15 research outputs found
Weak Fourier-Schur sampling, the hidden subgroup problem, and the quantum collision problem
Schur duality decomposes many copies of a quantum state into subspaces
labeled by partitions, a decomposition with applications throughout quantum
information theory. Here we consider applying Schur duality to the problem of
distinguishing coset states in the standard approach to the hidden subgroup
problem. We observe that simply measuring the partition (a procedure we call
weak Schur sampling) provides very little information about the hidden
subgroup. Furthermore, we show that under quite general assumptions, even a
combination of weak Fourier sampling and weak Schur sampling fails to identify
the hidden subgroup. We also prove tight bounds on how many coset states are
required to solve the hidden subgroup problem by weak Schur sampling, and we
relate this question to a quantum version of the collision problem.Comment: 21 page
Symmetric functions of qubits in an unknown basis
Consider an n qubit computational basis state corresponding to a bit string
x, which has had an unknown local unitary applied to each qubit, and whose
qubits have been reordered by an unknown permutation. We show that, given such
a state with Hamming weight |x| at most n/2, it is possible to reconstruct |x|
with success probability 1 - |x|/(n-|x|+1), and thus to compute any symmetric
function of x. We give explicit algorithms for computing whether or not |x| is
at least t for some t, and for computing the parity of x, and show that these
are essentially optimal. These results can be seen as generalisations of the
swap test for comparing quantum states.Comment: 6 pages, 3 figures; v2: improved results, essentially published
versio
Quantum Pseudoentanglement
Quantum pseudorandom states are efficiently constructable states which
nevertheless masquerade as Haar-random states to poly-time observers. First
defined by Ji, Liu and Song, such states have found a number of applications
ranging from cryptography to the AdS/CFT correspondence. A fundamental question
is exactly how much entanglement is required to create such states. Haar-random
states, as well as -designs for , exhibit near-maximal
entanglement. Here we provide the first construction of pseudorandom states
with only polylogarithmic entanglement entropy across an equipartition of the
qubits, which is the minimum possible. Our construction can be based on any
one-way function secure against quantum attack. We additionally show that the
entanglement in our construction is fully "tunable", in the sense that one can
have pseudorandom states with entanglement for any desired
function .
More fundamentally, our work calls into question to what extent entanglement
is a "feelable" quantity of quantum systems. Inspired by recent work of
Gheorghiu and Hoban, we define a new notion which we call "pseudoentanglement",
which are ensembles of efficiently constructable quantum states which hide
their entanglement entropy. We show such states exist in the strongest form
possible while simultaneously being pseudorandom states. We also describe
diverse applications of our result from entanglement distillation to property
testing to quantum gravity.Comment: 32 page
Weak Fourier-Schur Sampling, The Hidden Subgroup Problem, And The Quantum Collision Problem
Schur duality decomposes many copies of a quantum state into subspaces labeled by partitions, a decomposition with applications throughout quantum information theory. Here we consider applying Schur duality to the problem of distinguishing coset states in the standard approach to the hidden subgroup problem. We observe that simply measuring the partition (a procedure we call weak Schur sampling) provides very little information about the hidden subgroup. Furthermore, we show that under quite general assumptions, even a combination of weak Fourier sampling and weak Schur sampling fails to identify the hidden subgroup. We also prove tight bounds on how many coset states are required to solve the hidden subgroup problem by weak Schur sampling, and we relate this question to a quantum version of the collision problem. © Springer-Verlag Berlin Heidelberg 2007
Quantum Spectrum Testing
In this work, we study the problem of testing properties of the spectrum of a
mixed quantum state. Here one is given copies of a mixed state
and the goal is to distinguish whether 's
spectrum satisfies some property or is at least -far in
-distance from satisfying . This problem was promoted in
the survey of Montanaro and de Wolf under the name of testing unitarily
invariant properties of mixed states. It is the natural quantum analogue of the
classical problem of testing symmetric properties of probability distributions.
Here, the hope is for algorithms with subquadratic copy complexity in the
dimension . This is because the "empirical Young diagram (EYD) algorithm"
can estimate the spectrum of a mixed state up to -accuracy using only
copies. In this work, we show that given a
mixed state : (i) copies
are necessary and sufficient to test whether is the maximally mixed
state, i.e., has spectrum ; (ii)
copies are necessary and sufficient to test with
one-sided error whether has rank , i.e., has at most nonzero
eigenvalues; (iii) copies are necessary and
sufficient to distinguish whether is maximally mixed on an
-dimensional or an -dimensional subspace; and (iv) The EYD
algorithm requires copies to estimate the spectrum of
up to -accuracy, nearly matching the known upper bound. In
addition, we simplify part of the proof of the upper bound. Our techniques
involve the asymptotic representation theory of the symmetric group; in
particular Kerov's algebra of polynomial functions on Young diagrams.Comment: 70 pages, 6 figure