Abstract
The generation of certifiable randomness is the most fundamental information-theoretic task that meaningfully separates quantum devices from their classical counterparts. We propose a protocol for exponential certified randomness expansion using a single quantum device. The protocol calls for the device to implement a simple quantum circuit of constant depth on a 2D lattice of qubits. The output of the circuit can be verified classically in linear time, and is guaranteed to contain a polynomial number of certified random bits assuming that the device used to generate the output operated using a (classical or quantum) circuit of sub-logarithmic depth. This assumption contrasts with the locality assumption used for randomness certification based on Bell inequality violation and more recent proposals for randomness certification based on computational assumptions. Furthermore, to demonstrate randomness generation it is sufficient for a device to sample from the ideal output distribution within constant statistical distance. Our procedure is inspired by recent work of Bravyi et al. (Science 362(6412):308–311, 2018), who introduced a relational problem that can be solved by a constant-depth quantum circuit, but provably cannot be solved by any classical circuit of sub-logarithmic depth. We develop the discovery of Bravyi et al. into a framework for robust randomness expansion. Our results lead to a new proposal for a demonstrated quantum advantage that has some advantages compared to existing proposals. First, our proposal does not rest on any complexity-theoretic conjectures, but relies on the physical assumption that the adversarial device being tested implements a circuit of sub-logarithmic depth. Second, success on our task can be easily verified in classical linear time. Finally, our task is more noise-tolerant than most other existing proposals that can only tolerate multiplicative error, or require additional conjectures from complexity theory; in contrast, we are able to allow a small constant additive error in total variation distance between the sampled and ideal distributions.
Highlights
A fundamental point of departure between quantum mechanics and classical theory is that the former is non-deterministic: quantum mechanics, through the Born rule, posits the existence of experiments that generate intrinsic randomness
While a loophole-free implementation of a Bell test has been demonstrated [HBD+15,GVW+15,SMSC+15] it remains a challenging experimental feat, which leaves its promising applications wanting
We propose a different kind of experiment, or “test of quantumness”, for large but noisy quantum devices, that is inspired from recent work of Bravyi et al on the power of low-depth quantum circuits [BGK18]
Summary
A fundamental point of departure between quantum mechanics and classical theory is that the former is non-deterministic: quantum mechanics, through the Born rule, posits the existence of experiments that generate intrinsic randomness. We point out the existence of an efficiently sampleable distribution on inputs such that, for any classical low-depth circuit, we know that on average over the choice of an input the circuit returns an output that satisfies the relation with at most small probability This improvement follows using a simple extension of the arguments in [BGK17], and a similar improvement was independently derived by Bravyi et al in the final version of their paper [BGK18]. By adapting techniques from the area of randomness expansion from nonlocal games [AFDF+18] we are able to conclude that any sub-logarithmic-depth circuit, classical or quantum, that succeeds on our input distribution, must generate large amounts of entropy This guarantee holds even if the circuit only correctly computes a sufficiently large but constant fraction of outputs for the games. Neither of these results obtains an application to randomness expansion as in our Theorem 1.1
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