Trading locality for time: certifiable randomness from low-depth circuits

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 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 leads 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.

Average-Case Quantum Advantage with Shallow Circuits

“…Related works. A similar result has been recently (and independently) obtained by Coudron, Stark and Vidick and expanded into a framework for robust randomness expansion [CSV18]. The proof techniques are nevertheless different: [CSV18] constructs a problem hard for small-depth classical circuits by starting with a non-local game and showing how to plant a polynomial number of copies of the game into a graph.…”

“…A similar result has been recently (and independently) obtained by Coudron, Stark and Vidick and expanded into a framework for robust randomness expansion [CSV18]. The proof techniques are nevertheless different: [CSV18] constructs a problem hard for small-depth classical circuits by starting with a non-local game and showing how to plant a polynomial number of copies of the game into a graph. Our approach, on the other hand, starts with a graph and shows how to create from it a quantum state exhibiting global quantum correlations that cannot be simulated by small-depth classical circuits with bounded-fanin gates.…”

Interactive shallow Clifford circuits: quantum advantage against NC$^1$ and beyond

“…but it is a complicated theorem and it may help to keep an example in mind. For this section, the relevant setting of parameters 15 is G = H = F = C 2 P 2 , acting on the set of Pauli strings S = P 2 Z 2 by conjugation, U • P ∶= U P U † . Theorem 23.…”

“…Although constant-depth bounded fan-in classical circuits (i.e., NC 0 circuits) vs. constant-depth bounded fan-in quantum circuits (i.e., QNC 0 circuits) is a fair comparison, NC 0 is an extremely weak class of circuits, leaving lots of room for improvement. Indeed, there have been several followup papers which have strengthened the result by considering average-case versions of the problem [15,27,7], expanding the class of classical circuits [7], and adding noise [11]. Of particular relevance to this work is the result of Bene Watts, Kothari, Schaeffer, and Tal [7] which shows that even classical circuits with unbounded fan-in AND and OR gates (i.e., the circuit class AC 0 ) cannot cannot solve HLF.…”

Interactive quantum advantage with noisy, shallow Clifford circuits

“…The separation is based on the relation problem 1 associated with measuring the outputs of a shallow Clifford circuit, which they called the Hidden Linear Function (HLF) problem due to certain algebraic properties of the output. Furthermore, they show that NC 0 cannot even solve this problem on average, a result which was later strengthened in several ways [CSV18;Le 19;Ben+19]. Nevertheless, these works still assumed that the quantum circuit solving the task was noise free.…”