18 research outputs found
Test of Quantumness with Small-Depth Quantum Circuits
Recently Brakerski, Christiano, Mahadev, Vazirani and Vidick (FOCS 2018) have shown how to construct a test of quantumness based on the learning with errors (LWE) assumption: a test that can be solved efficiently by a quantum computer but cannot be solved by a classical polynomial-time computer under the LWE assumption. This test has lead to several cryptographic applications. In particular, it has been applied to producing certifiable randomness from a single untrusted quantum device, self-testing a single quantum device and device-independent quantum key distribution.
In this paper, we show that this test of quantumness, and essentially all the above applications, can actually be implemented by a very weak class of quantum circuits: constant-depth quantum circuits combined with logarithmic-depth classical computation. This reveals novel complexity-theoretic properties of this fundamental test of quantumness and gives new concrete evidence of the superiority of small-depth quantum circuits over classical computation
Trading locality for time: certifiable randomness from low-depth circuits
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 or computational assumptions. 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 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. 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.Comment: 36 pages, 2 figure
Parity vs. AC0 with simple quantum preprocessing
A recent line of work has shown the unconditional advantage of constant-depth
quantum computation, or , over ,
, and related models of classical computation. Problems
exhibiting this advantage include search and sampling tasks related to the
parity function, and it is natural to ask whether can be used
to help compute parity itself. We study -- a hybrid
circuit model where operates on measurement outcomes of a
circuit, and conjecture cannot
achieve correlation with parity. As evidence for this conjecture,
we prove:
When the circuit is ancilla-free, this model
achieves only negligible correlation with parity.
For the general (non-ancilla-free) case, we show via a connection
to nonlocal games that the conjecture holds for any class of postprocessing
functions that has approximate degree and is closed under restrictions,
even when the circuit is given arbitrary quantum advice. By
known results this confirms the conjecture for linear-size
circuits.
Towards a switching lemma for , we study
the effect of quantum preprocessing on the decision tree complexity of Boolean
functions. We find that from this perspective, nonlocal channels are no better
than randomness: a Boolean function precomposed with an -party nonlocal
channel is together equal to a randomized decision tree with worst-case depth
at most .
Our results suggest that while is surprisingly powerful for
search and sampling tasks, that power is "locked away" in the global
correlations of its output, inaccessible to simple classical computation for
solving decision problems.Comment: 26 pages. To appear in ITCS 2024. This revision: many typos fixed,
some statements clarifie
Device-independent and semi-device-independent entanglement certification in broadcast Bell scenarios
It has recently been shown that by broadcasting the subsystems of a bipartite
quantum state, one can activate Bell nonlocality and significantly improve
noise tolerance bounds for device-independent entanglement certification. In
this work we strengthen these results and explore new aspects of this
phenomenon. First, we prove new results related to the activation of Bell
nonlocality. We construct Bell inequalities tailored to the broadcast scenario,
and show how broadcasting can lead to even stronger notions of Bell nonlocality
activation. In particular, we exploit these ideas to show that bipartite states
admitting a local hidden-variable model for general measurements can lead to
genuine tripartite nonlocal correlations. We then study device-independent
entanglement certification in the broadcast scenario, and show through
semidefinite programming techniques that device-independent entanglement
certification is possible for the two-qubit Werner state in essentially the
entire range of entanglement. Finally, we extend the concept of EPR steering to
the broadcast scenario, and present novel examples of activation of the
two-qubit isotropic state. Our results pave the way for broadcast-based
device-dependent and semi-device-independent protocols.Comment: Updated appendices, 28 pages, 4 figure