Zvika Brakerski

Two fundamental properties of quantum states that quantum information theory explores are pseudorandomness and provability of destruction. We introduce the notion of quantum pseudorandom states with proofs of destruction (PRSPD) that combines both these properties. Like standard pseudorandom states (PRS), these are efficiently generated quantum states that are indistinguishable from random, but they can also be measured to create a classical string. This string is verifiable (given the secret key) and certifies that the state has been destructed. We show that, similarly to PRS, PRSPD can be constructed from any post-quantum one-way function. As far as the authors are aware, this is the first construction of a family of states that satisfies both pseudorandomness and provability of destruction. We show that many cryptographic applications that were shown based on PRS variants using quantum communication can be based on (variants of) PRSPD using only classical communication. This includes symmetric encryption, message authentication, one-time signatures, commitments, and classically verifiable private quantum coins.

A test of quantumness is a protocol that allows a classical verifier to certify (only) that a prover is not classical. We show that tests of quantumness that follow a certain template, which captures recent proposals such as (Kalai et al., 2022), can in fact do much more. Namely, the same protocols can be used for certifying a qubit, a building-block that stands at the heart of applications such as certifiable randomness and classical delegation of quantum computation. Certifying qubits was previously only known to be possible based on the hardness of the Learning with Errors problem and the use of adaptive hardcore (Brakerski et al., 2018). Our framework allows certification of qubits based only on the existence of post-quantum trapdoor claw-free functions, or on quantum fully homomorphic encryption. These can be instantiated, for example, from Ring Learning with Errors. On the technical side, we show that the quantum soundness of any such protocol can be reduced to proving a bound on a simple algorithmic task: informally, answering "two challenges simultaneously'' in the protocol. Our reduction formalizes the intuition that these protocols demonstrate quantumness by leveraging the impossibility of rewinding a general quantum prover. This allows us to prove tight bounds on the quantum soundness of (Kahanamoku-Meyer et al., 2021) and (Kalai et al., 2022), showing that no quantum polynomial-time prover can succeed with probability larger than cos^2(π/8)≈0.853. Previously, only an upper bound on the success probability of classical provers, and a lower bound on the success probability of quantum provers, were known. We then extend this proof of quantum soundness to show that provers that approach the quantum soundness bound must perform almost anti-commuting measurements. This certifies that the prover holds a qubit.

Abstract We present a garbling scheme for quantum circuits, thus achieving a decomposable randomized encoding scheme for quantum computation. Specifically, we show how to compute an encoding of a given quantum circuit and quantum input, from which it is possible to derive the output of the computation and nothing else. In the classical setting, garbled circuits (and randomized encodings in general) are a versatile cryptographic tool with many applications such as secure multiparty computation, delegated computation, depth-reduction of cryptographic primitives, complexity lower-bounds, and more. However, a quantum analogue for garbling general circuits was not known prior to this work. We hope that our quantum randomized encoding scheme can similarly be useful for applications in quantum computing and cryptography. To illustrate the usefulness of quantum randomized encoding, we use it to design a conceptually-simple zero-knowledge (ZK) proof system for the complexity class QMA. Our protocol has the so-called S format with a single-bit challenge, and allows the inputs to be delayed to the last round. The only previously-known ZK S-protocol for QMA is due to Broadbent and Grilo (FOCS 2020), which does not have the aforementioned properties.

Abstract In quantum copy-protection, an adversary who is given a quantum state computing a function f cannot produce two (possibly entangled) quantum states that each individually compute f. No constructions for copy-protection are known in the plain model. We consider a weaker notion, secure software leasing (SSL), where it is only impossible to produce two copies that can both compute f using the honest evaluation algorithm. We show the following: (1) SSL is possible for a subclass of evasive functions, assuming the existence of post-quantum indistinguishability obfuscators and hardness of LWE; (2) SSL is impossible in general, assuming hardness of LWE. The second statement has important implications for existing quantum-cryptographic notions: in particular, it implies the impossibility of quantum copy-protection for arbitrary unlearnable functions, and impossibility of quantum virtual-black-box obfuscation of classical circuits.

Efficiently sampling a quantum state that is hard to distinguish from a truly random quantum state is an elementary task in quantum information theory that has both computational and physical uses. This is often referred to as pseudorandom (quantum) state generator, or PRS generator for short. In existing constructions of PRS generators, security scales with the number of qubits in the states, i.e. the (statistical) security parameter for an n-qubit PRS is roughly n. Perhaps counter-intuitively, n-qubit PRS are not known to imply k-qubit PRS even for k<n. Therefore the question of \emph{scalability} for PRS was thus far open: is it possible to construct n-qubit PRS generators with security parameter m for all n, m. Indeed, we believe that PRS with tiny (even constant) n and large m can be quite useful. We resolve the problem in this work, showing that any quantum-secure one-way function implies scalable PRS. We follow the paradigm of first showing a \emph{statistically} secure construction when given oracle access to a random function, and then replacing the random function with a quantum-secure (classical) pseudorandom function to achieve computational security. However, our methods deviate significantly from prior works since scalable pseudorandom states require randomizing the amplitudes of the quantum state, and not just the phase as in all prior works. We show how to achieve this using Gaussian sampling.

Virtual black-box obfuscation is a strong cryptographic primitive: it encrypts a circuit while maintaining its full input/output functionality. A remarkable result by Barak et al. (Crypto 2001) shows that a general obfuscator that obfuscates classical circuits into classical circuits can- not exist. A promising direction that circumvents this impossibility result is to obfuscate classical circuits into quantum states, which would potentially be better capable of hiding information about the obfuscated circuit. We show that, under the assumption that learning-with-errors (LWE) is hard for quantum computers, this quantum variant of virtual black-box obfuscation of classical circuits is generally impossible. On the way, we show that under the presence of dependent classical auxiliary input, even the small class of classical point functions cannot be quantum virtual black-box obfuscated.

A proof of quantumness is a method for provably demonstrating (to a classical verifier) that a quantum device can perform computational tasks that a classical device with comparable resources cannot. Providing a proof of quantumness is the first step towards constructing a useful quantum computer. There are currently three approaches for exhibiting proofs of quantumness: (i) Inverting a classically-hard one-way function (e.g.\ using Shor’s algorithm). This seems technologically out of reach. (ii) Sampling from a classically-hard-to-sample distribution (e.g.\ BosonSampling). This may be within reach of near-term experiments, but for all such tasks known verification requires exponential time. (iii) Interactive protocols based on cryptographic assumptions. The use of a trapdoor scheme allows for efficient verification, and implementation seems to require much less resources than (i), yet still more than (ii). In this work we propose a significant simplification to approach (iii) by employing the random oracle heuristic. (We note that we do not apply the Fiat-Shamir paradigm.) We give a two-message (challenge-response) proof of quantumness based on any trapdoor claw-free function. In contrast to earlier proposals we do not need an adaptive hard-core bit property. This allows the use of smaller security parameters and more diverse computational assumptions (such as Ring Learning with Errors), significantly reducing the quantum computational effort required for a successful demonstration.

Fully homomorphic encryption schemes (FHE) allow to apply arbitrary efficient computation to encrypted data without decrypting it first. This allows a client to delegate computation to a computationally powerful server without compromising the privacy of the input. This notion was proposed by Rivest, Adleman and Dertouzos in 1978, but a first candidate was only proposed 30 years later by Gentry. Broadbent and Jeffery (2015) asked whether it is possible to construct a *quantum* analog, which will allow to apply an arbitrary quantumly efficient computation to (classical or quantum) encrypted data. The ultimate notion of Quantum FHE (QFHE) scheme will allow even to a classical client privately delegate computation to a quantum server. The talk will survey the exciting recent advancement on this question, through the first “ultimate” candidate proposed by Mahadev (2018) and beyond. Connections to other quantum-cryptographic primitives and open questions will also be discussed.