1,764 research outputs found
Polynomial Threshold Functions, AC^0 Functions and Spectral Norms
The class of polynomial-threshold functions is studied using harmonic analysis, and the results are used to derive lower bounds related to AC^0 functions. A Boolean function is polynomial threshold if it can be represented as a sign function of a sparse polynomial (one that consists of a polynomial number of terms). The main result is that polynomial-threshold functions can be characterized by means of their spectral representation. In particular, it is proved that a Boolean function whose L_1 spectral norm is bounded by a polynomial in n is a polynomial-threshold function, and that a Boolean function whose L_∞^(-1) spectral norm is not bounded by a polynomial in n is not a polynomial-threshold function. Some results for AC^0 functions are derived
A Full Characterization of Quantum Advice
We prove the following surprising result: given any quantum state rho on n
qubits, there exists a local Hamiltonian H on poly(n) qubits (e.g., a sum of
two-qubit interactions), such that any ground state of H can be used to
simulate rho on all quantum circuits of fixed polynomial size. In terms of
complexity classes, this implies that BQP/qpoly is contained in QMA/poly, which
supersedes the previous result of Aaronson that BQP/qpoly is contained in
PP/poly. Indeed, we can exactly characterize quantum advice, as equivalent in
power to untrusted quantum advice combined with trusted classical advice.
Proving our main result requires combining a large number of previous tools --
including a result of Alon et al. on learning of real-valued concept classes, a
result of Aaronson on the learnability of quantum states, and a result of
Aharonov and Regev on "QMA+ super-verifiers" -- and also creating some new
ones. The main new tool is a so-called majority-certificates lemma, which is
closely related to boosting in machine learning, and which seems likely to find
independent applications. In its simplest version, this lemma says the
following. Given any set S of Boolean functions on n variables, any function f
in S can be expressed as the pointwise majority of m=O(n) functions f1,...,fm
in S, such that each fi is the unique function in S compatible with O(log|S|)
input/output constraints.Comment: We fixed two significant issues: 1. The definition of YQP machines
needed to be changed to preserve our results. The revised definition is more
natural and has the same intuitive interpretation. 2. We needed properties of
Local Hamiltonian reductions going beyond those proved in previous works
(whose results we'd misstated). We now prove the needed properties. See p. 6
for more on both point
Signal Propagation, with Application to a Lower Bound on the Depth of Noisy Formulas
We study the decay of an information signal propagating through a series of noisy channels. We obtain exact bounds on such decay, and as a result provide a new lower bound on the depth of formulas with noisy components. This improves upon previous work of N. Pippenger and significantly decreases the gap between his lower bound and the classical upper bound of von Neumann. We also discuss connections between our work and the study of mixing rates of Markov chains
Approximate unitary -designs by short random quantum circuits using nearest-neighbor and long-range gates
We prove that -depth local random quantum circuits
with two qudit nearest-neighbor gates on a -dimensional lattice with n
qudits are approximate -designs in various measures. These include the
"monomial" measure, meaning that the monomials of a random circuit from this
family have expectation close to the value that would result from the Haar
measure. Previously, the best bound was due to
Brandao-Harrow-Horodecki (BHH) for . We also improve the "scrambling" and
"decoupling" bounds for spatially local random circuits due to Brown and Fawzi.
One consequence of our result is that assuming the polynomial hierarchy (PH)
is infinite and that certain counting problems are -hard on average,
sampling within total variation distance from these circuits is hard for
classical computers. Previously, exact sampling from the outputs of even
constant-depth quantum circuits was known to be hard for classical computers
under the assumption that PH is infinite. However, to show the hardness of
approximate sampling using this strategy requires that the quantum circuits
have a property called "anti-concentration", meaning roughly that the output
has near-maximal entropy. Unitary 2-designs have the desired anti-concentration
property. Thus our result improves the required depth for this level of
anti-concentration from linear depth to a sub-linear value, depending on the
geometry of the interactions. This is relevant to a recent proposal by the
Google Quantum AI group to perform such a sampling task with 49 qubits on a
two-dimensional lattice and confirms their conjecture that depth
suffices for anti-concentration. We also prove that anti-concentration is
possible in depth O(log(n) loglog(n)) using a different model
The Role of Correlated Noise in Quantum Computing
This paper aims to give an overview of the current state of fault-tolerant
quantum computing, by surveying a number of results in the field. We show that
thresholds can be obtained for a simple noise model as first proved in [AB97,
Kit97, KLZ98], by presenting a proof for statistically independent noise,
following the presentation of Aliferis, Gottesman and Preskill [AGP06]. We also
present a result by Terhal and Burkard [TB05] and later improved upon by
Aliferis, Gottesman and Preskill [AGP06] that shows a threshold can still be
obtained for local non-Markovian noise, where we allow the noise to be weakly
correlated in space and time. We then turn to negative results, presenting work
by Ben-Aroya and Ta-Shma [BT11] who showed conditional errors cannot be
perfectly corrected. We end our survey by briefly mentioning some more
speculative objections, as put forth by Kalai [Kal08, Kal09, Kal11]
Efficient Online Quantum Generative Adversarial Learning Algorithms with Applications
The exploration of quantum algorithms that possess quantum advantages is a
central topic in quantum computation and quantum information processing. One
potential candidate in this area is quantum generative adversarial learning
(QuGAL), which conceptually has exponential advantages over classical
adversarial networks. However, the corresponding learning algorithm remains
obscured. In this paper, we propose the first quantum generative adversarial
learning algorithm-- the quantum multiplicative matrix weight algorithm
(QMMW)-- which enables the efficient processing of fundamental tasks. The
computational complexity of QMMW is polynomially proportional to the number of
training rounds and logarithmically proportional to the input size. The core
concept of the proposed algorithm combines QuGAL with online learning. We
exploit the implementation of QuGAL with parameterized quantum circuits, and
numerical experiments for the task of entanglement test for pure state are
provided to support our claims
Implementing Unitary 2-Designs Using Random Diagonal-unitary Matrices
Unitary 2-designs are random unitary matrices which, in contrast to their Haar-distributed counterparts, have been shown to be efficiently realized by quantum circuits. Most notably, unitary 2-designs are known to achieve decoupling, a fundamental primitive of paramount importance in quantum Shannon theory. Here we prove that unitary 2-designs can be implemented approximately using random diagonal-unitaries
The battle of clean and dirty qubits in the era of partial error correction
When error correction becomes possible it will be necessary to dedicate a
large number of physical qubits to each logical qubit. Error correction allows
for deeper circuits to be run, but each additional physical qubit can
potentially contribute an exponential increase in computational space, so there
is a trade-off between using qubits for error correction or using them as noisy
qubits. In this work we look at the effects of using noisy qubits in
conjunction with noiseless qubits (an idealized model for error-corrected
qubits), which we call the "clean and dirty" setup. We employ analytical models
and numerical simulations to characterize this setup. Numerically we show the
appearance of Noise-Induced Barren Plateaus (NIBPs), i.e., an exponential
concentration of observables caused by noise, in an Ising model Hamiltonian
variational ansatz circuit. We observe this even if only a single qubit is
noisy and given a deep enough circuit, suggesting that NIBPs cannot be fully
overcome simply by error-correcting a subset of the qubits. On the positive
side, we find that for every noiseless qubit in the circuit, there is an
exponential suppression in concentration of gradient observables, showing the
benefit of partial error correction. Finally, our analytical models corroborate
these findings by showing that observables concentrate with a scaling in the
exponent related to the ratio of dirty-to-total qubits.Comment: 27 pages, 15 figures, (v2) minor change
- …