Since Theorem8.1only provides a lower bound on the memory time, a natural question is whether this bound is tight and, if so, what is the exact value of the constant coefficientc? To answer this
5 9 17 31 Linear system size 10
610
710
810
910
1010
1110
1210
13Memory time (a.u.) 0 1 2 3 4 5
exponent
+2.93β-10.52
4.5 5.0
β
13 17
25
L
exp(+0.78β-0.87)
Figure 8.1: The memory timeTmemvs. the system sizeL. In the upper inset is shown the exponent of the power law fit of Tmem for the first a few system sizes. It is clear that Tmem ∝ L2.93β−10.5 when L < L?, where L? is the optimal system size where Tmem reaches maximum. The data for β= 4.3,4.5,4.7,4.9,5.1,5.25 are shown.
4.0 4.2 4.4 4.6 4.8 5.0 5.2
Inverse Temperature (
β=2J/kT) 10
610
710
810
910
1010
1110
1210
13Maximum Memory time (a.u.)
exp( +1.69 β
2-5.05 β +7.81)
exp( +10.62 β -28.20)
0.3 0.0 0.3
Quadratic fit
Residual ( log T
mem)
4.1 4.6 5.1
β
0.3
0.0 0.3
Linear fit
Figure 8.2: The maximum memory time Tmem vs. the inverse temperature β. The memory time is maximized with respect to the system size. The logarithm of Tmem clearly follows a quadratic relation withβ as opposed to a linear one.
question, the memory time of the 3D cubic code has been computed numerically for a range ofβ’s andL’s. It should be emphasized that both Theorem8.1and our numerical simulation use thesame decoder at the read-out step. The numerical results strongly suggest that our analytical bound is tight up to constant coefficients for our renormalization group decoder. See Figure 8.1,8.2. It suggests thatTmem≈L2.93β−10.52as long asL≤L∗≈e0.78β−0.87. The number 0.78 in the estimate ofL∗ should not be taken too seriously, as one sees in the inset that the dependence ofL∗ onβ is hard to tell quantitatively. It is clear, however, thatL∗is increasing with the inverse temperature.
The interaction of the memory system with a thermal bath is simulated by Metropolis evolution.
As we wish to observe low temperature behavior we adopt continuous time algorithm by Bortz, Kalos, and Lebowitz (BKL) [132]. A pseudo-random number generation package RngStream by L’Ecuyer [133] was used. As before, the coupling constant in the Hamiltonian is set to J = 12 so a single defect has energy 1. Although the cubic code is inherently quantum, it is relevant to consider onlyX-type errors (bit flip) in the simulation, thanks to the duality of theX- andZ-type stabilizer generators of the cubic code. The simulation thus is purely classical. The errors are represented by a binary array of length 2L3, and the corresponding syndrome by a binary array of lengthL3.
The memory time is measured to be the first time when the memory becomes unreliable. There are two cases the memory is unreliable: either the broom algorithm fails to remove all the defects so we have to reinitialize the memory, or a nontrivial logical error is occurred. It is thus necessary in our simulation to keep track of the error operator during the time evolution. In fact, most of the time, it was the broom algorithm’s failure that made the memory unreliable. Nontrivial logical errors occurred only for very small system sizesL= 5,7.
It is too costly to decode the system every time it is updated. Alternatively, we have performed a trial decoding every fixed time interval
Tec= e4β 100
whereβ is the inverse temperature. Although the time evolution of the BKL algorithm is stochastic, a single BKL update typically advances time much smaller than Tec. So it makes sense to decode the system every Tec. The exponential factor appears naturally because BKL algorithm advances time exponentially faster asβ increases. It is to be emphasized that we do not alter the system by the trial decodings (a copy of the actual syndrome has been created for each trial decoding).
The system sizesL3 for the simulation are chosen such that the code space dimension is exactly 2, for which the complete list of logical operators is known. If the linear sizeLis ≤200, this is the case whenL is not a multiple of 2, 15, or 63 by Corollary 5.3.2. For these system sizes, to check whether a logical operator is nontrivial is to compute the commutation relation with the known nontrivial logical operators.
The measured memory time for a givenLandβis observed to follow an exponential distribution;
a memory system is corrupted with a certain probability given time interval. Specifically, the prob- ability that the measured memory time istis proportional toe−t/τ. Thus the memory time should be presented as the characteristic time of the exponential distribution. We choose the estimator for the characteristic time to be the sample average ¯T = n1Pn
i Ti. The deviation of the estimator will follow a normal distribution for large numbernof samples. We calculated the confidence interval to be the standard deviation of the samples divided by√
n. For eachL, 400 samples whenβ ≤5.0 and 100 samples whenβ >5.0 were simulated. The computation was performed on IBM Blue Gene/P using 512 cores located in IBM T. J. Watson Research Center, Yorktown Heights, New York. The result is summarized in Figure8.1,8.2.
Figure 8.2 clearly supports logTmem = cβ2+· · ·. Figure 8.1 demonstrates the power law for small system size:
Tmem∝L2.93β−10.5
We wish to relate some details of the model with the numerical coefficients. The rigorous analysis of the previous section, gives a relatively small coefficientcof the energy barrier for correctable errors by our RG decoder. However, we expect that the coefficient ofβ in the exponent is the same as the constantc that appear in the energy barrier
E=clog2R
to create an isolated defect separated from the other by a distanceR. This is based on an intuition that the outputP0 of the decoder would have roughly the same support as the real error P for the most of the time, provided that the error has energy barrier less than ∆ =clog2Ltqo. Thus, an error of energy barrier less than ∆ would be corrected by the decoder. Our empirical formula supports this intuition. It suggests thatc= 2.93 log 2 = 2.03∼2.
Indeed, we can illustrate explicitly an error path that separates a single defect from the rest by distance 2pduring which only 2p+ 4 defects are needed. Consider an error of weight 2 that creates 4 defects as shown in the top of Fig.8.3. We call it thelevel-0 hook. The bottom sequence depicts a process to create a configuration shown at the bottom-left, which we calllevel-1 hook. One sees that level-1 hook is similar with ratio 2 to level-0, and is obtained from level-0 with extra 2 defects.
One defines level-phooks hierarchically. We claim that a level-phook can be constructed from the vacuum using 2p+ 4 defects. The proof is by induction. The casep= 1 is treated in the diagrams.
Suppose we can construct level-phook using 2p+ 4 defects. Consider the 2nd, 4th, 6th, and 8thsteps in Fig.8.3. They can be viewed as a minuscule version of level-psteps that construct a level-(p+ 1) hook from the level-phooks. It requires at most 2p+ 4 + 2 defects to perform the level-pstep; this completes the induction.
Figure 8.3: Construction of a hook of level 2 from the vacuum. The grid diagram represents the position and the number of defects in the (x=z)-plane. For each transition, an operator of weight 1 is applied. The total number of defects never exceeds 6. From a level-0 hook (the second diagram in the sequence), a level-1 hook (the last in the sequence) is constructed using extra 2 defects.
It may not be obvious whether a high level hook corresponds to a nontrivial logical operator, but such a large hook is bad enough to make our decoder to fail.
Appendix A
Commutative algebra
We briefly review algebraic concepts and tools used in this thesis, mainly in Chapter 3 and 4.
There are many nice textbooks including those by Lang [60], Atiyah and MacDonald [80], and Eisenbud [76]. The book by Lang is a comprehensive textbook covering a wide range of topics in abstract algebra. The book by Atiyah and MacDonald explains commutative algebra that may look too concise, but precisely for this reason it is very useful as a reference. Examples are rare but essential. The book by Eisenbud is also on commutative algebra and is extensive. It covers more material than Atiyah-MacDonald. In particular, our summary of Gr¨obner basis follows Eisenbud.
The chapter on Gr¨obner basis appears in the middle of the book, but is relatively self-contained and elementary. Here, we will omit many proofs and not try to be fully rigorous. We explain theorems to the point where intuition can be developed. Rigorous proofs can be found in one of the three books.
We start by recalling definitions for abelian groups. An abelian groupGwith the identity element denoted by 0 is a set with an operation + :G×G→G such thatg+g0 =g0+g and 0 +g=g.
It is required forGto have inverses ofg denoted by−gsuch that g+ (−g) = 0. n-fold sum ofg is simply denoted asng, wheren∈Z. Given two abelian groupsGand H, we can form a direct sum G⊕H. It is the set of all tuples (g, h), whereg∈G, h∈H, and the group operation + is defined as (g, h) + (g0, h0) = (g+g0, h+h0). We can form adirect sumL
αGαof arbitrary family {Gα} of groups. It is the set of all indexed collections of group elements (gα) where only finitely manygα
are nonzero. The group operation is again defined component-wise. Thus, any element in the direct sum is a sum of finitely many gα ∈ Gα. A sum of two abelian groups can be defined if they are subgroups of a parent group. IfA, B≤Care subgroups, thesumA+Bis the group of all elements of C of form a+b where a∈A and b ∈B. Note that A+B ∼=A⊕B if and only if A∩B = 0.
Given a subgroupN ≤G, we can form a quotient groupG/N, the set of all equivalent classes under the equivalence relation [g] = [g0] iffg−g0 ∈ N. In commutative algebra, almost everything is an abelian group. On top of the abelian (additive) group structure, a new “multiplication” is added.
A.1 Rings and homomorphisms
The set of integers . . . ,−2,−1,0,1, . . . admits two operations, addition and multiplication. There is 0 that has no effect under addition, and 1 that has no effect under multiplication. One can always undo the addition because one can subtract a number. However, the multiplication is not invertible within the set of integers because fractions are not integers. One convenient thing is that the multiplication does not care about the order. A commutative ring is an abstraction of this structure. It is a set, in which one can add and subtract. A multiplication exists but is not in general invertible. An additive identity 0 exists, and a multiplicative identity 1 exists. The distribution law a(b+c) =ab+acis assumed, and the multiplication is commutativeab=ba. A ringRcan consists of a single element, in which caseR is called a zero ring, if and only if 0 = 1. Indeed, if a∈R and 1 = 0∈R, thena=a·1 =a·0 =a·(0 + 0) =a·(1 + 1) =a+a= 0. Examples of rings are abundant:
The set of all integers, the set of all complex numbers, the set of all square diagonal matrices of a fixed size, the set of polynomials, the set of all differentiable functions on a real line, the set of all continuous real-valued functions on a manifold, etc. Is the set of all even integers a ring? No. Some authors define rings to include this case where the multiplicative identity 1 is not provided, but we avoid this case. Any ring is with 1. Note that 1 is unique; if 10 is also a multiplicative identity, then 1 = 1·10= 10. The same is true for 0.
A ring is always understood in terms of relations with other rings. Given two rings A and B we consider a restricted class of maps between them. That is, we require that the map obeys the ring structure of the rings. f : A → B is a homomorphism if f(a+b) = f(a) +f(b) and f(ab) =f(a)f(b) for anya, b∈R. In addition, we assumef(0) = 0 andf(1) = 1. (“morph” means
“shape.”) The + or the omitted · between aand b in ab on the left-hand side are the operations defined inA, whereas those in the right-hand side are in B. The image of a homomorphismf is the subset ofB written asf(A) defined by{f(a)|a∈A}. Is the image of a homomorphism a ring?
Yes.
There is no point to speak of a map between two ringsA, B that is not a homomorphism. If we are going to ignore the ring structure, we would rather say the map between the “sets” A, B. We will simply say amap between ringsto mean a homomorphism. We note more terminologies: An endomorphism is a map from a ring into itself. An isomorphism is a map between two rings with a unique inverse. Anautomorphismis an isomorphism that is an endomorphism.
The ring of integers is so primitive in the following sense. LetA be an arbitrary ring. Consider a mapf :Z→A. f(n) =Pn
i=1f(1) andf(−n) =Pn
i=1(−f(1)) wheren >0. But,f(1) = 1 is the unique multiplicative identity. Therefore,f is completely determined, though we just requiredf be a homomorphism; there is a unique nonzero map fromZinto any ring. How many endomorphisms are there forZ?
The kernelof a map (homomorphism!) f is the subset of A written as kerf defined by {a∈ A|f(a) = 0}. It is easy to see that the kernel is closed under the addition and multiplication. Here, the closeness means that the result of the operation using two elements in a subset lies in the subset.
(It is pointless to speak of the closedness of an operation without reference to a subset.) There is one more important property as we discuss below.