Locally Accurate MPS Approximations for Ground States of One- Dimensional Gapped Local Hamiltonians

Locally accurate MPS approximations for ground states of one- dimensional gapped local Hamiltonians

Alexander M. Dalzell1 and Fernando G.S.L. Brandão1,2

1Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125, USA 2Google LLC, Venice, CA 90291, USA

A key feature of ground states of gapped local tion Group (DMRG) algorithm [36, 37], which aims to 1D Hamiltonians is their relatively low entangle- find a description of the ground state of local Hamilto- ment — they are well approximated by matrix nians on a one-dimensional chain of sites. DMRG has product states (MPS) with bond dimension scal- been an indispensable tool for research in many-body ing polynomially in the length N of the chain, physics, but the rationale for its empirical success did while general states require a bond dimension not become fully apparent until long after it was being scaling exponentially. We show that the bond widely used. Its eventual justification required two in- dimension of these MPS approximations can be gredients: first, that the method can be recast [20, 35] improved to a constant, independent of the chain as a variational algorithm that minimizes the energy length, if we relax our notion of approximation over the set of matrix product states (MPS), a tensor to be more local: for all length-k segments of network ansatz for 1D systems; and second, that the the chain, the reduced density matrices of our MPS ansatz set actually contains a good approximation approximations are -close to those of the ex- to the ground state, at least whenever the Hamiltonian act state. If the state is a ground state of a has a nonzero spectral gap. More specifically, Hastings gapped local Hamiltonian, the bond dimension [14] showed that ground states of gapped local Hamil- of the approximation scales like (k/)1+o(1), and tonians on chains with N sites can be approximated to at the expense of worse but still poly(k, 1/) scal- within trace distance by an MPS with bond dimen- ing of the bond dimension, we give an alternate sion (a measure of the complexity of the MPS) only construction with the additional features that it poly(N, 1/), exponentially smaller than what is needed can be generated by a constant-depth quantum to describe an arbitrary state on the chain. Even tak- circuit with nearest-neighbor gates, and that it ing into account these observations, DMRG is a heuris- applies generally for any state with exponen- tic algorithm and is not guaranteed to converge to the tially decaying correlations. For a completely global energy minimum as opposed to a local mini- general state, we give an approximation with mum; however, a recently developed alternative algo- bond dimension exp(O(k/)), which is exponen- rithm [2, 21, 26], sometimes referred to as the “Rigorous tially worse, but still independent of N. Then, RG” (RRG) algorithm, avoids this issue and provides a we consider the prospect of designing an algo- way one can guarantee finding an -approximation to rithm to find a local approximation for ground the ground state in poly(N, 1/) time. states of gapped local 1D Hamiltonians. When the Hamiltonian is translationally invariant, we These are extremely powerful results, but their value show that the ability to find O(1)-accurate local breaks down when the chain becomes very long. The arXiv:1903.10241v3 [quant-ph] 19 Sep 2019 approximations to the ground state in T (N) time bond dimension required to describe the ground state implies the ability to estimate the ground state grows relatively slowly, but it still diverges with N. energy to O(1) precision in O(T (N) log(N)) time. Meanwhile, if we run the RRG algorithm on longer and longer chains, we will eventually encounter an N too large to handle given finite computational re- 1 Introduction sources. Indeed, often we wish to learn something about the ground state in the thermodynamic limit In nature, interactions between particles act locally, mo- (N → ∞) but in this case these results no longer ap- tivating the study of many-body Hamiltonians consist- ply. Analogues of DMRG for the thermodynamic limit ing only of terms involving particles spatially near each [11, 22, 24, 31, 34, 38] — methods that, for example, op- other. An important method that has emerged from timize over the set of constant bond dimension “uniform this course of study is the Density Matrix Renormaliza- MPS” consisting of the same tensor repeated across the

Accepted in Quantum 2019-09-02, click title to verify. Published under CC-BY 4.0. 1 entire infinite chain — have been implemented with suc- that formally implied the existence of a uniform MPS cessful results, but these methods lack the second ingre- approximation of this type when the state is transla- dient that justified DMRG: it is not clear how large we tionally invariant and N = ∞, albeit without explicit must make the bond dimension to guarantee that the attention paid to the dependence of the bond dimen- set over which we are optimizing contains a good ap- sion on the locality k or approximation error , or to an proximation to the ground state. improvement therein when the state is the ground state Progress toward this ingredient can be found work by of a gapped Hamiltonian. Thus, we provide the missing Huang [17] (and later by Schuch and Verstraete [27]), ingredient for variational algorithms in the thermody- who showed that the ground state of a gapped local 1D namic limit as we show that a variational set over a MPS Hamiltonian can be approximated locally by a matrix with bond dimension independent in N and polynomial product operator (MPO) — a 1D tensor network object in 1/ contains a state that simultaneously captures all that corresponds to a (possibly mixed) density opera- the local properties of the ground state. tor as opposed to a quantum state vector — with bond We present two proofs of our claim about ground dimension independent of N and polynomial in the in- states of gapped Hamiltonians. The first yields supe- verse local approximation error. Here their approxi- rior scaling of the bond dimension, which grows asymp- mation sacrifices global fidelity with the ground state, totically slower than (k/)1+δ for any δ > 0; however, which decays exponentially with the chain length, in it constructs an MPS approximation |ψ˜i that is long- exchange for constant bond dimension, while retaining range correlated and non-injective. In contrast, the high fidelity with the ground state reduced density ma- second proof constructs an approximation that is in- trices on all segments of the chain with constant length. jective and can be generated by a constant-depth quan- In other words, the statistics for measurements of local tum circuit with nearest-neighbor gates, while retaining operators are faithfully captured by the MPO approx- poly(k, 1/) bond dimension. The latter construction imation, a notion of approximation that is often suffi- also follows merely from the assumption that the state cient in practice since many relevant observables, such has exponential decay of correlations. The proof idea as the individual terms in the Hamiltonian, are local. originates with a strategy first presented in [32] and However, the result does not provide the necessary in- constructs |ψ˜i by beginning with the true ground state gredient to justify infinite analogues of DMRG because |ψi and applying three rounds of operations: first, a MPO do not make a good ansatz class for variational set of unitaries that, intuitively speaking, removes the optimization algorithms. One can specify the matrix short-range entanglement from the chain; second, a se- elements for an MPO, but the resulting operator will quence of rank 1 projectors that zeroes-out the long- only correspond to a valid quantum state if it is posi- range entanglement; and third, the set of inverse uni- tive semi-definite, and verifying that this is the case is taries from step 1 to add back the short-range entan- difficult: it is NP-hard for finite chains, and in the limit glement. Intuitively, the method works because ground N → ∞ it becomes undecidable [19]. Thus, if we at- states of gapped Hamiltonians have a small amount of tempt to perform variational optimization over the set long-range entanglement. The non-trivial part is argu- of constant bond dimension MPO, we can be certain ing that the local properties are preserved even as the that our search space contains a good local approxima- small errors induced in step 2 accumulate to bring the tion to the ground state, but we have no way of restrict- global fidelity with the ground state to zero. The fact ing our search only to the set of valid quantum states; that |ψ˜i can be produced by a constant-depth quan- ultimately the minimal energy MPO we find may not tum circuit acting on an initial product state suggests correspond to any quantum state at all. the possibility of an alternative variational optimiza- In this work, we fix this problem by showing an anal- tion algorithm using the set of constant-depth circuits ogous result for MPS instead of MPO. We show that (a strict subset of constant-bond-dimension MPS) as for any gapped nearest-neighbor Hamiltonian on a 1D the variational ansatz. Additionally, we note that the chain with N sites, and for any parameters k and , disentangle-project-reentangle process that we utilize in there is an MPS representation of a state |ψ˜i with bond our proof might be of independent interest as a method dimension poly(k, 1/) such that the reduced density for truncating the bond dimension of MPS. We can matrix of |ψ˜ihψ˜| on any contiguous region of length k bound the truncation error of this method when the is -close in trace distance to that of the true ground state has exponentially decaying correlations. state. Importantly, the bond dimension is independent We also consider the question of whether these lo- of N. For general states (including ground states of cally approximate MPS approximations can be rigor- non-gapped local Hamiltonians), we give a construction ously found (à la RRG) more quickly than their globally with bond dimension that is also independent of N but approximate counterparts (and if they can be found at exponential in k/. This extends a previous result [8] all in the thermodynamic limit). We prove a reduction

Accepted in Quantum 2019-09-02, click title to verify. Published under CC-BY 4.0. 2 for ground states of translationally invariant Hamiltoni- Definition 1 (Matrix product state). A matrix prod- ans showing that finding approximations to local prop- uct state (MPS) |ηi on N sites of local dimension d erties to even a fixed O(1) precision implies being able (i) is specified by Nd matrices Aj with i = 1, . . . , d and to find an approximation to the ground state energy (i) j = 1,...,N. The matrices A1 are 1 × χ matrices and to O(1) precision with only O(log(N)) overhead. Since (i) strategies for estimating the ground state energy typi- AN are χ × 1 matrices, with the rest being χ × χ. The cally involve constructing a globally accurate approxi- state is defined as mation to the ground state, this observation gives us an d d intuition that it may not be possible to find the local X X (i1) (iN ) |ηi = ... A1 ...A |i1 . . . iN i . (2) approximation much more quickly than the global ap- N i1=1 iN =1 proximation, despite the fact that the bond dimensions required for the two approximations are drastically dif- The parameter χ is called the bond dimension of the ferent. MPS.

The same physical state has many diﬀerent MPS rep- 2 Background resentations, although one may impose a canonical form [25] to make the representation unique. The bond di- 2.1 One-dimensional local Hamiltonians mension of the MPS is a measure of the maximum amount of entanglement across any “cut” dividing the In this paper, we work exclusively with gapped nearest- state into two contiguous parts. More precisely, if we neighbor 1D Hamiltonians that have a unique ground perform a Schmidt decomposition on a state |ηi across state. Our physical system is a set of N sites, arranged every possible cut, the maximum number of non-zero in one dimension on a line with open boundary condi- Schmidt coeﬃcients (i.e. Schmidt rank) across any of tions (OBC), each with its own Hilbert space Hi of di- the cuts is equal to the minimum bond dimension we mension d. The Hamiltonian H consists of terms Hi,i+1 would need to exactly represent |ηi as an MPS [33]. that act non-trivially only on Hi and Hi+1: Thus to show a state has an MPS representation with a

N−1 certain bond dimension, it suffices to bound the Schd- X midt rank across all the cuts. This line of reasoning H = Hi,i+1. (1) i=1 shows that a product state, which has no entanglement, can be written as an MPS with bond dimension We will always require that Hi,i+1 be positive semi- 1. Meanwhile, a general state with any amount of en- definite and satisfy kHi,i+1k ≤ 1 for all i, where k·k tanglement can always be written as an MPS with bond is the operator norm. When this is not the case it is dimension dN/2. always possible to rescale H so that it is. We call H A cousin of matrix product states are matrix product translationally invariant if Hi,i+1 is the same for all i. operators (MPO). We will also always assume that H has a unique ground state |ψi with energy E and an energy gap ∆ > 0 to its Definition 2 (Matrix product operator). A matrix first excited state. We let ρ = |ψi hψ| refer to the (pure) product operator (MPO) σ on N sites of local dimension 2 (i) 2 density matrix representation of the ground state. For d is specified by Nd matrices Aj with i = 1, . . . , d any density matrix σ and subregion X of the chain, we (i) and j = 1,...,N. The matrices A1 are 1 × χ matrices let σ refer to Tr c (σ), the reduced density matrix of X X and A(i) are χ × 1 matrices, with the rest being χ × χ. σ after tracing out the complement Xc of X. N The operator is defined as Theorems1 and2 will make statements about effi- ciently approximating the ground state of such Hamil- d2 d2 tonians with matrix product states, and Theorem3 is X X (i1) (iN ) σ = ... A ...A σi ⊗ ... ⊗ σi , (3) a statement about algorithms that estimate the ground 1 N 1 N i1=1 iN =1 state energy E or approximate the expectation hψ| O |ψi of a local observable O in the ground state. d2 where {σi}i=1 is a basis for operators on a single site. The parameter χ is called the bond dimension of the 2.2 Matrix product states and matrix product MPO. operators However, MPO representations have the issue that (i) It is often convenient to describe states with one- specifying a set of matrices Aj does not always lead dimensional structure using the language of matrix to an operator σ that is positive semi-definite, which product states (MPS). is a requirement for the MPO to correspond to a valid

Accepted in Quantum 2019-09-02, click title to verify. Published under CC-BY 4.0. 3 quantum state. Checking positivity of an MPO in gen- Note that these quantities lack the property that (k) 0 (k) 0 0 1 eral is NP-hard for chains of length N and undecidable 0 = D (σ, σ ) = D1 (σ, σ ) implies σ = σ , but they for infinite chains [19]. do satisfy the triangle inequality. It is also clear that taking k = N recovers our notion of global distance: (N) 0 0 (N) 0 0 2.3 Notions of approximation D (σ, σ ) = D(σ, σ ) and D1 (σ, σ ) = D1(σ, σ ). We are interested in the existence of an MPS that ap- Definition 7 (Local approximation). We say a state σ proximates the ground state |ψi. We will have both a on a chain of N sites is a (k, )-local approximation to 0 (k) 0 global and a local notion of approximation, which we another state σ if D1 (σ, σ ) ≤ . define here. We will employ two different distance measures at different points in our theorems and proofs, the As we discuss in the next subsection, previous results ˜ purified distance [10, 29] and the trace distance. show that |ψi has a good global approximation |ψi that is an MPS with bond dimension that scales like a poly- Definition 3 (Purified distance). If σ and σ0 are two nomial in N. We will be interested in the question of normalized states on the same system, then what bond dimension is required when what we seek is merely a good local approximation. D(σ, σ0) = p1 − F(σ, σ0)2 (4)

0 2.4 Previous results is the purified√ distance between σ and σ , where F(σ, σ0) = Tr( σ1/2σ0σ1/2) denotes the fidelity between 2.4.1 Exponential decay of correlations and area laws σ and σ0. A key fact shown by Hastings [12] (see also [13, 15, Definition 4 (Trace distance). If σ and σ0 are two 23] for improvements and extensions) about nearest- normalized states on the same system, then neighbor 1D Hamiltonians with a non-zero energy gap is that the ground state |ψi has exponential decay of 1 1 D (σ, σ0) = kσ − σ0k = Tr(|σ − σ0|) (5) correlations. 1 2 1 2 Definition 8 (Exponential decay of correlations). A 0 is the trace distance between σ and σ . pure state σ = |ηi hη| on a chain of sites is said to Lemma 1 ([29]). have (t0, ξ)-exponential decay of correlations if for every t ≥ t0 and every pair of regions A and C separated by 0 0 p 0 at least t sites D1(σ, σ ) ≤ D(σ, σ ) ≤ 2D1(σ, σ ). (6)

0 0 0 Cor(A : C)|ηi We also note that D1(σ, σ ) = D(σ, σ ) if σ and σ are both pure. If the trace distance between ρ and σ is small := max Tr ((M ⊗ N)(σAC − σA ⊗ σC )) kMk,kNk≤1 then we would say σ is a good global approximation to ρ. We are also interested in a notion of distance that is ≤ exp(−t/ξ). (9) more local. The smallest ξ for which σ has (t0, ξ)-exponential de- Definition 5 (k-local purified distance). If σ and σ0 cay of correlations for some t0 is called the correlation are two normalized states on the same system, then the length of σ. 0 k-local purified distance between σ and σ is Lemma 2. If |ψi is the unique ground state of a Hamil- P (k) 0 0 tonian H = i Hi,i+1 with spectral gap ∆, then |ψi D (σ, σ ) = max D(σX , σX ), (7) X:|X|=k has (t0, ξ)-exponential decay of correlations for some t0 = O(1) and ξ = O(1/∆). where the max is taken over all contiguous regions X consisting of k sites. Proof. This statement is implied by Theorem 4.1 of [23].

Deﬁnition 6 (k-local trace distance). If σ and σ0 are two normalized states on the same system, then the k- While the exponential decay of correlations holds for local trace distance between σ and σ0 is lattice models in any spatial dimension, the other results we discuss are only known to hold in one dimension. (k) 0 0 D1 (σ, σ ) := max D1(σX , σX ), (8) X:|X|=k 1 To see this consider the simple counterexample where√ k = 2, 0 σ = |ηi hη|, σ √= |νi hν|, with |ηi = (|000i + |111i)/ 2, |νi = where the max is taken over all contiguous regions X (|000i − |111i)/ 2. In fact here hν|ηi = 0. This counterexample consisting of k sites. can be generalized to apply for any k.

Accepted in Quantum 2019-09-02, click title to verify. Published under CC-BY 4.0. 4 For example, in one dimension it has been shown that Crucially, if the local Hilbert space dimension d and ground states of gapped Hamiltonians obey an area law, the gap ∆ (or alternatively, the correlation length ξ) that is, the entanglement entropy of any contiguous re- are taken to be constant, then all three results read gion is bounded by a constant times the length of the χ = poly(N, 1/). boundary of that region, which in one dimension is just a constant. This statement was also first proven by 2.4.3 Existence of MPS approximations in the thermo- Hastings in [14] where it was shown that for any con- dynamic limit tiguous region X The aforementioned results, which describe explicit S(ρX ) ≤ exp(O(log(d)/∆)), (10) bounds on the bond dimension needed for good MPS which is independent of the number of sites in X, approximations, improved upon important prior work where S denotes the von Neumann entropy S(σ) = that characterized which states can be approximated −Tr(σ log σ). The area law has since been improved by MPS in the first place. Of course, any state on a [1,2] to finite chain can be exactly described by an MPS, but 3 the question of states living on the infinite chain is more S(ρX ) ≤ O˜(log (d)/∆), (11) subtle. In [9], the proper mathematical framework was ˜ where the O signifies a suppression of factors that scale developed to study MPS, which they call finitely corre- logarithmically with the quantity stated. lated states, in the limit of infinite system size, and in It was also discovered that an area law follows merely [8] it was shown that any translationally invariant state from the assumption of exponential decay of correla- on the infinite chain can be approximated arbitrarily tions in one dimension: if a pure state ρ has (t0, ξ)- well by a uniform (translationally invariant) MPS in the exponential decay of correlations, then it satisfies [5] following sense: for any translationally invariant pure state ρ there exists a net — a generalization of a se- S(ρX ) ≤ t0 exp(O˜(ξ log(d))). (12) quence — of translationally invariant MPS ρα for which 2.4.2 Efficient global MPS approximations the expectation value Tr(ραA) of any finitely supported observable A converges to Tr(ρA). An implication of The area law is closely related to the existence of an this is that if we restrict to observables A with support efficient MPS approximation to the ground state. To on only k contiguous sites, there exists a translation- make this implication concrete, one needs an area law ally invariant MPS that approximates the expectation using the α-Renyi entropy for some value of α with value of all A to arbitrarily small error . Thus, they es- 0 < α < 1 [32], where the Renyi entropy is given by tablished that local approximations for translationally α Sα(ρX ) = −Tr(log(ρX ))/(1 − α). An area law for the invariant states exist within the set of translationally von Neumann entropy (corresponding to α = 1) is not invariant MPS, but provided no discussion of the bond alone sufficient [28]. However, for all of the area laws dimension required for such an approximation, and did mentioned above, the techniques used are strong enough not explicitly consider the case where the state is the to also imply the existence of efficient MPS approxima- ground state of a gapped, local Hamiltonian. tions, and, moreover, area laws have indeed been shown Our Theorem1, which is stated in the following sec- for the α-Renyi entropy [16] with 0 < α < 1. tion, may be viewed as a generalization and improve- Hastings’ [14] original area law implied the existence ment on this work in several senses. Most importantly, of a global -approximation |ψ˜i for |ψi with bond di- we present a construction for which a bound on the mension bond dimension can be explicitly obtained. This bound log(d) scales like poly(k, 1/) when the state is a ground state O ∆ O˜ log(d) N χ = e ∆ . (13) of a gapped nearest-neighbor Hamiltonian, and exponentially in k/ when it is a general state. Furthermore, The improved area law in [1,2] yields a better scaling our method works for states on the finite chain that are for χ which is asymptotically sublinear in N: not translationally invariant, where it becomes unclear how the methods of this previous work would generalize. 3 ˜ log(d) ˜ log (d) O 1/4 O ∆ N (∆ log(N/)) χ = e . (14) 2.4.4 Constant-bond-dimension MPO local approximations Finally, the result implied only from exponential decay of correlations [5] is The problem of finding matrix product operator representations that capture all the local properties of a O˜(ξ log(d)) O˜(ξ log(d)) N state has been studied before. Huang [17] showed the χ = et0e . (15) existence of a positive semi-definite MPO ρχ with bond

Accepted in Quantum 2019-09-02, click title to verify. Published under CC-BY 4.0. 5 dimension 3 Statement of results

3 3/4 O˜ log (d) + log(d) log (1/) 3.1 Existence of local approximation χ = e ∆ ∆1/4 = (1/)o(1) (16) Theorem 1. Let |ψi be a state on a chain of N sites that is a (2, )-local approximation to the true ground of local dimension d. For any k and there exists an state ρ, where o(1) indicates the quantity approaches 0 MPS |ψ˜i with bond dimension at most χ such that as 1/ → ∞. Crucially, this is independent of the length ˜ of the chain N. Additionally, because the Hamilto- (1) |ψi is a (k, )-local approximation to |ψi χ nian is nearest-neighbor, we have Tr(Hρ ) − Tr(Hρ) ≤ O(k log(d)/) (N − 1), i.e., the energy per site (energy density) of (2) χ = e the state ρχ is within of the ground state energy den- provided that N is larger than some constant N0 = sity. Huang constructs this MPO explicitly and notes it O(k3/3) that is independent of |ψi. is a convex combination over pure states which them- If |ψi has (t0, ξ)-exponential decay of correlations, selves are MPS with bond dimension independent of then the bound on the bond dimension can be improved N. Thus, one of these MPS must have energy density to within of the ground state energy density. However, O˜(ξ log(d)) it is not guaranteed (nor is it likely) that one of these (2’) χ = et0e (k/3)O(ξ log(d)) constant-bond-dimension MPS is also a good local ap- 2 2 proximation to the ground state; thus our result may with N0 = O(k / ) + t0 exp(O˜(ξ log(d)) and if, addi- be viewed as an improvement on this front as we show tionally, |ψi is the unique ground state of a nearest- the existence not only of a low-energy-density constant- neighbor 1D Hamiltonian H with spectral gap ∆, it can bond-dimension MPS, but also one that is a good local be further improved to approximation to the ground state. 3 O˜ log (d) + log(d) log3/4(k/3) An alternative MPO construction achieving the same ∆ 1/4 (2”) χ = (k/)e ∆ task was later given in [27]. In this case, the MPO is a (k, )-local approximation to the ground state and has 2 2 3/4 with N0 = O(k / ) + O˜(log(d)/∆ ). Here χ is bond dimension asymptotically equivalent to (k/)1+o(1) where o(1) in-

3 3/4 3 dicates that the quantity approaches 0 as (k/) → ∞. O˜ log (d) + log(d) log (k/ ) ∆ ∆1/4 1+o(1) χ = (k/)e = (k/) . However, the state |ψ˜i that we construct in the (17) proof of Theorem1 is long-range correlated and cannot The idea they use is simple. They break the chain into be generated from a constant-depth quantum circuit. blocks of size l, which is much larger than k. On each Thus, while |ψ˜i is a good local approximation to the block they construct a constant bond dimension MPO ground state |ψi of H, it is not the exact ground state that closely approximates the reduced density matrix of of any gapped local 1D Hamiltonian. the ground state on that block, which is easy since each Next, we show that it remains possible to approxi- block has constant length and they must make only a mate the state even when we require the approximation constant number of bond truncations to the exact state. to be produced by a constant-depth quantum circuit; The tensor product of these MPO will be an MPO on the scaling of the bond dimension is faster in k and 1/, the whole chain that is a good approximation on any but it is still polynomial. length-k region that falls within one of the larger length l blocks, but not on a region that crosses the boundary Theorem 2. Let |ψi be a state on a chain of N sites of between blocks. To remedy this, they take the mixture local dimension d. If |ψi has (t0, ξ)-exponential decay of MPO formed by considering all l translations of the of correlations, then, for any k and , there is an MPS boundaries between the blocks. Now as long as l is |ψ˜i with bond dimension at most χ such that much larger than k, any region of length k will only span the boundary between blocks for a small fraction (1) |ψ˜i is a (k, )-local approximation to |ψi of the MPO that make up this mixture, and the MPO O˜(ξ log(d)) O(ξ2 log2(d)) will be a good local approximation. (2) χ = et0e k/2 This same idea underlies our proof of Theorem1, with the complication that we seek a pure state approxima- (3) |ψ˜i can be prepared from the state |0i⊗N by a tion and cannot take a mixture of several MPS. Instead, quantum circuit that has depth O˜(χ2) and consists we combine the translated MPS in superposition, which only of unitary gates acting on neighboring pairs of brings new but manageable challenges. qubits

Accepted in Quantum 2019-09-02, click title to verify. Published under CC-BY 4.0. 6 If, additionally, |ψi is the unique ground state of a smaller. However, the following result challenges the nearest-neighbor 1D Hamiltonian with spectral gap ∆, validity of this intuition, at least in the case that the then the bound on the bond dimension can be improved Hamiltonian is translationally invariant, by showing a to relationship between the problem of ﬁnding a local ap- 4 proximation and the problem of estimating the energy ˜ log (d) log(d) O 2 O (2’) χ = e ∆ k/2 ∆ density. The sort of constant-depth quantum circuit that can Problem 1 (Estimating energy density). Given a generate the state |ψ˜i in Theorem2 is shown in Figure nearest-neighbor translationally invariant 1D Hamilto- 1. Proof summaries as well as full proofs of Theorems nian H on N ≥ 2 sites and error parameter , produce 1 and2 appear in Section4. an estimate u˜ such that |u−u˜| ≤ where u = E/(N −1) We also note that, unlike Theorem1, Theorem2 does is the ground state energy density. not require that the chain be longer than some threshold N0; the statement holds regardless of the chain length, Problem 2 (Approximating local properties). Given a although this should be considered a technical detail nearest-neighbor translationally invariant Hamiltonian and not an essential aspect of the constructions. H, an error parameter δ, and an operator O whose support is contained within a contiguous region of length k, produce an estimate v˜ such that |v − v˜| ≤ δ, where v = hψ| O |ψi /kOk is the expectation value of the operator O/kOk in the ground state |ψi of H.

Problem1 is the restriction of Problem2 to the case where k = 2 and the operator O is the energy interaction term. Thus, there is a trivial reduction from Problem1 to Problem2 with δ = . However, the next theorem, whose proof is presented in Section4, states a much more powerful reduction.

Theorem 3. Suppose one has an algorithm that solves Problem2 for any single-site ( k = 1) operator O and δ = 0.9 in f(∆, d, N) time, under the promise that the Figure 1: Constant-depth quantum circuit that constructs Hamiltonian H has spectral gap at least ∆. Here d de- the (k, )-local approximation |ψ˜i in Theorem2 starting notes the local dimension of H and N the length of the from the initial state |0i⊗N . It is drawn here as a depth- chain. Then there is an algorithm for Problem1 under 2 circuit where unitaries Uj act on segments consisting of the same promise that runs in time O(ξ2 log(d) log(k/2)) contiguous sites. Each of these unitaries could themselves be decomposed into a sequence of nearest- min(2∆, (N − 1), 2) f , 2d, N O(log(1/)). (18) neighbor gates with depth poly(k, 1/). 12

Estimating the energy density to precision is equiva- 3.2 Reduction from estimating energy density to lent to measuring the total energy to precision (N −1), finding local properties so the quantity min(2∆, (N − 1), 2) is equivalent to the global energy resolution, twice the gap, or two, The previously stated results show that there exists a whichever is smallest. Thus, one may take = O(1/N) state that is both a (k, )-local approximation and an and understand the theorem as stating that finding local MPS with bond dimension poly(k, 1/). They say noth- properties to within O(1) precision can be done at most ing of the algorithmic complexity required to find a O(log(N)) faster than finding an estimate to the total (k, )-local approximation. The proofs describe how to ground state energy to O(1) precision. If local proper- construct the local approximation from a description ties can be found in time independent of N (i.e. there is of the exact ground state, but following this strategy an N-independent upper bound to f), then the ground would require first finding a description of the exact state energy can be estimated to O(1) precision in time ground state (or perhaps a global approximation to it). O(log(N)), which would be optimal since the ground One might hope that a different strategy would allow state energy scales extensively with N, and Ω(log(N)) the local approximation to be found much more quickly time would be needed simply to write down the output. than the global approximation, since the bond dimen- Another way of understanding the significance of the sion needed to represent the approximation is much theorem is in the thermodynamic limit. Here it states

Accepted in Quantum 2019-09-02, click title to verify. Published under CC-BY 4.0. 7 that if one could estimate expectation values of local ob- In both proofs we will also utilize the well-known fact servables in the thermodynamic limit to O(1) precision that Schmidt ranks cannot differ by more than a factor in some finite amount of time (for constant ∆ and d), of d between any two neighboring cuts on the chain. then one could compute the ground state energy den- Lemma 4. Any state σ on a bipartite system AB sity of such Hamiltonians to precision in O(log(1/)) AB satisfies the following relations. time. This would be an exponential speedup over the best-known algorithm for computing the energy density rank(σAB)rank(σB) ≥ rank(σA) (21) given in [17], which has runtime poly(1/). Taking the rank(σ )rank(σ ) ≥ rank(σ ) (22) contrapositive, if one could show that poly(1/) time is AB A B necessary for computing the energy density, this would rank(σA)rank(σB) ≥ rank(σAB) (23) imply that Problem2 with δ = O(1) is in general un- Proof. We can purify σAB with an auxiliary system C computable in the thermodynamic limit, even given the into the state |ηi. We can let σ = |ηi hη| and note that promise that the input Hamiltonian is gapped. It is rank(σAB) = rank(σC ). Thus each of these three equa- already known that Problem2 is uncomputable when tions say the same thing with permutations of A, B, there is no such promise [3]. and C. We will show the first equation. Write Schmidt It is not clear whether a O(log(1/)) time algorithm decomposition for computing the energy density is possible. The rank(σAB ) poly(1/) algorithm in [17] works even when the Hamil- X tonian is not translationally invariant, but it is not im- |ηi = λj |νjiAB ⊗ |ωjiC (24) mediately apparent to us how one might exploit trans- j=1 lational invariance to yield an exponential speedup. and then decompose |νji to find

rank(σAB ) rank(σB ) X X 4 Proofs |ηi = λjγjk |τjkiA ⊗ |µkiB ⊗ |ωjiC , j=1 k=1 4.1 Important lemmas for Theorems1 and2 (25) rank(σB ) where {|µki}k=1 are the eigenvectors of σB. This The pair of lemmas stated here are utilized in both The- shows that the support of σ is spanned by the orem1 and Theorem2. The first lemma captures the A set of |τjki and thus its rank can be at most essence of the area laws stated previously, and will be rank(σ )rank(σ ). essential when we want to bound the error incurred by AB B truncating a state along a certain cut. Corollary 1. If |ηi is a state on a chain of N sites with local dimension d, and the Schmidt rank of |ηi across Lemma 3 (Area laws [5], [1]). If σ = |ψi hψ| has (t0, ξ)- the cut between sites m and m+1 is χ, then the Schmidt exponential decay of correlations then for any χ and any rank of |ηi across the cut between sites m0 and m0 + 1 0 region of the chain A, there is a state σÃ with rank at is at most χd|m−m |. most χ such that Proof. Without loss of generality, assume m ≤ m0. The log(χ) reduced density matrix of |ηi hη| on sites [m+1, m0] has D(σ , σ˜ ) ≤ C exp − , (19) 0 A A 1 8ξ log(d) rank at most d|m −m| since this is the dimension of the entire Hilbert space on that subsystem. Meanwhile the where C1 = exp(t0 exp(O˜(ξ log(d)))) is a constant inde- rank of the reduced density matrix on sites [1, m] is χ. pendent of N. So by the previous lemma, the rank over sites [1, m0] is 0 If σ is the unique ground state of a nearest-neighbor at most χd|m −m|. Hamiltonian with spectral gap ∆, then this can be improved to 4.2 Proof of Theorem1 !! ∆1/3 log4/3(χ) First we state and prove a lemma that will be essential D(σ , σ˜ ) ≤ C exp −O˜ , (20) A A 2 4/3 for showing the first part of Theorem1. Then we pro- log (d) vide a proof summary of Theorem1, followed by its full proof. 8/3 where C2 = exp(O˜(log (d)/∆)). Lemma 5. Given two quantum systems A and B and Proof. The first part follows from the main theorem of states τA on A and τB on B, there exists a state σAB [5]. The second follows from the 1D area law presented on the joint system AB such that σA = τA, σB = τB, in [1], and log(d) dependence explicitly stated in [2]. and rank(σAB) ≤ max(rank(τA), rank(τB)).

Accepted in Quantum 2019-09-02, click title to verify. Published under CC-BY 4.0. 8 Proof. We’ll apply an iterative procedure. For round 1 let α1 = τA and β1 = τB. In round j write spectral decomposition

aj X αj = λj,i |sj,ii hsj,i|A (26) i=1

bj X βj = µj,i |rj,ii hrj,i|B , (27) i=1 where aj and bj are the ranks of states αj and βj, Figure 2: Schematic overview of the proof of Theorem1. Many eigenvectors {s }aj and {r }bj form orthonormal j,i i=1 j,i i=1 states |φ i are constructed with staggered divisions between bases of the Hilbert spaces of systems A and B, respec- j regions Mj,i of length l, then the |φj i are summed in super- aj bj tively, and eigenvalues {λj,i}i=1 and {µj,i}i=1 are non- position. Properties supported within a length-k region X are decreasing with increasing index i (i.e. smallest eigen- faithfully captured by |φj i for values of j such that X does values first). Then define not overlap the boundaries between regions Mj,i. Most values of j qualify under this criteria as long as l is much larger than min(aj ,bj ) X q k. Additional structure is defined (the regions Bj,j0 in Part 1, |uji = min(λj,i, µj,i) |sj,iiA ⊗ |rj,iiB , (28) and Bj,i in Part 2) in order to force hφj |φj0 i = δjj0 , but this i=1 structure is not reflected on the schematic. which may not be a normalized state. Define recursion relation Proof summary of Theorem1. In Ref. [27], an MPO αj+1 = αj − TrB(|uji huj|) that is a (k, )-local approximation to a given state |ψi was formed by dividing the chain into many length-l βj+1 = βj − TrA(|uji huj|) (29) segments, tensoring together a low-bond-dimension ap- and repeat until round m when αm+1 = βm+1 = 0. Let proximation of the exact state reduced to each segment, m and then summing over (as a mixture) translations of X locations for the divisions between the segments. We σAB = |uji huj| . (30) j=1 follow the same idea but for pure states: for each integer j = 0, . . . , l − 1, we divide the state into many Clearly rank(σAB) ≤ m. We claim that m ≤ length-l segments and create a pure state approxima- max(rank(τ ), rank(τ )). To show this we note that A B tion |φji that captures any local properties that are supported entirely within one of the segments. Then aj+1 + bj+1 ≤ max(aj, bj). (31) ˜ to form |ψi, we sum in superposition over all the |φji, We can see this is true by inspecting the ith term in the where each |φji has boundaries between segments oc- Schmidt decomposition of |uji in Eq. (28), and noting curring in different places (see Figure2). Thus, for any that either its reduced density matrix on system A is length-k region X, a large fraction of the terms |φji λj,i |sj,ii hsj,i| or its reduced density matrix on system B in the superposition are individually good approxima- is µj,i |rj,ii hrj,i| (or both). So when the reduced density tions on region X. The fact that a small fraction of the matrices of |uji huj| are subtracted from αj and βj to terms are not necessarily a good approximation creates form αj+1 and βj+1 in Eqs. (29), each of the min(aj, bj) additional, but small, error in the local approximation. terms causes the combined rank aj+1 +bj+1 to decrease In order to avoid interaction between different terms by at least one in comparison to aj + bj. in the superposition, we add additional structure to This alone implies that m ≤ max(a1, b1) + 1, since make the |φji in the superposition exactly orthogonal to by Eq. (31), the sequence {aj + bj}j must decrease one another. In our construction for general states, this by at least min(a1, b1) after the first round, and then additional structure consists of a set of disjoint, sparsely by at least 1 in every other round, reaching 0 when distributed, single-site regions Bj,j0 , one for each pair of 0 0 j = max(a1, b1) + 1. However, we can also see that integers j 6= j with 0 ≤ j, j < l. We force |φji to be the last round must see a decrease by at least 2, be- the pure state |0i and |φj0 i to be |1i when reduced to cause it is impossible for am = 0 and bm = 1 or vice Bj,j0 to guarantee that hφj|φj0 i = 0. Our construction versa (since Tr(αj) must equal Tr(βj) for all j). Thus for states that have exponential decay of correlations is m ≤ max(a1, b1). similar: we define a series of regions Bj,i and for each 0 Moreover, Eqs. (29) and (30) imply that σA = α1 = pair (j, j ) force |φji and |φj0 i to have orthogonal sup- τA and σB = β1 = τB. ports when reduced to one of these regions.

Accepted in Quantum 2019-09-02, click title to verify. Published under CC-BY 4.0. 9 Our approach for constructing |φji differs if |ψi is inition is possible as long as their are sufficiently many 3 a general state (Part 1), or if it is a state that either sites: N ≥ N0 = O(l ). It can be verified that since 0 has exponentially decaying correlations or (addition- 0 ≤ j, j ≤ l − 1, the distance between any Bj,j0 and 0 00 000 ally) is the ground state of a nearest-neighbor Hamil- Bj00,j000 for any distinct pairs (j, j ) and (j , j ) is at tonian (Part 2). In the latter case, we examine each least l. For each j, let Bj = ∪j06=jBj,j0 ∪ Bj0,j. For each length-l segment individually and truncate the bonds j, i, let Aj,i = Mj,i \ Mj,i ∩ Bj. Thus, in most cases of the exact state on all but a few of the rightmost sites Aj,i = Mj,i since Bj is a relatively small set of sites. n−1 c within that segment. The area law implies these trunca- Let Aj = ∪i=0 Aj,i = Bj , the complement of Bj. tions have minimal effect. We use those few rightmost The state |φji will have the form sites to purify the mixed state on the rest of the seg- O ment. Then |φji is a tensor product over pure states |φji = |Qji ⊗ |0i ⊗ |1i , (32) Aj Bj,j0 Bj0,j on each of the segments. The bond dimension can be j06=j bounded within each segment of |φji which is sufficient to bound the bond dimension of |ψ˜i. where |0i and |1i are two of the d computational basis This does not work for general states because without states located on a single site. In other words |φji is a an area law, bond truncations result in too much error, product state over all single site regions Bj,j0 with some and without the truncations we do not have enough other (yet to be specified) state |Qji on the remainder room to purify the state. For a general state, there of the chain Aj. is simply too much entropy in the length-l segment to To construct |Qji, we apply Lemma5 iteratively fully absorb with only a few sites at the edge of the as follows. We let σ1 = ρAj,j+2 (the reduced ma- segment. Instead, we have the various segments absorb trix of the exact state ρ on region Aj,j+2). We each other’s entropy by developing a procedure to engi- combine σ1 with ρAj,j+3 using Lemma5 to form a neer entanglement between different length-l segments state σ2 on region Aj,j+2Aj,j+3 such that rank(σ2) ≤ that exactly preserves the reduced density matrix on max(rank(σ1), rank(ρAj,j+3 )). For any j, i, the rank of 2l each segment and keeps the Schmidt rank constant (al- the state on any region Aj,i is less than d since any re- beit exponential in k/) across any cut. The crux of gion contains at most 2l sites. So if we apply this process this procedure is captured in Lemma5. iteratively, forming σp+1 by combining σp and the state on region Aj,j+p+2 (j + p + 2 is taken mod n), then we 2l Proof of Theorem1. Part 1: Items (1) and (2) end up with a state σn−2 with rank at most d defined over all of Aj except Aj,j and Aj,j+1. Since by construc- First we consider the case of a general state |ψi and con- 2 ˜ tion Bj contains no sites with index smaller than 2l and struct an approximation |ψi satisfying items (1) and (2). 2 A different construction is given in Part 2 to show items Aj,j and Aj,j+1 are contained within the first 2l sites, (2’) and (2”), but it is similar in approach. Throughout we have Aj,j = Mj,j and Aj,j+1 = Mj,j+1 meaning each this proof, we use a convention where sites are num- of these two regions each contain l sites and the total dimension of the Hilbert space over Aj,jAj,j+1 is at least bered 0 to N − 1, which differs from the rest of the 2l paper. d . Thus we may use regions Aj,j and Aj,j+1 to purify the state σ . We let |Q i be any such purification. We choose integer l to be specified later. We require n−2 j The key observation is that the state |φ i, as defined l > k. To construct |ψ˜i, we will first construct states j by Eq. (32), will get any local properties exactly correct |φ i for j = 0, . . . , l − 1 and then sum these states in j as long as they are supported entirely within a segment superposition. Any reference to the index j will be A for some i 6= j, j + 1. As long as l is large, this will taken mod l. Consider a fixed value of j. To con- j,i be the case for most length-k regions, but it will not struct |φji we break the chain into n = bN/lc blocks n−1 be the case for some regions that cross the boundaries where n − 1 blocks {Mj,i}i=1 have length exactly l and between regions Aj,i or for regions that contain one of the final block Mj,0 has length at least l and less than the single site regions B 0 or B 0 . 2l. We arrange the blocks so that the leftmost site of j,j j ,j To fix this we sum in superposition over the states block M is site j; thus block M contains the sites j,1 j,i |φ i for each value of j. The motivation to do this is so [j + l(i − 1), j + li − 1] for i = 1, . . . , n − 1, and the last j that every length-k region will be contained within A block M “rolls over” to include sites at both ends of j,i j,0 for some value of i in most, but not all, of the terms the chain: sites [j + l(n − 1),N − 1] and [0, j − 1].A in the superposition. We will show that most is good schematic of the arrangement is shown in Figure2. Any enough. We let reference to the index i will be taken mod n. 2 We also define l − l single-site blocks on the chain l−1 0 1 0 0 ˜ X which we label Bj,j for all pairs j 6= j . Bj,j consists of |ψi = √ |φji . (33) 2 0 0 2 the site with index 3l (j + j ) + l(j − j ) + 2l . This def- l j=0

Accepted in Quantum 2019-09-02, click title to verify. Published under CC-BY 4.0. 10 We note that hφj|φj0 i = δjj0 since |φji is simply |0i To show item (2), we bound the Schmidt rank of the ˜ when reduced to region Bj,j0 and |1i when reduced to state |ψi across every cut that bipartitions the chain ˜ region Bj0,j, while |φj0 i is |1i when reduced to region into two contiguous regions. Since |ψi is a superposi- ˜ Bj,j0 and |0i when reduced to Bj0,j. Thus |ψi is nor- tion over l terms |φji, the Schmidt rank can be at most malized: l times greater than that of an individual |φji. Fix some 1 X hψ˜|ψ˜i = hφ |φ 0 i = 1. (34) value of j and some cut of the chain at site s. Since |φji l j j j,j0 is a product state between regions Aj and Bj, and moreover it is a product state on each individual site in B , This completes the construction of the approxima- j we may ignore B when calculating the Schmidt rank tion. We now wish to show it has the desired properties. j (it has no entanglement), and focus merely on |Qji . To show item (1), we compute the (local) distance from Aj |ψi to |ψ˜i. Let ρ˜ = |ψ˜ihψ˜| and consider an arbitrary We constructed the state |Qji by building up mixed length-k region X. We may write states σp on region Aj,j+2 ...Aj,j+p+1 until p = n − 2, then purified with the remaining two regions. Each σp 2l D1(ρX , ρ˜X ) has rank(σp) ≤ d . Now consider an integer b with   1 ≤ b ≤ n − 1 and b 6= j, j + 1. Denote σ = σ and l−1 l−1 n−2 1 1 X X note that =  TrXc (|φji hφj0 |) − TrXc (|ψi hψ|) . 2 l j=0 j0=0 1 rank(σ ) (35) Aj,1...Aj,b ≤ rank(σAj,j+2...Aj,b )rank(σAj,j+2...Aj,0 ) 0 First we examine terms in the sum for which j 6= j . 4l = rank(σb−j−1)rank(σn−j−1) ≤ d , (38) Since Bj,j0 and Bj0,j are separated by at least l sites c and l > k, X must include either Bj,j0 or Bj0,j (or where the first inequality follows from Lemma4. both). Since |φji and |φj0 i have orthogonal support on Moreover, we may choose b such that the cut be- both those regions, and at least one of them is traced tween Aj,b and Aj,b+1 falls within l sites of site s. We out, the term vanishes. find that the region Aj,1 ...Aj,b differs from the region Thus we have containing sites [0, s] by at most l sites at each edge. Thus the Schmidt rank on the region left of the cut can l−1 1 1 X be at most d2l larger than that of A ...A (Corol- D1(ρX , ρ˜X ) = TrXc (|φji hφj| − |ψi hψ|) j,1 j,b 2 l 6l j=0 lary1), giving a bound of d for the the Schmidt rank 1 ˜ of |φji. This implies the Schmidt rank of |ψi is at l−1 1 X most ld6l, which proves item (2). This applies when- ≤ D1 (ρX , TrXc (|φji hφj|)) . (36) 3 3 3 l ever N ≥ N0 = O(l ) = O(k / ), a bound which must j=0 be satisfied in order for the chain to be long enough to For a particular j, there are two cases. Case 1 in- fit all the regions Bj,j0 as defined above. This completes cludes values of j for which X falls completely within Part 1. the Aj,i for some i with i 6= j, j + 1. For these values of j the term vanishes because the reduced density matrix Part 2: Items (2’) and (2”) of |φji hφj| on X is exactly ρX . Case 2 includes all other values of j. For this to be the case, either X spans the This construction is mostly similar to the previous boundary between two regions Mj,i and Mj,i+1 (at most one with a few key differences. We choose integers l, t, 0 k − 1 different values of j), X contains a site Bj,j0 or and χ to be specified later. We require t be even and 0 2 Bj0,j for some j (at most 2 values of j, since the separa- l ≥ 2k, l ≥ 2t. We assume N ≥ N0 = 2l . We also tion between sites Bj,j0 implies only one may lie within require that d ≥ 4. If this is not the case, we coarse- X), or X is contained within Aj,j or Aj,j+1 (at most 2 grain the system by combining neighboring sites, and values of j). In this case, the term will not necessarily henceforth we assume d ≥ 4. be close to zero, but we can always upper bound the As in Part 1, to construct |ψ˜i, we will first construct trace distance by 1. The number of terms in the sum states |φji for j = 0, . . . , l−1 and then sum these states that qualify as Case 2 is therefore at most k + 3, and in superposition. Consider a fixed value of j. To con- the total error can be bounded: struct |φji we break the chain into n = bN/lc blocks k + 3 and we arrange them exactly as in Part 1. D1(ρX , ρ˜X ) ≤ . (37) Now the construction diverges from Part 1: the state l |φji will be a product state over each of these blocks Choosing l = (k + 3)/ shows item (1), that |ψ˜i is a |φ i = |φ i ⊗ ... ⊗ |φ i (39) (k, )-local approximation to |ψi. j j,0 Mj,0 j,n−1 Mj,n−1

Accepted in Quantum 2019-09-02, click title to verify. Published under CC-BY 4.0. 11 0 with states |φj,ii for i = 0, . . . , n−1 that we now specify. as long as χ is large enough that the 2l − t truncations The idea is to create a state |φj,ii that has nearly do not have much effect on the reduced density matrix the same reduced density matrix as |ψi on the leftmost there. Thus we will choose our parameters so that χ0 is l − t sites of region Mj,i. It uses the rightmost t sites to large (but independent of N), such that l is much larger purify the reduced density matrix on the leftmost l − t than t and k (so that most regions fall within a region sites. First we denote the leftmost l − t sites of block Aj,i), and such that t is large enough that it is possible Mj,i by Aj,i = [j+l(i−1), j+li−t−1] and the rightmost to purify states on Aj,i onto Mj,i. But, as in Part 1, we t sites by Bj,i = [j +li−t, j +li−1] (or appropriate “roll have the issue that some regions will not be contained over” definitions when i = 0). We write |ψi as an exact entirely within region Aj,i for some i. MPS with exponential bond dimension and form |ψj,ii We again deal with this issue by summing in super- by truncating to bond dimension χ0 (i.e. projecting onto position: the span of the right Schmidt vectors associated with l−1 ˜ 1 X the largest χ0 Schmidt coefficients, then normalizing the |ψi = √ |φji . (43) l j=0 state) at the cut to the left of region Aj,i, every cut within Aj,i, and at the cut to the right of Aj,i, for a To complete the construction we also must specify total of at most 2l−t truncations (recall the final region χ02−1 the orthonormal states {|rj,i,zi}z=0 defined on the t Mj,0 may have as many as 2l − 1 sites). We denote the sites in region B . We choose a set that satisfies the (j,i) j,i pure density matrix of this state by ρ = |ψj,ii hψj,i|. following requirements. We can bound the effect of these truncations using the area law given by Lemma3: (1) The reduced density matrix of |rj,i,zi on any single √ (j,i) χ0 site among the leftmost t/2 sites of Bj,i (recall we D(ρ , ρ) ≤ 2l − t , (40) have assumed t is even) is entirely supported on 0 where χ is the cost (in purified distance) of a single basis states 1,..., bd/2c. truncation, given by the right hand side of Eq. (19), (2) The reduced density matrix on any single site or by Eq. (20) in the case |ψi is the ground state of a among the rightmost t/2 sites is entirely supported gapped nearest-neighbor 1D Hamiltonian. These trun- on basis states bd/2c + 1, . . . , d. cations were not possible in Part 1 because we could 0 not invoke the area law for general states. (3) Let j = j + i mod l, and let i0 = i mod l. If Because of the truncations, we can express the re- t ≤ i0 ≤ l − t then for all z, |rj,i,zi is orthogonal to (j,i) duced density matrix ρ as a mixture of χ02 pure the support of the reduced density matrix of |φj0 i Aj,i χ02−1 on region Bj,i. states {|φj,i,zi}z=0 each of which can be written as 0 an MPS with bond dimension χ : We assess how large t must be for it to be possible

χ02−1 to satisfy these three conditions. The third item specif- (j,i) X ically applies only for values of i that lead to values of ρ = pj,i,z |φj,i,zi hφj,i,z| , (41) Aj,i j0 that are at least t away from j (modulo l) so that z=0 the purifying system Bj,i does not overlap with Bj0,i0 χ02−1 for any i0. The support of the reduced density ma- for some probability distribution {pj,i,z}z=0 . We now (j,i) trix of any |φj0 i on region Bj,i has dimension at most form |φj,ii by purifying ρ onto the region Mj,i using Aj,i χ02. Thus, if the dimension of B is more than 2χ02 the space B , which contains t sites, as the purifying j,i j,i it will always be possible to choose an orthonormal set subspace: χ02−1 {|rj,i,zi}z=0 satisfying the third condition. The first χ02−1 and second conditions cut the accessible part of the local X √ |φ i = p |φ i ⊗ |r i , (42) dimension of the purifying system in half, so a purifica- j,i Mj,i j,i,z j,i,z Aj,i j,i,z Bj,i z=0 tion that satisfies all three conditions will be possible if bd/2ct ≥ 2χ02. Any choice of set that meets all three χ02−1 where the set of states {|rj,i,zi}z=0 is an orthonormal conditions is equally good for our purposes. set defined on region Bj,i. This purification will only be We now demonstrate that the three conditions imply possible if the dimension of Bj,i is sufficiently large, and that for any pair (j, j0) there are regions of the chain we comment later on this fact, as well as how exactly on which the supports of the reduced density matrices χ02−1 to choose the set {|rj,i,zi}z=0 . of states |φji and |φj0 i are orthogonal. If it is the case 0 0 The key observation is that state |φji will get any lo- that j − j mod l < t or j − j mod l < t, then for 0 cal properties approximately correct as long as they are every i the region Bj,i overlaps with Bj0,i0 for some i . supported entirely within a segment Aj,i for some i, and Because Bj,i 6= Bj0,i0 , there will be some site that is

Accepted in Quantum 2019-09-02, click title to verify. Published under CC-BY 4.0. 12 in the right half of one of the two regions, but in the can always upper bound the trace distance by 1. The left half of the other, and items 1 and 2 imply that the number of terms in the sum that qualify as Case 1 is two states will be orthogonal when reduced to this site. of course bounded by l (there are only l terms) and the If this is not the case, then as long as there is some number of Case 2 terms is at most t + k − 1. Thus the i for which j0 = j + i mod l, then item 3 implies the total error can be bounded: orthogonality of the supports of |φji and |φj0 i. In fact because n ≥ 2l, there will be at least 2 such values of D1(ρX , ρ˜X ) i. We conclude that hφ |φ 0 i = δ 0 , which implies that 1 √ 0 j j jj ≤ (l 2l − tχ + (t + k − 1)) |ψ˜i is normalized as shown by the computation l √ 0 ≤ 2lχ + (k + t)/l. (47) 1 X hψ˜|ψ˜i = hφ |φ 0 i = 1. (44) l j j j,j0 We will choose parameters so this quantity is less than . Parameters l and t will be related to χ0 as follows. We have now shown how to deﬁne the approximation |ψ˜i and discussed the conditions for the parameters t, t = log(2χ02)/ log(bd/2c) (48) χ0, and d that make the construction possible. Now l = 2(k + t)/ (49) we assess the error in the approximation (locally). Let ˜ ˜ ρ˜ = |ψihψ| and consider an arbitrary length-k region X. If |ψi is known only to have exponentially decaying cor- We may write relations, then we choose

D1(ρX , ρ˜X ) 0 p 3/2 log(χ ) = 16ξ log(d) log(16C1 ξ log(d)(k + 3)/ ),   l−1 l−1 (50) 1 1 X X ˜ =  TrXc (|φji hφj0 |) − TrXc (|ψi hψ|) . where C1 = exp(t0 exp(O(ξ log(d)))) is the constant 2 l 0 j=0 j0=0 from Eq. (19). We note that t ≤ 3 log(χ ), so we can 1 (45) bound

p 0 For the same reason that led to the conclusion 4(k + t)C1 − log(χ ) D1(ρX , ρ˜X ) ≤ √ e 8ξ log(d) + hφj|φj0 i = δjj0 , we can conclude that TrXc (|φji hφj0 |) = 2

δjj0 , so long as k is smaller than l/2. To see that this p 0 log(χ0) 4(k + 3) log(χ )C1 − holds, it is suﬃcient to show that there is a region ≤ √ e 8ξ log(d) + 2 lying completely outside of X with the property that p 0 0 64ξ log(d)(k + 3)C1 − log(χ ) |φji and φj share no support on the region. Since ≤ √ e 16ξ log(d) + k = |X| ≤ l/2 and |Bj,i| = t ≤ l/2, for any j, X can 2 overlap the region Bj,i for at most one value of i. We ≤ + = , (51) showed before that for any j there would be at least 2 2 two values of i for which a subregion of B has this j,i where in the third line we have used the (crude) bound property, implying one of them must lie outside X. √ u ≤ eu with u = log(χ0)/(16ξ log(d)). Thus we have In the case that |ψi is known to be the ground state

l−1 of a gapped local Hamiltonian, we may choose 1 1 X D1(ρX , ρ˜X ) = TrXc (|φji hφj| − |ψi hψ|) √ 2 l 0 ˜ −1/4 3/4 3/2 j=0 log(χ ) = O(∆ log(d) log (C2 k + 3/ )), 1 (52) l−1 1 X where C = exp(O˜(log8/3(d)/∆) is the constant in ≤ D (ρ , Tr c (|φ i hφ |)) . (46) 2 l 1 X X j j j=0 Eq. (20) and the same analysis will follow. This proves item (1) of the theorem for the construction in Part 2. For a particular j, there are two cases. Case 1 oc- Items (2’) and (2”) assert that |ψ˜i can be written curs if X falls completely within the Aj,i for some i, in as an MPS with constant bond dimension, which we which case the only error is due to the 2l − t trunca- now show. Each state |φji is a product state with pure 0 tions to bond dimension χ . Since the trace norm D1 state |φj,ii on each each block Mj,i, and |φj,ii has bond 02 is smaller than the puriﬁed distance D (Lemma1), the dimension at most 2χ . Thus, if we cut the state |φji √ 0 contribution for these values of j is at most 2l − tχ . at a certain site, the bond dimension will be at most 02 Case 2 includes values of j for which X does not fall 4χ (recall that block Mj,0 may have sites at both ends completely within a region Aj,i for any i. In this case, of the chain and can contribute to the bond dimension). ˜ the term will not necessarily be close to zero, but we Since |ψi is a sum over |φji for l values of j, the bond

Accepted in Quantum 2019-09-02, click title to verify. Published under CC-BY 4.0. 13 dimension χ of |ψ˜i is at most 4lχ02. For the case of 4.3 Proof of Theorem2 exponentially decaying correlations, this evaluates to In this section, we first state a pair of lemmas that will 2 3 16ξ log(d) be essential for the proof of Theorem2, then we give χ = 4l(256C1 ξ log(d)(k + 3)/ ) O˜(ξ log(d)) a proof summary of Theorem2, and finally we provide ≤ et0e (k/3)O(ξ log(d)), (53) the full proof of Theorem2. First, an important and well-known tool we use is proving item (2’), and for the case of ground state of Uhlmann’s theorem [30], which expresses the fact that gapped Hamiltonian, we find if two states are close, their purifications will be equally √ close up to a unitary acting on the purifying auxiliary χ = 4l exp(O˜(∆−1/4 log(d) log3/4(C k + 3/3/2))) 2 space. 3 ˜ log (d) log(d) 3/4 3 O ∆ + 1/4 log (k/ ) ≤ (k/)e ∆ , (54) Lemma 6 (Uhlmann’s theorem [30]). Suppose τA and σA are states on system A. Suppose B is an auxiliary proving item (2”), where the second factor is asymptot- system and |T iAB and |SiAB are purifications of τA and o(1) ically (k/) . For completeness, we note that if we σA, respectively. Then combined neighboring sites because d < 4, we can now uncombine them possibly incurring a factor of 2 or 3 q 2 increase in the bond dimension, which has no effect on D(τA, σA) = min 1 − |hS| (IA ⊗ U) |T i | , (57) U AB AB the stated asymptotic forms for χ. These results hold as long as N ≥ 2l2, which trans- where the min is taken over unitaries on system B. 2 2 ˜ lates to N ≥ O(k / ) + t0 exp(O(ξ log(d))) in the Second, we prove the following essential statement case of exponentially decaying correlations, and N ≥ about states with exponential decay of correlations. O(k2/2) + O˜(log(d)/∆3/4) in the case that |ψi is a ground state of a local Hamiltonian. This completes Lemma 7. If L and R are disjoint regions of a 1D the proof. lattice of N sites and the state τ = |ηi hη| has (t0, ξ)- exponential decay of correlations, then Now we demonstrate that the state |ψ˜i constructed in 0 the proof of Theorem1, Part 2, is long-range correlated. D(τLR, τL ⊗ τR) ≤ C3 exp(−dist(L, R)/ξ ) (58) Given an integer m, consider the pair of regions A = whenever dist(L, R) ≥ t , where ξ0 = 16ξ2 log(d) and [0, l(l − 1) − 1] and C = [l(l − 1 + m),N − 1], which are 0 C = exp(t exp(O˜(ξ log(d)))). separated by ml sites. Assume n ≥ 2l + m, so that A 3 0 If τ is the unique ground state of a gapped nearest- and C both contain at least l2 sites. Define the following neighbor 1D Hamiltonian with spectral gap ∆, then this operators. can be improved to

Q1 = |φ0,1i hφ0,1| ⊗ ... ⊗ |φ0,li hφ0,l| 0 D(τLR, τL ⊗ τR) ≤ C4 exp(−dist(L, R)/ξ ) (59) Q2 = |φ0,l+mi hφ0,l+m| ⊗ ... whenever dist(L, R) ≥ Ω(log4(d)/∆2), where ξ0 = ... ⊗ |φ0,n−1i hφ0,n−1| ⊗ |φ0,0i hφ0,0| (55) 3 O(1/∆) and C4 = exp(O˜(log (d)/∆)). The operator Q1 is supported on A and Q2 is supported For pure states σ, we call σ a Markov chain for the on C. Since A and C each contain blocks M for at least i tripartition L/M/R if σLR = σL ⊗ σR. Thus Lemma l values of i, conditions (1),√ (2), and (3) above imply 7 states that exponential decay of correlations implies ˜ ˜ that Q1|ψi = Q2|ψi = |φ0i / l. Thus that the violation of the Markov condition, as measured by the purified distance (or alternatively, trace distance) Cor(A : C) ≥Tr((Q ⊗ Q )(˜ρ − ρ˜ ⊗ ρ˜ )) |ψ˜i 1 2 AC A C decays exponentially with the size of M. =Tr(Q ⊗ Q |ψ˜ihψ˜|) 1 2 Proof of Lemma7. The goal is to show that an expo- ˜ ˜ ˜ ˜ − Tr(Q1|ψihψ|)Tr(Q2|ψihψ|) nential decay of correlations in τ = |ηi hη| implies that 2 =1/l − 1/l . (56) τLR is close to τL ⊗ τR. We will do this by truncating the rank of τ on the region L to form σ, arguing that The choice of l is independent of the chain length σLR is close to σL ⊗ σR, and finally using the triangle N, so the above quantity is independent of N and in- inequality to show the same holds for τ. dependent of the parameter m measuring the distance Lemma3 says that there is a state σL with rank χ between A and C. Thus, the correlation certainly does defined on region L such that not decay exponentially in the separation between the − log(χ) regions. D(τL, σL) ≤ C1e 8ξ log(d) . (60)

Accepted in Quantum 2019-09-02, click title to verify. Published under CC-BY 4.0. 14 In fact, the choice of σL of rank χ that minimizes the and also such that kTL,jk, kTR,jk ≤ 1 (see Lemma 20 of distance to τL is the state PLτL/Tr(PLτL) where PL [5] for full justiﬁcation of this). Then we may write is the projector onto the eigenvectors of τL associated with the largest χ eigenvalues. Accordingly, we de- kσLR − σL ⊗ σRk1  2   ﬁne a normalization constant q = 1/Tr(PLτL) and let χ √ X |νi = qPL |ηi and σ = |νi hν| = qPLτPL be normal- ≤ max Tr  TL,j ⊗ TR,j (σLR − σL ⊗ σR) kT k≤1 ized states. j=1

We ﬁrst need to show that χ2 X Cor(L : R) := max Tr((A ⊗ B)(σ − σ ⊗ σ )) ≤ max Tr ((TL,j ⊗ TR,j)(σLR − σL ⊗ σR)) |νi LR L R kT k,kT k≤1 kAk,kBk≤1 j=1 L,j R,j (61) 2 ≤ χ Cor(L : R)|νi is small, given only that Cor(L : R)|ηi is small. Sup- 2 2 pressing tensor product symbols, we can write ≤ 6χ Cor(L : R)|ηi ≤ 6χ exp(−dist(L, R)/ξ) (66)

kσLR − σL ⊗ σRk1 If τ is the unique ground state of a gapped Hamil- tonian, then we may use the second part of Lemma3, = max Tr(T (σLR − σL ⊗ σR)) kT k≤1 and bound

= max Tr(PLTPL(σLR − σL ⊗ σR)), (64) kT k≤1 D(τLR, τL ⊗ τR) where the second equality follows from the fact that PL 1/3 4/3 ﬁxes the state σ. We can perform a Schmidt decom- −O˜ ∆ log (χ) √ log4/3(d) −O(∆dist(L,R)) ≤ 3C2e + 6χe . (71) position of the operator PLTPL into a sum of at most 2 χ terms which are each a product operator across the Here we can choose L/R partition 3 ! log(d)(1 + log(3C )) 4 χ2 log(χ) = O 2 + ∆dist(L, R) , X ∆1/4 PLTPL = TL,j ⊗ TR,j (65) j=1 (72)

Accepted in Quantum 2019-09-02, click title to verify. Published under CC-BY 4.0. 15 † and then each term is small enough to make the bound apply the reverse unitaries Ui that we applied in step 1. D(τLR, τL ⊗ τR) The projection step is the cause of the error between 5/3 4/3 |ψi and |ψ˜i. The number of projections is O(N), but O˜ ∆ dist(L,R) ≤ e log4/3(d) the error accrued locally is only constant. This follows

3/4 from the fact that the projectors are rank 1, so once we √ O log(d) log (3eC2) + 6e ∆1/4 e−O(∆dist(L,R)) apply projector Πi, region Mj is completely decoupled 0 from Mj0 when j < i and j > i. Thus any additional 3 ˜ log (d) O ∆ projections, which act only on the regions to the right of ≤ e e−O(∆dist(L,R)) (73) Mi, have no effect on the reduced density matrix on Mj (except to reduce its norm). Using this logic, we show as long as dist(L, R) ≥ Ω(log4(d)/∆2), so that the first that the number of errors that actually affect the state term in the second line is dominated by the second term. locally on a region X is proportional to the number of Also note we have used log(C ) = O˜(log8/3(d)/∆) in the 2 sites in X, and not the number of sites in the whole last line. This proves the second part of the lemma. chain. To make this error less than for any region of at most k sites, we can choose l = O(ξ0 log(k/)). After step 2, the state is a product state on blocks of l sites each, and in step 3, unitaries are applied that couple neighboring blocks, so the maximum Schmidt rank across any cut cannot exceed dl. This yields the Figure 3: Schematic for the proof of Theorem2. The chain scaling for χ. is divided into regions Mi of length l, which are themselves The result is improved when |ψi is the ground state of L R divided into left and right halves Mi and Mi . The state a gapped local Hamiltonian by using the improved area |ψ˜i is constructed by starting with |ψi, applying unitaries that law [1,2] in the proof of Lemma7 and in the truncation act only on M for each i to disentangle the state across the i of the states ρL and ρR before purifying into |αi and |βi. L R Mi /Mi cut, projecting onto a product state across those cuts, Finally, it can be seen that |ψ˜i is formed by a and finally applying the inverse unitaries on regions Mi. constant-depth quantum circuit with two layers of unitaries, where each unitary acts on l qubits. The first layer prepares the product state over blocks of length Proof summary for Theorem2. First, we make the fol- l that is attained after applying projections in step 2, lowing observation about tripartitions of the chain into and the second layer applies the inverse unitaries from contiguous regions L, M, and R: since |ψi has expo- step 3. Each unitary in this circuit can be decomposed nential decay of correlations, the quantity D(ρ , ρ ⊗ LR L into a sequence of nearest-neighbor gates with depth ρ ) is exponentially small in |M|/ξ0, where ξ0 = R O˜(d2l). O(ξ2 log(d)). This is captured in Lemma7 and requires the area law result from [5]. One can truncate ρL and Proof of Theorem2. We fix an even integer l, which we |M|/2 ρR to bond dimension d , incurring small error, then will specify later, and divide the N sites of the chain purify ρL into |αi using the left half of region M as the into n + 2 segments of length l, which we label, from purifying auxiliary space and ρR into |βi using the right left to right: M0,M1,...,Mn+1. If N does not divide l half. Since |αi ⊗ |βi and |ψi are nearly the same state evenly, then we allow segment Mn+1 to have fewer than after tracing out M, there is a unitary U acting only on l sites. For i ∈ [1, n] let Li be the sites to the left of M that nearly disentangles |ψi across the central cut of region Mi and let Ri be the sites to the right of region M, with U |ψi ≈ |αi⊗|βi (Uhlmann’s theorem, Lemma Mi. 6). Lemma7 tells us that, since |ψi has (t0, ξ)- The proof constructs the approximation |ψ˜i by ap- exponential decay of correlations, for any i ∈ [1, n], plying three steps of operations on the exact state |ψi. ρLiRi is close to ρLi ⊗ ρRi when l is much larger than n+1 0 First, the chain is broken up into many regions {Mi}i=0 ξ ; that is, of length l, and disentangling unitaries Ui as described 0 D(ρLiRi , ρLi ⊗ ρRi ) ≤ C3 exp(−l/ξ ) (74) above are applied to each region Mi in parallel. The state is close to, but not exactly, a product state across whenever l ≥ t . We also choose χ = dl and for each 0 √ the center cut of each region M . To make it an exact i define ρ0 and ρ0 , each with rank at most χ by i Li Ri product state, the second step is to apply rank-1 pro- taking A = Li and A = Ri in Lemma3. Thus we have jectors Πi onto the right half of the state across each 0 D(ρL , ρ ) ≤ C1 exp(−l/(16ξ)), (75) of these cuts, starting with the leftmost cut and work- i Li D(ρ , ρ0 ) ≤ C exp(−l/(16ξ)). (76) ing our way down the chain. Then, the third step is to Ri Ri 1

Accepted in Quantum 2019-09-02, click title to verify. Published under CC-BY 4.0. 16 Then by the triangle inequality we have Claim 1. If l ≥ ξ00 log(3C), then

0 0 D(ρ , ρ ⊗ ρ ) kΠnUn ... Π1U1 |ψik= 6 0. (81) LiRi Li Ri 0 ≤ D(ρL R , ρL ⊗ ρR ) + D(ρL ⊗ ρR , ρ ⊗ ρR ) i i i i i i Li i This allows us to define 0 0 0 + D(ρL ⊗ ρRi , ρL ⊗ ρR ) i i i Π U ... Π U |ψi = D(ρ , ρ ⊗ ρ ) + D(ρ , ρ0 ) + D(ρ , ρ0 ) ˜ n n 1 1 LiRi Li Ri Li Li Ri Ri |φi = . (82) kΠnUn ... Π1U1 |ψik ≤ C exp(−l/ξ00), (77) Note that, to put our proof in line with what is de- 00 0 where C ≤ 2C1 + C3, ξ = max(ξ , 16ξ), whenever l ≥ scribed in the introduction and proof summary earlier, t0. we may act with all the unitaries prior to the projectors Note that |ψi can be viewed as a purification LiMiRi if we conjugate the projectors of ρLiRi with Mi the purifying auxiliary system. Divide L R |φ˜i ∝ Π0 ... Π0 U ...U |ψi , (83) region Mi in half, forming Mi (left half) and Mi (right n 1 n 1 half). See Figure3 for a schematic. Each of these sub- † l/2 0 † systems has total dimension d and thus can act as the where Πi = Un ...Ui+1ΠiUi+1 ...Un, which still only 0 0 acts on the region R . purifying auxiliary system for ρ or ρ . Let |αii L i Li Ri LiMi 0 This can be compared with the state |φi defined by be a purification of ρ and |βii R be a purification Li Mi Ri 0 applying the disentangling operations without project- of ρ . Thus, |αii⊗|βii, which is defined over the entire Ri ing: original chain, is a purification of ρ0 ⊗ ρ0 . Li Ri |φi = U ...U |ψi . (84) Uhlmann’s theorem (Lemma6) shows how these pu- n 1 rifications are related by a unitary on the purifying aux- We claim that |φ˜i is a good local approximation for |φi iliary system: for each Mi with i ∈ [1, n], there is a (and delay the proof for clarity of argument). unitary Ui acting non-trivially on region Mi and as the Claim 2. For any integer k0, |φ˜i is a (k0, 0)-local ap- identity on the rest of the chain such that Ui |ψi is very proximation to |φi with 0 = Cpk0/l + 3 exp(−l/ξ00). close to |αii ⊗ |βii, a product state across the cut be- L R Next we can define tween Mi and Mi . In other words, Ui disentangles Li from R , up to some small error, by acting only on M . i i ˜ † † ˜ Formally we say that |ψi = Un ...U1 |φi, (85) q which parallels the relationship 2 hα | L ⊗ hβ | R U |ψi = 1 − δ , (78) i LiM i M Ri i LiMiRi i i i † † |ψi = Un ...U1 |φi . (86) 00 where δi ≤ C exp(−l/ξ ) for all i. An equivalent way to write this fact is Now suppose X is a contiguous region of the chain of length k. Then there is a region X0 of the chain of q 0 2 length at most k = k + 2l that contains X and is made Ui |ψiL M R = 1 − δi |αiiL M L ⊗ |βiiM RR 0 0 i i i i i i i up of regions Mj where j ∈ [a , b ]. Then 0 + δi |φii L R (79) LiMi Mi Ri ˜ ˜ 0 kTrXc (|ψihψ| − |ψi hψ|)k1 where |φii is a normalized state orthogonal to |αii⊗|βii. ˜ ˜ We can define the projector ≤ kTrX0c (|ψihψ| − |ψi hψ|)k1 ˜ ˜ = kTrX0c (|φihφ| − |φi hφ|)k1 Π = I L ⊗ |β i hβ | R , (80) i LiM i i M Ri i i ≤ Cpk/l + 5 exp(−l/ξ00) √ whose rank is 1 when considered as an operator acting ≤ C 6k exp(−l/ξ00), (87) R only on Mi Ri. We notice that Π U |ψi is a product state i i LiMiRi where the third line follows from the fact that |φi and L R across the Mi /Mi cut and has a norm close to 1. Sup- |ψi are related by a unitary that does not couple region pose we alternate between applying disentangling oper- X0 and region X0c, and the fourth line follows from ations Ui and projections Πi onto a product state as we Claim2. √ 00 move down the chain. Each Πi will reduce the norm If we choose l = max(t0, ξ log(3C k/)), then the of the state, but we claim the norm will never vanish requirements of Claim1 are satisfied, and we can see completely (and delay the proof for later for clarity of from Eq. (87) that |ψ˜i is a (k, )-local approximation to argument). |ψi, item (1) of the theorem.

Accepted in Quantum 2019-09-02, click title to verify. Published under CC-BY 4.0. 17 Item (2) states that |ψ˜i can be written as an MPS other, a factor only logarithmic in the total depth. This with constant bond dimension. This can be seen by proves the theorem. the following logic. The Schmidt rank of |φ˜i across any cut M L/M R is 1, as discussed in the proof of Claim ˜ i i Proof of Claim1. We prove by induction. Let |φji = 2 (see Eq. (91)), since the projector Πi projects onto ΠjUj ... Π1U1 |ψi. Note that ΠiUi |ψi is non-zero for all a product state across that cut and unitaries Uj with ˜ i, so in particular |φ1i is non-zero. Furthermore we note j > i act trivially across the cut. By Corollary1, this ˜ R L that, if it is non-zero, |φji can be written as a product implies that the Schmidt rank across any cut Mi /Mi+1 0 state α ⊗ |βji R for some unnormalized but j L M L M Rj can be at most dl/2. Acting with the inverse unitaries j j j † 0 ˜ ˜ non-zero state αj , and the reduced density matrix of Uj on |φi to form |ψi preserves the Schmidt rank across ˜ 0 ˜ R L |φji on Rj is ρ . If we assume |φji is non-zero then the cut Mi /Mi+1, since none couple both sides of the Rj cut. Because any cut is at most distance l/2 from some we can write M R/M L cut, the Schmidt rank across an arbitrary i i+1 ˜ ˜ cut can be at most a factor of dl/2 greater, again by |φj+1i = Πj+1Uj+1|φji 0 Corollary1, meaning the maximum Schmidt rank across = αj ⊗ Πj+1Uj+1 |βji (90) l any cut of |ψi is χ = d . √ Given our choice of l = ξ00 log(C 6k/), we find that and the state can be represented by an MPS with bond di- ˜ 2 0 2 mension k|φj+1ik = k αj ⊗ Πj+1Uj+1 |βjik 0 2 2 √ 00 00 = k α k kΠ U |β ik χ = ( 6C)ξ log(d)(k/2)ξ log(d)/2 j j+1 j+1 j 0 2 † O˜(ξ log(d)) 2 2 = k α k Tr(Πj+1Uj+1 |βji hβj| U Πj+1) = ee (k/2)O(ξ log (d)). (88) j j+1 = k α0 k2Tr(Π U ρ0 U † Π ) j j+1 j+1 Rj j+1 j+1 This proves item (2). Note that, in the case that our ≥ k α0 k2Tr(Π U ρ U † Π ) choice of l exceeds N, it is not possible to form the j j+1 j+1 Rj j+1 j+1 construction we have described. However, in this case − k α0 k2kρ − ρ0 k j Rj Rj 1 dl will exceed dN and we may take |ψ˜i = |ψi, which ≥ k α0 k2Tr(Π U ρU † Π ) is a local approximation for any k and and has bond j j+1 j+1 j+1 j+1 − k α0 k2kρ − ρ0 k dimension in line with item (2) or (2’). j Rj Rj 1 Item (2’) follows by using the same equations with 0 2 2 ≥ k α k kΠj+1Uj+1 |ψik C = 2C + C = exp(O˜(log3(d)/∆)) and ξ00 = O(1/∆). j 2 4 0 2 0 4 2 − k α k kρ − ρ k For Lemma7 to apply, we must have l ≥ Ω(log (d)/∆ ), j Rj Rj 1 but this will be satisfied for sufficiently large choices of 0 2 00 2 ≥ k αj k (1 − C exp(−l/ξ )) 2 k/ . Thus the final analysis yields 0 2 00 − k αj k C exp(−l/ξ ) 4 2 χ = eO˜(log (d)/∆ )(k/2)O(log(d)/∆). (89) > 0

Item (3) states that |ψ˜i can be formed from a low- as long as l ≥ ξ00 log(3C). depth quantum circuit. In the proof of Claim 2, we ˜ show how the state |φi is a product state across divi- L R L R ˜ Proof of Claim2. First, consider the cut Mi /Mi dur- sions Mi /Mi , as in Eq. (91). Thus the state |φi can ˜ ⊗N ing the formation of the state |φi. When the projector be created from |0i by acting with non-overlapping Π is applied, the state becomes a product state across R L i unitaries on regions Mi Mi+1 in parallel. Each of these this cut. The remaining operators are U and Π with ˜ j j unitaries is supported on l sites. Then, |ψi is related j > i, and thus they have no eﬀect on the Schmidt rank to |φ˜i by another set of non-overlapping unitaries sup- L R ˜ across the Mi /Mi cut, meaning |φi is a product state ported on l sites, as shown in Eq. (85). We conclude across each of these cuts, or in other words that |ψ˜i can be created from the trivial state by two layers of parallel unitary operations where each unitary ˜ |φi = |φ1iM M L ⊗ |φ2iM RM L ⊗ ... is supported on l sites, as illustrated in Figure1. In [6], 0 1 1 2 ... ⊗ |φni R L ⊗ |φn+1i R . (91) it is shown how any l-qudit unitary can be decomposed Mn−1Mn Mn Mn+1 into O(d2l) = O(χ2) two-qudit gates, with no need for ancillas. We can guarantee that these gates are all spa- Given an integer k and a contiguous region X of tially local by spending at most depth O(l) perform- length k, we can ﬁnd integers a and b such that Y = R L ing swap operations to move any two sites next to each Ma Ma+1 ...Mb−1Mb contains X and |b−a| ≤ k/l +2.

Accepted in Quantum 2019-09-02, click title to verify. Published under CC-BY 4.0. 18 Then 4.4 Proof of Theorem3 ˜ ˜ First we state a lemma that will do most of the legwork TrY c (|φihφ|) needed for Theorem3. Then we prove Theorem3. = |φa+1i hφa+1| ⊗ ... ⊗ |φbi hφb| Lemma 8. Suppose, for j = 0, 1, H(j) is a translation- † † ∝ Tr L R ΠbUb ... ΠaUa |ψi hψ| U Πa ...U Πb . LaMa Mb Rb a b ally invariant Hamiltonian defined on a chain of length (92) N and local dimension d. Further suppose that |ψ(j)i is (j) (j) the unique ground state of H with energy E0 , and The advantage here is that all of the Ui and Πi for which let ∆(j) be the spectral gap of H(j). We may form a new i 6∈ [a, b] have disappeared. On the other hand, we have chain with local dimension 2d by adding an ancilla qubit to each site of the chain. Then there is a Hamiltonian TrY c (|φi hφ|) K defined on this chain such that † † = Tr c (U ...U |ψi hψ| U ...U ) Y n 1 1 n (1) The ground state energy of K is † † = TrL M LM RR Ub ...Ua |ψi hψ| Ua ...U . (93) a a b b b K 1 (j) E0 = min E0 . (100) 3 j Note that, since (0) (1) q (2) If E < E , then the ground state of K is 2 0 0 0 Ui |ψi = 1 − δi |αii ⊗ |βii + δi |φii , (94) N (0) (1) (0) 0 A ⊗ |ψ i and if E0 < E0 , then the ground state is 1N ⊗ |ψ(1)i, where A refers to the N we can say that A ancilla registers collectively. 0 ΠiUi |ψi = Ui |ψi − δi(I − Πi) |φii , (95) (0) (1) (3) If E0 < E0 , then the spectral gap of K is at least (0) (1) (0) (1) (0) and thus min(∆ ,E0 −E0 , 1)/3 and if E0 < E0 , then (1) (0) the spectral gap of K is at least min(∆ ,E0 − ΠbUb ... ΠaUa |ψi (1) E0 , 1)/3. = Ub ...Ua |ψi − Proof of Lemma8. Note that a variant of this lemma b X is employed in [3,4,7] to show the undecidability of δ (Π U ... Π U )(I − Π ) φ0 j b b j+1 j+1 j j certain properties of translationally invariant Hamilto- j=a nians. 0 ≡ Ub ...Ua |ψi − δ |φ i , (96) Since H(j) is translationally invariant, it is specified (j) q by its single interaction term Hi,i+1: Pb 2 0 where δ ≤ j=a δj and normalized |φ i is normalized. N−1 This implies (j) X (j) H = Hi,i+1. (101) † † i=1 |hψ| Ua ...Ub ΠbUb ... ΠaUa |ψi| p 2 ≥ 1 − δ , (97) The Hamiltonian K will be defined over a new chain kΠ ...U †Π U ... Π U |ψik a b b b a a where we attach to each site an ancilla qubit, increas- 0 which shows that D1(τ, τ ) ≤ δ, where ing the local Hilbert space dimension by a factor of 2. We refer to the ancilla associated with site i by the † † τ = Ub ...Ua |ψi hψ| Ua ...Ub subscript Ai, and we refer to the collection of ancillas together with the subscript A. Operators or states Π U ... Π U |ψi hψ| U †Π ...U †Π τ 0 = b b a a a a b b (98) without a subscript are assumed to act on the original † 2 kΠa ...Ub ΠbUb ... ΠaUa |ψik d-dimensional part of the local Hilbert spaces. Let N−1 and hence X K = Ki,i+1, (102) ˜ ˜ D1(TrXc (|φi hφ|), TrXc (|φihφ|)) i=1 ˜ ˜ ≤ D1(TrY c (|φi hφ|), TrY c (|φihφ|)) where 0 0 1 (0) ≤ D1(τY , τY ) ≤ D1(τ, τ ) ≤ δ K = H ⊗ |00i h00| (103) i,i+1 i,i+1 AiAi+1 p 00 3 ≤ C k/l + 3 exp(−l/ξ ). (99) 1 + H(1) ⊗ |11i h11| (104) 3 i,i+1 AiAi+1 This holds for any region X of length k, so this proves + Ii,i+1 ⊗ (|01i h01| + |10i h10|) (105) the claim. AiAi+1

Accepted in Quantum 2019-09-02, click title to verify. Published under CC-BY 4.0. 19 with Ii,i+1 denoting the identity operation on sites i and bound the energy of the state |φi, i.e., the quantity i+1. In this form it is clear that K is a nearest-neighbor translationally invariant Hamiltonian and that each in- hφ| K |φi teraction term has operator norm 1 (a requirement un- 2N −2 2N −2 der our treatment of 1D Hamiltonians). The picture X X ∗ = αxαy hx|A hηx| K |yiA |ηyi we get is that if two neighboring ancillas are both |0i, x=1 y=1 (0) (1) N N N then Hi,i+1/3 is applied, if both are |1i then Hi,i+1/3 2 −2 2 −2 2 −1 N−1 is applied, and if the ancillas are diﬀerent, then I is X X X ∗ X i,i+1 = αxαy hx|zi hz|yiA hηx| Kz,i |ηyi applied. Following this intuition, we can rewrite K as x=1 y=1 z=0 i=1 follows. 2N −2 N−1 X 2 X = |αx| hηx| Kx,i |ηxi . (109) x=1 i=1 2N −1 N−1 ! X X K = |xi hx|A ⊗ Kx,i , (106) We make the following claim: x=0 i=1 Claim 3. For any state |ηi, and any 1 ≤ a < b ≤ N

b−2 X (j) b − a (j) where the ﬁrst sum is over all settings of the ancillas hη| H |ηi ≥ E − 1. (110) i,i+1 N − 1 0 and the operator Kx,i acts on the non-ancilla portion i=a of sites i and i + 1, with Proof of Claim3. First we prove it in the case that M := b − a divides N. Let region Y refer to sites  (0) [a, b − 1], let ρ = TrY c (|ηi hη|), and let σ = ρ ⊗ ... ⊗ ρ Hi,i+1/3 if xi = xi+1 = 0  (1) be N/M copies of ρ which covers all N sites. Then Kx,i = Hi,i+1/3 if xi = xi+1 = 1 (107) I if x 6= x b−2 i,i+1 i i+1 N X (j) Tr(H(j)σ) = hη| H |ηi M i,i+1 i=a N/M−1 (j) X (j) We analyze the spectrum of K. If H has eigenval- + hη| HkM,kM+1 |ηi (j) (j) ues En with corresponding eigenvectors |φn i (where k=1 (j) b−2 En is non-decreasing with increasing integers n), then N X (j) N N (0) N (1) ≤ hη| Hi,i+1 |ηi + − 1 , (111) the states 0 ⊗ |φn i and 1 ⊗ |φn i are eigen- M M A A i=a (0) (1) states of K with eigenvalues En /3 and En /3, respec- where the last line follows from the fact that the inter- tively. (j) action strength kHi,i+1k ≤ 1. Moreover, by the varia- Recall that eigenvectors of a Hamiltonian span the (j) (j) tional principle, Tr(H σ) ≥ E0 . These observations whole Hilbert space over which the Hamiltonian is de- together yield ﬁned. Therefore, the eigenvectors of K listed above span the entire sectors of the Hilbert space associated b−2 N N X (j) (j) with the ancillas set to 0 A or 1 A. Suppose |φi is hη| Hi,i+1 |ηi ≥ (M/N)(E0 + 1) − 1, (112) another eigenvector of K. Since it is orthogonal to all i=a of the previously listed eigenvectors, |φi can be written which implies the statement of the claim. Now suppose M does not divide N. We decompose N = sM + r for non-negative integers s and r < M. 2N −2 X Let σ = ρ⊗...⊗ρ⊗|νi hν| where there are s copies of ρ |φi = αx |xiA ⊗ |ηxi (108) PN−1 (j) x=1 and |νi is the exact ground state of i=N−r+1 Hi,i+1, which has energy Er. Then

i0+M0−2 P 2 X (j) for some set of complex coeﬃcients αx with x|αx| = (j) Tr(H σ) ≤ s hη| Hi,i+1 |ηi + Er + s. (113) 1 and some set of normalized states |ηxi. The sum ex- i=a plicitly leaves out the x = 0 = 0N and x = 2N − 1 = 1N binary strings because these states lie in the subspace Here we invoke the variational principle twice. First, PN−1 (j) spanned by eigenstates already listed. We wish to lower note that the expectation value of i=N−r+1 Hi,i+1 in

Accepted in Quantum 2019-09-02, click title to verify. Published under CC-BY 4.0. 20 (j) the state |φ0 i (the exact ground state of the whole Proof of Theorem3. We begin by specifying a family of (j) Hamiltonians, parameterized by t ∈ [0, 1/2] and deﬁned chain) is exactly (r − 1)E0 /(N − 1). Since |νi is the exact ground state of that Hamiltonian, Er must over a chain of length N with local dimension d. be smaller than this quantity. Second, as before, N−1 (j) (j) Tr(H σ) ≥ E0 . Combining these observations yields Z X H (t) = Ii ⊗Ii+1 −(1−t) |0i h0|i ⊗|0i h0|i+1 (117) i0+M−1 i=1 X (j) 1 (j) r − 1 hη| H |ηi ≥ E 1 − − 1 i,i+1 s 0 N − 1 Z i=i The ground state of H (t) is the trivial product state 0 ⊗N M |0i with ground state energy t(N − 1), and thus en- = E(j) − 1. (114) N − 1 0 ergy density t. The interaction strength is bounded by 1 and the spectral gap is 1 − t ≥ 1/2. Now we construct an algorithm for Problem1. We are Now we use this claim to complete the proof of given H as input, with associated parameters N and d, Lemma8. For any binary string x we can associate and a lower bound ∆ on the spectral gap. Let the true ground state energy for H be E and let u = E/(N − 1). a sequence of indices 1 = i0 < i1 < . . . < im < im+1 = We choose a value of s between 0 and 1, and we ap- N + 1 such that xi = xi for all k = 1, . . . , m and k ply Lemma8 to construct a Hamiltonian K combining all i in the interval [ik, ik+1 − 1]. Moreover we require Hamiltonians H/2 and HZ (s/2). K acts on N sites, xi 6= xi . In other words, x can be decomposed into k−1 k has local dimension 2d, and has spectral gap at least substrings of consecutive 0s and consecutive 1s, with ij representing the index of the “domain wall” that sep- min(∆, |s − u|(N − 1), 1)/6. We are given a procedure arates a substring of 0s from a substring of 1s. The to solve Problem2 with δ = 0.9 for a single site, i.e. we parameter m is the number of domain walls. Using this can estimate the expectation value of any single site ob- K (j) servable in the ground state of K. If s < u, the true notation, and letting E = minj E /3 we can rewrite 0 0 reduced density matrix of K will have its ancilla bits N−1 m ik+1−2 all set to 1. If s > u the reduced density matrix corre- X X X 1 (xi ) hη | K |η i = hη | H k |η i x x,i x x 3 i,i+1 x sponds to the reduced density matrix of H with all its i=1 k=0 i=ik ancilla bits set to 0. Thus we can choose our single site m X operator to be the ZA operator that has eigenvalue 1 + hη | I |η i x ik−1,ik x for states whose ancilla bit is |0i and eigenvalue −1 for k=1 states whose ancilla bit is |1i. If we have a procedure m X ik+1 − ik (xi ) 1 to determine hψ| Z |ψi to precision 0.9 then we can ≥ E j − + m A 3(N − 1) 0 3 determine the setting of one of the ancilla bits in the k=0 ground state and thus determine whether u is larger or 2m − 1 ≥ EK + . (115) smaller than s. The time required to make this determi- 0 3 nation is f(min(∆, (N − 1)|s − u|, 1)/6, 2d, N). Because For any x other than 0N and 1N , there is at least one we have control over s, we can use this procedure to domain wall and m ≥ 1. Thus we can say binary search for the value of u. We assume we are given a lower bound on ∆ but since we do not know hφ| K |φi ≥ EK + 1/3. (116) 0 u a priori, we have no lower bound on |s − u|, so we We have shown that any state orthogonal to the states may not know how long to run the algorithm for Prob- N (0) N (1) lem2 in each step of the binary search. If our desired 0 A ⊗ |φn i and 1 A ⊗ |φn i will have energy at least 1/3 larger than the lowest energy state of the sys- precision is , we will impose a maximum runtime of tem. Without loss of generality, suppose E(0) ≤ E(1). f(min(∆, (N − 1)/2, 1)/6, 2d, N) for each step. Thus, 0 0 if we choose a value of s for which |s−u| < /2, the out- Then, the ground state energy is EK = E(0)/3 and the 0 0 put of this step of the binary search may be incorrect. ground state is 0N ⊗ |φ(0)i (note that in the state- A 0 After such a step, our search window will be cut in half E (0) (0) and the correct value of u will no longer be within the ment of the Lemma we have ψ = φ0 ). The ﬁrst window. However, u will still lie within /2 of one edge excited state is either 0N ⊗ |φ(0)i, 1N ⊗ |φ(1)i, or A 1 A 0 of the window. Throughout the binary search, some el- N lies outside the sector associated with ancillas 0 and ement of the search window will always lie within /2 of N 1 , whichever has lowest energy. The three cases lead u, so if we run the search until the window has width (0) (1) (0) to spectral gaps of ∆ /3, (E0 − E0 )/3, and some- and output the value u˜ in the center of the search win- thing larger than 1/3 (due to Eq. (116)), respectively. dow, we are guaranteed that |u − u˜| ≤ . The number This proves all three items of the Lemma. of steps required is O(log(1/)) and the time for each

Accepted in Quantum 2019-09-02, click title to verify. Published under CC-BY 4.0. 21 step is f(min(2∆, (N − 1), 2)/12, 2d, N), yielding the locality of the gates to be O(log(k/2)), then the circuit statement of the theorem. can have depth 2, as in Figure1. If we require the locality of the gates be only a constant, say 2, then the circuit can have depth poly(k, 1/). 5 Discussion The fact that the approximation is injective perhaps makes the latter method more powerful. Injective MPS Our results paint an interesting landscape of the are the exact ground states of some local gapped Hamil- complexity of approximating ground states of gapped tonian [9, 25]. Additionally, non-injective MPS form a nearest-neighbor 1D Hamiltonians locally. On the one set of measure zero among the entire MPS manifold, hand, we show all k-local properties of the ground state so variational algorithms that explore the whole mani- can be captured by an MPS with only constant bond fold are most compatible with an injective approxima- dimension, an improvement over the poly(N) bond di- tion. In fact, since the approximation can be generated mension required to represent the global approximation. from a constant-depth circuit, the result justifies a more This constant scales like a polynomial in k and 1/, restricted variational ansatz using states of that form. when parameters like ∆, ξ, and d are taken as con- This ansatz could provide several advantages over MPS stants. On the other hand, we give evidence that, at in terms of number of parameters needed and ability least for the case where the Hamiltonian is translation- to quickly calculate local observables, like the energy ally invariant, finding the local approximation may not density. However, algorithms that perform variational offer a significant speedup over finding the global ap- optimization of the energy density generally suffer from proximation: we have shown that the ability to find two issues, regardless of the ansatz they use. First, they even a constant-precision estimate of local properties do not guarantee convergence to the global minimum would allow one to learn a constant-precision estimate within the ansatz set, and second, even when they do of the ground state energy with only O(log(N)) over- find the global minimum, the output does not neces- head. This reduction does not allow one to learn any sarily correspond to a good local approximation. This global information about the state besides the ground stems from the fact that a state that is -close to the state energy, so it falls short of giving a concrete rela- ground state energy density may actually be far, even tionship between the complexity of the global and local orthogonal, to the actual ground state. Therefore, even approximations. Nonetheless, the reduction has con- a brute-force optimization over the ansatz set cannot be crete consequences. In particular, at least one of the guaranteed to give any information about the ground following must be true about translationally invariant state, other than its energy density. gapped Hamiltonians on chains of length N: This leaves open many questions regarding the algo- (1) The ground state energy can be estimated to O(1) rithmic complexity of gapped local 1D Hamiltonians. precision in O(log(N)) time. For the general case on a finite chain, can one find a local approximation to the ground state faster than (2) Local properties of the ground state cannot be es- the global approximation? For translationally invari- timated to O(1) precision in time independent of ant chains, can one learn the ground state energy to N. O(1) precision in O(log(N)) time, and can one learn local properties in time independent of the chain length? In particular, the second item, if true, would seem to Relatedly, in the thermodynamic limit, can one learn imply that, in the translationally invariant case when an -approximation to the ground state energy density N → ∞, local properties cannot be estimated at all. in O(log(1/)) time, and can one learn local properties Indeed, it is when the chain is very long, or when we at all? These are interesting questions to consider in are considering the thermodynamic limit directly that future work. our results are most relevant. In the translationally in- We would like to conclude by drawing the reader’s variant case as N → ∞, our first proof method (Theo- attention to independent work studying the same prob- rem1) yields a local approximation that is a translation- lem by Huang [18], which appeared simultaneously with ally invariant MPS. However, the MPS is non-injective our own. and the state is a macroscopic superposition on the infinite chain. Thus the bulk tensors alone do not uniquely define the state and specification of a boundary tensor Acknowledgments at infinity is also required [31, 38]. Our second proof method (Theorem2), on the other hand, yields a peri- We thank Thomas Vidick for useful discussions about odic MPS (with period O(log(k/2))) that is injective this work and its algorithmic implications. AMD grate- and can be constructed by a constant-depth quantum fully acknowledges support from the Dominic Orr Fel- circuit made from spatially local gates. If we allow the lowship and the National Science Foundation Graduate

Accepted in Quantum 2019-09-02, click title to verify. Published under CC-BY 4.0. 22 Research Fellowship Program under Grant No. DGE- tices. Phys. Rev. Lett., 107:070601, Aug 2011. DOI: 1745301. This work was supported by NSF and Sam- 10.1103/PhysRevLett.107.070601. sung. The Institute for Quantum Information and Mat- [12] M. B. Hastings. Lieb-Schultz-Mattis in higher di- ter (IQIM) is an NSF Physics Frontiers Center. mensions. Phys. Rev. B, 69:104431, Mar 2004. DOI: 10.1103/PhysRevB.69.104431. References [13] M. B. Hastings. Locality in quantum and markov dynamics on lattices and networks. Phys. Rev. [1] I. Arad, A. Kitaev, Z. Landau, and U. Vazirani. Lett., 93:140402, Sep 2004. DOI: 10.1103/Phys- An area law and sub-exponential algorithm for 1D RevLett.93.140402. systems. arXiv preprint arXiv:1301.1162, 2013. [14] M. B. Hastings. An area law for one-dimensional [2] I. Arad, Z. Landau, U. Vazirani, and T. Vidick. quantum systems. Journal of Statistical Me- Rigorous RG algorithms and area laws for low en- chanics: Theory and Experiment, 2007(08): ergy eigenstates in 1D. Communications in Math- P08024–P08024, aug 2007. DOI: 10.1088/1742- ematical Physics, 356(1):65–105, Nov 2017. ISSN 5468/2007/08/p08024. 1432-0916. DOI: 10.1007/s00220-017-2973-z. [15] M. B. Hastings and T. Koma. Spectral gap [3] J. Bausch, T. Cubitt, A. Lucia, and D. Perez- and exponential decay of correlations. Communi- Garcia. Undecidability of the spectral gap in one cations in Mathematical Physics, 265(3):781–804, dimension. arXiv preprint arXiv:1810.01858, 2018. Aug 2006. ISSN 1432-0916. DOI: 10.1007/s00220- [4] J. Bausch, T. S. Cubitt, A. Lucia, D. Perez-Garcia, 006-0030-4. and M. M. Wolf. Size-driven quantum phase transi- [16] Y. Huang. Area law in one dimension: Degener- tions. Proceedings of the National Academy of Sci- ate ground states and Renyi entanglement entropy. ences, 115(1):19–23, 2018. ISSN 0027-8424. DOI: arXiv preprint arXiv:1403.0327, 2014. 10.1073/pnas.1705042114. [17] Y. Huang. Computing energy density in one di- [5] F. G. S. L. Brandão and M. Horodecki. Ex- mension. arXiv preprint arXiv:1505.00772, 2015. ponential decay of correlations implies area law. [18] Y. Huang. Approximating local properties by ten- Communications in Mathematical Physics, 333(2): sor network states with constant bond dimension. 761–798, Jan 2015. ISSN 1432-0916. DOI: arXiv preprint arXiv:1903.10048, 2019. 10.1007/s00220-014-2213-8. [19] M. Kliesch, D. Gross, and J. Eisert. Matrix- [6] G. K. Brennen, S. S. Bullock, and D. P. product operators and states: NP-hardness and O’Leary. Eﬃcient circuits for exact-universal undecidability. Phys. Rev. Lett., 113:160503, Oct computation with qudits. Quantum Informa- 2014. DOI: 10.1103/PhysRevLett.113.160503. tion and Computation, 6(4-5), 2006. URL [20] A. Klümper, A. Schadschneider, and J. Zittartz. http://www.rintonpress.com/journals/ Matrix product ground states for one-dimensional qiconline.html#v6n45. spin-1 quantum antiferromagnets. Europhysics [7] T. S. Cubitt, D. Perez-Garcia, and M. M. Wolf. Letters (EPL), 24(4):293–297, nov 1993. DOI: Undecidability of the spectral gap. Nature, 528 10.1209/0295-5075/24/4/010. (7581):207, 2015. DOI: 10.1038/nature16059. [8] M. Fannes, B. Nachtergaele, and R. F. Werner. [21] Z. Landau, U. Vazirani, and T. Vidick. A Abundance of translation invariant pure states on polynomial time algorithm for the ground state quantum spin chains. Letters in Mathematical of one-dimensional gapped local Hamiltoni- Physics, 25(3):249–258, Jul 1992. ISSN 1573-0530. ans. Nature Physics, 11(7):566, 2015. DOI: DOI: 10.1007/BF00406552. 10.1038/NPHYS3345. [9] M. Fannes, B. Nachtergaele, and R. F. Werner. [22] I. P. McCulloch. Inﬁnite size density matrix Finitely correlated states on quantum spin chains. renormalization group, revisited. arXiv preprint Communications in Mathematical Physics, 144(3): arXiv:0804.2509, 2008. 443–490, Mar 1992. ISSN 1432-0916. DOI: [23] B. Nachtergaele and R. Sims. Lieb-Robinson 10.1007/BF02099178. bounds and the exponential clustering theorem. [10] A. Gilchrist, N. K. Langford, and M. A. Nielsen. Communications in Mathematical Physics, 265 Distance measures to compare real and ideal quan- (1):119–130, Jul 2006. ISSN 1432-0916. DOI: tum processes. Phys. Rev. A, 71:062310, Jun 2005. 10.1007/s00220-006-1556-1. DOI: 10.1103/PhysRevA.71.062310. [24] S. Östlund and S. Rommer. Thermodynamic [11] J. Haegeman, J. I. Cirac, T. J. Osborne, limit of density matrix renormalization. Phys. I. Pižorn, H. Verschelde, and F. Verstraete. Time- Rev. Lett., 75:3537–3540, Nov 1995. DOI: dependent variational principle for quantum lat- 10.1103/PhysRevLett.75.3537.

Accepted in Quantum 2019-09-02, click title to verify. Published under CC-BY 4.0. 23 [25] D. Perez-García, F. Verstraete, M. M. Wolf, and 10345–10356, Oct 1993. DOI: 10.1103/Phys- J. I. Cirac. Matrix product state representations. RevB.48.10345. Quantum Information and Computation, 7(5-6): [38] V. Zauner-Stauber, L. Vanderstraeten, M. T. Fish- 401–430, 2007. URL http://www.rintonpress. man, F. Verstraete, and J. Haegeman. Variational com/journals/qiconline.html#v7n56. optimization algorithms for uniform matrix prod- [26] B. Roberts, T. Vidick, and O. I. Motrunich. Im- uct states. Phys. Rev. B, 97:045145, Jan 2018. plementation of rigorous renormalization group DOI: 10.1103/PhysRevB.97.045145. method for ground space and low-energy states of local Hamiltonians. Phys. Rev. B, 96:214203, Dec 2017. DOI: 10.1103/PhysRevB.96.214203. [27] N. Schuch and F. Verstraete. Matrix product state approximations for infinite systems. arXiv preprint arXiv:1711.06559, 2017. [28] N. Schuch, M. M. Wolf, F. Verstraete, and J. I. Cirac. Entropy scaling and simulability by matrix product states. Phys. Rev. Lett., 100:030504, Jan 2008. DOI: 10.1103/PhysRevLett.100.030504. [29] M. Tomamichel, R. Colbeck, and R. Ren- ner. A fully quantum asymptotic equiparti- tion property. IEEE Transactions on Informa- tion Theory, 55(12):5840–5847, Dec 2009. DOI: 10.1109/TIT.2009.2032797. [30] A. Uhlmann. The “transition probability” in the state space of a ?-algebra. Reports on Mathematical Physics, 9(2):273 – 279, 1976. ISSN 0034-4877. DOI: 10.1016/0034-4877(76)90060-4. [31] L. Vanderstraeten, J. Haegeman, and F. Ver- straete. Tangent-space methods for uniform matrix product states. SciPost Phys. Lect. Notes, page 7, 2019. DOI: 10.21468/SciPostPhysLectNotes.7. [32] F. Verstraete and J. I. Cirac. Matrix product states represent ground states faithfully. Phys. Rev. B, 73:094423, Mar 2006. DOI: 10.1103/Phys- RevB.73.094423. [33] G. Vidal. Efficient classical simulation of slightly entangled quantum computations. Phys. Rev. Lett., 91:147902, Oct 2003. DOI: 10.1103/Phys- RevLett.91.147902. [34] G. Vidal. Classical simulation of infinite-size quantum lattice systems in one spatial dimension. Phys. Rev. Lett., 98:070201, Feb 2007. DOI: 10.1103/PhysRevLett.98.070201. [35] A. Weichselbaum, F. Verstraete, U. Schollwöck, J. I. Cirac, and J. von Delft. Variational matrix- product-state approach to quantum impurity models. Phys. Rev. B, 80:165117, Oct 2009. DOI: 10.1103/PhysRevB.80.165117. [36] S. R. White. Density matrix formulation for quantum renormalization groups. Phys. Rev. Lett., 69:2863–2866, Nov 1992. DOI: 10.1103/Phys- RevLett.69.2863. [37] S. R. White. Density-matrix algorithms for quantum renormalization groups. Phys. Rev. B, 48:

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