30 research outputs found
The Equivalence of Sampling and Searching
In a sampling problem, we are given an input x, and asked to sample
approximately from a probability distribution D_x. In a search problem, we are
given an input x, and asked to find a member of a nonempty set A_x with high
probability. (An example is finding a Nash equilibrium.) In this paper, we use
tools from Kolmogorov complexity and algorithmic information theory to show
that sampling and search problems are essentially equivalent. More precisely,
for any sampling problem S, there exists a search problem R_S such that, if C
is any "reasonable" complexity class, then R_S is in the search version of C if
and only if S is in the sampling version. As one application, we show that
SampP=SampBQP if and only if FBPP=FBQP: in other words, classical computers can
efficiently sample the output distribution of every quantum circuit, if and
only if they can efficiently solve every search problem that quantum computers
can solve. A second application is that, assuming a plausible conjecture, there
exists a search problem R that can be solved using a simple linear-optics
experiment, but that cannot be solved efficiently by a classical computer
unless the polynomial hierarchy collapses. That application will be described
in a forthcoming paper with Alex Arkhipov on the computational complexity of
linear optics.Comment: 16 page
A single-shot measurement of the energy of product states in a translation invariant spin chain can replace any quantum computation
In measurement-based quantum computation, quantum algorithms are implemented
via sequences of measurements. We describe a translationally invariant
finite-range interaction on a one-dimensional qudit chain and prove that a
single-shot measurement of the energy of an appropriate computational basis
state with respect to this Hamiltonian provides the output of any quantum
circuit. The required measurement accuracy scales inverse polynomially with the
size of the simulated quantum circuit. This shows that the implementation of
energy measurements on generic qudit chains is as hard as the realization of
quantum computation. Here a ''measurement'' is any procedure that samples from
the spectral measure induced by the observable and the state under
consideration. As opposed to measurement-based quantum computation, the
post-measurement state is irrelevant.Comment: 19 pages, transition rules for the CA correcte
Classical simulation of Yang-Baxter gates
A unitary operator that satisfies the constant Yang-Baxter equation
immediately yields a unitary representation of the braid group B n for every . If we view such an operator as a quantum-computational gate, then
topological braiding corresponds to a quantum circuit. A basic question is when
such a representation affords universal quantum computation. In this work, we
show how to classically simulate these circuits when the gate in question
belongs to certain families of solutions to the Yang-Baxter equation. These
include all of the qubit (i.e., ) solutions, and some simple families
that include solutions for arbitrary . Our main tool is a
probabilistic classical algorithm for efficient simulation of a more general
class of quantum circuits. This algorithm may be of use outside the present
setting.Comment: 17 pages. Corrected error in proof of Theorem
Hybrid quantum-classical and quantum-inspired classical algorithms for solving banded circulant linear systems
Solving linear systems is of great importance in numerous fields. In
particular, circulant systems are especially valuable for efficiently finding
numerical solutions to physics-related differential equations. Current quantum
algorithms like HHL or variational methods are either resource-intensive or may
fail to find a solution. We present an efficient algorithm based on convex
optimization of combinations of quantum states to solve for banded circulant
linear systems whose non-zero terms are within distance of the main
diagonal. By decomposing banded circulant matrices into cyclic permutations,
our approach produces approximate solutions to such systems with a combination
of quantum states linear to , significantly improving over previous
convergence guarantees, which require quantum states exponential to . We
propose a hybrid quantum-classical algorithm using the Hadamard test and the
quantum Fourier transform as subroutines and show its PromiseBQP-hardness.
Additionally, we introduce a quantum-inspired algorithm with similar
performance given sample and query access. We validate our methods with
classical simulations and actual IBM quantum computer implementation,
showcasing their applicability for solving physical problems such as heat
transfer.Comment: 21 pages, 12 figure
The power of fixing a few qubits in proofs
What could happen if we pinned a single qubit of a system and fixed it in a particular state? First, we show that this leads to difficult static questions about the ground-state properties of local Hamiltonian problems with restricted types of terms. In particular, we show that the pinned commuting and pinned stoquastic Local Hamiltonian problems are quantum-Merlin-Arthur–complete. Second, we investigate pinned dynamics and demonstrate that fixing a single qubit via often repeated measurements results in universal quantum computation with commuting Hamiltonians. Finally, we discuss variants of the ground-state connectivity (GSCON) problem in light of pinning, and show that stoquastic GSCON is quantum-classical Merlin-Arthur–complete