6 research outputs found
Quantum Random Access Codes for Boolean Functions
An random access code (RAC) is an encoding of
bits into bits such that any initial bit can be recovered with probability
at least , while in a quantum RAC (QRAC), the bits are encoded into
qubits. Since its proposal, the idea of RACs was generalized in many different
ways, e.g. allowing the use of shared entanglement (called
entanglement-assisted random access code, or simply EARAC) or recovering
multiple bits instead of one. In this paper we generalize the idea of RACs to
recovering the value of a given Boolean function on any subset of fixed
size of the initial bits, which we call -random access codes. We study and
give protocols for -random access codes with classical (-RAC) and quantum
(-QRAC) encoding, together with many different resources, e.g. private or
shared randomness, shared entanglement (-EARAC) and Popescu-Rohrlich boxes
(-PRRAC). The success probability of our protocols is characterized by the
\emph{noise stability} of the Boolean function . Moreover, we give an
\emph{upper bound} on the success probability of any -QRAC with shared
randomness that matches its success probability up to a multiplicative constant
(and -RACs by extension), meaning that quantum protocols can only achieve a
limited advantage over their classical counterparts.Comment: Final version to appear in Quantum. Small improvements to Theorem 2
Quantum algorithm for robust optimization via stochastic-gradient online learning
Optimization theory has been widely studied in academia and finds a large
variety of applications in industry. The different optimization models in their
discrete and/or continuous settings has catered to a rich source of research
problems. Robust convex optimization is a branch of optimization theory in
which the variables or parameters involved have a certain level of uncertainty.
In this work, we consider the online robust optimization meta-algorithm by
Ben-Tal et al. and show that for a large range of stochastic subgradients, this
algorithm has the same guarantee as the original non-stochastic version. We
develop a quantum version of this algorithm and show that an at most quadratic
improvement in terms of the dimension can be achieved. The speedup is due to
the use of quantum state preparation, quantum norm estimation, and quantum
multi-sampling. We apply our quantum meta-algorithm to examples such as robust
linear programs and robust semidefinite programs and give applications of these
robust optimization problems in finance and engineering.Comment: 21 page
Constant-depth circuits for Uniformly Controlled Gates and Boolean functions with application to quantum memory circuits
We explore the power of the unbounded Fan-Out gate and the Global Tunable
gates generated by Ising-type Hamiltonians in constructing constant-depth
quantum circuits, with particular attention to quantum memory devices. We
propose two types of constant-depth constructions for implementing Uniformly
Controlled Gates. These gates include the Fan-In gates defined by
for
and , where is a Boolean function. The first of our
constructions is based on computing the one-hot encoding of the control
register , while the second is based on Boolean analysis and
exploits different representations of such as its Fourier expansion. Via
these constructions, we obtain constant-depth circuits for the quantum
counterparts of read-only and read-write memory devices -- Quantum Random
Access Memory (QRAM) and Quantum Random Access Gate (QRAG) -- of memory size
. The implementation based on one-hot encoding requires either
ancillae and Fan-Out gates or
ancillae and Global Tunable gates. On the other hand, the
implementation based on Boolean analysis requires only Global Tunable gates
at the expense of ancillae.Comment: 50 pages, 10 figures. Comments are welcom
Quantum algorithm for stochastic optimal stopping problems with applications in finance
The famous least squares Monte Carlo (LSM) algorithm combines linear least
square regression with Monte Carlo simulation to approximately solve problems
in stochastic optimal stopping theory. In this work, we propose a quantum LSM
based on quantum access to a stochastic process, on quantum circuits for
computing the optimal stopping times, and on quantum techniques for Monte
Carlo. For this algorithm, we elucidate the intricate interplay of function
approximation and quantum algorithms for Monte Carlo. Our algorithm achieves a
nearly quadratic speedup in the runtime compared to the LSM algorithm under
some mild assumptions. Specifically, our quantum algorithm can be applied to
American option pricing and we analyze a case study for the common situation of
Brownian motion and geometric Brownian motion processes.Comment: 45 pages; v2: title slightly changed, typos fixed, references adde