139 research outputs found

### Quantum Recommendation Systems

A recommendation system uses the past purchases or ratings of $n$ products by
a group of $m$ users, in order to provide personalized recommendations to
individual users. The information is modeled as an $m \times n$ preference
matrix which is assumed to have a good rank-$k$ approximation, for a small
constant $k$.
In this work, we present a quantum algorithm for recommendation systems that
has running time $O(\text{poly}(k)\text{polylog}(mn))$. All known classical
algorithms for recommendation systems that work through reconstructing an
approximation of the preference matrix run in time polynomial in the matrix
dimension. Our algorithm provides good recommendations by sampling efficiently
from an approximation of the preference matrix, without reconstructing the
entire matrix. For this, we design an efficient quantum procedure to project a
given vector onto the row space of a given matrix. This is the first algorithm
for recommendation systems that runs in time polylogarithmic in the dimensions
of the matrix and provides an example of a quantum machine learning algorithm
for a real world application.Comment: 22 page

### A Quantum Interior Point Method for LPs and SDPs

We present a quantum interior point method with worst case running time
$\widetilde{O}(\frac{n^{2.5}}{\xi^{2}} \mu \kappa^3 \log (1/\epsilon))$ for
SDPs and $\widetilde{O}(\frac{n^{1.5}}{\xi^{2}} \mu \kappa^3 \log
(1/\epsilon))$ for LPs, where the output of our algorithm is a pair of matrices
$(S,Y)$ that are $\epsilon$-optimal $\xi$-approximate SDP solutions. The factor
$\mu$ is at most $\sqrt{2}n$ for SDPs and $\sqrt{2n}$ for LP's, and $\kappa$ is
an upper bound on the condition number of the intermediate solution matrices.
For the case where the intermediate matrices for the interior point method are
well conditioned, our method provides a polynomial speedup over the best known
classical SDP solvers and interior point based LP solvers, which have a worst
case running time of $O(n^{6})$ and $O(n^{3.5})$ respectively. Our results
build upon recently developed techniques for quantum linear algebra and pave
the way for the development of quantum algorithms for a variety of applications
in optimization and machine learning.Comment: 32 page

### Increasing the power of the verifier in Quantum Zero Knowledge

In quantum zero knowledge, the assumption was made that the verifier is only
using unitary operations. Under this assumption, many nice properties have been
shown about quantum zero knowledge, including the fact that Honest-Verifier
Quantum Statistical Zero Knowledge (HVQSZK) is equal to Cheating-Verifier
Quantum Statistical Zero Knowledge (QSZK) (see [Wat02,Wat06]).
In this paper, we study what happens when we allow an honest verifier to flip
some coins in addition to using unitary operations. Flipping a coin is a
non-unitary operation but doesn't seem at first to enhance the cheating
possibilities of the verifier since a classical honest verifier can flip coins.
In this setting, we show an unexpected result: any classical Interactive Proof
has an Honest-Verifier Quantum Statistical Zero Knowledge proof with coins.
Note that in the classical case, honest verifier SZK is no more powerful than
SZK and hence it is not believed to contain even NP. On the other hand, in the
case of cheating verifiers, we show that Quantum Statistical Zero Knowledge
where the verifier applies any non-unitary operation is equal to Quantum
Zero-Knowledge where the verifier uses only unitaries.
One can think of our results in two complementary ways. If we would like to
use the honest verifier model as a means to study the general model by taking
advantage of their equivalence, then it is imperative to use the unitary
definition without coins, since with the general one this equivalence is most
probably not true. On the other hand, if we would like to use quantum zero
knowledge protocols in a cryptographic scenario where the honest-but-curious
model is sufficient, then adding the unitary constraint severely decreases the
power of quantum zero knowledge protocols.Comment: 17 pages, 0 figures, to appear in FSTTCS'0

### Quantum Clock Synchronization with one qubit

The clock synchronization problem is to determine the time difference T
between two spatially separated parties. We improve on I. Chuang's quantum
clock synchronization algorithm and show that it is possible to obtain T to n
bits of accuracy while communicating only one qubit in one direction and using
an O(2^n) frequency range. We also prove a quantum lower bound of \Omega(2^n)
for the product of the transmitted qubits and the range of frequencies, thus
showing that our algorithm is optimal.Comment: LaTeX, 5 page

### Statistical Zero Knowledge and quantum one-way functions

One-way functions are a very important notion in the field of classical
cryptography. Most examples of such functions, including factoring, discrete
log or the RSA function, can be, however, inverted with the help of a quantum
computer. In this paper, we study one-way functions that are hard to invert
even by a quantum adversary and describe a set of problems which are good such
candidates. These problems include Graph Non-Isomorphism, approximate Closest
Lattice Vector and Group Non-Membership. More generally, we show that any hard
instance of Circuit Quantum Sampling gives rise to a quantum one-way function.
By the work of Aharonov and Ta-Shma, this implies that any language in
Statistical Zero Knowledge which is hard-on-average for quantum computers,
leads to a quantum one-way function. Moreover, extending the result of
Impagliazzo and Luby to the quantum setting, we prove that quantum
distributionally one-way functions are equivalent to quantum one-way functions.
Last, we explore the connections between quantum one-way functions and the
complexity class QMA and show that, similarly to the classical case, if any of
the above candidate problems is QMA-complete then the existence of quantum
one-way functions leads to the separation of QMA and AvgBQP.Comment: 20 pages; Computational Complexity, Cryptography and Quantum Physics;
Published version, main results unchanged, presentation improve

- …