Learning with Errors is one of the fundamental problems in computational learning theory and has in the last years become the cornerstone of post-quantum cryptography. In this work, we study the quantum sample complexity of Learning with Errors and show that there exists an efficient quantum learning algorithm (with polynomial sample and time complexity) for the Learning with Errors problem where the error distribution is the one used in cryptography. While our quantum learning algorithm does not break the LWE-based encryption schemes proposed in the cryptography literature, it does have some interesting implications for cryptography: first, when building an LWE-based scheme, one needs to be careful about the access to the public-key generation algorithm that is given to the adversary; second, our algorithm shows a possible way for attacking LWE-based encryption by using classical samples to approximate the quantum sample state, since then using our quantum learning algorithm would solve LWE. Finally, we extend our results and show quantum learning algorithms for three related problems: Learning Parity with Noise, Learning with Rounding and Short Integer Solution.

The problem of reliably certifying the outcome of a computation performed by a quantum device is rapidly gaining relevance. We present two protocols for a classical verifier to verifiably delegate a quantum computation to two non-communicating but entangled quantum provers. Our protocols have near-optimal complexity in terms of the total resources employed by the verifier and the honest provers, with the total number of operations of each party, including the number of entangled pairs of qubits required of the honest provers, scaling as O(g log g) for delegating a circuit of size g. This is in contrast to previous protocols, whose overhead in terms of resources employed, while polynomial, is far beyond what is feasible in practice. Our first protocol requires a number of rounds that is linear in the depth of the circuit being delegated, and is blind, meaning neither prover can learn the circuit or its input. The second protocol is not blind, but requires only a constant number of rounds of interaction.Our main technical innovation is an efficient rigidity theorem that allows a verifier to test that two entangled provers perform measurements specified by an arbitrary m-qubit tensor product of singlequbit Clifford observables on their respective halves of m shared EPR pairs, with a robustness that is independent of m. Our two-prover classical-verifier delegation protocols are obtained by combining this rigidity theorem with a single-prover quantum-verifier protocol for the verifiable delegation of a quantum computation, introduced by Broadbent (Theory of Computing, 2018).

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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 ǫ, 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.We also provide two small side results of potential interest. First, we are able to generalize our result by showing that deciding if a uniform stoquastic Local Hamiltonian has negligible or constant frustration can be also solved in NP. Additionally, our work reveals a new MA-complete problem which we call SetCSP, stated in terms of classical constraints on strings of bits, which we define in the appendix. As far as we know this is the first (arguably) natural MA-complete problem stated in non-quantum CSP language.Looking at this problem through the Theoretical Computer Science lens, Kitaev defined the k-Local Hamiltonian problem [KSV02, AN02], whose input is a Hamiltonian H on an n particle system given as a sum of m local terms, each of them acting non-trivially on at most k out of the n particles (the term local only refers to the fact that k is assumed to be small, there are no geometrical restrictions on the interaction). We are also given as input two parameters, α and β. We then ask if the smallest eigenvalue of H is smaller than α, or all states have energy larger than β. The hardness of the Local Hamiltonian problem depends on the input promise gap defined as β − α; Kitaev showed that the problem is QMA-complete for some inverse polynomial promise gap [KSV02, AN02].Bravyi, DiVincenzo, Oliveira and Terhal [BDOT08] asked how the problem behaves when the terms are restricted such that their off-diagonal elements are all non-positive, a property that they named "stoquastic" (as a combination of the words stochastic and quantum). This property implies a lot of structure on groundstates (See Lemma 3.2), and in physics it is associated with the lack of the "sign problem", in which case one can associate with the Hamiltonian a classical Markov Chain Monte Carlo experiment and study it (See [BDOT08]); such systems are considered far easier than general Hamiltonians.[BBT06] showed that the stoquastic Local Hamiltonia...

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