Exponential separation between shallow quantum circuits and unbounded fan-in shallow classical circuits

Abstract

Recently, Bravyi, Gosset, and König (Science, 2018) exhibited a search problem called the 2D Hidden Linear Function (2D HLF) problem that can be solved exactly by a constant-depth quantum circuit using bounded fan-in gates (or QNC 0 circuits), but cannot be solved by any constant-depth classical circuit using bounded fan-in AND, OR, and NOT gates (or NC 0 circuits). In other words, they exhibited a search problem in QNC 0 that is not in NC 0 .We strengthen their result by proving that the 2D HLF problem is not contained in AC 0 , the class of classical, polynomial-size, constant-depth circuits over the gate set of unbounded fan-in AND and OR gates, and NOT gates. We also supplement this worst-case lower bound with an average-case result: There exists a simple distribution under which any AC 0 circuit (even of nearly exponential size) has exponentially small correlation with the 2D HLF problem. Our results are shown by constructing a new problem in QNC 0 , which we call the Relaxed Parity Halving Problem, which is easier to work with. We prove our AC 0 lower bounds for this problem, and then show that it reduces to the 2D HLF problem.As a step towards even stronger lower bounds, we present a search problem that we call the Parity Bending Problem, which is in QNC 0 /qpoly (QNC 0 circuits that are allowed to start with a quantum state of their choice that is independent of the input), but is not even in AC 0 [2] (the class AC 0 with unbounded fan-in XOR gates).All the quantum circuits in our paper are simple, and the main difficulty lies in proving the classical lower bounds. For this we employ a host of techniques, including a refinement of Håstad's switching lemmas for multi-output circuits that may be of independent interest, the Razborov-Smolensky AC 0 [2] lower bound, Vazirani's XOR lemma, and lower bounds for non-local games. * Massachusetts Institute of Technology. abenewat@mit.edu.

Interactive quantum advantage with noisy, shallow Clifford circuits

“…In this section, we review the noisy relaxation of the 1-Round Graph State Measurement Problem of [Bra+20] called the noisy extension. We will revisit the main results of that paper, and show how their separation between noisy QNC 0 and NC 0 can be extended to a separation between noisy QNC 0 and AC 0 using the results of Bene Watts et al [Ben+19]. The results of this section are not independent from our results for interactive Clifford simulation tasks since the noisy extension will play a critical role there, as well.…”

“…It was shown in [Ben+19] that there is a problem that is solved with certainty by a QNC 0 circuit but is hard for AC 0 circuits to solve. The problem, a promise version of 1-Round Graph State Measurement Problem that they call the Relaxed Parity Having Problem (RPHP), is a relation problem with inputs uniformly chosen from a set P n ⊆ {0, 1} n for all n that is solved with certainty by a constant-depth classically-controlled Clifford circuit.…”

“…Lemma 17 (Theorem 26 in [Ben+19]). Any probabilistic AC 0 circuit of size s and depth d cannot solve RPHP with probability exceeding…”

“…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.…”

“…As a warm-up, we first consider the separation of Bene Watts, Kothari, Schaeffer, and Tal [Ben+19], which shows that constant-depth classical circuits with unbounded fan-in gates (AC 0 ) cannot solve the HLF problem on average. 2 Combining this result with the general error-correction recipe for relation problems, we arrive at the following result: Theorem 1.…”