravity will be assumed. Based on joint work with Bill Fefferman and Umesh Vazirani.

Predicting properties of complex, large-scale quantum systems is essential for developing quantum technologies. We present an efficient method for constructing an approximate classical description of a quantum state using very few measurements of the state. This description, called a classical shadow, can be used to predict many different properties: order log M measurements suffice to accurately predict M different functions of the state with high success probability. The number of measurements is independent of the system size, and saturates information-theoretic lower bounds. Moreover, target properties to predict can be selected after the measurements are completed. We support our theoretical findings with extensive numerical experiments. We apply classical shadows to predict quantum fidelities, entanglement entropies, two-point correlation functions, expectation values of local observables, and the energy variance of many-body local Hamiltonians. The numerical results highlight the advantages of classical shadows relative to previously known methods. This is joint work with Hsin-Yuan Huang and John Preskill.

My talk has two purposes. First, I will review the problems quantum tomography is designed to solve, its relation to other verification protocols, and the tools that have been proposed for this task. In a completely independent second part, I will report on recent work (around [arXiv:1712.08628]) on the representation theory of the Clifford group and its applications to property testing. In particular, I'll explain how "stabilizerness" and "Cliffordness" are properties of pure states and unitaries that can be tested from a system-size independent number of copies.

Recovering an unknown Hamiltonian or a Linbladian from measurements is an increasingly important task for certification of noisy quantum devices and simulators. Recent works have succeeded in recovering the Hamiltonian of an isolated quantum system with local interactions from long-ranged correlators of a single eigenstate. Here we generalize these works to allow for the recovery of the Hamlitonian or a Linbladian from their steady states, including, for example, the Gibbs state at a finite temperature. Our approach takes advantage of the non-commuteness of the underlying dynamics to derive non-trivial local constraints between the local reduced density matrices and the local generators of dynamics. This enables us to learn a local patch of the system from local observations only on that patch, even though the overall state might be globally entangled. Surprisingly, there are cases in which our algorithm can be exponentially faster than the classical problem of learning a Boltzmann machine, or, more generally, a graphical model. Finally, our approach can also be easily extended to the case of learning a system from its dynamics. Based on joint works with E. Bairy, N. Lindner, Chu Guo and D. Poletti: 1. Phys. Rev. Lett. 122, 020504 (2019), arXiv:1907.11154 2. New J. Phys. 22 032001 (2020), arXiv:1807.04564 

The hardness of the Learning with Errors (LWE) problem is by now a cornerstone of the cryptographic landscape, allowing to construct cryptographic schemes with properties unknown under other assumptions, and being conjectured to be resilient to quantum attacks. LWE is essentially the task of solving a noisy system of random linear equations over uniformly random secret variables (“the LWE secret”), evaluated modulo some integer. In applications the secret variables usu- ally correspond to the secret key of the cryptographic scheme. It is therefore of great importance to understand what happens when the secret variables are not sampled uniformly (but still have some entropy). This is relevant for settings where an adversary manages to obtain partial information on the secret (a.k.a key leakage), for various theoretical ap- plications, and also for practical use where for efficiency or convenience it is easier to sample the secret from some non-uniform distribution. This so called “Entropic LWE” problem has been studied in a number of works, starting with Goldwasser et al. (ICS 2010). However, so far it was only known how to prove the hardness of Entropic LWE for secret distributions supported inside a ball of small radius. In this work we resolve the hardness of Entropic LWE with arbitrary long secrets, in the following sense. We show an entropy bound that guarantees the security of arbitrary Entropic LWE. This bound is higher than what is required in the ball-bounded setting, but we show that this is essentially tight. Tightness is shown unconditionally for highly-composite moduli, and using black-box impossibility for arbitrary moduli. Technically, we show that the entropic hardness of LWE relies on a simple to describe lossiness property of the distribution of secrets itself. This is simply the probability of recovering a random sample from this distribution s, given s + e, where e is Gaussian noise (i.e. the quality of the distribution of secrets as an error correcting code for Gaussian noise). We hope that this characterization will make it easier to derive entropic LWE results more easily in the future. We also use our techniques to show new results for the ball-bounded setting, essentially showing that under a strong enough assumption even polylogarithmic entropy suffices.