51,818 research outputs found
An average-case depth hierarchy theorem for Boolean circuits
We prove an average-case depth hierarchy theorem for Boolean circuits over
the standard basis of , , and gates.
Our hierarchy theorem says that for every , there is an explicit
-variable Boolean function , computed by a linear-size depth- formula,
which is such that any depth- circuit that agrees with on fraction of all inputs must have size This
answers an open question posed by H{\aa}stad in his Ph.D. thesis.
Our average-case depth hierarchy theorem implies that the polynomial
hierarchy is infinite relative to a random oracle with probability 1,
confirming a conjecture of H{\aa}stad, Cai, and Babai. We also use our result
to show that there is no "approximate converse" to the results of Linial,
Mansour, Nisan and Boppana on the total influence of small-depth circuits, thus
answering a question posed by O'Donnell, Kalai, and Hatami.
A key ingredient in our proof is a notion of \emph{random projections} which
generalize random restrictions
Bounding quantum gate error rate based on reported average fidelity
Remarkable experimental advances in quantum computing are exemplified by
recent announcements of impressive average gate fidelities exceeding 99.9% for
single-qubit gates and 99% for two-qubit gates. Although these high numbers
engender optimism that fault-tolerant quantum computing is within reach, the
connection of average gate fidelity with fault-tolerance requirements is not
direct. Here we use reported average gate fidelity to determine an upper bound
on the quantum-gate error rate, which is the appropriate metric for assessing
progress towards fault-tolerant quantum computation, and we demonstrate that
this bound is asymptotically tight for general noise. Although this bound is
unlikely to be saturated by experimental noise, we demonstrate using explicit
examples that the bound indicates a realistic deviation between the true error
rate and the reported average fidelity. We introduce the Pauli distance as a
measure of this deviation, and we show that knowledge of the Pauli distance
enables tighter estimates of the error rate of quantum gates.Comment: New Journal of Physics Fast Track Communication. Gold open access
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Consistency of circuit lower bounds with bounded theories
Proving that there are problems in that require
boolean circuits of super-linear size is a major frontier in complexity theory.
While such lower bounds are known for larger complexity classes, existing
results only show that the corresponding problems are hard on infinitely many
input lengths. For instance, proving almost-everywhere circuit lower bounds is
open even for problems in . Giving the notorious difficulty of
proving lower bounds that hold for all large input lengths, we ask the
following question: Can we show that a large set of techniques cannot prove
that is easy infinitely often? Motivated by this and related
questions about the interaction between mathematical proofs and computations,
we investigate circuit complexity from the perspective of logic.
Among other results, we prove that for any parameter it is
consistent with theory that computational class , where is one of
the pairs: and , and , and
. In other words, these theories cannot establish
infinitely often circuit upper bounds for the corresponding problems. This is
of interest because the weaker theory already formalizes
sophisticated arguments, such as a proof of the PCP Theorem. These consistency
statements are unconditional and improve on earlier theorems of [KO17] and
[BM18] on the consistency of lower bounds with
Measurement cost of metric-aware variational quantum algorithms
Variational quantum algorithms are promising tools for near-term quantum
computers as their shallow circuits are robust to experimental imperfections.
Their practical applicability, however, strongly depends on how many times
their circuits need to be executed for sufficiently reducing shot-noise. We
consider metric-aware quantum algorithms: variational algorithms that use a
quantum computer to efficiently estimate both a matrix and a vector object. For
example, the recently introduced quantum natural gradient approach uses the
quantum Fisher information matrix as a metric tensor to correct the gradient
vector for the co-dependence of the circuit parameters. We rigorously
characterise and upper bound the number of measurements required to determine
an iteration step to a fixed precision, and propose a general approach for
optimally distributing samples between matrix and vector entries. Finally, we
establish that the number of circuit repetitions needed for estimating the
quantum Fisher information matrix is asymptotically negligible for an
increasing number of iterations and qubits.Comment: 17 pages, 3 figure
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