987 research outputs found
Lower Bounds on the Bounded Coefficient Complexity of Bilinear Maps
We prove lower bounds of order for both the problem to multiply
polynomials of degree , and to divide polynomials with remainder, in the
model of bounded coefficient arithmetic circuits over the complex numbers.
These lower bounds are optimal up to order of magnitude. The proof uses a
recent idea of R. Raz [Proc. 34th STOC 2002] proposed for matrix
multiplication. It reduces the linear problem to multiply a random circulant
matrix with a vector to the bilinear problem of cyclic convolution. We treat
the arising linear problem by extending J. Morgenstern's bound [J. ACM 20, pp.
305-306, 1973] in a unitarily invariant way. This establishes a new lower bound
on the bounded coefficient complexity of linear forms in terms of the singular
values of the corresponding matrix. In addition, we extend these lower bounds
for linear and bilinear maps to a model of circuits that allows a restricted
number of unbounded scalar multiplications.Comment: 19 page
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
General guarantees for randomized benchmarking with random quantum circuits
In its many variants, randomized benchmarking (RB) is a broadly used
technique for assessing the quality of gate implementations on quantum
computers. A detailed theoretical understanding and general guarantees exist
for the functioning and interpretation of RB protocols if the gates under
scrutiny are drawn uniformly at random from a compact group. In contrast, many
practically attractive and scalable RB protocols implement random quantum
circuits with local gates randomly drawn from some gate-set. Despite their
abundance in practice, for those non-uniform RB protocols, general guarantees
under experimentally plausible assumptions are missing. In this work, we derive
such guarantees for a large class of RB protocols for random circuits that we
refer to as filtered RB. Prominent examples include linear cross-entropy
benchmarking, character benchmarking, Pauli-noise tomography and variants of
simultaneous RB. Building upon recent results for random circuits, we show that
many relevant filtered RB schemes can be realized with random quantum circuits
in linear depth, and we provide explicit small constants for common instances.
We further derive general sample complexity bounds for filtered RB. We show
filtered RB to be sample-efficient for several relevant groups, including
protocols addressing higher-order cross-talk. Our theory for non-uniform
filtered RB is, in principle, flexible enough to design new protocols for
non-universal and analog quantum simulators.Comment: 77 pages, 3 figures. Accepted for a talk at QIP 202
Approximate F_2-Sketching of Valuation Functions
We study the problem of constructing a linear sketch of minimum dimension that allows approximation of a given real-valued function f : F_2^n - > R with small expected squared error. We develop a general theory of linear sketching for such functions through which we analyze their dimension for most commonly studied types of valuation functions: additive, budget-additive, coverage, alpha-Lipschitz submodular and matroid rank functions. This gives a characterization of how many bits of information have to be stored about the input x so that one can compute f under additive updates to its coordinates.
Our results are tight in most cases and we also give extensions to the distributional version of the problem where the input x in F_2^n is generated uniformly at random. Using known connections with dynamic streaming algorithms, both upper and lower bounds on dimension obtained in our work extend to the space complexity of algorithms evaluating f(x) under long sequences of additive updates to the input x presented as a stream. Similar results hold for simultaneous communication in a distributed setting
Computational Distinguishability of Quantum Channels
The computational problem of distinguishing two quantum channels is central
to quantum computing. It is a generalization of the well-known satisfiability
problem from classical to quantum computation. This problem is shown to be
surprisingly hard: it is complete for the class QIP of problems that have
quantum interactive proof systems, which implies that it is hard for the class
PSPACE of problems solvable by a classical computation in polynomial space.
Several restrictions of distinguishability are also shown to be hard. It is
no easier when restricted to quantum computations of logarithmic depth, to
mixed-unitary channels, to degradable channels, or to antidegradable channels.
These hardness results are demonstrated by finding reductions between these
classes of quantum channels. These techniques have applications outside the
distinguishability problem, as the construction for mixed-unitary channels is
used to prove that the additivity problem for the classical capacity of quantum
channels can be equivalently restricted to the mixed unitary channels.Comment: Ph.D. Thesis, 178 pages, 35 figure
Approximate degree in classical and quantum computing
In this book, the authors survey what is known about a particularly natural notion of approximation by polynomials, capturing pointwise approximation over the real numbers.FG-2022-18482 - Alfred P. Sloan Foundation; CNS-2046425 - National Science Foundation; CCF-1947889 - National Science FoundationAccepted manuscrip
Concrete resource analysis of the quantum linear system algorithm used to compute the electromagnetic scattering cross section of a 2D target
We provide a detailed estimate for the logical resource requirements of the
quantum linear system algorithm (QLSA) [Phys. Rev. Lett. 103, 150502 (2009)]
including the recently described elaborations [Phys. Rev. Lett. 110, 250504
(2013)]. Our resource estimates are based on the standard quantum-circuit model
of quantum computation; they comprise circuit width, circuit depth, the number
of qubits and ancilla qubits employed, and the overall number of elementary
quantum gate operations as well as more specific gate counts for each
elementary fault-tolerant gate from the standard set {X, Y, Z, H, S, T, CNOT}.
To perform these estimates, we used an approach that combines manual analysis
with automated estimates generated via the Quipper quantum programming language
and compiler. Our estimates pertain to the example problem size N=332,020,680
beyond which, according to a crude big-O complexity comparison, QLSA is
expected to run faster than the best known classical linear-system solving
algorithm. For this problem size, a desired calculation accuracy 0.01 requires
an approximate circuit width 340 and circuit depth of order if oracle
costs are excluded, and a circuit width and depth of order and
, respectively, if oracle costs are included, indicating that the
commonly ignored oracle resources are considerable. In addition to providing
detailed logical resource estimates, it is also the purpose of this paper to
demonstrate explicitly how these impressively large numbers arise with an
actual circuit implementation of a quantum algorithm. While our estimates may
prove to be conservative as more efficient advanced quantum-computation
techniques are developed, they nevertheless provide a valid baseline for
research targeting a reduction of the resource requirements, implying that a
reduction by many orders of magnitude is necessary for the algorithm to become
practical.Comment: 37 pages, 40 figure
Group-theoretic error mitigation enabled by classical shadows and symmetries
Estimating expectation values is a key subroutine in many quantum algorithms.
However, near-term implementations face two major challenges: a limited number
of samples to learn a large collection of observables, and the accumulation of
errors in devices without quantum error correction. To address these challenges
simultaneously, we develop a quantum error-mitigation strategy which unifies
the group-theoretic structure of classical-shadow tomography with symmetries in
quantum systems of interest. We refer to our protocol as "symmetry-adjusted
classical shadows," as it mitigates errors by adjusting estimators according to
how known symmetries are corrupted under those errors. As a concrete example,
we highlight global symmetry, which manifests in fermions as
particle number and in spins as total magnetization, and illustrate their
unification with respective classical-shadow protocols. One of our main results
establishes rigorous error and sampling bounds under readout errors obeying
minimal assumptions. Furthermore, to probe mitigation capabilities against a
more comprehensive class of gate-level errors, we perform numerical experiments
with a noise model derived from existing quantum processors. Our analytical and
numerical results reveal symmetry-adjusted classical shadows as a flexible and
low-cost strategy to mitigate errors from noisy quantum experiments in the
ubiquitous presence of symmetry.Comment: 45 pages, 13 figures. Typos corrected and references updated.
Open-source code available at
https://github.com/zhao-andrew/symmetry-adjusted-classical-shadow
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