41 research outputs found
Stationary Phase Method in Discrete Wigner Functions and Classical Simulation of Quantum Circuits
One of the lowest-order corrections to Gaussian quantum mechanics in
infinite-dimensional Hilbert spaces are Airy functions: a uniformization of the
stationary phase method applied in the path integral perspective. We introduce
a ``periodized stationary phase method'' to discrete Wigner functions of
systems with odd prime dimension and show that the gate is the
discrete analog of the Airy function. We then establish a relationship between
the stabilizer rank of states and the number of quadratic Gauss sums necessary
in the periodized stationary phase method. This allows us to develop a
classical strong simulation of a single qutrit marginal on qutrit
gates that are followed by Clifford evolution, and show that
this only requires quadratic Gauss sums. This outperforms
the best alternative qutrit algorithm (based on Wigner negativity and scaling
as for precision) for any number of
gates to full precision
Discrete Wigner Function Derivation of the Aaronson-Gottesman Tableau Algorithm
The Gottesman-Knill theorem established that stabilizer states and operations
can be efficiently simulated classically. For qudits with dimension three and
greater, stabilizer states and Clifford operations have been found to
correspond to positive discrete Wigner functions and dynamics. We present a
discrete Wigner function-based simulation algorithm for odd- qudits that has
the same time and space complexity as the Aaronson-Gottesman algorithm. We show
that the efficiency of both algorithms is due to the harmonic evolution in the
symplectic structure of discrete phase space. The differences between the
Wigner function algorithm and Aaronson-Gottesman are likely due only to the
fact that the Weyl-Heisenberg group is not in for and that qubits
have state-independent contextuality. This may provide a guide for extending
the discrete Wigner function approach to qubits
Improved Strong Simulation of Universal Quantum Circuits
We find a scaling reduction in the stabilizer rank of the twelve-qubit
tensored gate magic state. This lowers its asymptotic bound to for multi-Pauli measurements on magic states, improving over the
best previously found bound of . We numerically demonstrate
this reduction. This constructively produces the most efficient strong
simulation algorithm of the Clifford+ gateset to relative or multiplicative
error. We then examine the cost of Pauli measurement in terms of its Gauss sum
rank, which is a slight generalization of the stabilizer rank and is a lower
bound on its asymptotic scaling. We demonstrate that this lower bound appears
to be tight at low -counts, which suggests that the stabilizer rank found at
the twelve-qubit state can be lowered further to and we
prove and numerically show that this is the case for single-Pauli measurements.
Our construction directly shows how the reduction at qubits is iteratively
based on the reduction obtained at , , , and qubits. This explains
why novel reductions are found at tensor factors for these number of qubit
primitives, an explanation lacking previously in the literature. Furthermore,
in the process we observe an interesting relationship between the T gate magic
state's stabilizer rank and decompositions that are Clifford-isomorphic to a
computational sub-basis tensored with single-qubit states that produce minimal
unique stabilizer state inner products -- the same relationship that allowed
for finding minimal numbers of unique Gauss sums in the odd-dimensional qudit
Wigner formulation of Pauli measurements
Octonions and Quantum Gravity through the Central Charge Anomaly in the Clifford Algebra
We derive a theory of quantum gravity containing an AdS isometry/qubit
duality. The theory is based on a superalgebra generalization of the enveloping
algebra of the homogeneous AdS spacetime isometry group and is isomorphic
to the complexified octonion algebra through canonical quantization. Its first
three quaternion generators correspond to an -quantized AdS embedded
spacetime and its remaining four non-quaternion generators to a -quantized
embedding Minkowski spacetime. The quaternion algebra's expression after
a monomorphism into the complexified Clifford algebra produces a
two-dimensional conformal operator product expansion with a central charge
anomaly, which results in an area-law scaling satisfying the
holographic principle and defines an "arrow of time". This relationship allows
us to extend the theory through supersymmetry- and conformal-breaking transformations of the embedding to produce dS and dS spacetimes
and derive a resolution to the black hole information paradox with an explicit
mechanism. Unlike string theory, the theory is background-independent and
suggests that our local dS spacetime is the largest possible
AdS Vacuum State from Four Minkowski Vacuum States
We show that a tensor product of four specific Minkowski vacuum
states is a self-consistent vacuum state for an infinite set of
three-dimensional anti-de Sitter (AdS) spacetimes if their parity and
time-reversal symmetry are broken in a particular way. The infinite set
consists of pairs of all AdS with non-zero unique scalar curvatures.Comment: 3 pages + appendices, 1 figur