117 research outputs found
Quantum Hypothesis Testing with Group Structure
The problem of discriminating between many quantum channels with certainty is
analyzed under the assumption of prior knowledge of algebraic relations among
possible channels. It is shown, by explicit construction of a novel family of
quantum algorithms, that when the set of possible channels faithfully
represents a finite subgroup of SU(2) (e.g., ) the
recently-developed techniques of quantum signal processing can be modified to
constitute subroutines for quantum hypothesis testing. These algorithms, for
group quantum hypothesis testing (G-QHT), intuitively encode discrete
properties of the channel set in SU(2) and improve query complexity at least
quadratically in , the size of the channel set and group, compared to
na\"ive repetition of binary hypothesis testing. Intriguingly, performance is
completely defined by explicit group homomorphisms; these in turn inform simple
constraints on polynomials embedded in unitary matrices. These constructions
demonstrate a flexible technique for mapping questions in quantum inference to
the well-understood subfields of functional approximation and discrete algebra.
Extensions to larger groups and noisy settings are discussed, as well as paths
by which improved protocols for quantum hypothesis testing against structured
channel sets have application in the transmission of reference frames, proofs
of security in quantum cryptography, and algorithms for property testing.Comment: 22 pages + 9 figures + 3 table
Multivariable quantum signal processing (M-QSP): prophecies of the two-headed oracle
Recent work shows that quantum signal processing (QSP) and its multi-qubit
lifted version, quantum singular value transformation (QSVT), unify and improve
the presentation of most quantum algorithms. QSP/QSVT characterize the ability,
by alternating ans\"atze, to obliviously transform the singular values of
subsystems of unitary matrices by polynomial functions; these algorithms are
numerically stable and analytically well-understood. That said, QSP/QSVT
require consistent access to a single oracle, saying nothing about computing
joint properties of two or more oracles; these can be far cheaper to determine
given an ability to pit oracles against one another coherently.
This work introduces a corresponding theory of QSP over multiple variables:
M-QSP. Surprisingly, despite the non-existence of the fundamental theorem of
algebra for multivariable polynomials, there exist necessary and sufficient
conditions under which a desired stable multivariable polynomial transformation
is possible. Moreover, the classical subroutines used by QSP protocols survive
in the multivariable setting for non-obvious reasons, and remain numerically
stable and efficient. Up to a well-defined conjecture, we give proof that the
family of achievable multivariable transforms is as loosely constrained as
could be expected. The unique ability of M-QSP to obliviously approximate joint
functions of multiple variables coherently leads to novel speedups
incommensurate with those of other quantum algorithms, and provides a bridge
from quantum algorithms to algebraic geometry.Comment: 23 pages + 4 figures + 10 page appendix (added background information
on algebraic geometry; publication in Quantum
Circuit Transformations for Quantum Architectures
Quantum computer architectures impose restrictions on qubit interactions. We propose efficient circuit transformations that modify a given quantum circuit to fit an architecture, allowing for any initial and final mapping of circuit qubits to architecture qubits. To achieve this, we first consider the qubit movement subproblem and use the ROUTING VIA MATCHINGS framework to prove tighter bounds on parallel routing. In practice, we only need to perform partial permutations, so we generalize ROUTING VIA MATCHINGS to that setting. We give new routing procedures for common architecture graphs and for the generalized hierarchical product of graphs, which produces subgraphs of the Cartesian product. Secondly, for serial routing, we consider the TOKEN SWAPPING framework and extend a 4-approximation algorithm for general graphs to support partial permutations. We apply these routing procedures to give several circuit transformations, using various heuristic qubit placement subroutines. We implement these transformations in software and compare their performance for large quantum circuits on grid and modular architectures, identifying strategies that work well in practice
Spacetime-Efficient Low-Depth Quantum State Preparation with Applications
We propose a novel deterministic method for preparing arbitrary quantum
states. When our protocol is compiled into CNOT and arbitrary single-qubit
gates, it prepares an -dimensional state in depth and spacetime
allocation (a metric that accounts for the fact that oftentimes some ancilla
qubits need not be active for the entire circuit) , which are both
optimal. When compiled into the gate set, we show
that it requires asymptotically fewer quantum resources than previous methods.
Specifically, it prepares an arbitrary state up to error in depth
and spacetime allocation ,
improving over and ,
respectively. We illustrate how the reduced spacetime allocation of our
protocol enables rapid preparation of many disjoint states with only
constant-factor ancilla overhead -- ancilla qubits are reused
efficiently to prepare a product state of -dimensional states in depth
rather than , achieving effectively constant
depth per state. We highlight several applications where this ability would be
useful, including quantum machine learning, Hamiltonian simulation, and solving
linear systems of equations. We provide quantum circuit descriptions of our
protocol, detailed pseudocode, and gate-level implementation examples using
Braket
Does qubit connectivity impact quantum circuit complexity?
Some physical implementation schemes of quantum computing can apply two-qubit
gates only on certain pairs of qubits. These connectivity constraints are
commonly viewed as a significant disadvantage. For example, compiling an
unrestricted -qubit quantum circuit to one with poor qubit connectivity,
such as a 1D chain, usually results in a blowup of depth by and size
by . It is appealing to conjecture that this overhead is unavoidable -- a
random circuit on qubits has two-qubit gates in each layer and
a constant fraction of them act on qubits separated by distance .
While it is known that almost all -qubit unitary operations need quantum
circuits of depth and size to realize with
all-to-all qubit connectivity, in this paper, we show that all -qubit
unitary operations can be implemented by quantum circuits of depth
and size even under {1D chain} qubit connectivity constraint.
We extend this result and investigate qubit connectivity in three directions.
First, we consider more general connectivity graphs and show that the circuit
size can always be made as long as the graph is connected. For circuit
depth, we study -dimensional grids, complete -ary trees and expander
graphs, and show results similar to the 1D chain. Second, we consider the case
when ancillary qubits are available. We show that, with ancilla, the circuit
depth can be made polynomial, and the space-depth trade-off is not impaired by
connectivity constraints unless we have exponentially many ancillary qubits.
Third, we obtain nearly optimal results on special families of unitaries,
including diagonal unitaries, 2-by-2 block diagonal unitaries, and Quantum
State Preparation (QSP) unitaries, the last being a fundamental task used in
many quantum algorithms for machine learning and linear algebra
Technology 2002: the Third National Technology Transfer Conference and Exposition, Volume 1
The proceedings from the conference are presented. The topics covered include the following: computer technology, advanced manufacturing, materials science, biotechnology, and electronics
Quantum Computing for Fusion Energy Science Applications
This is a review of recent research exploring and extending present-day
quantum computing capabilities for fusion energy science applications. We begin
with a brief tutorial on both ideal and open quantum dynamics, universal
quantum computation, and quantum algorithms. Then, we explore the topic of
using quantum computers to simulate both linear and nonlinear dynamics in
greater detail. Because quantum computers can only efficiently perform linear
operations on the quantum state, it is challenging to perform nonlinear
operations that are generically required to describe the nonlinear differential
equations of interest. In this work, we extend previous results on embedding
nonlinear systems within linear systems by explicitly deriving the connection
between the Koopman evolution operator, the Perron-Frobenius evolution
operator, and the Koopman-von Neumann evolution (KvN) operator. We also
explicitly derive the connection between the Koopman and Carleman approaches to
embedding. Extension of the KvN framework to the complex-analytic setting
relevant to Carleman embedding, and the proof that different choices of complex
analytic reproducing kernel Hilbert spaces depend on the choice of Hilbert
space metric are covered in the appendices. Finally, we conclude with a review
of recent quantum hardware implementations of algorithms on present-day quantum
hardware platforms that may one day be accelerated through Hamiltonian
simulation. We discuss the simulation of toy models of wave-particle
interactions through the simulation of quantum maps and of wave-wave
interactions important in nonlinear plasma dynamics.Comment: 42 pages; 12 figures; invited paper at the 2021-2022 International
Sherwood Fusion Theory Conferenc
Multi-vehicle rover testbed using a new indoor positioning sensor
Thesis (M. Eng.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2003.Includes bibliographical references (p. 63-64).This thesis describes the design and implementation of an indoor multi-rover testbed using a high-precision indoor positioning sensor system. The system model for the rover is derived and used in a Kalman filter that estimates each rover's velocity given the position measurements. These state estimates are then used in high-level path planning and control algorithms. Single and multi-rover tests are performed using single and multi-waypoint path plans. The results from these tests demonstrate the correctness of the system model, estimator, and control algorithms. This testbed serves as a platform for future multi-vehicle coordinated control experiments.by Chris Sae-Hau.M.Eng
- …