Counting points in algebraic varieties

3.4 Ground-state degeneracy

3.4.2 Counting points in algebraic varieties

It is important that the factor ring

R/bL=F2[x1, . . . , xD]/ (x^L₁ −1, . . . , x^L_D−1)

is finite dimensional as a vector space over F², and hence is Artinian. In fact, dim_F2R/bL =L^D. This ring appears also in [70]. Due to the following structure theorem of Artinian rings,K(L) can be explicitly analyzed by the localizations.

Proposition 3.4.3(Chapter 8 of [80], Section 2.4 of [76]). LetSbe an Artinian ring. (For example, S is a homomorphic image of a polynomial ring over finitely many variables with coefficients in a field F, and is finite dimensional as a vector space over F.) Then, there are only finitely many maximal ideals ofS, and

S∼=M

m

S_m

where the sum is over all maximal ideals m ofS andS_m is the localization ofS atm.

The following calculation tool is sometimes useful. Recall that a group algebra is equipped with a non-degenerate scalar product hv, wi= tr(vw). This scalar product naturally extends to a direct¯

sum of group algebras.

Lemma 3.4.4. Let F be a field, and S =F[Λ] be the group algebra of a finite abelian groupΛ. If N is a submodule ofSⁿ, then the dual vector spaceN^∗ is vector-space isomorphic toSⁿ/N^⊥, where

⊥is with respect to the scalar product h·,·i.

Proof. Consider φ : Sⁿ 3 x 7→ h·, xi ∈ N^∗. The map φ is surjective since the scalar product is non-degenerate andSⁿ is a finite dimensional vector space. The kernel ofφis preciselyN^⊥. Corollary 3.4.5. Put 2k= dim_F2K(L). Then,

k=qL^D−dim_F2imσL= dim_F2kerL−qL^D. Further, ifq=t, then

k= dim_F2kerσL= dim_F2cokerL.

The first formula is a rephrasing of the fact that the number of encoded qubits is the total number of qubits minus the number of independent stabilizer generators [40, 41].

Proof. PutS=R/bL. Ifv1, . . . , vtdenote the columns ofσL, we have

kerσ^†_L=λ_qker_L=\

i

v_i^⊥= X

i

Sv_i

!^⊥

= (imσ_L)^⊥. (3.3)

Hence, dim_F2kerL= dim_F2S^2q−dim_F2imσL.Since dim_F2S =L^DandK(L) = kerL/imσL, the first claim follows.

Since imσL ∼=S^t/kerσL, if t=q, we havek= dim_F2kerσL by the first claim. From Eq. (3.3), we conclude thatk= dim_F2S^t/imσ_L^† = dim_F2cokerL.

We will apply these formulas in Section5.3and Example4.3.

The characteristic dimension is related to the rate at which the degeneracy increases as the system size increases in the following sense. Recall that 2k = dim_F2K(L) and the ground-state degeneracy is 2^k.

Lemma 3.4.6. Suppose2-L. LetF^a be the algebraic closure ofF². IfN is the number of maximal ideals in F^a⊗_F2R that contain bL+Iq(σ), then

N ≤dim_F2K(L)≤2qN.

Proof. We replace the ground fieldF2 withF^a. Any maximal ideal of an Artinian ringF^a[x^±1_i ]/b_L is of form m = (x₁−a₁, . . . , x_D−a_D) where a^L_i = 1 by Nullstellensatz. Since 2- L, we see that (bL)m=mm and that (R/bL)m∼=F^a is the ground field. (See the proof of Corollary3.4.2.)

Now,Iq(σ)+bL⊆miffIq(σ)m+(bL)m⊆mm= (bL)miffIq(σ) becomes zero overRm/(bL)m∼=F^a iff 1≤dim_F^aK(L)m≤2q. Since by Proposition3.4.3,K(L) is a finite direct sum of localized ones, we are done.

Lemma 3.4.7. LetI be an ideal such that dimR/I=d. We have dim_F2R/(I+bL)≤cL^d

for allL≥1 and some constant c independent ofL.

Proof. We replace the ground field with its algebraic closure F^a. Write ˜xi for the image of xi in R/I. By Noether normalization theorem [76, Theorem 13.3], there existy1, . . . , yd∈R/I such that R/I is a finitely generated module overF^a[y1, . . . , yd]. Moreover, one can chooseyi=PD

j=1Mijx˜j

for some rank dmatrixM whose entries are in F^a. MakingM into the reduced row echelon form, we may assumeyi= ˜xi+P

j>daijx˜j for each 1≤i≤d.

LetS=F^a[z1, . . . , zD] be a polynomial ring inD variables. Letφ:S→R/(I+bL) be the ring homomorphism such thatzi7→yifor 1≤i≤dandzj 7→x˜j ford < j≤D. By the choice ofyi,φis clearly surjective. Consider the idealJ of S generated by the initial terms of kerφwith respect to the lexicographical monomial order in whichz₁ ≺ · · · ≺z_D. Since ˜x_j is integral over F[y₁, . . . , y_d], the monomial ideal J containszⁿ_j^j for some positive n_j ford < j ≤D. Here, n_j is independent of L. Sincez_i^L∈J for 1≤i≤d, we conclude that

dim_FaR/(I+b_L) = dim_FaS/J ≤L^d·n_d+1n_d+2· · ·n_D

by Macaulay theorem [76, Theorem 15.3].

Corollary 3.4.8. If 2-L, andd= dimR/Iq(σ)is the characteristic dimension of a code Hamilto- nian, then

dim_F2K(L)≤cL^d for some constant cindependent of L.

Proof. If J = bL +I(σ), N in Lemma 3.4.6 is equal to dim_F^aF^a ⊗R/radJ. This is at most dim_F^aF^a⊗R/J = dim_F2R/J.

Lemma 3.4.9. Let dbe the characteristic dimension. There exists an infinite set of integers {Li}

such that

dim_F2K(L_i)≥L_i^d/2

Proof. We replace the ground field with its algebraic closure F^a. Let p⁰ ⊇ I(σ) be a prime of R of codimension D−d. Let p be the contraction (pull-back) of p⁰ in the polynomial ring S = F^a[x₁, . . . , x_D]. Since the set of all primes of R is in one-to-one correspondence with the set of primes inS that does not include monomials, it follows thatphas codimensionD−dand does not contain any monomials. LetV denote the affine variety defined byp= (g1, . . . , gn). Sincepcontains no monomials,V is not contained in any hyperplanesxi= 0 (i= 1, . . . , D).

LetA1 be a finite subfield ofF^a that contains all the coefficients ofgi, soV can be defined over A1. Let An ⊆ F^a be the finite extension fields of A1 of extension degree n. Put Ln = |An| −1.

For any subfieldA ofF^a, let us say a point ofV is rational over Aif its coordinates are inA. The numberN⁰(L_n) of points (a_i)∈V satisfyinga^L_iⁿ= 1 is precisely the number of the rational points ofV overA_n that are not contained in the hyperplanesx_i = 0. SinceI(σ)⊆p⁰, the number N in Lemma3.4.6is at leastN⁰(L_n). It remains to showN⁰(L_n)≥L^d_n/2 for all sufficiently largen.

This follows from the result by Lang and Weil [81], which states that the number of points of a projective variety of dimensiondthat are rational over a finite field ofmelements ism^d+O

m^d−12 asymptotically in m. Since Lang-Weil theorem is for projective variety and we are with an affine varietyV, we need to subtract the number of points in the hyperplanesxi = 0 (i= 0,1, . . . , D) from the Zariski closure of V. The subvarieties in the hyperplanes, being closed, have strictly smaller dimensions, and we are done.