59 research outputs found

### Commuting Pauli Hamiltonians as maps between free modules

We study unfrustrated spin Hamiltonians that consist of commuting tensor
products of Pauli matrices. Assuming translation-invariance, a family of Hamiltonians
that belong to the same phase of matter is described by a map between modules over
the translation-group algebra, so homological methods are applicable. In any dimension
every point-like charge appears as a vertex of a fractal operator, and can be isolated with
energy barrier at most logarithmic in the separation distance. For a topologically ordered
system in three dimensions, there must exist a point-like nontrivial charge. A connection
between the ground state degeneracy and the number of points on an algebraic set is
discussed. Tools to handle local Clifford unitary transformations are given

### An invariant of topologically ordered states under local unitary transformations

For an anyon model in two spatial dimensions described by a modular tensor
category, the topological S-matrix encodes the mutual braiding statistics, the
quantum dimensions, and the fusion rules of anyons. It is nontrivial whether
one can compute the S-matrix from a single ground state wave function. Here, we
define a class of Hamiltonians consisting of local commuting projectors and an
associated matrix that is invariant under local unitary transformations. We
argue that the invariant is equivalent to the topological S-matrix. The
definition does not require degeneracy of the ground state. We prove that the
invariant depends on the state only, in the sense that it can be computed by
any Hamiltonian in the class of which the state is a ground state. As a
corollary, we prove that any local quantum circuit that connects two ground
states of quantum double models (discrete gauge theories) with non-isomorphic
abelian groups, must have depth that is at least linear in the system's
diameter.
As a tool for the proof, a manifestly Hamiltonian-independent notion of
locally invisible operators is introduced. This gives a sufficient condition
for a many-body state not to be generated from a product state by any small
depth quantum circuit; this is a many-body entanglement witness.Comment: revtex 11pt, 43 pages, (v2) minor change (v3) ref. added. To appear
in Commun. Math. Phy

### Codes and Protocols for Distilling $T$, controlled-$S$, and Toffoli Gates

We present several different codes and protocols to distill $T$,
controlled-$S$, and Toffoli (or $CCZ$) gates. One construction is based on
codes that generalize the triorthogonal codes, allowing any of these gates to
be induced at the logical level by transversal $T$. We present a randomized
construction of generalized triorthogonal codes obtaining an asymptotic
distillation efficiency $\gamma\rightarrow 1$. We also present a Reed-Muller
based construction of these codes which obtains a worse $\gamma$ but performs
well at small sizes. Additionally, we present protocols based on checking the
stabilizers of $CCZ$ magic states at the logical level by transversal gates
applied to codes; these protocols generalize the protocols of 1703.07847.
Several examples, including a Reed-Muller code for $T$-to-Toffoli distillation,
punctured Reed-Muller codes for $T$-gate distillation, and some of the check
based protocols, require a lower ratio of input gates to output gates than
other known protocols at the given order of error correction for the given code
size. In particular, we find a $512$ T-gate to $10$ Toffoli gate code with
distance $8$ as well as triorthogonal codes with parameters
$[[887,137,5]],[[912,112,6]],[[937,87,7]]$ with very low prefactors in front of
the leading order error terms in those codes.Comment: 28 pages. (v2) fixed a part of the proof on random triorthogonal
codes, added comments on Clifford circuits for Reed-Muller states (v3) minor
chang

### Local stabilizer codes in three dimensions without string logical operators

We suggest concrete models for self-correcting quantum memory by reporting
examples of local stabilizer codes in 3D that have no string logical operators.
Previously known local stabilizer codes in 3D all have string-like logical
operators, which make the codes non-self-correcting. We introduce a notion of
"logical string segments" to avoid difficulties in defining one dimensional
objects in discrete lattices. We prove that every string-like logical operator
of our code can be deformed to a disjoint union of short segments, and each
segment is in the stabilizer group. The code has surface-like logical operators
whose partial implementation has unsatisfied stabilizers along its boundary.Comment: 18 pages, 12 figures; clarified intermidiate steps in the proo

### Invertible subalgebras

We introduce invertible subalgebras of local operator algebras on lattices.
An invertible subalgebra is defined to be one such that every local operator
can be locally expressed by elements of the inveritible subalgebra and those of
the commutant. On a two-dimensional lattice, an invertible subalgebra hosts a
chiral anyon theory by a commuting Hamiltonian, which is believed not to be
possible on a full local operator algebra. We prove that the stable equivalence
classes of $D$-dimensional invertible subalgebras form an abelian group under
tensor product, isomorphic to the group of all $D+1$ dimensional QCA modulo
blending equivalence and shifts.
In an appendix, we consider a metric on the group of all QCA on infinite
lattices and prove that the metric completion contains the time evolution by
local Hamiltonians, which is only approximately locality-preserving. Our metric
topology is strictly finer than the strong topology.Comment: 29 pages, 2 figure

### Localization from superselection rules in translation invariant systems

We study a translation invariant spin model in a three-dimensional regular
lattice, called the cubic code model, in the presence of arbitrary extensive
perturbations. Below a critical perturbation strength, we show that most states
with finite energy are localized; the overwhelming majority of such states have
energy concentrated around a finite number of defects, and remain so for a time
that is near-exponential in the distance between the defects. This phenomenon
is due to an emergent superselection rule and does not require any disorder. An
extensive number of local integrals of motion for these finite energy sectors
are identified as well. Our analysis extends more generally to systems with
immobile topological excitations.Comment: 7.5+1pages, 2 figure

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