The main idea of encoding quantum information in a topologically ordered system is similar. A requirement for a system to be useful for topological quantum computing is that the system must have locally indistinguishable eigenstates of the same energy.
Topological order
A pair of particles can be created from a ground state by applying "spin flip" operators X along a string on the dual lattice. A pair of particles can be created from a ground state by applying "phase flip" operators Z along a string on the primary lattice.
String operators
Consider a region R as large as possible, on which nothing observable can resolve the subspace of the ground state. If there were a three-dimensional model in which all surface-like operators act on the subspace of the ground state, they must intersect along a one-dimensional line.
Quantum codes and a new model
We note that the existence of string operators implies that point-like excitations at the ends of the string are generally mobile. It seems that the local indistinguishability of the subspace from the ground state is the most mathematically viable definition of the topological order.
Summary of chapters
Here, the energy barrier of a fault is defined in the same way as above: the height of the lowest hill in the energy landscape on any path connecting a ground state to the state affected by the fault. A pair of binary string entries describes a one-qubit component in the tensor product expression.
Additive/stabilizer codes
Inductively, one can completely eliminate all entries of M. The equationTTλqT =λq now implies that N = idq. By focusing on the action on the code space by the logical operators, one realizes that different logical operators can have the same action.
Cleaning lemma
The purification lemma for subsystem codes relates the number of independent bare logical operators supported on the set of qubitsM to the number of independent dressed logical operators supported on the complementary set Mc. For subsystem code, let gbare(M) be the number of independent nontrivial bare Boolean operators supported on M, and let g(M) be the number of independent nontrivial dressed Boolean operators supported on M, i.e.
Operator trade-off for local subsystem codes
Pauli space on a group
There is a natural effect of Λ on the Pauli group modulo phase factors induced from the group effect of Λ on itself by multiplication. If R=F2[Λ] is the group algebra with the multiplicative identity denoted by 1, the modulo phase factors of the Pauli group acquire a structure of an R module.
Local Hamiltonians on groups
If σ is a 2q×t matrix whose columns generate a submoduleS⊆P, then the commutation value onS always vanishes if and only if.
Excitations
The matrix can be viewed as a generalization of the parity check matrix of the standard theory of classical or error correcting quantum codes when the translation structure is given. In the Pauli module, they are expressed as ghi, where hi is the ith column of σ.
Equivalent Hamiltonians
Symplectic transformations
Asymplectic transformationT is an automorphism of the Pauli module induced by a unitary operator on the system of qubits. Only the unitary operator on the physical Hilbert space that respects the translation can bring about a sympectic transformation.
Coarse-graining
Tensoring ancillas
Topological order
Accordingly, a Gr¨obner base is defined for generating the initial terms of the given module in each of the cones. We will deal with three different types of 'dimensions': The first is the spatial dimension D, which has an obvious physical meaning.
Ground-state degeneracy
Condition for degenerate Hamiltonians
It is necessary and sometimes desirable not to invert all non-zero elements for the localization to be useful. An important fact about the localization is that a module is zero if and only if its localization at every prime ideal is zero.
Counting points in algebraic varieties
Therefore dimF2kerL= dimF2S2q−dimF2imσL. Since dimF2S =LDogK(L) = kerL/imσL, the first statement follows. The number N0(Ln) of points (ai)∈V satisfying ringaLin= 1 is exactly the number of the rational points of V overAn that are not contained in hyperplanenexi = 0.
Fractal operators and topological charges
It is a construction consisting of single-qubit operators. on can be expressed as “on=R s”, the sum of all elements in the recipe. Their model has generating matrix σ=. The statement is a simple generalization of Newman and Moore's construction.
Two dimensions
By Proposition 4.2.2, the initial terms of the syzygy generators (S-polynomials) τij for {gi} lack the variablex1. It follows that |a|+|b| ≥N.2 Therefore, the length of the string segment carrying a charge is exponential in the interaction range.
Three dimensions
Since the radical of an ideal is the intersection of all primes that contain it [80, Proposition 1.8], the claim is proved. Since there are 6 in total, the largest possible number, we conclude that K(L) is 6-dimensional, i.e., the number of encoded qubits is 3 when the linear dimensions are even.
Discussion
According to Corollary 4.3.3[47], the characteristic dimension of the cubic code should be 1 and the degeneracy should generally increase with the size of the system. An immediate consequence of the no-strings rule in the case of a translation-invariant Hamiltonian code is that there are infinitely many charges.
Search for models
The no-strings rule may seem too strong than necessary; why do we need an upper bound by alinear function. A necessary condition for the no-strings rule is that any fractal generator must not consist of two terms.
Cubic code
Since the conditions used in the search for the model were only necessary conditions for the no-strings rule. To summarize, except for the special point (1,1,1) ∈ F3 in the affine space, every point in the algebraic set.
Real-space renormalization of the cubic code
The topological load module is the torsional part of the virtual excitation module, taking into account the trivial load module, i.e. the torsional submodule of the coker. The generating matrixσA for the stabilizer module of the cubic code transforms under the coarse grain by blocking two locations in the x-direction axis.
Thermal partition function of the cubic code
Since the trace of a non-identity Pauli operator is zero, only the term in the expansion of the product proportional to the identity contributes to Z. Remember that every defect in S can be associated with some elementary cube of the grid.
Superlinear code distance
Consider the shortest network of N paths in C◦ connecting the four cities su1, m, un, s. The length of the route network is the number of cities in the route network.). Decoder performance is measured by how close the corrupted state is restored to the original coded state.
Renormalization group decoder
At each levelp, the RG decoder divides the set of defects into connected clusters—connected components of a graph with vertices representing defects and edges connecting pairs of defects separated by the distance 2p or less (not shown). Each neutral cluster containing even number of defects (blue) is annihilated by applying a local Pauli operator supported in the smallest rectangular box that encloses the cluster.
Cluster decomposition
Defects in fields B, B0 belong to the same component if and only if there is a pair u∈B,v∈B0 of defects such that d(u, v)≤r. According to the δ table, we can conclude by calculating the decomposition in time O((L/r)D) as in the case of r= 1.
Gr¨ obner basis and broom algorithm
The syndromes due to weight three errors are used to push the defects to the lower left corner. We continue with ZI and shift all defects in the anterior plane of B to the left vertical line and the bottom horizontal line.
Threshold theorem for topological stabilizer codes
If the length m of the piece decomposition of an error E satisfies Qm+1< Ltqo, then E is corrected by the RG decoder. Thus, on the finite system of linear size L, the probability of the occurrence of a level m-piece is bounded above by LDpm.
Benchmark of the decoder
From the degeneracy formula of the cubic code in Chapter 5, we know that there are system sizes where the number of encoded qubits is 2. Perhaps this is already suggested by the smooth thermal partition function of the cubic code presented in Section 5.5.
Storage scheme
Finally, let us emphasize that quantum self-correction is technically different from the thermal stability of topological phases, see e.g. Let us note that the Davies weak coupling limit was adopted as a model of the thermal dynamics in most of the previous works with a rigorous analysis of quantum self-correction; see, for example.
Properties of the Lindbladian
Let L∗ be the additive linear map of L (the one describing time evolution in the Heisenberg picture) with respect to Hilbert-Schmidt inner product.
Analysis of thermal errors
Let f be the maximum energy barrier of the Pauli operators that appear in the expansion of the quantum jump operators Aα,ω or A†α,ωAα,ω of the equation. Suppose that the error-correction algorithm s 7→ Pec(s) corrects any Pauli error P with an energy barrier less than m+ 2f.
Correctability of errors with an energy barrier
Calling the RG decoder on the syndrome S(P) returns a corrective operatorPec such that P Pec is a stabilizer. The full hierarchy of the RG decoder is not necessary to correct the error with the low energy barrier.
A lower bound on memory time
The upper bound on the storage error can easily be translated into a lower bound on the memory time. If the development time is large enough such that ρ(t)≈ρβ, the coded information cannot be retrieved from ρ(t), since ρβ does not depend on the initial state.
Numerical simulation
Thus the memory time should be represented as the characteristic time of the exponential distribution. "form.") The + or · omitted between a and b in ab on the left side are the operations defined in A, while those on the right side are in B.
Ideals and modules
An ideal is a subset of R such that it is a module over R. 2 Note that R itself is an R-module via the multiplication within R. The above example R⊕R is an R-module. A Noetherian ringR is a ring which is Noetherian as an R-module, that is, the a.c.c is satisfied with respect to the ideals of R.
Gr¨ obner basis
Equipped with the sort, given a dividendf and a set of divisors, (Step-1) should be able to match the leading term and kill it, "shrinking" the leading term of the dividend. Step-2) Stop when reduction becomes impossible. This means that X consists of multiples of a finite number of monomials of G. Therefore, a minimal element of G, a finite set, is a minimal element of X. The statement can be reformulated as any decreasing sequence of monomials is stationary or a strictly decreasing sequence of monomials is finite.
Localization
4Rigorously, the ring of fractions is a collection of equivalence classes of R×U; (a, p) = (b, q) if and only if there exists∈ Usuch thats(aq−bp) = 0. Note that every prime ideal of U−1R is an extended ideal of a prime ideal ofR that does not intersect U .
Determinantal ideal
The transformed matrix M0 will have a non-zero entry in the first row in the first column. Since any matrix, not necessarily square, can be brought into Smith normal form by reversible transformations, it follows that the elementary divisors are complete invariants, i.e., the elementary divisors are the same for two matrices if and only if the two matrices are connected by left and right invertible matrix multiplication.
Finite fields
Ground state degeneracy of the fractional quantum Hall states in the presence of a random potential and on Riemann surfaces of high genus. Interplay between topological order and spin glassiness in the toric code under random magnetic fields.
