1,233 research outputs found
A Quasi-Random Approach to Matrix Spectral Analysis
Inspired by the quantum computing algorithms for Linear Algebra problems
[HHL,TaShma] we study how the simulation on a classical computer of this type
of "Phase Estimation algorithms" performs when we apply it to solve the
Eigen-Problem of Hermitian matrices. The result is a completely new, efficient
and stable, parallel algorithm to compute an approximate spectral decomposition
of any Hermitian matrix. The algorithm can be implemented by Boolean circuits
in parallel time with a total cost of Boolean
operations. This Boolean complexity matches the best known rigorous parallel time algorithms, but unlike those algorithms our algorithm is
(logarithmically) stable, so further improvements may lead to practical
implementations.
All previous efficient and rigorous approaches to solve the Eigen-Problem use
randomization to avoid bad condition as we do too. Our algorithm makes further
use of randomization in a completely new way, taking random powers of a unitary
matrix to randomize the phases of its eigenvalues. Proving that a tiny Gaussian
perturbation and a random polynomial power are sufficient to ensure almost
pairwise independence of the phases is the main technical
contribution of this work. This randomization enables us, given a Hermitian
matrix with well separated eigenvalues, to sample a random eigenvalue and
produce an approximate eigenvector in parallel time and
Boolean complexity. We conjecture that further improvements of
our method can provide a stable solution to the full approximate spectral
decomposition problem with complexity similar to the complexity (up to a
logarithmic factor) of sampling a single eigenvector.Comment: Replacing previous version: parallel algorithm runs in total
complexity and not . However, the depth of the
implementing circuit is : hence comparable to fastest
eigen-decomposition algorithms know
Quantum Tokens for Digital Signatures
The fisherman caught a quantum fish. "Fisherman, please let me go", begged
the fish, "and I will grant you three wishes". The fisherman agreed. The fish
gave the fisherman a quantum computer, three quantum signing tokens and his
classical public key. The fish explained: "to sign your three wishes, use the
tokenized signature scheme on this quantum computer, then show your valid
signature to the king, who owes me a favor".
The fisherman used one of the signing tokens to sign the document "give me a
castle!" and rushed to the palace. The king executed the classical verification
algorithm using the fish's public key, and since it was valid, the king
complied.
The fisherman's wife wanted to sign ten wishes using their two remaining
signing tokens. The fisherman did not want to cheat, and secretly sailed to
meet the fish. "Fish, my wife wants to sign ten more wishes". But the fish was
not worried: "I have learned quantum cryptography following the previous story
(The Fisherman and His Wife by the brothers Grimm). The quantum tokens are
consumed during the signing. Your polynomial wife cannot even sign four wishes
using the three signing tokens I gave you".
"How does it work?" wondered the fisherman. "Have you heard of quantum money?
These are quantum states which can be easily verified but are hard to copy.
This tokenized quantum signature scheme extends Aaronson and Christiano's
quantum money scheme, which is why the signing tokens cannot be copied".
"Does your scheme have additional fancy properties?" the fisherman asked.
"Yes, the scheme has other security guarantees: revocability, testability and
everlasting security. Furthermore, if you're at sea and your quantum phone has
only classical reception, you can use this scheme to transfer the value of the
quantum money to shore", said the fish, and swam away.Comment: Added illustration of the abstract to the ancillary file
- …