Neural-Network Quantum States

Transcription

1 Neural-Network Quantum States A Lecture for the Machne Learnng and Many-Body Physcs workshop Guseppe Carleo 1 June , Bejng 1 Insttute for Theoretcal Physcs, ETH Zurch, Wolfgang-Paul-Str. 27, 8093 Zurch, Swtzerland

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3 Chapter 1 Quantum Mechancs as a Machne Learnng Problem Every machne learnng approach has two fundamental ngredents. 1. The machne: typcally an artfcal neural-network, t s a hghly dmensonal (non-lnear) functon F (x; p 1... p Np ) of the parameters p 1... p Np 2. The learnng: the parameters p are learned on the bass of a stochastc optmzaton, that mnmzes some average loss functon L (p) on a dataset x 1, x 2... x Ns. For example L(x ) = F (x ; p) y,n the supervsed learnng settng wth expected labels y. On the other hand, the central goal n quantum mechancs s to fnd a soluton to Schroednger equaton H Ψ = E Ψ, (1.1) for = 0, 1,... and E 0 < E 1 <.... How can we reduce quantum mechancs then to a machne learnng problem? Frst of all, I wll address the requrement 2, whch has been done by poneers n computatonal quantum physcs lke Bll McMllan, n the 60s. Then, I wll address requrement 1, whch has been done nstead only very recently, thus completng the connecton between machne learnng and quantum mechancs. 1.1 Varatonal Monte Carlo To satsfy requrement 2, we need to transform ths egenvalue problem (1.1) nto a stochastc optmzaton problem. To acheve ths, we start from an alternatve formulaton of Schroednger s equaton, based on the varatonal prncple. In partcular, consder the energy functonal: E[Ψ] = Ψ H Ψ E 0, (1.2) where Ψ s some arbtrary physcal state, and E 0 s the exact ground-state energy of the Hamltonan H. From the varatonal theorem t s then clear that one can fnd the 3

4 exact ground-state wave-functon as the soluton of the optmzaton problem: Ψ 0 = argmn Ψ E[Ψ]. (1.3) For an arbtrary state Ψ, however t s seldom possble to compute analytcally the energy functonal, snce t nvolves ntegrals over a hgh-dmensonal space. To solve ths problem, n the 60s McMllan realzed that the energy functonal can be computed stochastcally. [McMllan1965] In partcular, the Varatonal Monte Carlo method s rooted nto the observaton that expectaton values lke (1.2) can be wrtten as statstcal averages over a sutable probablty dstrbuton. Let us assume that our Hlbert space s spanned by the many-body kets x. These n practce depend on the system n exam. For example n the case of spns 1/2 we would typcally have x = σ1, z σ2, z... σn z, for second-quantzed fermons x = n 1, n 2,... n N, for partcles n contnuous space x = r 1, r 2,... r N. The only dfference s of course that n the frst two cases one has a dscrete set of quantum numbers, whereas n the latter case the degrees of freedom are contnuos. In both cases we wll denote sums over the Hlbert space wth dscrete sums, although one should always bear n mnd that n the case of contnuous varables these sums must be nterpreted as ntegrals. In partcular we wll use the closure relaton x x x = Stochastc Estmates of Propertes Usng the closure relaton, we can rewrte a generc quantum expectaton value of some operator O as x,x Ψ x x O x x Ψ = x Ψ x x Ψ (1.4) Ψ O Ψ Ψ Ψ There can be, n general, two cases: = x,x Ψ (x)o xx Ψ(x ) x Ψ(x) 2. (1.5) 1. The operator O s dagonal n the computatonal bass,.e. O xx = δ xx O(x). Then Ψ O Ψ x = Ψ(x) 2 O(x) (1.6) Ψ Ψ x Ψ(x) 2 O, (1.7) where... denote statstcal expectaton values over the probablty dstrbuton Π(x) = Ψ(x) 2. In other words, n ths case quantum expectaton values are completely equvalent to averagng over Hlbert-space states sampled accordng to the square-modulus of the wave-functon. 2. The operator O s off-dagonal n the computatonal bass. Then, we can defne an auxlary dagonal operator (often called, n a somehow msleadng fashon, local operator or estmator) O loc (x) = Ψ(x ) O xx Ψ(x), (1.8) x 4

5 such that t s easly proven that Ψ O Ψ Ψ Ψ = x Ψ(x) 2 O loc (x) x Ψ(x) 2 (1.9) O loc. (1.10) For any observable, then, we can always compute expectaton values over arbtrary wave-functons as statstcal averages. In the case of off-dagonal operators, t should be notced that the sum x O xx Ψ(x ), s extended to the tny porton of the Hlbert Ψ(x) space for whch x s such that O xx 0. For the great majorty of physcal observables, and for a gven x, the number of elements x connected by those matrx elements s polynomal n the system sze, thus the summaton can be carred systematcally. Ths has to be contrasted nstead to the summatons n x Ψ(x) 2, where one typcally has an exponentally large number of possble values of x on whch to perform the summaton, and therefore cannot be done by brute-force. The powerful dea of the Varatonal Monte Carlo, s therefore to replace these sums over exponentally many states, wth a statstcal average over a large but fnte set of states sampled accordng to the probablty dstrbuton Π(x). We therefore have a way to compute, stochastcally, the expectaton value of all the propertes of nterest. For example we mght want to compute the expectaton value of σ x for a spn system, the expectaton value of c c j for fermons, or even the expectaton value of the nteracton energy W ee ( r 1... r N ) for our electronc structure problems Energy An mmedate corollary of the prevously presented scheme, s that also the expectaton value of the Hamltonan H (whch s tself a generc off-dagonal operator) can be computed usng the estmator (1.10). Hstorcally, the local estmator assocated to the Hamltonan s called local energy : E loc (x) = Ψ(x ) H xx Ψ(x). (1.11) x 1.2 Stochastc Varatonal Optmzaton The fnal goal we want to acheve here s to optmze the varatonal energy. In practce, we assume that the wave functon depends on some (possbly mllons of) parameters p = p 1,... p M. We have seen that the expectaton value of the energy can be wrtten as a statstcal average of the form H E loc. (1.12) It s easy to show that also the gradent of the energy can be wrtten under to form of the expectaton value of some stochastc varable. In partcular, defne D k (x) = p k Ψ(x) Ψ(x), (1.13) 5

6 then pk H = pk x,x Ψ (x)h xx Ψ(x ) x Ψ(x) 2 = x,x Ψ (x)h xx D k (x )Ψ(x ) x,x Ψ (x)d + k (x)h xx Ψ(x ) x x Ψ(x) 2 Ψ(x) 2 x,x Ψ (x)h xx Ψ(x ) x Ψ(x) 2 x Ψ(x) 2 (D k (x) + D k (x)) x Ψ(x) 2 x,x Ψ (x) Ψ = H (x ) xx D k(x ) Ψ(x ) 2 + x,x Ψ(x) 2 H xx Dk (x ) Ψ(x ) Ψ(x) x Ψ(x) 2 x H Ψ(x) 2 (D k (x) + Dk (x)) x Ψ(x) 2 E loc Dk E loc Dk + cc. (1.14) We can therefore compactly wrte pk H G k, wth the gradent estmator beng G k (x) = 2Re [(E loc (x) E loc ) D k(x)]. (1.15) Zero-Varance Property One of the most nterestng feature of the energy and energy-gradent estmators sofar presented s that they have the so-called zero-varance property: ther statstcal fluctuatons are exactly zero when samplng from the exact ground-state wave-functon. Let us consder for example var(e loc ) = E 2 loc E loc 2 = Ψ(x) 2 E loc (x) 2 H 2 x = H x,x1 Ψ(x 1 ) H x,x2 Ψ(x 2 ) H 2 x x 1 x 2 = Ψ(x 1 ) H x,x1 H x,x2 Ψ(x 2 ) H 2 x 1 x x 2 = H 2 H 2, (1.16) where we have assumed for smplcty that the wave-functon s real. Therefore the varance of the local energy s an mportant physcal quantty: the energy varance. It s easy to see that f Ψ s an egenstate of H then H 2 = H 2 = E0, 2 and var(e loc ) = 0,.e. the statstcal fluctuatons completely vansh. Ths property s very mportant snce t also mples that, n a sense to be specfed below, the closer we get to the ground-state, the less fluctuatons we have on the quantty we want to mnmze, the energy. 6

7 1.2.2 Stochastc Gradent Descent The gradent descent method s the smplest optmzaton scheme, where at each teraton the varatonal parameters are modfed accordng to p +1 k = p k η pk H, (1.17) where η s a (small) parameter called the learnng rate n the machne learnng communty. An mportant dfference wth respect to the non-stochastc (determnstc) gradent descent approach, s that now we only have stochastc averages of the gradent whch s therefore subjected to nose. Let us assume for smplcty that all the components of the gradent are subjected to the same amount of gaussan nose wth varance σ,.e. pk H = Normal ( G k, σ). (1.18) We can then compare Eq. 7 to the dscretzed Langevn equaton: ( p +1 k = p k δ t G k + Normal 0, ) 2δ t T, (1.19) where δ t s a small tme step. Ths equaton samples the Boltzmann dstrbuton Π B (p 1... p M ) = e H T, (1.20) whch n the lmt T 0 would converge to the varatonal ground-state,.e. to mn p H (p). We therefore see that the varance of the gradent corresponds to the effectve temperature as σ 2 = var(ḡk) = 2T/δ t (1.21) η = δ t. (1.22) Snce we want to fnd the varatonal ground state, we should have a scheme n whch the temperature s gradually decreased at each optmzaton step,.e. T 1 > T 2 > T 3..., as n the smulated annealng optmzaton protocol. The frst thng we notce s that σ 2 1 N s, decreases lke the number of samples n the Markov chan, therefore T = ηvar(ḡk) (1.23) 2 η var(g k ) (1.24) N s and convenent ways to reduce the temperature are ether to reduce the learnng rate: η() = η 0 / + 1 or to ncrease the number of samples wth the teraton count. Durng the optmzaton however f often happens that f we are close enough to the ground-state soluton var(g k ) 0. Indeed, t s easy to show that for an exact egenstate the statstcal fluctuatons of the gradent are exactly vanshng,.e. var(g k ) = 0. In practce then, even a constant number of samples and a fxed (small) η are suffcent to converge to the ground-state, provded that one checks durng the optmzaton that the value of the effectve temperature (1.23) s actually gong to zero as expected. 7

8 1.3 Example: Jastrow Factors Let us gve a specfc example of varatonal states, we consder now a system of nteractng partcles n contnuous space, for whch the most general Hamltonan s H = 2 2m N r 2 + V 1 ( r ) + <j V 2 ( r, r j ), (1.25) where V 1 and V 2 are generc one and two-body nteracton potental. We now defne the exact Jastrow-Feenberg expanson for the many-body state: [ Ψ p ( r 1,... r N ) = Ψ 0 ( r 1,... r N ) exp J 1 ( r ) + 1 J 2 ( r, r j ) + 2 j J p ( r 1, r 2... r p ), (1.26) p! p where Ψ 0 ( r 1,... r N ) s some parameter-ndependent wave-functon, and the varatonal parameters are the functons J 1 ( r),j 2 ( r, r ),... J p ( r 1, r 2... r p ). These expanson s clearly exact when p = N, however n practce one observes convergence to the exact groundstate much sooner, and typcally p = 2, 3 are enough to obtan very accurate results One-dmensonal trapped partcles As a smple exercse one can consder sngle-partcle, one-dmensonal Hamltonans of trapped partcles, for whch V 1 (x) s an even functon of x and V 2 = 0 (non-nteractng partcles). In ths case (for symmetry reasons) one can wrte the functon expanson J 1 (x) = p 1 x 2 + p 2 x , where p 1,p 2 etc are the parameters to be determned varatonally 1 Ψ(x) 2 In ths case t s easy to show that Ψ(x) = J x2 1 (x) + J 1(x) 2. (1.27) E loc (x) = 2 ( J 1 (x) + J 2m 1(x) 2) + V 1 (x) (1.28) D k (x) = x 2k. (1.29) 8

9 Chapter 2 Neural-Network Quantum States In the frst part of ths lecture we have rephrased the problem of fndng a ground state n terms of a stochastc optmzaton problem. To really take advantage of the potentaltes of machne learnng, however t s stll necessary to accomplsh task 1, n our prevous lst: we need to defne a sutable machne to solve our learnng problem. Ths s what we have done n our recent work. [CarleoTroyer2017] 2.1 Wave-Functon as a Neural Network The fundamental problem wth the stochastc optmzaton problem descrbed before s that, n prncple, to acheve the exact ground state energy one needs to consder exponentally many parameters. To see ths pont, consder the case of N spn 1/2 partcles, then the exact ground-state wave-functon s fully specfed by the 2 N ampltudes x Ψ = Ψ(x), (2.1) for all the possble values of x = σ1σ z 2 z... σn z. Ths task however s clearly unfeasble when the number of partcles N s too large. For example, one can do a back-of-the-envelope calculaton to show that only storng the wave-functon for more than 100 spns would requre a number of atoms larger than what can be found on our planet! However, ths exponental complexty s not necessary a lmtng factor. In ths case we can ndeed thnk of usng the ablty of artfcal neural networks to compress hgh-dmensonal data nto a low-dmensonal representaton. The startng pont s to ask a sutable neural network to compute the wave-functon ampltudes. Formally, we then set: Ψ(x) = F (x; p 1, p 2... p Np ), (2.2) where F s the output of a sutably chosen artfcal neural network, dependng on a set of parameters p. 9

10 2.2 Restrcted Boltzmann Machnes The choce of the specfc neural network used to represent the wave-functon s arbtrary, provded that t s reasonably expressve (.e. that n the lmt of large N p we can always recover the exact wave-functon). A convenent choce s the so-called Restrcted Boltzmann Machne (RBM), whch s defned as: F rbm (σ1, z σ2, z... σn) z = [ exp W j σ z h j + h j b j + ] σ z a, (2.3) {h} j j where the network parameters are W,a, and b. Ths archtecture corresponds to the partton functon of a gas of M hdden unts (h j ) connected to the physcal spns (σ z ). Snce the connectons are allowed only between hdden and vsble unts, but not between hdden unts, nor between vsble unts, ths archtecture s called restrcted. Because of ths restrcton, however t s easy to compute F explctly. Indeed [ exp W j σ z h j + h j b j + ] σ z a = (2.4) {h} j j e σz a [ ] Π j exp W j σ z h j + h j b j = (2.5) {h} [ ] [ e σz a Π j (exp W j σ z + b j + exp ]) W j σ z b j [ ] e σz a Π j 2 cosh W j σ z + b j = (2.6). (2.7) Because the wave-functon, n general, can be complex valued, also the weghts n ths expresson should be taken complex. It s easy to convnce one-self that f ths s the case than the wave-functon takes arbtrary complex values. 2.3 An example mplementaton Durng the lecture I wll show an example mplementaton of the stochastc optmzaton algorthm for neural-network RBM states. In partcular, I wll consder the transversefled Isng hamltonan n 1D: H = h σ x J σ z σ z +1, (2.8) wth perodc boundary condtons over a rng of L stes. To smplfy thngs, and knowng that the ground-state wave-functon n ths case s postve defnte, I wll consder the followng quantum state [TGC2017]: Ψ(σ1, z σ2, z... σn) z = F rbm (σ1, z σ2, z... σn z ), (2.9) 10

11 where the specfc RBM taken here contans only real-valued parameters. An advantage of ths formulaton s that samplng from Ψ(σ z 1, σ z 2,... σ z N ) 2 = F rbm (σ z 1, σ z 2,... σ z N ) s partcularly easy, snce t can been done usng alternate Gbbs samplng Gbbs Samplng Gbbs samplng s a specal case of the Metropols-Hastngs algorthm (see Appendx A). When the RBM has only real-valued parameters, then one can nterpret the quantty [ P (σ, h) = exp W j σ z h j + h j b j + ] σ z a, (2.10) j j as a jont probablty densty (apart from a global normalzaton) of the physcal and hdden unts. [FscherIgel2014] The dea of alternate Gbbs samplng s then to devse a two step Markov-chan samplng wth transton probabltes: T σ ((σ, h) (σ, h)) = P (σ, h) σ P (σ, h) = P (σ h) (2.11) T h ((σ, h) (σ, h )) = P (σ, h ) h P (σ, h ) = P (h σ). (2.12) The acceptance probablty for these two type of moves can be readly computed usng the Metropols-Hastng acceptance rule: ( A(x x ) = mn 1, Π(x ) Π(x) T ) (x x), (2.13) T (x x ) where n one case spn confguraton are changed, x =(σ, h) and n the other case hdden varable confguraton are changed, thus x =(σ, h ). For example, n the frst case the acceptance probablty reads: { A((σ, h) (σ, h)) = mn 1, P (σ, h) P (σ, h) T } σ((σ, h) (σ, h)) T σ ((σ, h) (σ, h)) { = mn 1, P (σ, h) P (σ, h) P (σ h) } P (σ h) = 1, (2.14) where n the last lne we have used the fact that P (σ h) P (σ h) = P (σ,h) σ P (σ,h) P (σ,h) σ P (σ,h) = P (σ, h) P (σ, h). (2.15) The same reasonng can be done for moves that change the hdden unts only, and one gets an acceptance of 1 as well. The mportant pont s that P (σ h) and P (h σ) can be computed exactly for an RBM. 11

12 For example, we have that: P (h σ) = P (σ, h) h P (σ, h ) = e σz a Π j exp [ W jσ z h j + b j h j ] e σz a Πj 2 cosh [ W jσ z + b, (2.16) j] and each hdden varable has a probablty whch s ndependent on the value of the other hdden varables. We have: P (h j = 1 σ) = Logstc(2θ j ) (2.17) P (h j = 1 σ) = Logstc( 2θ j ), (2.18) where Logstc(x) = 1 1+exp( x) and θ j = W jσ z + b j. A smlar expresson can be derved also for the other condtonal probablty, whch reads: P (h σ) = = P (σ, h) P (h) e [ j h jb j ] Π exp j W jσ z h j + a σ e [ j h jb j ], (2.19) Π 2 cosh j W jh z j + a and each spn varable has a probablty whch s ndependent on the value of the other spn varables. We then have: P (σ = 1 h) = Logstc(2γ ) (2.20) P (σ = 1 h) = Logstc( 2γ ), (2.21) where γ = j W jh z j +a. Proposng spn and hdden varable confguratons accordng to the Gbbs transton probablty s therefore very easy and conssts n the followng: 1. Generate N random numbers r [0, 1). 2. Set the -th spn wth probablty P (σ = 1 h) = Logstc(2γ ),.e. f P (σ = 1 h) > r then set σ = 1 otherwse σ = Generate M random number l j [0, 1). 4. Set the j-th hdden unt wth probablty P (h j = 1 σ) = Logstc(2θ j ),.e. f P (h j = 1 σ) > l j then set h j = 1 otherwse h j = 1. Repeatng these steps N s tmes, we then generate spn confguratons whch are sampled from Ψ(σ1, z σ2, z... σn z ) 2 = F rbm (σ1, z σ2, z... σn z ) Eq. (2.3). Notce that ths scheme s rather easy to mplement snce we do not need to perform a Metropols-Hastngs test at each step of the Markov chan, gven that all moves are accepted (see Appendx A for detals). 12

13 2.3.2 Computng the local energy For a gven spn confguraton, we also need to compute the local energy: E loc (σ) = ψ(σ ) H σ,σ ψ(σ). (2.22) σ For the transverse-feld Isng model, the sum runs over the N+1 confguratons σ (0) = σ and σ (k) = σ z 1 σ z k... σz N, wth H σ,σ = J σz σ z +1 and H σ,σ (k>0) = h. Ths sum can then be computed n polynomal tme, and t s effcently done pre-computng the values of the angles θ j. In partcular, ψ(σ (k)) ψ(σ) = e 2a kσ k cosh(θ j 2σ k W jk ) Π j. (2.23) cosh(θ j ) Computng the varatonal dervatves The varatonal dervatves can also be computed effcently and read: D k (σ) = p k Ψ(σ) Ψ(σ), (2.24) D a (σ) = 1 2 σz, (2.25) D bj (σ) = 1 2 tanh(θ j), (2.26) D Wj (σ) = 1 2 tanh(θ j)σ z. (2.27) 13

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15 Bblography [McMllan1965] Wllam L. McMllan, Phys. Rev. 138, A442 (1965) [FscherIgel2014] Asja Fscher, and Chrstan Igel. Tranng Restrcted Boltzmann Machnes: An Introducton. Pattern Recognton 47, 25-39, [CarleoTroyer2017] Guseppe Carleo, and Matthas Troyer. Solvng the quantum many-body problem wth artfcal neural networks. Scence 355, , [TGC2017] Gacomo Torla, Guglelmo Mazzola, Juan Carrasqulla, Matthas Troyer, Roger Melko, and Guseppe Carleo. arxv ,

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17 Appendx A Samplng Methods Durng the lecture we have establshed a fundamental connecton between quantum mechancs and statstcal samplng. For ths mappng to be effcent, we need an effcent way of samplng from the probablty dstrbuton Π(x) = Ψ(x) 2. In partcular the goal s to generate N s samples x (1), x (2),... x (Ns) such that we can estmate expectaton values as averages over those samples: A.0.1 O loc 1 O loc (x () ). N s Markov Chan and Detaled Balance (A.1) A Markov chan s completely specfed by the transton probablty T (x () x (+1) ),.e. gven a sample x (), we transton to the next element of the chan wth probablty T. The transton probablty (as all well-defne probabltes) must always be normalzed: x T (x x ) = 1. We would lke to devse a Markov chan process such that Π mc (x) = Π(x),.e. that the probablty wth whch a gven state x appears n the chan s equal to desred probablty we want to sample from. An mportant condton for ths to happen s that the probablty dstrbuton Π mc (x) s statonary,.e. all states along the chan should be dstrbuted accordng to the same probablty, and ths should not change along the chan. A suffcent condton for ths to happen s that Π(x)T (x x ) = Π(x )T (x x), (A.2) whch s called detaled balance equaton. Ths condton bascally enforces statonarty (also called mcro-reversblty) n the chan: the probablty of beng n a gven state x and of dong a transton to another state x must be equal to the reverse process, startng from x and transtonng to x. A.0.2 The Metropols-Hastngs Algorthm There exst many possble transton probabltes that satsfy the detaled balance condton (A.2), however the most famous choce s certanly the Metropols-Hastngs pre- 17

18 scrpton. In ths case, we separate the transton process nto two steps: T (x x ) = T (x x )A(x x ), (A.3).e. we frst propose a state wth some (smple) probablty dstrbuton T (x x ) we can easly sample from, and then accept or reject the new state x as the next element of the chan wth probablty A(x x ). Usng the detaled balance condton, we see that the acceptance probablty must satsfy: A(x x ) A(x x) = Π(x ) Π(x) T (x x) T (x x ). (A.4) A possble acceptance that satsfes ths condton s: ( A(x x ) = mn 1, Π(x ) Π(x) T ) (x x). (A.5) T (x x ) Notce that ths acceptance probablty satsfes (A.4), snce f Π(x ) T (x x) < 1 then Π(x) T (x x ) Π(x) T (x x ) > 1, Π(x ) T (x x) A(x x) = 1 and (A.4) s trvally verfed. The same reasonng can be appled for the case Π(x ) T (x x) > 1. Π(x) T (x x ) The Metropols-Hastng Algorthm can be then summarzed n the followng steps: 1. Generate a random state x drawng from the (smple) transton probablty T (x () x ). 2. Compute the quantty R = Π(x ) Π(x () ) T (x x () ) T (x () x ). (A.6) 3. Draw a unformly dstrbuted random number η [0, 1). 4. If R > η, accept the new states,.e. x (+1) = x. Otherwse, the followng state n the chan stays the current one: x (+1) = x (). Notce that steps 2-4 are necessary to decde whether to accept or reject the proposed state accordng to the Metropols probablty (A.5). 18

19 Appendx B Estmatng Errors and Auto-Correlaton Tmes Snce Markov chans are generated transtonng from a state to the next one, t s natural to expect that adjacent ponts n the chan wll be statstcally correlated. To quantfy ths noton of correlaton more precsely, let us frst consder the Markov chan estmate for the expectaton value of a gven functon: ḡ ns = 1 n s g, n s (B.1) where we have used the short-hand g g(x () ). The law of large numbers states that ḡ ns n s x and the central lmt theorem says that ḡ ns Π(x)g(x), (B.2) s a random varable normally dstrbuted, Prob(ḡ ns ) = Normal(ḡ, σ 2 ), (B.3) wth expected value ḡ and varance σ 2 = var(ḡ ns ), where the varance s computed over dfferent realzatons of the Markov chan. It explctly reads ( ) 1 var(ḡ ns ) = var g n s = 1 g n 2 g j 1 g s n 2 g j j s j ( ) = 1 1 ( g 2 n s n g 2) + 2 ( g g j g g j ) s n s j=+1 = 1 n s ) var(g 0 ) + 2 ( g 0 g j g 0 g j ) (1 jns, (B.4) n s j=1 where we assumed that the Markov chan s statonary,.e. var(g ) does not depend on the ndex and the same for the covarance. Therefore var(ḡ ns ) = 1 n s var(g 0 )2τ nt, (B.5) 19

20 havng defned the ntegrated auto-correlaton tme as τ nt = 1 n s 2 + ) ( g 0 g j g 0 g j ) (1 jns. (B.6) j=1 We therefore see that unless the Markov chan samples are completely uncorrelated (.e. g s g j g s g j = 0) the statstcal error on the estmator ḡ ns s ncreased by the postve factor τ nt. A way to correctly estmate the ntegrated autocorrelaton tme s through the correlaton functon ρ(j) = g 0g j g 2 g 2 g 2, (B.7) and a numercally stable estmate of the correlaton tme s gven by τ nt 1 j cut 2 + ρ(j), j=1 (B.8) where j cut s chosen for numercal stablty as the frst j such that ρ(j max ) < 0. In practce, gven a sequence of estmates g 1,... g ns = g, then the correlaton functon can be effcently estmated wth a sequence of Fast Fourer Transforms and ts nverses: A = F F T (g ḡ), (B.9) B = AA, (B.10) ρ = F F T 1 (B) g 2 g 2. (B.11) 20