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 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.

Depth-efficient proofs of quantumness

“…There, the authors proposed a protocol for certifiable random-number generation with constant depth quantum circuits. The first difference with respect to our work is that [CSV21] do not base the soundness of their protocol on the classical intractability of some computational problem, such as LWE. Instead, the protocol assumes that the "prover" generating the randomness is a circuit of sub-logarithmic depth (showing that sub-logarithmic classical circuits would not succeed in this task).…”

“…In terms of constant quantum depth constructions, it is interesting to contrast our work to that of [CSV21]. There, the authors proposed a protocol for certifiable random-number generation with constant depth quantum circuits.…”

“…Instead, the protocol assumes that the "prover" generating the randomness is a circuit of sub-logarithmic depth (showing that sub-logarithmic classical circuits would not succeed in this task). The second difference is that our protocols require interleaving constant depth quantum circuits with logarithmic depth classical computation, whereas the protocol in [CSV21] only requires the application of a constant depth quantum circuit. Finally, our protocols are interactive, whereas [CSV21] is not.…”

Quantum Advantage with Shallow Circuits Under Arbitrary Corruption

“…There are several results related to this separation. Coudron, Stark and Vidick [17] and Le Gall [21] showed a similar separation in the average case setting, instead of the worst case setting considered in the original version of [12]: there exists a relation problem such that constant-depth quantum circuits can solve the relation on all inputs, but any O(log n) depth randomized bounded fan-in classical circuits cannot solve it on most inputs with high probability. Bene Watts et al [8] showed that a similar separation holds against classical circuits using unbounded fan-in gates.…”

Test of Quantumness with Small-Depth Quantum Circuits

“…Under assumptions such as the non-collapse of the polynomial hierarchy or the hardness of (appropriate versions of) the permanent, strong evidence of the superiority of weak classes of quantum circuits has been obtained from the 2000s [2,3,4,6,12,13,14,18,19,20,33,39]. A recent breakthrough by Bravyi, Gosset and König [10], further strengthened by subsequent works [5,11,17,21], showed an unconditional separation between the computational powers of quantum and classical small-depth circuits by exhibiting a computational task that can be solved by constant-depth quantum circuits but requires logarithmic depth for classical circuits. A major shortcoming, however, is that logarithmic-depth classical computation is a relatively weak complexity class.…”