Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userConstructing quantum codes from any classical code and their embedding in ground space of local Hamiltonians
67
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We introduce a framework for constructing a quantum error correcting code from any classical error correcting code. This includes CSS codes and goes beyond the stabilizer formalism to allow quantum codes to be constructed from classical codes that are not necessarily linear or self-orthogonal (Fig. 1). We give an algorithm that explicitly constructs quantum codes with linear distance and constant rate from classical codes with a linear distance and rate. As illustrations for small size codes, we obtain Steanes $7-$qubit code uniquely from Hammings [7,4,3] code, and obtain other error detecting quantum codes from other explicit classical codes of length 4 and 6. Motivated by quantum LDPC codes and the use of physics to protect quantum information, we introduce a new 2-local frustration free quantum spin chain Hamiltonian whose ground space we analytically characterize completely. By mapping classical codewords to basis states of the ground space, we utilize our framework to demonstrate that the ground space contains explicit quantum codes with linear distance. This side-steps the Bravyi-Terhal no-go theorem because our work allows for more general quantum codes beyond the stabilizer and/or linear codes. We hesitate to call this an example of {it subspace} quantum LDPC code with linear distance.
rate research
Read More
Traditional quantum physics solves ground states for a given Hamiltonian, while quantum information science asks for the existence and construction of certain Hamiltonians for given ground states. In practical situations, one would be mainly interested in local Hamiltonians with certain interaction patterns, such as nearest neighbour interactions on some type of lattices. A necessary condition for a space $V$ to be the ground-state space of some local Hamiltonian with a given interaction pattern, is that the maximally mixed state supported on $V$ is uniquely determined by its reduced density matrices associated with the given pattern, based on the principle of maximum entropy. However, it is unclear whether this condition is in general also sufficient. We examine the situations for the existence of such a local Hamiltonian to have $V$ satisfying the necessary condition mentioned above as its ground-state space, by linking to faces of the convex body of the local reduced states. We further discuss some methods for constructing the corresponding local Hamiltonians with given interaction patterns, mainly from physical points of view, including constructions related to perturbation methods, local frustration-free Hamiltonians, as well as thermodynamical ensembles.
In this work, we make a connection between two seemingly different problems. The first problem involves characterizing the properties of entanglement in the ground state of gapped local Hamiltonians, which is a central topic in quantum many-body physics. The second problem is on the quantum communication complexity of testing bipartite states with EPR assistance, a well-known question in quantum information theory. We construct a communication protocol for testing (or measuring) the ground state and use its communication complexity to reveal a new structural property for the ground state entanglement. This property, known as the entanglement spread, roughly measures the ratio between the largest and the smallest Schmidt coefficients across a cut in the ground state. Our main result shows that gapped ground states possess limited entanglement spread across any cut, exhibiting an area law behavior. Our result quite generally applies to any interaction graph with an improved bound for the special case of lattices. This entanglement spread area law includes interaction graphs constructed in [Aharonov et al., FOCS14] that violate a generalized area law for the entanglement entropy. Our construction also provides evidence for a conjecture in physics by Li and Haldane on the entanglement spectrum of lattice Hamiltonians [Li and Haldane, PRL08]. On the technical side, we use recent advances in Hamiltonian simulation algorithms along with quantum phase estimation to give a new construction for an approximate ground space projector (AGSP) over arbitrary interaction graphs.
Hilbert space combines the properties of two fundamentally different types of mathematical spaces: vector space and metric space. While the vector-space aspects of Hilbert space, such as formation of linear combinations of state vectors, are routinely used in quantum mechanics, the metric-space aspects of Hilbert space are much less exploited. Here we show that a suitable metric stratifies Fock space into concentric spheres. Maximum and minimum distances between wave functions are derived and geometrically interpreted in terms of this metric. Unlike the usual Hilbert-space analysis, our results apply also to the reduced space of only ground-state wave functions and to that of particle densities, each of which forms a metric space but not a Hilbert space. The Hohenberg-Kohn mapping between densities and ground-state wave functions, which is highly complex and nonlocal in coordinate description, is found, for three different model systems, to be very simple in metric space, where it is represented by a monotonic mapping of vicinities onto vicinities. Surprisingly, it is also found to be nearly linear over a wide range of parameters.
Kitaevs quantum double models in 2D provide some of the most commonly studied examples of topological quantum order. In particular, the ground space is thought to yield a quantum error-correcting code. We offer an explicit proof that this is the case for arbitrary finite groups. Actually a stronger claim is shown: any two states with zero energy density in some contractible region must have the same reduced state in that region. Alternatively, the local properties of a gauge-invariant state are fully determined by specifying that its holonomies in the region are trivial. We contrast this result with the fact that local properties of gauge-invariant states are not generally determined by specifying all of their non-Abelian fluxes -- that is, the Wilson loops of lattice gauge theory do not form a complete commuting set of observables. We also note that the methods developed by P. Naaijkens (PhD thesis, 2012) under a different context can be adapted to provide another proof of the error correcting property of Kitaevs model. Finally, we compute the topological entanglement entropy in Kitaevs model, and show, contrary to previous claims in the literature, that it does not depend on whether the log dim R term is included in the definition of entanglement entropy.
Ground state counting plays an important role in several applications in science and engineering, from estimating residual entropy in physical systems, to bounding engineering reliability and solving combinatorial counting problems. While quantum algorithms such as adiabatic quantum optimization (AQO) and quantum approximate optimization (QAOA) can minimize Hamiltonians, they are inadequate for counting ground states. We modify AQO and QAOA to count the ground states of arbitrary classical spin Hamiltonians, including counting ground states with arbitrary nonnegative weights attached to them. As a concrete example, we show how our method can be used to count the weighted fraction of edge covers on graphs, with user-specified confidence on the relative error of the weighted count, in the asymptotic limit of large graphs. We find the asymptotic computational time complexity of our algorithms, via analytical predictions for AQO and numerical calculations for QAOA, and compare with the classical optimal Monte Carlo algorithm (OMCS), as well as a modified Grovers algorithm. We show that for large problem instances with small weights on the ground states, AQO does not have a quantum speedup over OMCS for a fixed error and confidence, but QAOA has a sub-quadratic speedup on a broad class of numerically simulated problems. Our work is an important step in approaching general ground-state counting problems beyond those that can be solved with Grovers algorithm. It offers algorithms that can employ noisy intermediate-scale quantum devices for solving ground state counting problems on small instances, which can help in identifying more problem classes with quantum speedups.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
