Lifetime of topological quantum memories in thermal environment

Home Search Collections Journals About Contact us My IOPscience Lifetime of topological quantum memories in thermal environment This content has been downloaded from IOPscience. Please scroll down to see the full text. 2013 New J. Phys. 15 025027 (http://iopscience.iop.org/1367-2630/15/2/025027) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 211.138.121.37 This content was downloaded on 13/10/2013 at 20:31 Please note that terms and conditions apply. Lifetime of topological quantum memories in thermal environment Abbas Al-Shimary1 , James R Wootton2,3 and Jiannis K Pachos1,3 1 School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, UK 2 Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland E-mail: j.k.pachos@leeds.ac.uk and james.wootton@unibas.ch New Journal of Physics 15 (2013) 025027 (13pp) Received 27 September 2012 Published 19 February 2013 Online at http://www.njp.org/ doi:10.1088/1367-2630/15/2/025027 Abstract. Here we investigate the effect lattice geometry has on the lifetime of two-dimensional topological quantum memories. Initially, we introduce various lattice patterns and show how the error-tolerance against bit-flips and phase-flips depends on the structure of the underlying lattice. Subsequently, we investigate the dependence of the lifetime of the quantum memory on the structure of the underlying lattice when it is subject to a finite temperature. Importantly, we provide a simple effective formula for the lifetime of the memory in terms of the average degree of the lattice. Finally, we propose optimal geometries for the Josephson junction implementation of topological quantum memories. 3 Authors to whom any correspondence should be addressed. Content from this work may be used under the terms of the Creative Commons Attribution-NonCommercialShareAlike 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. New Journal of Physics 15 (2013) 025027 1367-2630/13/025027+13$33.00 © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft 2 Contents 1. Introduction 2. The two-dimensional toric code 3. Error model and simulations 4. Degree as a measure of connectivity 5. Numerical results on the relationship between d, pc and τ 6. Josephson-junction arrays 7. Conclusion Acknowledgments References 2 3 5 7 8 10 12 12 12 1. Introduction Topologically ordered systems are promising candidates for quantum memories. The main idea of Kitaev in [1] was to use spin (or qubit) Hamiltonians with topologically ordered ground states to encode quantum information [2]. Logical qubits encoded into the ground state of such systems become virtually resilient against environmentally induced local static perturbations to their Hamiltonian [3]. In the case of unknown perturbations, one may need to incorporate at least one round of error correction at the readout stage [4]. The simplest such memory, known as the toric code, is considered as the testbed for studying the properties of topological order against various models of errors. For concreteness, we shall restrict our study to this model. In the presence of probabilistic errors, robustness against perturbations is no longer sufficient to ensure self-correction [5]. Assume independent probabilistic bit-flip (Pauli X operator) and phase-flip (Pauli Z operator) errors during the preparation of the initial state on the physical qubits of the toric code. Retrieval of encoded information is possible only if the relative number of X and Z errors is below a certain threshold. The error threshold of the toric code for the optimal decoding is related to the phase transition point of the random bond Ising model and found to be approximately 11% [5]. Dependence of the phase transition point on the lattice geometry has been studied for various lattices in [6, 7]. Consider now a topological memory coupled to a thermal bath. If the system is initially in a pure state then the thermal environment will start introducing errors. After a certain time one expects that a sufficiently large number of errors will be accumulated causing the memory to fail [8]. This time is known as the memory lifetime τ [9]. Errors in topological systems, such as the toric code, take the form of topological defects known as anyons [2]. A finite temperature can cause anyons to be created, propagated or annihilated. A fundamental flaw of two-dimensional topological memories that rules out selfcorrection under probabilistic errors is the lack of energy barriers that could suppress diffusion of anyons over large distances. For instance, consider a process that involves a creation of anyon pair from the ground state, a transport of one anyon along a non-contractible loop on the torus, and a final annihilation of the pair. This process enacts a non-trivial transformation on the subspace of ground states. However, it can be implemented by a stream of local errors at a constant energy cost [8, 10]. New Journal of Physics 15 (2013) 025027 (http://www.njp.org/) 3 Av Bf Figure 1. The toric code defined for a general lattice structure. The plaquette, or face operator B f acts non-trivially on the qubits of the face f . The vertex operator Av acts non-trivially on the qubits that neighbour vertex v. In fact, Alicki et al [11] demonstrated that in two spatial dimensions, the toric code cannot serve as a self-correcting quantum memory. They showed that the relaxation time towards the equilibrium state is a constant independent of the lattice size. The physical mechanism behind this instability is the absence of interactions between the anyonic thermal excitations. Indeed, [9, 12, 24] demonstrated self-correction properties of two-dimensional topological systems by allowing long-range interactions between anyons. It is worth mentioning that in the infinite temperature limit, it has been shown that the lifetime increases at most logarithmically with N [13]. The main goal of this paper is to investigate the possibility of hindering the diffusion of anyons by considering the toric code on general, non-regular lattices. By localizing anyons to certain regions on the lattice, they are less likely to travel far apart, leading to better error correction properties. However, deforming the surface code breaks the symmetry between the X and Z error correction properties, creating an asymmetry in the error threshold values and their corresponding lifetimes [6]. This is because stabilizer operators are no longer symmetric under the lattice duality transformation (i.e. exchange of vertices and faces with each other). As an application, we demonstrate how such lattice geometries can be employed to enhance topological systems with intrinsically biased couplings due to physical implementation such as Josephson-junction arrays. 2. The two-dimensional toric code The starting point of our study is Kitaev’s two-dimensional toric code [1]. Consider a general lattice L(E, V, F), as shown in figure 1, with periodic boundary conditions (a torus), where E, V and F are sets of edges, vertices and faces, respectively. A qubit is associated with each edge. The stabilizer operators of the code on the lattice L are defined for each vertex v ∈ V and face New Journal of Physics 15 (2013) 025027 (http://www.njp.org/) 4 (a) (b)) (c) (d)) Figure 2. The ‘reduced square’ and ‘union’ lattices with their dual lattices. The dual lattice is obtained from the primal by exchanging faces and vertices with each other. f ∈ F, respectively, as O ALv = Xi , i∈E v B Lf = O Z j, j∈E f where X i and Z j denote Pauli operators on the ith and jth qubits respectively, and E v and E f indicate the sets of edges originating from vertex v and surrounding the face f respectively [6]. We define lattice L to be the primal lattice. Its dual lattice, denoted by L̄( Ē, V̄ , F̄), is defined by exchanging the vertices and faces of L with each other. The stabilizer space, where information is stored, is composed of states which satisfy ALv |ψi = |ψi and B Lf |ψi = |ψi for all v and f . Violations of these stabilizers on vertices or faces are associated with different types of anyons. The so-called e anyons reside on vertices and m anyons on faces. Alternatively, the error syndromes of e and m anyons are associated with the vertices on the primal and dual lattices, respectively. Hence, the thermal diffusion of the e and m anyons depends on the connectivity of the primal and dual lattices, respectively. Error correction corresponds to a pairwise annihilation of anyons which does not form topologically non-contractable loops. The toric code has been so far intensively investigated mainly on regular lattices [5, 6]. In the case of the square lattice, which is a regular self-dual lattice L = L̄, assuming independent errors, the toric code can be mapped to a random bond Ising model, resulting in symmetric error tolerances to e and m anyons and a threshold value of 11% [5]. When the toric code is coupled to a thermal bath, one observes a sharp transition in time known as the memory lifetime, τ . For the square lattice, the lifetime of the primal and dual lattices is the same and for T /J = 0.3 it approximately equals τ ≈ 5.8 [9], where T is the temperature and J is the single anyon excitation energy. Our goal is to study the toric code beyond regular self-dual lattices. By considering such lattices, one can break the symmetry between properties of X and Z error-correction. For example, consider the toric code on the reduced square lattice, as shown in figure 2. In this case L 6= L̄ and hence error-tolerances and lifetimes are no longer symmetric. Intuitively, e anyons New Journal of Physics 15 (2013) 025027 (http://www.njp.org/) 5 are more localized because the primal lattice is less connected leading to a better anyon pairing. On the other hand, m anyons on the dual lattice are more sparse because the dual lattice is more connected allowing anyon pairs to travel far apart. Indeed, the error-tolerance thresholds of the primal and dual lattices are 13% and 8%, respectively. Furthermore, the lifetimes of the lattices are 6.8 and 4.7, respectively. It is worth mentioning that in general there are two types of topological quantum error correction codes, surface codes [1] and colour codes [14], both of which are Calderbank–Shor–Steane (CSS) codes [15]. In both cases, the threshold values under perfect syndrome measurement have been calculated to be approximately 11% [5, 16]. In the case of colour codes, their performance have been investigated among different lattice geometries and their threshold result in similar values 11% [17]. This is because colour codes are symmetric under a Hadamard transformation. On the other hand, surface codes are not symmetric under a Hadamard transformation, hence it is possible to break the symmetry between Z and X error correction. 3. Error model and simulations In order to determine the lifetime of the memory against thermal noise, it is necessary to simulate the resulting dynamics. In this section we show how to model the classical dynamics of the system. We model the interaction of the system with a thermal environment by coupling each spin to a bath which can introduce spin errors in the initial state |ψ0 i, the ground state of the system [11, 18]. In the limit of weak coupling, the rate for the probabilities pψ to be in state |ψi is described by a coupled rate equations X in out ṗψ = [γi,ψ pzi (ψ) (t) − γi,ψ pψ (t)]. (1) i Q Here, pψ (t) is the time-dependent probability to find the system in the state |ψi = k∈ψ Z k |ψ0 i where ψ is the set of affected spin indices in |ψi. Similarly, pzi (ψ) (t) is the probability to be in out in the state Z i |ψi. Finally, γi,ψ = γ (−ωi , ψ) and γi,ψ = γ (ωi , ψ) are the transition rates to arrive or leave the state |ψi, where ω = ψ − zi (ψ) is the energy difference between the states |ψi and Z i |ψi. The rates γ (ω) describe transition probabilities between states with energy difference ω. A standard expression for γ (ω) can be obtained from a spin-boson model and reads γ (ω) = 2κn ωn , 1 − e−βω (2) where κn is a constant with units 1/energyn setting the time scale, β = 1/kB T , with T being the temperature of the bath and kB denoting Boltzmann’s constant which we will set to one [19, 20]. For n = 1, the bath is called ‘Ohmic’, whereas for n > 2 it is called ‘super-Ohmic’. In this work we will only consider the former case. The relevant rates for our simulation of the toric code are γ (0) (rate for an anyon to hop to a free neighbouring site), γ (−2J ) (rate to create an anyon pair) and γ (2J ) = γ (−2J )e2β J (rate to annihilate a pair of adjacent anyons, obtained from the detailed balance property). An important feature of the Ohmic case is that γ (0) 6= 0, thus direct hopping to neighbouring sites is always allowed. It is impossible to solve equation (1) analytically for meaningful system sizes, because the number of states pψ grows exponentially with lattice size. We thus stochastically simulate New Journal of Physics 15 (2013) 025027 (http://www.njp.org/) 6 1 (b) 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 <Zec> <Zec> (a) 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 1 2 3 4 5 γ(0)t 6 7 8 9 10 0 0 1 2 3 4 5 γ(0)t 6 7 8 9 10 Figure 3. Decay of the logical Z operator in the non-interacting toric code for the square lattice (left) and reduced square lattice (right) for grid sizes L = 16, 32, 64, 128 and L = 15, 30, 60, 120, respectively. The transition becomes sharper as the system size increases. All curves are ensemble averages over 104 runs. We have used T /J = 0.3, γ (0) = γ (2J ) and applied error correction at the readout stage. the system and obtain the quantities of interest. A single iteration of the simulation at time t consists P of (i) record the relevant parameters of the system, (ii) calculate the total spin flip rate R = i γ (ωi,ψ ), where ψ is the current state of the system, (iii) draw the time it takes for the next spin-flip to occur from an exponential distribution with rate R and (iv) calculate individual spin-flip probabilities pi = γ (ωi,ψ )/R and flip a spin at random accordingly. After some initially specified time has been reached, we stop and have obtained a single iteration. We also apply an error correction scheme at the read out stage. The error correction scheme consists of pairing up all detected anyons and then annihilating them by connecting each pair with a string of errors from one partner to another. The pairing is usually chosen such that all anyons are annihilated with the smallest total number of single-spin operators. This is known as the minimal-weight perfect matching and can be found in polynomial time with the help of Blossom algorithm [21]. In order to find the true matching with minimal weight, one in principle would need to choose the set of edges that include all connections from every anyon to every other. This forms a complete graph with number of edges that grows quadratically with n. Hence, the overall scaling of the matching algorithm becomes formidable for large n. Instead, as a good approximation we perform a Delaunay triangulation using Triangle [22], so that searching is performed only on anyons that are close to each other resulting in a number of edges that is linear in the number of anyons. Figure 3 illustrates the decay of the logical Z operator in the non-interacting toric code for the square and reduced square lattices. New Journal of Physics 15 (2013) 025027 (http://www.njp.org/) 7 Figure 4. Examples of a family of lattices obtained by replacing each edge of the square lattice with two and three sub-edges respectively. In general one replaces each edge of the square lattice with n-sub-edges. 4. Degree as a measure of connectivity There are a great deal of different lattices on which the toric code may be defined with many distinct characteristics. However, much of their behaviour with respect to the propagation of errors depends on a single property, the connectivity of the lattice [6]. The average degree of a lattice is a simple and effective measure for the connectivity. Lattices with high connectivity will have a large average degree and vice versa. Consider a periodic connected planar lattices with a finite number K such that a unit cell of the lattice can have vertices of various degrees d1 , d2 , . . . , d K . The degree of a vertex is the number of edges originating from the vertex. The average degree of the lattice is defined as a weighted average q= K X wi di , (3) i=1 where the weights w = (w1 , . . . , w K ) are the proportions in which each node appears within a unit cell. Furthermore, by using Euler’s formula for planar graphs, one can show that for any periodic lattice 2 2 + = 1, (4) q q̄ where q̄ is the average degree of the dual lattice L̄. For example, the triangular lattice has an average degree q = 6, its dual, the hexagonal lattice has an average degree q̄ = 3. Intuitively, one expects a highly connected lattice (large q) to be fragile against Z errors, resulting in a relatively small error-tolerance threshold. On the other hand, by equation (4), its dual lattice will be less connected (small q̄) and more robust against X errors, resulting in a relatively bigger error-tolerance threshold. To understand how the threshold probability pc varies with the average degree q let us consider the following argument. Intuitively, one would expect that error-correction is likely to fail when the density of anyons exceeds some constant threshold independent of the lattice geometry. For small errors, the density of errors in a region surrounding a given vertex with degree q is pc × q, this implies that pc ∼ 1/q since the density is constant. To gain a deeper insight into how the threshold probability varies with average degree, especially in the q → ∞ limit, we will consider the class of lattices given in figure 4. These are based on the square lattice, but replace each edge with n so-called sub-edges. Each edge therefore contains n spins, New Journal of Physics 15 (2013) 025027 (http://www.njp.org/) 8 rather than just one as before, and each vertex has a degree of 4n. The effect of a Z error on any of the n spins per edge is identical and indistinguishable to that on any other. Each changes the anyonic occupancy of the two vertices connected by the edge. A net error on the edge therefore only occurs when an odd number of single spin errors are applied, since even numbers of errors cancel each other out. If the probability of a single spin Z error is p, the probability of a net error on the edge is therefore 1 P( p) = (1 − (1 − 2 p)n ). (5) 2 Error correction on the n-sub-edge lattice is exactly the same as that on the square lattice, except that P is used as the error probability per edge. The threshold probability for the n-subedge lattice, pcn , is therefore that for which P( pcn ) = pc , where pc ≈ 0.11 is the normal threshold probability for the square lattice. Rearranging equation (5), we find 1 pcn = (1 − (1 − 2 pc )1/n ) 2 (6) 1 = (1 − (1 − 2 pc )4/q ) 2 (7) 4 1 = (1 − e q ln(1−2 pc ) ) 2 (8) = (−2/q) ln(1 − 2 pc ) ≈ O(1/q). (9) Here we have used the fact that n = q/4. This shows that the threshold probability will decay as O(1/q) as q → ∞. This decay is consistent with that seen in figure 5. For the dual lattices with 2 q̄n = 2 + (q̄0 − 1/n), a similar argument based on the √ Chernoff bound p0 = e−2n( pn −1/2) can be used to show that as 1/(q̄ − 2) → ∞, pc ≈ 1/2 − O( q̄ − 2). In practice, we will be working with reasonable values of q and q̄, i.e. 3 6 q, q̄ 6 6, rather than q → ∞ and q̄ → 2. 5. Numerical results on the relationship between d, pc and τ Now that all our tools have been introduced, we may now begin to a study of how the error tolerance thresholds and thermal lifetimes of lattices depend on their connectivity. We focus on concrete examples of reasonable lattices, i.e. those for which no vertex or plaquette operator need act on an non-physically high number of spins. Figure 5 shows the error-tolerances for various lattices as a function of the degree of the lattice. For a square lattice where L = L̄, the error-tolerances of the primal and dual lattices are symmetric. One can break this symmetry by considering non-self-dual lattices. From equation (4), we know that for such lattices, there is an asymmetry in the average degrees of the primal and dual lattices. This in turn results in an asymmetry in the error-tolerance thresholds of the primal and dual lattices. The data in figure 5 confirm our earlier intuition that lattices with small average degrees (i.e. less connectivity) will have better localization properties and consequently high error-tolerance thresholds. On the other hand, lattices with large average degrees (i.e. low connectivity) will have less localization properties and consequently small New Journal of Physics 15 (2013) 025027 (http://www.njp.org/) 9 (a) (b) 9.5 0.18 Square − Analytical Square − Simulation Reduced Square − Simulation Reduced Square − Analytical Union − Analytical Union − Simulation 9 8.5 0.16 8 0.14 7.5 7 pc Lifetime 0.12 6.5 0.1 6 5.5 0.08 5 0.06 4.5 4 0.04 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 Average Degree 0.27 0.28 0.29 0.3 0.31 0.32 0.33 T/J Figure 5. Left: simulation data of error-tolerance probability thresholds for lattices with various average degrees. We have focused on realistic lattice geometries with average degree 3 6 q 6 6. Right: analytical data obtained from equation (14) versus simulation data for the lifetime of the quantum memory for various values of T /J and for various lattice patterns. This comparison reveals that equation (14) provides a good description of the system’s lifetime. error-tolerance thresholds. One can also express the error-tolerance threshold in terms of the average degree by considering the best-fit curve to the data points in figure 5, for 3 6 q 6 6 (µq + ν) pc ≈ , (10) q −2 where µ = 0.0231 and ν = 0.111 are the fitting parameters. The data in table 1 show the lifetime of the memory for several lattices at different values of T /J . Clearly, the higher the temperature T , the smaller the lifetime. Furthermore, for any lattice with L 6= L̄, the lifetimes of the L and L̄ are not the same. Consequently any formula for the lifetime must take into account the structure of the underlying lattice, as well as the coupling constant J and the temperature of the system T . This because for any non-self-dual lattice the lifetime must exhibit an asymmetry and varies with J and T . Let N be the number of diffusing anyons present in the system, the fraction p of spins affected by Z error after a time t is estimated as εtN p= 2 , (11) L New Journal of Physics 15 (2013) 025027 (http://www.njp.org/) 10 Table 1. Data for primal and dual, square, reduced square and union lattices. The second column lists the average degree of each lattice. The third column lists error-tolerance thresholds obtained from simulations. The table also compares the lifetimes of the quantum memories as a function of T /J . Lattice Average degree pc Square Primal (4.0) Reduced square Union T /J = 0.28 0.29 0.3 0.31 0.32 0.1 7.3 6.4 5.7 5 4.6 Dual (4.0) Primal (3.56) Dual (4.57) 0.1 0.13 0.08 7.3 9.0 6.1 6.4 7.7 5.3 5.7 6.8 4.7 5 6.0 4.1 4.6 5.4 3.7 Primal (4.44) Dual (3.63) 0.09 0.11 6.8 8 5.8 7.2 5.0 6.5 4.55 5.65 4.15 5.0 where ε is some constant that may or may not depend on the lattice structure. The memory fails when p is larger than some critical value pc . This gives a lifetime τ for the memory pc τ = (e J/T + 1), (12) ε where we have replaced the density of anyons by the Fermi–Dirac equilibrium distribution N /L 2 = (e J/T + 1)−1 . It turns out that the constant ε is actually independent of the lattice size and structure and that ε ≈ 1/2. Therefore τ ≈ 2 pc (e J/T + 1). (13) This formula gives an excellent approximation for the memory lifetime when the error-tolerance threshold is known. By using equation (10), one can rewrite this formula in terms of the average degree q 2(µq + ν) J/T τ≈ (e + 1). (14) q −2 Thus, by simply knowing the average degree of a lattice at a given T /J , one can estimate the lifetime of the quantum memory. Figure 5 compares the lifetimes of the memory obtained using equation (13) with those obtained by direct simulation. Despite the approximation used in deriving equation (13), the lifetimes obtained from the two approaches are very close. An important conclusion we draw from the above analysis is that for a given value of T /J , one cannot hope to improve the lifetimes of both the primal and dual lattice, relative to the square lattices, solely by changing the lattice geometry. However, under a given temperature T , if one consider physical systems with biased couplings J for the vertex and the plaquette interactions, then one can find an optimal lattice structure that improves the lifetimes of both the primal and dual lattices relative to the square lattice under the same temperature T . 6. Josephson-junction arrays In this section we demonstrate that for systems with biased couplings, it is possible to improve the coherence time (the minimum of τp and τd ) of the quantum memory relative to the square lattice by optimizing lattice geometry. We demonstrate this by considering a Josephson-junction New Journal of Physics 15 (2013) 025027 (http://www.njp.org/) 11 15 10 Lifetime T=0.002 T=0.003 5 T=0.004 0 3.0 3.2 3.4 3.6 3.8 4.0 4.2 Average Degree Figure 6. The coherence time (minimum of τp and τd ) of the quantum memory obtained from equations (19) and (20) as a function of the average degree for x = 1.01. From the top, each curve represent the coherence time at temperature T = 0.002, 0.003, 0.004. The circles are the values of the coherence time obtained from direct simulations. implementation of the quantum memory [23]. The relations that relate the charging and the Josephson energies, E C and E J , to the couplings are given by p 3/4 1/4 Jv = E J E C exp(−qα E J /E C ), (15) Jf = EJ , q̄ (16) where the subscripts v and f refer to the vertex and face operators respectively, α = 1.61 is a constant and q and q̄ are related by equation (4) [23]. Let x be the ratio x = E J /E C , for x > 1 the system is said to be in the toric code regime. Using this we can simplify the above relations √ Jv /E C = x 3/4 exp(−qα x), (17) x J f /E C = . q̄ New Journal of Physics 15 (2013) 025027 (http://www.njp.org/) (18) 12 These formulas relate the couplings to the structure of the lattice. We can substitute equations (17) and (18) in the lifetime formula to obtain formulas for the lifetimes of the primal and dual lattices in terms of q and x  3/4  √ 2(µq + ν) x −(1.61q x) τp ≈ exp e , (19) q −2 T x  2(µq̄ + ν) exp q̄ . (20) q̄ − 2 T Thus, for a fixed x, by optimizing the degree of the lattice q one hopes to be able to improve the coherence time of the memory relative to the square lattice. For example, using these formulas, for x = 1.01, we calculated the lifetimes as a function of the degree for different temperatures, figure 6. It is evident that along a given curve, say T = 0.003, one can achieve a better coherence time for a Josephson-junction quantum memory by considering, say, a hexagonal lattice (q = 3) rather than a square lattice (q = 4). τd ≈ 7. Conclusion We have studied the effect of deforming lattice geometry on the error-tolerance and lifetime of the quantum memory. We have considered the toric code on non-regular lattices obtained by removing or adding edges to the square lattice. Unlike the square lattice, these lattices are non-self-dual, hence, the connectivity and the error-tolerance properties of the primal and the dual lattices are no longer the same. We have shown that the average degree of a lattice is a good measure for its connectivity and by considering families of lattices we have shown that the two quantitates are closely related. Lattices with small average degree will be more robust against errors and vice versa. We have also studied the lifetime of these lattices and provided a formula for calculating the lifetime of the memory at a given temperature by using the average degree. Finally, we have shown that for systems with biased couplings such as Josephson-junction arrays, it is possible to improve the coherence time of the memory relative to the square lattice by deforming the lattice geometry. 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