A polynomial time algorithm for the ground state of one-dimensional gapped local Hamiltonians

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1 A polynomal tme algorthm for the ground state of one-dmensonal gapped local Hamltonans Zeph Landau Umesh Vazran Thomas Vdck Aprl 16, 2015 Low entanglement approxmatons of the ground state Defnton 1. Gven a vector v H, by a Schmdt decomposton across the (, + 1) cut we shall mean a decomposton v = D j=1 λ j a j b j wth { a j } (respectvely { b j } ) a famly of orthonormal vectors of H [1,] (respectvely H [+1,n] ) and wth λ j λ j+1 > 0 for all 1 j D. The vectors a j wll be called the left Schmdt vectors across that cut, and the vectors b j the rght Schmdt vectors; D s the Schmdt rank across the cut. The followng lemma follows from the 1D area law [1]. Although we wll only need a polynomal bound on the Schmdt rank, we state the lemma usng the best known parameters [2, Secton 7]. Lemma 5. For any constant c > 0 there s a constant C 1 such that for every n there s a vector v wth Schmdt rank bounded by exp(c(ln n) 3/4 ε 1/4 ) across every cut such that Γ v 1 n c. The operaton of trmmng a state across a cut removng Schmdt vectors assocated wth the smallest Schmdt coeffcents wll be used repeatedly by our algorthm. Defnton 2. Gven a state v Hwth Schmdt decomposton v = j λ j a j b j across the (, + 1) cut and an nteger D, defne trm D v := D j=1 λ j a j b j. 1 The followng well-known lemma states that among all vectors wth Schmdt rank D across a certan cut, trm D v provdes the closest approxmaton to v. Lemma 6 (Eckart-Young theorem). Let v Hhave Schmdt decomposton v = λ a v across the (, + 1) cut. Then for any nteger D the vector v = trm D v / trm D v s such that v v w v for any unt w of Schmdt rank at most D across the -th cut. Computer Scence Dvson, Unversty of Calforna, Berkeley. Supported by ARO Grant W911NF , NSF Grant CCF and Templeton Foundaton Grant Department of Computng and Mathematcal Scences, Calforna Insttute of Technology. Part of ths work was completed whle the author was vstng UC Berkeley. Supported by the Natonal Scence Foundaton under Grant No and by the Mnstry of Educaton, Sngapore under the Ter 3 grant MOE2012-T We note an ambguty n the defnton of trm D v n the case of degeneraces among the Schmdt decomposton. In our analyss t wll never matter whch egenvectors assocated wth the same egenvalue are kept. NATURE PHYSICS Macmllan Publshers Lmted. All rghts reserved

2 We wll requre the exstence of close approxmatons to the ground state that have constant Schmdt rank across a gven cut (and polynomal across the others). Lemma 7 ([1, 2]). For any cut (, + 1) and any constant δ, there exsts a constant B δ = exp(o(1/ε log 3 d log 1/δ)) such that the state v := trm B δ Γ / trm B δ Γ has the property that Γ v 1 δ. Lemma 8. Let δ > 0 be such that δ(1 + 1/ε) 1 2 and w a unt vector wth energy no larger than ε 0 + δ. Then v := trm Bδ w / trm Bδ w has energy no larger than ε δ. Proof. By Lemma 14, Γ w 1 δ/ε. Let u := trm B δ Γ / trm B δ Γ. Snce by Lemma 7, Γ u 1 δ, Lemma 15 mples w u 1 δ(1 + 1/ε) =1 δ. The Eckart-Young theorem (see Lemma 6) therefore mples that trm B δ w 2 1 δ. Set w 0 = trm B δ w, w 1 = w w 0. We have w H w = w 0 H H w 0 + w 1 H H w 1 + w H w, and t follows that w 0 H w 0 ε 0 + δ w 1 H H w 1 + w 0 H w 0 w H w. Usng the fact that w 0 H w 0 w H w = w 0 H w 0 w + w 0 w H w 2 w w 0 2 δ along wth the lower bound of w 1 H H w 1 (ε 0 1) w 1 2 (ε 0 1)δ we have: w 0 H w 0 ε 0 (1 δ )+δ + 2 δ. Ths mples: f δ 1/2. v H v ε 0 + δ + 2 δ 1 δ ε δ, Corollary 1. For any cut (, + 1) and any constant δ, there exsts a constant B δ such that there exsts a state v wth Schmdt rank B δ that has energy at most ε δ as well as the property that Γ v 1 δ. Proof. Settng v = trm B δ Γ / trm B δ Γ, the result follows from Lemma 7 and Lemma Extenson Proof of Lemma 1 from Methods. By defnton S (1) = d S 1. Clearly, the bond dmensons of vectors n S (1) are no larger than that of vectors n S 1. Gven a wtness v for S 1 wth Schmdt decomposton across the ( 1, ) cut v = j λ j s j t j, decompose the frst qudt of t j on the computatonal bass as t j = d 1 k=0 k t jk. Then clearly v = j,k λ j s j k t jk s also a wtness for S (1). 2. Sze trmmng Boundary contractons The followng clam (specfcally part 3) shows how the boundary contracton can be used to decompose the problem of fndng an approxmate ground state nto ndependent left and rght subproblems: 2 NATURE PHYSICS Macmllan Publshers Lmted. All rghts reserved

3 SUPPLEMENTARY INFORMATION Lemma 9 (Glung). Gven a densty matrx σ on the space H L C B and a state v = B j=1 λ j a j b j on H L H R the densty matrx σ := U v σu v on H L H R satsfes the followng propertes: 1. tr HR (σ )=tr C B(σ), 2. tr [1,, 1] (σ ) tr [1,, 1] ( v v ) 1 = tr [1,..., 1] (σ) cont(v) 1, 3. tr(σ H) tr(σh L )+tr( v v (H R + H )) + n tr [1,..., 1] (σ) cont(v) 1. Proof. 1. Clear, snce U v s untary. 2. We have tr [1,, 1] (σ ) tr [1,, 1] ( v v ) 1 = tr [1,, 1] (σ v v ) 1 = tr [1,, 1] (U v σ U v U v v v )U v 1 = tr [1,..., 1] (σ) cont(v) Wrte tr(σ H)=tr(σ H L )+tr(σ (H + H R )). By the frst tem, tr(σ H L )=tr(σh L ). By the second tem, tr(σ (H + H R )) tr( v v (H + H R )) = (tr [1,..., 1] σ cont(v))(h + H R )) n tr [1,..., 1] (σ) cont(v) 1. We show that the set S (2) defned n the second step of the algorthm s a (, p 1 (n), q(n)b, 1/12)-vable set, for some polynomal q(n). The key observaton s that, condtoned on the exstence of a state w n Span{S (1) } H R havng both low energy and low bond dmenson, the soluton σ of the sze trmmng convex program (cf. (2) n Methods) for an X suffcently close to the boundary contracton of w allows for the easy computaton of the left Schmdt vectors of a good approxmaton to the ground state. Ths s shown n the followng lemma; the subsequent Lemma 11 establshes the exstence of w. Lemma 10. Suppose there exsts a state w n Span{S (1) } H R of bond dmenson B cε havng energy at most ε c ε. Let X be the element of the net N that s closest to cont(w) and let σ be the soluton to the sze trmmng convex program. Let u = j u j j be the leadng egenvector of σ. Then there exst orthonormal vectors { b j H R } such that u := j u j b j has energy at most ε 0 + ε/12 and u Γ 1 1/12. Proof of Lemma 10. Apply Lemma 9 to σ and w to conclude that the energy of σ = U w σu w can be upper bounded as follows tr(σ H) tr(σh L )+tr( w w (H R + H )) + n tr [1,..., 1] (σ) cont(w) 1 ε c ε + c ε ε c ε, (1) where we used the optmalty of σ to bound tr(σh L ) tr( w w H L ); ndeed, ls(w) tself s a feasble soluton to the sze trmmng convex program. Let v j be the egenvectors of σ, wth correspondng egenvalues λ 1 λ Bcε. From (1) we get that j J λ 2 j 1/2 where J = {j : tr(h v j v j ) ε c ε }. 3 NATURE PHYSICS Macmllan Publshers Lmted. All rghts reserved

4 But snce by defnton 14 c ε < ε/12 < ε/2, any v j wth energy less than ε c ε must satsfy v j Γ 2 > 1/2. Thus there can only be one such v j = v 1, and λ 2 1 > 1/2. Lettng u := U w v 1, u s the leadng egenvector of σ and has energy at most ε 0 + ε/12. Applyng Lemma 14 to u := v 1 establshes u Γ 1 1/12. In order to apply the prevous lemma, we need to establsh ts hypothess: that there exsts a vector w wth small bond dmenson and low energy that les n Span{S (1) } H R. Lemma 11. There exsts w n Span{S (1) } H R wth bond dmenson B cε and energy bounded by ε c ε. Proof. Let v be the wtness for S (1) beng a (, c ε /n)-vable set. Snce v s (c ε /n)-close to Γ and H has norm at most n, ts energy v s upper bounded by ε 0 + c ε. Applyng Lemma 8 to v we get a state w := trm Bcε v / trm Bcε v wth energy bounded by ε c ε ; moreover the left Schmdt vectors of w stll le n Span{S (1) }. We note that snce the set S (1) contans vectors specfed usng polynomal-sze MPS, for any X a polynomal-sze representaton for the optmal soluton σ to the sze trmmng convex program, (2) from Methods, can be computed effcently. For ths, we frst compute an orthonormal bass { f k } for Span{S (1) }. Vectors n ths bass van be represented as lnear combnatons of vectors n S (1). The varables of the convex program wll be the polynomally many coeffcents of σ on the f k f l ; to express the objectve functon as a functon of these varables t suffces to compute each f k H j f l, whch can be done effcently by expandng the f k on the vectors of S (1) and evaluatng the resultng expresson by usng the MPS representatons of the latter. The constrants can also be expressed as convex functons of the varables by pre-computng all f k H j f l. The remanng steps rely on the sngular value decomposton whch can be performed effcently as well. Proof of Lemma 2 from Methods. Together, Lemmas 10 and 11 establsh that the vector u from Lemma 10 s a wtness for S (2) wth error For every element of the net N, step 2. of the algorthm generates at of p 1 (n) =B cε N wth most B cε vectors to be added to S (2) yeldng a bound on the cardnalty of S (2) N =(Bd/η) O((Bd)2). Fnally snce S (2) Span{S (1) } t s clear that each vector n S (2) descrpton wth bond dmenson bounded by the product of S (1) elements of S (1),.e. dsb. 3. Bond Trmmng has an MPS and the maxmal bond dmenson of the Lemma 12. Gven a unt vector v wth D non-zero Schmdt coeffcents across the (, + 1) cut, for any u t holds that trm D/ε (u) v u v ε. Proof. Denote by λ 1 λ 2... the Schmdt coeffcents of u. We proceed by contradcton: assume w = u trm D/ε (u) has the property that w v > ε. Snce v has only D non-zero Schmdt coeffcents, by the Eckart-Young theorem (Lemma 6) ths last condton mples that D =1 λ D/ε + 2 > ε. 4 4 NATURE PHYSICS Macmllan Publshers Lmted. All rghts reserved

5 SUPPLEMENTARY INFORMATION Usng that the Schmdt coeffcents are decreasng, we get D/ε j=1 λ 2 j 1 D ε λ 2 D/ε +j > 1, j=1 a contradcton. We also show that trmmng a state across a gven bond does not ncrease the Schmdt rank across any of the other bonds. Lemma 13. For any nteger m the Schmdt rank of trm m(u) s no larger than the Schmdt rank of u across any cut (j, j + 1). Proof. Wthout loss of generalty, assume j. Wrtng the Schmdt decomposton across cut (, + 1) as u = λ α β notce that ( trm m(u) = Id m k=1 ) β k β k u. Snce ths operator only acts strctly to the rght of the (j, j + 1) cut, t cannot ncrease the Schmdt rank across that cut. Proof of Lemma 3 from Methods. Based on the analyss of the prevous step of the algorthm, from Lemma 12 we know that there exsts a wtness u for S (2) such that u Γ 1 1/12; furthermore u = ls(u ) s a member of the set S (1). Usng Lemma 15 we get that v u 1 5/24. Applyng Lemmas 12 and 13 to u yelds that the successve trmmng of u for each of the bonds 1,..., to Schmdt rank p 2 (n) results n a state w wth v w v u n/(48n) 1 11/48. Applyng Lemma 15 once more, Γ w 1 2(11/48 + 1/48) =1/2. Fnally, observe that the left Schmdt vectors of w are dentcal to the left Schmdt vectors of the state obtaned from applyng the successve trmmng procedure to u = ls(u ) nstead. 4. Error reducton The samplng AGSP We frst state the followng, whch s mpled by the varant of the Matrx-valued Chernoff bound [3, Theorem 1.6]. Theorem 1 (Matrx-valued Chernoff bound). Let X be d d..d. matrx random varables such that E[X ]=X, X X R, and σ 2 := max{e((x X)(X X) ), E((X X) (X X))}. Then for all ntegers l and t 0, ( 1 Pr l l k=1 ) X X t 2de lt2 2σ 2 +2Rt/3. (2) NATURE PHYSICS Macmllan Publshers Lmted. All rghts reserved

6 Proof of Proposton 1 from Methods. We apply Theorem 1 wth X = C m P I, X = A and t = 1 q(n), notng the bounds R = C m + 1, σ 2 (C m + 1) 2. Usng these n (2) yelds 1. wth probablty at least l 1 2d exp( 4C 2m q(n) 2 ). Choosng m as prescrbed and notng that C 2m = n O( ε 0 ) ε (where the constant n the exponent may depend on the degree of q) leads to the probablty of falure bounded by exp( l/(n O(ε0/ε) q(n) 2 )) whch for the specfed choce of l s exponentally small and n partcular can be made less than n 3 /2 wth an approprate choce of constants. Property 2 follows from elementary probablty: lettng Y j to be a random varable countng the number of tmes P j appears n a gven term, Y j has mean m n and varance bounded by m n and thus ( Yj Pr m m ) a e Ω(a2). n n For a proper choce of a = O( log(nlm)) the probablty s bounded by 1 2n 3 lm probablty of every projecton P j appearng no more than (a m n + m n below by 1 1 2n 3. Wth the prescrbed choces of a and m, a m. By a unon bound, the ) tmes n any term of K s bounded n + m n s upper bounded by O((ε 0/ε) log n). We note that the exponental dependence of the exponent c(ɛ) on the ground state energy that s stated n our man theorem s due to the dependence of the parameters m, l and κ n Proposton 1 on 1/ε. Huang [4] suggested a varant of our algorthm n whch our AGSP s replaced by a constructon based on deas of Hastngs [1] and Osborne [5], leadng to an mproved dependence on 1/ε that allows for an algorthm that runs n polynomal tme for any value of the ground state energy. Proof of Lemma 4 from Methods. Aplyng Proposton 1 from Methods wth the polynomal q(n)/2, we obtan that for a proper choce of the parameters m and l, and wth probablty at least 1 1/n 3, the samplng AGSP K wll have the desred propertes. Note that only P acts across the boundary cut (, + 1) and that we can decompose t as the sum of d 2 terms as P = d j,k=1 E j F k. Snce furthermore P appears no more than κ log(n) tmes n each term n K (for some κ = O( ε 0 ε )), usng the decomposton of P wthn each term P I of K we can wrte K = d 2κ log n l j=1 A j B j (3) as the sum of polynomally many terms wth A j actng only to the left of the cut and B j actng to the rght. Defne L := {A j } and set S := {A j s : A j L, s S}. Lettng v = λ j a j b j be a wtness for S, we get Γ Av Av 1 c ε/(2q(n)). Gven that K A c ε /(2q(n)), we get Γ Kv Kv 1 c Kv ε/q(n), so that Kv s a wtness for S achevng the clamed error. Each step of the constructon: generatng the randomness needed for K, the computaton of the A j and the constructon of S can be done effcently snce there are only a polynomal number of terms nvolved, and the matrx product operator representatons of the A j have polynomal sze bond dmenson and can be effcently computed. As a result, the set S has sze a fxed polynomal tmes that of S, and the bond dmenson of vectors n S s also a fxed polynomal tmes the bond dmenson of vectors n S. 6 NATURE PHYSICS Macmllan Publshers Lmted. All rghts reserved

7 SUPPLEMENTARY INFORMATION 5. Auxlary lemma The followng two smple lemma are used repeatedly. Lemma 14. Suppose a state v has energy v H v ε 0 + δ, for some 0 δ ε. Then v Γ 1 δ/ε. Proof. Wrte v = λ Γ + 1 λ 2 Γ for some unt vector Γ orthogonal to Γ. v has energy whch gves λ 2 1 δ/ε, hence λ 1 δ/ε. ε 0 + δ v H v λ 2 ε 0 +(1 λ 2 ) ε 1, Lemma 15. Let 0 δ, δ 1 and v, v and w be states such that v w 1 δ and v w 1 δ. Then v v 1 2(δ + δ ). Proof. We have v v v Γ v Γ ( (1 v Γ 2 )(1 v Γ 2 ) ) 1/2 =(1 δ)(1 δ ) 2 δδ 1 2(δ + δ ). References [1] M. B. Hastngs, An area law for one-dmensonal quantum systems, Journal of Statstcal Mechancs: Theory and Experment, vol. 2007, no. 08, p. P08024, [2] I. Arad, A. Ktaev, Z. Landau, and U. Vazran, An area law and sub-exponental algorthm for 1D systems, n Proceedngs of the 4th Innovatons n Theoretcal Computer Scence (ITCS), [3] J. A. Tropp, User-frendly tal bounds for sums of random matrces, Foundatons of Computatonal Mathematcs, vol. 12, pp , [4] Y. Huang, A polynomal-tme algorthm for approxmatng the ground state of 1D gapped Hamltonans, tech. rep., arxv: , [5] T. J. Osborne, Effcent approxmaton of the dynamcs of one-dmensonal quantum spn systems, Phys. Rev. Lett., vol. 97, p , Oct NATURE PHYSICS Macmllan Publshers Lmted. All rghts reserved