Aarthi Sundaram

Quantum access to arbitrary classical data encoded in unitary black-box oracles underlies interesting data-intensive quantum algorithms, such as machine learning or electronic structure simulation. The feasibility of these applications depends crucially on gate-efficient implementations of these oracles, which are commonly some reversible versions of the boolean circuit for a classical lookup table. We present a general parameterized architecture for quantum circuits implementing a lookup table that encompasses all prior work in realizing a continuum of optimal tradeoffs between qubits, non-Clifford gates, and error resilience, up to logarithmic factors. Our architecture assumes only local 2D connectivity, yet recovers results, with the appropriate parameters, poly-logarithmic error scaling. We also identify novel regimes, such as simultaneous sublinear scaling in all parameters. These results enable tailoring implementations of the commonly used lookup table primitive to any given quantum device with constrained resources.

Quantum cryptography is known for enabling functionalities that are unattainable using classical information alone. Recently, Secure Software Leasing (SSL) has emerged as one of these areas of interest. Given a target circuit C from a circuit class, SSL produces an encoding of C which enables the evaluation of C, and also enables a verify procedure, by which the originator of the software becomes convinced that the software is returned --- meaning that the recipient has relinquished the possibility of any further use of the software. Clearly, such functionality is unachievable using classical information alone, since it is impossible to prevent a user from keeping a copy of the software. Recent results have shown the achievability of SSL using quantum information for a class of functions called compute-and-compare (these are a generalization of the well-known point functions). These prior works, however, all make use of setup or computational assumptions. Here, we show that SSL is achievable for compute-and-compare circuits without any assumptions. Our technique is a generic reduction from any quantum message authentication code to such an SSL scheme. Along the way, we also show that point functions can be copy-protected without any assumptions, for a security definition that involves one honest and one malicious evaluator.

Abstract We establish the first general connection between the design of quantum algorithms and circuit lower bounds. Specifically, let C be a class of polynomial-size concepts, and suppose that C can be learned in the PAC model under the uniform distribution with membership queries, and with error 1/2 - c by a time T quantum algorithm. We prove that if (c^2 * T) << 2^n/n, then BQE is not contained in C, where BQE = BQTIME[2^{O(n)}] is an exponential-time analogue of BQP. This result is optimal in both c and T since it is not hard to learn any class C of functions in (classical) time T = 2^n (with no error) or in quantum time T = poly(n) with error at most 1/2 - \Omega(2^{-n/2}) via Fourier sampling. In other words, even a marginal improvement on these generic learning algorithms would lead to major consequences in complexity theory. Our proof builds on several works in learning theory, pseudorandomness, and computational complexity, and crucially, on a connection between non-trivial classical learning algorithms and circuit lower bounds established by Oliveira and Santhanam (CCC 2017). Extending their approach to quantum learning algorithms turns out to create significant challenges. To achieve that, we show among other results how pseudorandom generators imply learning-to-lower-bound connections in a generic fashion, construct the first conditional pseudorandom generator secure against uniform quantum computations, and extend the local list-decoding algorithm of Impagliazzo, Jaiswal, Kabanets and Wigderson (SICOMP 2010) to quantum circuits via a delicate analysis. We believe that these contributions are of independent interest and might find other applications.

Quantum cryptography is known for enabling functionalities that are unattainable using classical information alone. Recently, Secure Software Leasing (SSL) has emerged as one of these areas of interest. Given a target circuit C from a circuit class, SSL produces an encoding of C that enables a recipient to evaluate C, and also enables the originator of the software to verify that the software has been returned --- meaning that the recipient has relinquished the possibility of any further use of the software. Clearly, such a functionality is unachievable using classical information alone, since it is impossible to prevent a user from keeping a copy of the software. Recent results have shown the achievability of SSL using quantum information for a class of functions called compute-and-compare (these are a generalization of the well-known point functions). These prior works, however all make use of setup or computational assumptions. Here, we show that SSL is achievable for compute-and-compare circuits without any assumptions. Our technique involves the study of quantum copy-protection, which is a notion related to SSL, but where the encoding procedure inherently prevents a would-be quantum software pirate from splitting a single copy of an encoding for C into two parts, each of which enables a user to evaluate C. We show that point functions can be copy-protected without any assumptions, for a novel security definition involving one honest and one malicious evaluator; this is achieved by showing that from any quantum message authentication code, we can derive such an honest-malicious copy-protection scheme. We then show that a generic honest-malicious copy-protection scheme implies SSL; by prior work, this yields SSL for compute-and-compare functions.