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

A key feature of ground states of gapped local 1D Hamiltonians is their relatively low entanglement --- they are well approximated by matrix product states (MPS) with bond dimension scaling polynomially (as opposed to exponentially) in the length $N$ of the chain. We show that the bond dimension of the approximation can be improved to a constant, independent of the chain length, if we relax our notion of approximation to be more local. For all length-$k$ segments of the chain, the reduced density matrices of our approximation are $\epsilon$-close to those of the exact ground state, and the bond dimension of the approximation scales like $(k/\epsilon)^{1+o(1)}$. At the expense of worse but still $\text{poly}(k,1/\epsilon)$ scaling of the bond dimension, we construct an approximation with the additional features that it can be generated by a constant-depth quantum circuit with nearest-neighbor gates, and that it applies generally for any state with exponentially decaying correlations. Then, we consider the prospect of designing an algorithm to find the local approximation. When the Hamiltonian is translationally invariant, we show that the ability to find $O(1)$-accurate local approximations in $T(N)$ time implies the ability to estimate the ground state energy to $O(1)$-precision in $O(T(N)\log(N))$ time.


I. INTRODUCTION
In nature, interactions between particles act locally, motivating the study of many-body Hamiltonians consisting only of terms involving particles spatially near each other. An important method that has emerged from this course of study is the Density Matrix Renormalization Group (DMRG) algorithm [1,2], which aims to find a description of the ground state of local Hamiltonians on a one-dimensional chain of sites. DMRG has been an indispensable tool for research in many-body physics, but the rationale for its empirical success did not become fully apparent until long after it was being widely used. Its eventual justification required two ingredients: first, that the method can be recast [3,4] as a variational algorithm that minimizes the energy over the set of matrix product states (MPS), a tensor network ansatz for 1D systems; and second, that the MPS ansatz set actually contains a good approximation to the ground state, at least whenever the Hamiltonian has a nonzero spectral gap. More specifically, Hastings [5] showed that ground states of gapped local Hamiltonians on chains with N sites can be approximated to within trace distance ǫ by an MPS with bond dimension (a measure of the complexity of the MPS) only poly(N, 1/ǫ), exponentially smaller than what is needed to describe a generic state. Even taking into account these observations, DMRG is a heuristic algorithm and is not guaranteed to converge to the global energy minimum as opposed to a local minimum; however, a recently developed alternative algorithm [6][7][8], sometimes referred to as the "Rigorous RG" (RRG) algorithm, avoids this issue and provides a way one can guarantee finding an ǫ-approximation to the ground state in poly(N, 1/ǫ) time.
These are extremely powerful results, but their value breaks down when the chain becomes very long. The bond dimension required to describe the ground state grows relatively slowly, but it still diverges with N . 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 resources. Indeed, often we wish to learn something about the ground state in the thermodynamic limit (N → ∞) but in this case these results no longer apply. Analogues of DMRG for the thermodynamic limit [9][10][11][12][13][14] -methods that, for example, optimize over the set of constant bond dimension "uniform MPS" consisting of the same tensor repeated across the entire infinite chain -have been implemented with successful results, but these methods lack the second ingredient that justified DMRG: it is not clear how large we must make the bond dimension to guarantee that the set over which we are optimizing contains a good approximation to the ground state.
Progress toward this ingredient can be found in work by Huang [15] (and later by Schuch and Verstraete [16]), who showed that the ground state of a gapped local 1D Hamiltonian can be approximated locally by a matrix product operator (MPO) -a 1D tensor network object that corresponds to a (possibly mixed) density operator as opposed to a quantum state vector -with bond dimension independent of N . Here their approximation sacrifices global fidelity with the ground state, which decays exponentially with the chain length, in exchange for constant bond dimension, while retaining high fidelity with the ground state reduced density matrices on all segments of the chain with constant length. In other words, the statistics for measurements of local operators are faithfully captured by the MPO approximation, a notion of approximation that is often sufficient in practice since many relevant observables, such as the individual terms in the Hamiltonian, are local. However, the result does not provide the necessary ingredient to justify infinite analogues of DMRG because MPO do not make a good ansatz class for variational optimization algorithms. One can specify the matrix elements for an MPO, but the resulting operator will only correspond to a valid quantum state if it is positive semidefinite, and verifying that this is the case is difficult: it is NP-hard for finite chains, and in the limit N → ∞ it becomes undecidable [17]. Thus, if we attempt to perform variational optimization over the set of constant bond dimension MPO, we can be certain that our search space contains a good local approximation to the ground state, but we have no way of restricting our search only to the set of valid quantum states; ultimately the minimal energy MPO we find may not correspond to any quantum state at all.
In this work, we fix this problem by showing an analogous result for MPS instead of MPO. We show that for any gapped nearest-neighbor Hamiltonian on a 1D chain with N sites, and for any parameters k and ǫ, there is an MPS representation of a state |ψ with bond dimension poly(k, 1/ǫ) such that the reduced density matrix of |ψ ψ | on any contiguous region of length k is ǫ-close in trace distance to that of the true ground state. Importantly, the bond dimension is independent of N . Thus, we provide the missing ingredient for variational algorithms in the thermodynamic limit as we show that a variational set over constant bond dimension MPS contains a state that simultaneously captures all the local properties of the ground state.
We present two proofs of this fact. The first yields superior scaling of the bond dimension, which grows asymptotically slower than (k/ǫ) 1+δ for any δ > 0; however, it constructs an MPS approximation |ψ that is long-range correlated and non-injective. In contrast, the second proof constructs an approximation that is injective and can be generated by a constant-depth quantum circuit with nearest-neighbor gates, while retaining poly(k, 1/ǫ) bond dimension. The latter construction also follows merely from the assumption that the state has exponential decay of correlations. The proof idea originates with a strategy first presented in [18] and constructs |ψ by beginning with the true ground state |ψ and applying three rounds of operations: first, a set of unitaries that, intuitively speaking, removes the short-range entanglement from the chain; second, a sequence of rank 1 projectors that zeroes-out the long-range entanglement; and third, the set of inverse unitaries from step 1 to add back the short-range entanglement. Intuitively, the method works because ground states of gapped Hamiltonians have a small amount of long-range entanglement. The non-trivial part is arguing that the local properties are preserved even as the small errors induced in step 2 accumulate to bring the global fidelity with the ground state to zero. The fact that |ψ can be produced by a constant-depth quantum circuit acting on an initial product state suggests the possibility of an alternative variational optimization algorithm using the set of constant-depth circuits (a strict subset of constant-bonddimension MPS) as the variational ansatz. Additionally, we note that the disentangle-project-reentangle process that we utilize in our proof might be of independent interest as a method for truncating the bond dimension of MPS. We can bound the truncation error of this method when the state has exponentially decaying correlations.
We also consider the question of whether these locally approximate MPS approximations can be rigorously found (à la RRG) more quickly than their globally approximate counterparts (and if they can be found at all in the thermodynamic limit). We prove a reduction for translationally invariant Hamiltonians showing that finding approximations to local properties to even a fixed O(1) precision implies being able to find an approximation to the ground state energy to O(1) precision with only O(log(N )) overhead. Since strategies for estimating the ground state energy typically involve constructing a globally accurate approximation to the ground state, this observation gives us an intuition that it may not be possible to find the local approximation much more quickly than the global approximation, despite the fact that the bond dimensions required for the two approximations are drastically different.

A. One-dimensional local Hamiltonians
In this paper we work exclusively with gapped nearestneighbor 1D Hamiltonians that have a unique ground state. Our physical system is a set of N sites, arranged in one dimension on a line with open boundary conditions (OBC), each with its own Hilbert space H i of dimension d.
Hamiltonian H consists only of terms H i,i+1 that act non-trivially only on H i and H i+1 .
We will always require that H i,i+1 be positive semidefinite and satisfy H i,i+1 ≤ 1 for all i, where · is the operator norm. When this is not the case it is always possible to rescale H so that it is. We call H translationally invariant if H i,i+1 is the same for all i. We will also always assume that H has a unique ground state |ψ with energy E and an energy gap ∆ > 0 to its first excited state. We let ρ = |ψ ψ| refer to the (pure) density matrix representation of the ground state. For any density matrix σ and subregion X of the chain, we let σ X refer to Tr X c (σ), the reduced density matrix of σ after tracing out the complement X c of X.

B. Matrix product states and matrix product operators
It is often convenient to describe states with onedimensional structure using the language of matrix product states (MPS).
Definition 1 (Matrix product state). A matrix product state (MPS) |η on N sites of local dimension d is specified by N d matrices A N are χ × 1 matrices, with the rest being χ × χ. The state is defined as The parameter χ is called the bond dimension of the MPS.
The same physical state has many different MPS representations, although one may impose a canonical form [19] to make the representation unique. The bond dimension of the MPS is a measure of the maximum amount of entanglement across any "cut" dividing the state into two contiguous parts. More precisely, if we perform a Schmidt decomposition on a state |η across every possible cut, the maximum number of non-zero Schmidt coefficients (i.e. Schmidt rank) across any of the cuts is equal to the minimum bond dimension we would need to exactly represent |η as an MPS [20]. Thus to show a state has an MPS representation with a certain bond dimension, it suffices to bound the Schdmidt rank across all the cuts. This line of reasoning shows that a product state, which has no entanglement, can be written as an MPS with bond dimension 1. Meanwhile, an arbitrary state with any amount of entanglement can always be written as an MPS with bond dimension d N/2 .
A cousin of matrix product states are matrix product operators.
Definition 2 (Matrix product operator). A matrix product operator (MPO) σ on N sites of local dimension d is specified by N d 2 matrices A N are χ × 1 matrices, with the rest being χ × χ. The operator is defined as is a basis for operators on a single site. The parameter χ is called the bond dimension of the MPO.
However, MPO representations have the issue that specifying a set of matrices A (i) j does not always lead to an operator σ that is positive semi-definite, which is a requirement for the MPO to correspond to a valid quantum state. Checking positivity of an MPO in general is NP-hard for chains of length N and undecidable for infinite chains [17].

C. Notions of approximation
We are interested in the existence of an MPS that approximates the ground state |ψ . We will have both a global and a local notion of approximation, which we define here. We will employ two different distance measures at different points in our theorems and proofs, the purified distance and the trace distance.
Definition 4 (Trace distance). If σ and σ ′ are two normalized states on the same system, then is the trace distance between σ and σ ′ .
If the purified distance between ρ and σ is small then we would say σ is a good global approximation to ρ. We are also interested in a notion of distance that is more local.
Definition 5 (k-local purified distance). If σ and σ ′ are two normalized states on the same system, then the klocal purified distance between σ and σ ′ is where the max is taken over all contiguous regions X consisting of k sites.
Definition 6 (k-local trace distance). If σ and σ ′ are two normalized states on the same system, then the klocal trace distance between σ and σ ′ is where the max is taken over all contiguous regions X consisting of k sites.
Note that these quantities lack the property that 0 = D (k) (σ, σ ′ ) = D (k) 1 (σ, σ ′ ) implies σ = σ ′ [34], but they do satisfy the triangle inequality. It is also clear that taking k = N recovers our notion of global distance: Definition 7 (Local approximation). We say a state σ on a chain of N sites is a (k, ǫ)-local approximation to As we discuss in the next subsection, previous results show that |ψ has a good global approximation |ψ which is an MPS with bond dimension that scales like a polynomial in N . We will be interested in the question of what bond dimension is required when what we seek is merely a good local approximation.

Exponential decay of correlations and area laws
A key fact shown by Hastings [22,23] about nearestneighbor 1D Hamiltonians with a non-zero energy gap is that the ground state |ψ has exponential decay of correlations.
Definition 8 (Exponential decay of correlations). A pure state σ = |η η| on a chain of sites is said to have (t 0 , ξ)-exponential decay of correlations if for every t ≥ t 0 and every pair of regions A and C separated by at least t sites We call ξ the correlation length of |η .
Lemma 2 ( [22,23]). If |ψ is the unique ground state of a Hamiltonian H = i H i,i+1 with spectral gap ∆, then |ψ has (t 0 , ξ)-exponential decay of correlations for some While the exponential decay of correlations holds for lattice models in any spatial dimension, the other results we discuss are only known to hold in one dimension.
For example, in one dimension it has been shown that ground states of gapped Hamiltonians obey an area law, that is, the entanglement entropy of any contiguous region is bounded by a constant times the length of the boundary of that region, which in one dimension is just a constant. This statement was also first proven by Hastings in [24] where it was shown that for any contiguous region X which is independent of the number of sites in X, where S denotes the von Neumann entropy S(σ) = −Tr(σ log σ).
The area law has since been improved [7,25] to where theÕ signifies a suppression of factors that scale logarithmically with the quantity stated. It was also discovered that an area law follows merely from the assumption of exponential decay of correlations in one dimension: if a pure state σ has (t 0 , ξ)-exponential decay of correlations, then it satisfies [26] S(ρ X ) ≤ t 0 exp(Õ(ξ log(d))) The area law is closely related to the existence of an efficient MPS approximation to the ground state. To make this implication concrete, one needs an area law using the α-Renyi entropy for some value of 0 < α < 1 [18], where the Renyi entropy is given by S α (ρ X ) = −Tr(log(ρ α X ))/(1 − α). An area law for the von Neumann entropy (corresponding to α = 1) is not alone sufficient [27]. However, for all of the previously stated area laws, the methods are strong enough to also give the existence of efficient MPS approximations. Hastings' [24] original area law implied the existence of a global ǫ-approximation |ψ for |ψ with bond dimension The improved area law in [7,25] yields a better scaling for χ which is asymptotically sublinear in N χ = eÕ Finally, the result implied only from exponential decay of correlations [26] is Crucially, if the local Hilbert space dimension d and the gap ∆ (or alternatively, the correlation length ξ) are taken to be constant, then all three results read χ = poly(N, 1/ǫ).

Constant-bond-dimension MPO local approximations
The problem of finding matrix product operator representations that capture all the local properties of a state has been studied before. Huang [15] showed the existence of a positive semi-definite MPO ρ χ with bond dimension that is a (2, ǫ)-local approximation to the true ground state ρ, where o(1) indicates the quantity approaches 0 as 1/ǫ → ∞. Crucially, this is independent of the length of the chain N . Additionally, because the Hamiltonian is nearest-neighbor, we have Tr(Hρ χ )− Tr(Hρ) ≤ (N − 1)ǫ, i.e., the energy per site (energy density) of the state ρ χ is within ǫ of the ground state energy density. Huang constructs this MPO explicitly and notes it is a convex combination over pure states which themselves are MPS with bond dimension independent of N . Thus, one of these MPS must have energy density within ǫ of the ground state energy density. However, it is not guaranteed (nor is it likely) that one of these constant-bond-dimension MPS is also a good local approximation to the ground state; thus our result may be viewed as an improvement on this front as we show the existence not only of a lowenergy-density constant-bond-dimension MPS, but also one that is a good local approximation to the ground state.
An alternative MPO construction achieving the same task was later given in [16]. In this case, the MPO is a (k, ǫ)-local approximation to the ground state and has bond dimension The idea they use is simple. They break the chain into blocks of size l, which is much larger than k. On each block they construct a constant bond dimension MPO that closely approximates the reduced density matrix of the ground state on that block, which is easy since each block has constant length and they must make only a constant number of bond truncations to the exact state. The tensor product of these MPO will be an MPO on the whole chain that is a good approximation on any length-k region that falls within one of the larger length l blocks, but not on a region that crosses the boundary between blocks. To remedy this, they take the mixture of MPO formed by considering all l translations of the boundaries between the blocks. Now as long as l is much larger than k, any region of length k will only span the boundary between blocks for a small fraction of the MPO that make up this mixture, and the MPO will be a good local approximation.
This same idea underlies our proof of Theorem 1, with the complication that we seek a pure state approximation and cannot take a mixture of several MPS. Instead, we combine the translated MPS in superposition, which brings new but manageable challenges.

III. STATEMENT OF RESULTS
A. Existence of local approximation Theorem 1. Let |ψ be a state on a chain of N sites of local dimension d. If |ψ has (t 0 , ξ)-exponential decay of correlations, then, for any k and ǫ, there exists an MPS |ψ with bond dimension at most χ such that (1) |ψ is a (k, ǫ)-local approximation to |ψ Additionally, if |ψ is the unique ground state of a nearest-neighbor 1D Hamiltonian with spectral gap ∆, then the bound on the bond dimension can be improved to which is asymptotically equivalent to (k/ǫ) 1+o (1) where o(1) indicates that the quantity approaches 0 as (k/ǫ) → ∞.
However, the state |ψ that we construct in the proof of Theorem 1 is long-range correlated and cannot be generated from a constant-depth quantum circuit. Thus, while |ψ is a good local approximation to the ground state of H, it is not the exact ground state of any gapped local 1D Hamiltonian.
Next, we show that it remains possible to approximate the state even when we require the approximation to be produced by a constant-depth quantum circuit; the scaling of the bond dimension is faster in k and 1/ǫ, but it is still polynomial.
Theorem 2. Let |ψ be a state on a chain of N sites of local dimension d. If |ψ has (t 0 , ξ)-exponential decay of correlations, then, for any k and ǫ, there exists an MPS |ψ with bond dimension at most χ such that (1) |ψ is a (k, ǫ)-local approximation to |ψ (3) |ψ can be prepared from the state |0 ⊗N by a quantum circuit consisting only of unitaries acting on neighboring qubits with depthÕ(χ 2 ).
Additionally, if |ψ is the unique ground state of a nearest-neighbor 1D Hamiltonian with spectral gap ∆, then the bound on the bond dimension can be improved to Proof summary. First, we make the following observation about tripartitions of the chain into contiguous regions L, M , and R: since |ψ has exponential decay of correlations, the quantity . This is captured in Lemma the same state after tracing out M , there is a unitary U acting only on M that nearly disentangles |ψ across the central cut of M , with U |ψ ≈ |α ⊗ |β (Uhlmann's theorem).
The proof constructs the approximation |ψ by applying three steps of operations on the exact state |ψ . First, the chain is broken up into many regions {M i } n+1 i=0 of length l, and disentangling unitaries U i as described above are applied to each region M i in parallel. The state is close to, but not exactly, a product state across the center cut of each region M i . To make it an exact product state, the second step is to apply rank-1 projectors Π i onto the right half of the state across each of these cuts, starting with the leftmost cut and working our way down the chain. Then, the third step is to apply the reverse unitaries U † i that we applied in step 1. The projection step is the cause of the error between |ψ and |ψ . The number of projections is O(N ), but the error accrued locally is only constant. This follows from the fact that the projectors are rank 1, so once we apply projector Π i , region M j is completely decoupled from M j ′ when j < i and j ′ > i. Thus any additional projections, which act only on the regions to the right of M i , have no effect on the reduced density matrix on M j (except to reduce its norm). Using this logic, we show that the number of errors that actually affect the state locally on a region X is proportional to the number of sites in X, and not the number of sites in the whole chain. To make this error less than ǫ for any region of at most k sites, we can choose l = O(ξ ′ log(k/ǫ)).
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 d l . This yields the scaling for χ.
The result is improved when |ψ is the ground state of a gapped local Hamiltonian by using the improved area law [7,25] in the proof of Lemma 4 and in the truncation of the states ρ L and ρ R before purifying into |α and |β .
Finally, it can be seen that |ψ is formed by a constantdepth 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 l that is attained after applying projections in step 2, and the second layer applies the inverse unitaries from step 3. Each unitary in this circuit can be decomposed into a sequence of nearestneighbor gates with depthÕ(d 2l ).

B. Reduction from estimating energy density to finding local properties
The previously stated results show that there exists a state that is both a (k, ǫ)-local approximation and an MPS with bond dimension poly(k, 1/ǫ). They say nothing of the algorithmic complexity required to find a (k, ǫ)local approximation. The proofs describe how to construct the local approximation from a description of the exact ground state, but following this strategy would re-quire first finding a description of the exact ground state (or perhaps a global approximation to it). One might hope that a different strategy would allow the local approximation to be found much more quickly than the global approximation, since the bond dimension needed to represent the approximation is much smaller. However, the result we now present challenges the validity of this intuition, at least in the case that the Hamiltonian is translationally invariant, by showing a relationship between the problem of finding a local approximation and the problem of estimating the energy density.
Problem 1 (Estimating energy density). Given a nearest-neighbor translationally invariant 1D Hamiltonian H on N ≥ 2 sites and error parameter ǫ, produce an estimateũ such that |u −ũ| ≤ ǫ where u = E/(N − 1) is the ground state energy density.
Problem 2 (Approximating local properties). Given a nearest-neighbor translationally invariant Hamiltonian H, an error parameter δ, and an operator O whose support is contained within a contiguous region of length k, produce an estimateṽ such that |v Problem 1 is the restriction of Problem 2 to the case where k = 2 and the operator O is the energy interaction term. Thus, there is a trivial reduction from Problem 1 to Problem 2 with δ = ǫ. However, the next theorem states a much more powerful reduction.
Estimating the energy density to precision ǫ is equivalent to measuring the total energy to precision ǫ(N − 1), so the quantity min(2∆, (N − 1)ǫ, 4) is equivalent to the global energy resolution, twice the gap, or four, whichever is smallest. Thus, one may take ǫ = O(1/N ) and understand the theorem as stating that finding local properties to within O(1) precision can be done at most O(log(N )) faster than finding an estimate to the total ground state energy to O(1) precision. If local properties can be found in time independent of N (i.e. there is an N -independent upper bound to f ), then the ground state energy can be estimated to O(1) precision in time O(log(N )), which would be optimal since the ground state energy scales extensively with N , and O(log(N )) time would be needed simply to write down the output.
Another way of understanding the significance of the theorem is in the thermodynamic limit. Here it states that if one could estimate expectation values of local observables in the thermodynamic limit to O(1) precision in some finite amount of time (for constant ∆ and d), then one could compute the energy density of such Hamiltonians to precision ǫ in O(log(1/ǫ)) time. This would be an exponential speedup over the best-known algorithm for computing the energy density given in [15], which has runtime poly(1/ǫ). Taking the contrapositive, if one could show that poly(1/ǫ) time is necessary for computing the energy density, this would imply that Problem 2 with δ = O(1) is in general uncomputable in the thermodynamic limit, even given the promise that the input Hamiltonian is gapped. It is already known that Problem 2 is uncomputable when there is no such promise [28].
It is not clear whether a O(log(1/ǫ)) time algorithm for computing the energy density is possible. The poly(1/ǫ) algorithm in [15] works even when the Hamiltonian is not translationally invariant, but it is not immediately apparent to us how one might exploit translational invariance to yield an exponential speedup.

A. Important Lemmas
Before we get into the proofs of our main results we state a few lemmas. First, we will need to use the area law result for 1D states.
Lemma 3 (Area laws [26], [25]). If σ = |ψ ψ| has (t 0 , ξ)-exponential decay of correlations then for any χ and any region of the chain A, there is a stateσ A with rank at most χ such that where C 1 = exp(exp(Õ(ξ log(d)))) is a constant independent of N . If σ is the unique ground state of a nearest-neighbor Hamiltonian with spectral gap ∆, then this can be improved to where Proof. The first part follows from the main theorem of [26]. The second follows from the 1D area law presented in [25], and log(d) dependence explicitly stated in [7].
Lemma 4. If L and R are disjoint regions of a 1D lattice of N sites and the state τ = |η η| has (t 0 , ξ)-exponential decay of correlations, then where ξ ′ = 16ξ 2 log(d) and C 3 = exp(exp(Õ(ξ log(d)))) whenever dist(L, R) ≥ t 0 . If τ is the unique ground state of a gapped nearestneighbor 1D Hamiltonian with spectral gap ∆, then this can be improved to We prove this lemma later in the section. For pure states σ, we call σ a Markov chain for the tripartition L/M/R if σ LR = σ L ⊗ σ R . Thus the lemma states that exponential decay of correlations implies that the violation of the Markov condition, as measured by the purified distance (or alternatively, trace distance) decays exponentially with the size of M .
Next, an important and well-known tool we use is Uhlmann's theorem, which expresses the fact that if two states are close, their purifications will be equally close up to a unitary acting on the purifying auxiliary space.
Lemma 5 (Uhlmann's theorem). Suppose τ A and σ A are states on system A. Suppose B is an auxiliary system and |T AB and |S AB are purifications of τ A and σ A , respectively. Then where the min is taken over unitaries on system B.
We will also utilize the well-known fact that Schmidt ranks cannot differ by more than a factor of d between any two neighboring cuts on the chain. Lemma 6. If |η is a state on a chain of N sites with local dimension d, and the Schmidt rank of |η across the cut between sites m and m + 1 is χ, then the Schmidt rank of |η across the cut between sites m ′ and m ′ + 1 is at most χd |m−m ′ | .
Proof. Suppose w.l.o.g. that m ′ > m. We may Schmidt decompose |η across cut m producing χ terms. We can proceed to Schmidt decompose the right vector in each of these terms into a sum over at most d m−m ′ terms. We interpret this as a decomposition of |η across cut m ′ with at most χd |m−m ′ | terms. While this decomposition is not necessarily a Schmidt decomposition, the Schmidt decomposition can only have fewer terms, making this quantity an upper bound to the Schmidt rank at cut m ′ .
Finally, we state an important lemma we will use in our discussion of algorithms for finding local properties. We provide the proof later in the section. Lemma 7. Suppose, for j = 0, 1, H (j) is a translationally invariant Hamiltonian defined on a chain of length N and local dimension d. Further suppose that |ψ (j) is the unique ground state of H (j) with energy E (j) 0 , and let ∆ (j) be the spectral gap of H (j) . We may form a new chain with local dimension 2d by adding an ancilla qubit to each site i of the chain. Then there is a Hamiltonian K defined on this chain such that (1) The ground state energy of K is 0 , then the ground state is 1 N A ⊗ |ψ (1) , where A refers to the N ancilla registers collectively.
Proof. We choose integers l, t, and χ ′ to be specified later. We require t be even and l ≥ 2k. If N is not a multiple of l, or if it is less than 2l 2 , then we add sites to the chain and corresponding non-interacting terms to the Hamiltonian so that the ground state on the added part of the chain is a trivial product state. Henceforth we assume N is a multiple of l and N ≥ 2l 2 . We also require that d ≥ 4. If this is not the case, we coarse-grain the system by combining neighboring sites, and henceforth we assume d ≥ 4.
To construct |ψ , we will first construct states |φ j for j = 0, . . . , l − 1 and then sum these states in superposition. Consider a fixed value of j. To construct |φ j we break the chain into n = N/l blocks {M j,i } n−1 i=0 of length l. We arrange the blocks so that the leftmost site of block M j,1 is site j; thus block M j,i contains the sites [j + l(i − 1), j + li − 1] for i = 1, . . . , n − 1, and the last block M j,0 "rolls over" to include sites at both ends of the chain: sites [j + N − l, N − 1] and [0, j − 1]. The state |φ j will be a product state over all these blocks |φ j = |φ j,0 Mj,0 ⊗ . . . ⊗ |φ j,n−1 Mj,n−1 (25) with states |φ j,i for i = 0, . . . , n − 1 that we now specify. The idea is to create a state |φ j,i that has nearly the same reduced density matrix as |ψ on the leftmost l − t sites of region M j,i . It uses the rightmost t sites to purify the reduced density matrix on the leftmost l − t sites. First we denote the leftmost l − t sites of block M j,i by A j,i = [j + l(i − 1), j + li − t − 1] and the rightmost t sites by B j,i = [j + li − t, j + li − 1] (or appropriate "roll over" definitions when i = 0). We write |ψ as an exact MPS with exponential bond dimension and form |ψ j,i by truncating to bond dimension χ ′ (i.e. projecting onto the span of the right Schmidt vectors associated with the largest χ ′ Schmidt coefficients, then normalizing the state) at the cut to the left of region A j,i , every cut within A j,i , and at the cut to the right of A j,i , for a total of l − t + 1 truncations. We denote the pure density matrix of this state by ρ (j,i) = |ψ j,i ψ j,i |. We can bound the effect of these truncations using the area law given by Lemma 3: where ǫ χ ′ is the cost (in purified distance) of a single truncation, given by the right hand side of Eq. (19), or by Eq. (20) in the case |ψ is known to be the ground state of a gapped nearest-neighbor 1D Hamiltonian.
Because of the truncations, we can express the reduced density matrix ρ (j,i) Aj,i as a mixture of χ ′2 pure states each of which can be written as an MPS with bond dimension χ ′ : for some probability distribution {p j,i,z } χ ′2 −1 z=0 . We now form |φ j,i by purifying ρ (j,i) Aj,i onto the region M j,i using the space B j,i , which contains t sites, as the purifying subspace: where the set of states {|r j,i,z } χ ′2 −1 z=0 is an orthonormal set defined on region B j,i . This purification will only be possible if the dimension of B j,i is sufficiently large, and we comment later on this fact, as well as how exactly to choose the set {|r j,i,z } χ ′2 −1 z=0 . The key observation is that state |φ j will get any local properties approximately correct as long as they are supported entirely within a segment A j,i for some i, and as long as χ ′ is large enough that the l − t + 1 truncations do not have much effect on the reduced density matrix there. Thus we will choose our parameters so that χ ′ is large (but independent of N ), such that l is much larger than t (so that most regions fall within a region A j,i ), and such that t is large enough that it is possible to purify states on B j,i onto M j,i . But we still have the issue that some regions will not be contained entirely within region A j,i for some i.
To fix this we sum in superposition over the states |φ j for each value of j. The motivation to do this is so that every region will be contained within A j,i for some value of i in most, but not all, of the terms in the superposition. We will show that most is good enough. We let To complete the construction we also must specify the orthonormal states {|r j,i,z } χ ′2 −1 z=0 defined on the t sites in region B j,i . We choose a set that satisfies the following requirements: (1) The reduced density matrix of |r j,i,z on any single site among the leftmost t/2 sites of B j,i (recall we have assumed t is even) is entirely supported on basis states 1, . . . , ⌊d/2⌋ (2) The reduced density matrix on any single site among the rightmost t/2 sites is entirely supported on basis states ⌊d/2⌋ + 1, . . . , d (3) Let j ′ = j + i mod l, and let i 0 = i mod l. If t ≤ i 0 ≤ l − t then for all z, |r j,i,z is orthogonal to the support of the reduced density matrix of |φ j ′ on region B j,i .
We assess how large t must be for it to be possible to satisfy these three conditions. The third item specifically applies only for values of i that lead to values of j ′ that are at least t away from j (modulo l) so that the purifying system B j,i does not overlap with B j ′ ,i ′ for any i ′ . The support of the reduced density matrix of any |φ j ′ on region B j,i has dimension at most χ ′2 . Thus, if the dimension of B j,i is more than 2χ ′2 it will always be possible to choose an orthonormal set {|r j,i,z } χ ′2 −1 z=0 satisfying the third condition. The first and second conditions cut the accessible part of the local dimension of the purifying system in half, so a purification that satisfies all three conditions will be possible if ⌊d/2⌋ t ≥ 2χ ′2 . Any choice of set that meets all three conditions is equally good for our purposes.
We now demonstrate that the three conditions imply that for any pair (j, j ′ ) there are regions of the chain on which the supports of the reduced density matrices of states |φ j and |φ j ′ are orthogonal. If it is the case that j − j ′ mod l < t or j ′ − j mod l < t, then for every i the region B j,i overlaps with B j ′ ,i ′ for some i ′ . Because B j,i = B j ′ ,i ′ , there will be some site that is in the right half of one of the two regions, but in the left half of the other, and items 1 and 2 imply that the two states will be orthogonal when reduced to this site. If this is not the case, then as long as there is some i for which j ′ = j + i mod l, then item 3 implies the orthogonality of the supports of |φ j and |φ j ′ . In fact because n ≥ 2l, there will be at least 2 such values of i. We conclude that φ j |φ j ′ = δ jj ′ . which implies that |ψ is normalized as shown by the computation We have now shown how to define the approximation |ψ and discussed the conditions for the parameters t, χ ′ , and d that make the construction possible. Now we assess the error in the approximation (locally). Letρ = |ψ ψ |.
For any region X D 1 (ρ X ,ρ X ) For the same reason that led to the conclusion φ j |φ j ′ = δ jj ′ , we can conclude that Tr X c (|φ j φ j ′ |) = δ jj ′ , so long as k is smaller than l/2. To see that this holds, it is sufficient to show that there is a region lying completely outside of X with the property that |φ j and φ ′ j share no support on the region. Since k = |X| ≤ l/2, for any j, X can overlap the region B j,i for at most one value of i. We showed before that for any j there would be at least two values of i for which a subregion of B j,i has this property, implying one of them must lie outside X.
Thus we have For a particular j, there are two cases. Case 1 occurs if X falls completely within the A j,i for some i, in which case the only error is due to the l − t + 1 truncations to bond dimension χ ′ . Since the trace norm D 1 is smaller than the purified distance D (Lemma 1), the contribution of for these values of j is at most √ l − t + 1ǫ χ ′ . Case 2 includes values of j for which X does not fall completely within a region A j,i for any i. In this case, the term will not necessarily be close to zero, but we can always upper bound the trace distance by 1. The number of terms in the sum that qualify as Case 1 is l − t − k + 1 and the number of Case 2 terms is t + k − 1. Thus the total error can be bounded: We will choose parameters so this quantity is less than ǫ. Parameters l and t will be related to χ ′ as t = log(2χ ′2 )/ log(⌊d/2⌋) (33) l = 2(k + t)/ǫ If |ψ is known only to have exponentially decaying correlations, then we choose log(χ ′ ) = 16ξ log(d) log(C 1 128ξ log(d)(k + 3)/ǫ 3/2 ) where C 1 = exp(exp(Õ(ξ log(d)))) is the constant from Eq. (19). We note that t ≤ 3 log(χ ′ ), so we can bound where in the third line we have used the (crude) bound √ u ≤ e u with u = log(χ ′ )/(16ξ log(d)).
In the case that |ψ is known to be the ground state of a gapped local Hamiltonian, we may choose where C 2 = exp(Õ(log 8/3 (d)/∆) is the constant in Eq. (20) and the same analysis will follow. This proves item (1) of the theorem. Item (2) and (2') assert that |ψ can be written as an MPS with constant bond dimension, which we now show. Each state |φ j is a product state with pure state |φ j,i on each each block M j,i , and |φ j,i has bond dimension at most 2χ ′2 . Thus, if we cut the state |φ j at a certain site, the bond dimension will be at most 4χ ′2 (recall that block M j,0 may have sites at both ends of the chain and can contribute to the bond dimension). Since |ψ is a sum over |φ j for l values of j, the bond dimension χ of |ψ is at most 4lχ ′2 . For the case of exponentially decaying correlations, this evaluates to proving item (2), and for case of ground state of gapped Hamiltonian, we find proving item (2'), where the second factor is asymptotically (k/ǫ) o (1) . This completes the proof.
The operator Q 1 is supported on A and Q 2 is supported on C. Since A and C each contain blocks M i for at least l values of i, conditions (1), (2), and (3) above imply that The choice of l is independent of the chain length N , so the above quantity is independent of N and independent of the parameter m measuring the distance between A and C. Thus, the correlation certainly does not decay exponentially in the separation between the regions.

C. Proof of Theorem 2
Proof. We fix an even integer l, which we will specify later, and divide the N sites of the chain into n + 2 segments of length l, which we label, from left to right: M 0 , M 1 , . . . , M n+1 . If N does not divide l evenly, then we allow segment M n+1 to have fewer than l sites. For i ∈ [1, n] let L i be the sites to the left of region M i and let R i be the sites to the right of region M i .
Lemma 4 tells us that, since |ψ has (t 0 , ξ)-exponential decay of correlations, for any i ∈ [1, n], ρ LiRi is close to ρ Li ⊗ ρ Ri when l is much larger than ξ ′ : whenever l ≥ t 0 . We also choose χ = d l and for each i define ρ ′ Li and ρ ′ Ri , each with rank at most √ χ by taking A = L i and A = R i in Lemma 3. Thus we have Then by the triangle inequality we have Note that |ψ LiMiRi can be viewed as a purification of ρ LiRi with M i the purifying auxiliary system. Divide region M i in half, forming M L i (left half) and M R i (right half). Each of these subsystems has total dimension d l/2 and thus can act as the purifying auxiliary system for ρ ′ Li or ρ ′ Ri . Let |α i LiM L i be a purification of ρ ′ Li and |β i M R i Ri be a purification of ρ ′ Ri . Thus, |α i ⊗|β i , which is defined over the entire original chain, is a purification Uhlmann's theorem (Lemma 5) shows how these purifications are related by a unitary on the purifying auxilary system: for each M i with i ∈ [1, n], there is a unitary U i acting non-trivially on region M i and as the identity on the rest of the chain such that U i |ψ is very close to |α i ⊗ |β i , a product state across the cut between M L i and M R i . In other words, U i disentangles L i from R i , up to some small error, by acting only on M i . Formally we say that where δ i ≤ C exp(−l/ξ ′′ ) for all i. An equivalent way to write this fact is where |φ ′ i is a normalized state orthogonal to |α i ⊗ |β i . We can define the projector whose rank is 1 when considered as an operator acting only on M R i R i . We notice that Π i U i |ψ LiMiRi is a product state across the M L i /M R i cut and has a norm close to 1. Suppose we alternate between applying disentangling operations U i and projections Π i onto a product state as we move down the chain. Each Π i will reduce the norm of the state, but we claim the norm will never vanish completely (and delay the proof for later).
This allows us to define Note that, to put our proof in line with what is described in the introduction and proof summary earlier, we may act with all the unitaries prior to the projectors if we conjugate the projectors . . U † n , which still only acts on the region R i .
This can be compared with the state |φ defined by applying the disentangling operations without projecting: We claim that |φ is a good local approximation for |φ .
We delay the proof for clarity of argument. Next we can define which parallels the relationship Now suppose X is a contiguous region of the chain of length k. Then there is a region X ′ of the chain of length at most k ′ = k + 2l that contains X and is made up of where the third line follows from the fact that |φ and |ψ are related by a unitary that does not couple region X ′ and region X ′c , and the fourth line follows from Claim 2.
Item (2) states that |ψ can be written as an MPS with constant bond dimension. This can be seen by the following logic. The Schmidt rank of |φ across any cut M L i /M R i is 1, as discussed in the proof of Claim 2 (see Eq. (59)), since the projector Π i projects onto a product state across that cut and unitaries U j with j > i act trivially across the cut. By Lemma 6, this implies that the Schmidt rank across any cut M R i /M L i+1 can be at most d l/2 . Acting with the inverse unitaries U † j on |φ to form |ψ preserves the Schmidt rank across the cut M R i /M L i+1 , since none couple both sides of the cut. Because any cut is at most distance l/2 from some M R i /M L i+1 cut, the Schmidt rank across an arbitrary cut can be at most a factor of d l/2 greater, again by Lemma 6, meaning the maximum Schmidt rank across any cut of |ψ is χ = d l .
Given our choice of l = ξ ′′ log(C √ 6k/ǫ), we find that the state can be represented by an MPS with bond dimension This proves item (2). Item (2') follows by using the same equations with C = 2C 3 + C 4 = exp(Õ(log 8/3 (d)/∆)) and ξ ′′ = O((log(d)/∆) 4/3 ). Thus the final analysis yields Item (3) states that |ψ can be formed from a lowdepth quantum circuit. In the proof of Claim 2, we show how the state |φ is a product state across divisions M L i /M R i , as in Eq. (59). Thus the state |φ can be created from |0 ⊗N by acting with non-overlapping unitaries on regions M R i M L i+1 in parallel. Each of these unitaries is supported on l sites. Then, |ψ is related to |φ by another set of non-overlapping unitaries supported on l sites, as shown in Eq. (53). We conclude that |ψ can be created from the trivial state by two layers of parallel unitary operations where each unitary is supported on l sites. In [29], it is shown how any l-qudit unitary can be decomposed into O(d 2l ) = O(χ 2 ) two-qudit gates, with no need for ancillas. We can guarantee that these gates are all spatially local by spending at most depth O(l) performing swap operations to move any two sites next to each other, a factor only logarithmic in the total depth. This proves the theorem.
Proof of Claim 1. We prove by induction. Let |φ j = Π j U j . . . Π 1 U 1 |ψ . Note that Π i U i |ψ is non-zero for all i, so in particular |φ 1 is non-zero. Furthermore we note that, if it is non-zero, |φ j can be written as a product state α ′ j LjM L j ⊗ |β j M R j Rj for some unnormalized but non-zero state α ′ j , and the reduced density matrix of |φ j on R j is ρ ′ Rj . If we assume |φ j is non-zero then we can write and as long as l ≥ ξ ′′ log(3C).
Proof of Claim 2. First, consider the cut M L i /M R i during the formation of the state |φ . When the projector Π i is applied, the state becomes a product state across this cut. The remaining operators are U j and Π j with j > i, and thus they have no effect on the Schmidt rank across the M L i /M R i cut, meaning |φ is a product state across each of these cuts, or in other words (59) Given an integer k and a contiguous region X of length k, we can find integers a and b such that The advantage here is that all of the U i and Π i for which i ∈ [a, b] have disappeared. On the other hand, we have Note that, since we can say that and thus where δ ≤ b j=a δ 2 j and normalized |φ ′ is normalized. This implies and hence This holds for any region X of length k, so this proves the claim.

D. Proof of Theorem 3
Proof. We begin by specifying a family of Hamiltonians, parameterized by t ∈ [0, 1/2] and defined over a chain of length N with local dimension d.
The ground state of H Z (t) is the trivial product state |0 ⊗N with ground state energy t(N −1), and thus energy density t. The interaction strength is bounded by 1 and the spectral gap is 1 − t ≥ 1/2. Now we construct an algorithm for Problem 1. We are given H as input, with associated parameters N and d, and a lower bound ∆ on the spectral gap. Let the true ground state energy for H be E and let u = E/(N − 1). We choose a value of s between 0 and 1, and we apply Lemma 7 to construct a Hamiltonian K combining Hamiltonians H/2 and H Z (s/2). K acts on N sites, has local dimension 2d, and has spectral gap at least min(∆, |s − u|(N − 1), 2)/6. We are given a procedure to solve Problem 2 with δ = 0.9 for a single site, i.e. we can estimate the expectation value of any single site observable in the ground state of K. If s < u, the true reduced density matrix of K will have its ancilla bits all set to 1. If s > u the reduced density matrix corresponds to the reduced density matrix of H with all its ancilla bits set to 0. Thus we can choose our single site operator to be the Z A operator that has eigenvalue 1 for states whose ancilla bit is |0 and eigenvalue −1 for states whose ancilla bit is |1 . If we have a procedure to determine ψ| Z A |ψ to precision 0.9 then we can determine the setting of one of the ancilla bits in the ground state and thus determine whether u is larger or smaller than s. The time required to make this determination is f (min(∆, (N − 1)|s − u|, 2)/6, 2d, N ). Because we have control over s, we can use this procedure to binary search for the value of u. We assume we are given a lower bound on ∆ but since we do not know u a priori, we have no lower bound on |s − u|, so we may not know how long to run the algorithm for Problem 2 in each step of the binary search. If our desired precision is ǫ, we will impose a maximum runtime of f (min(∆, (N − 1)ǫ/2, 2)/6, 2d, N ) for each step. Thus, if we choose a value of s for which |s − u| < ǫ/2, the output of this step of the binary search may be incorrect. After such a step, our search window will be cut in half and the correct value of u will no longer be within the window. However, u will still lie within ǫ/2 of one edge of the window. Throughout the binary search, some element of the search window will always lie within ǫ/2 of u, so if we run the search until the window has width ǫ and output the valueũ in the center of the search window, we are guaranteed that |u −ũ| ≤ ǫ. The number of steps required is O(log(1/ǫ)) and the time for each step is f (min(2∆, (N − 1)ǫ, 4)/12, 2d, N ), yielding the statement of the theorem. E. Proof of Lemma 4 Proof. The goal is to show that an exponential decay of correlations in τ = |η η| implies that τ LR is close to τ L ⊗ τ R . We will do this by truncating the rank of τ on the region L to form σ, arguing that σ LR is close to σ L ⊗ σ R , and finally using the triangle inequality to show the same holds for to τ . Lemma 3 says that there is a state σ L with rank χ defined on region L, such that In fact, the choice of σ L of rank χ that minimizes the distance to τ L is the state P L τ L /Tr(P L τ L ) where P L is the projector onto the eigenvectors of τ L associated with the largest χ eigenvalues. Accordingly, we define a normalization constant q = 1/Tr(P L τ L ) and let |ν = √ qP L |η and σ = |ν ν| = qP L τ P L be normalized states.
We first need to show that (69) is small, given only that Cor(L : R) |η is small. Suppressing tensor product symbols, we can write Tr((AB)(σ LR − σ L σ R )) = ν| AB |ν − ν| A |ν ν| B |ν = q η| P L ABP L |η − q 2 η| P L AP L |η η| P L BP L |η = q η| (P L AP L )B |η − q 2 η| P L AP L |η η| P L B |η = q η| (P L AP L )B |η −q 2 η| P L AP L |η ( η| P L B |η − η| P L |η η| B |η ) −q 2 η| P L AP L |η η| P L |η η| B |η = q ( η| (P L AP L )B |η − η| P L AP L |η η| B |η ) −q 2 η| P L AP L |η ( η| P L B |η − η| P L |η η| B |η ) from which we can conclude Cor(L : R) |ν ≤ (q + q 2 )Cor(L : R) |η The normalization constant q is 1/(1−D(τ L , σ L ) 2 ) which will be close to 1 as long as χ is sufficiently large. If we choose log(χ) = 8ξ log(d)(1 + log(C 1 )) or larger, then q will certainly be smaller than 2 and q + q 2 ≤ 6. The combination of the fact that σ L has small rank and that σ has small correlations between L and R will allow us to show that σ LR is close to σ L ⊗ σ R . We do this by invoking Lemma 20 of [26], although we reproduce the argument below. We can express the trace norm as where the second equality follows from the fact that P L fixes the state σ. We can perform a Schmidt decomposition of the operator P L T P L into a sum of at most χ 2 terms which are each a product operator across the L/R partition and also such that T L,j , T R,j ≤ 1 (see Lemma 20 of [26] for full justification of this). Then we may write as long as χ ≥ 8ξ log(d)(1 + log(C)) and dist(L, R) ≥ t 0 . Moreover the purified distance is bounded by the square root of the trace norm of the difference (Lemma 1), allowing us to say Then, by the triangle inequality, we can bound We can choose log(χ) = 8ξ log(d)(1 + log(3C 1 )) + dist(L, R)/(4ξ) (73) Then each term can be bounded so that which proves the first part of the Lemma. If τ is the unique ground state of a gapped Hamiltonian, then we may use the second part of Lemma 3, and bound Here we can choose log(χ) = O log(d)(1 + log(3C 2 )) 3 4 ∆ 1/4 + (∆dist(L, R)) 3 4 and then each term is small enough to make the bound which proves the second part of the lemma.
F. Proof of Lemma 7 Proof. Note that a variant of this lemma is employed in [28,30,31] to show the undecidability of certain properties of translationally invariant Hamiltonians.
Since H (j) is translationally invariant, it is specified by its single interaction term H (j) i,i+1 : The Hamiltonian K will be defined over a new chain where we attach to each site an ancilla qubit, increasing the local Hilbert space dimension by a factor of 2. We refer to the ancilla associated with site i by the subscript A i , and we refer to the collection of ancillas together with the subscript A. Operators or states without a subscript are assumed to act on the original d-dimensional part of the local Hilbert spaces. Let where with I i,i+1 denoting the identity operation on sites i and i + 1. In this form it is clear that K is a nearest-neighbor translationally invariant Hamiltonian and that each interaction term has operator norm 1 (a requirement under our treatment of 1D Hamiltonians). The picture we get is that if two neighboring ancillas are both |0 , then H i,i+1 /3 is applied, and if the ancillas are different, then I i,i+1 is applied. Following this intuition, we can rewrite K as follows where the first sum is over all settings of the ancillas and the operator K x,i acts on the non-ancilla portion of sites i and i + 1, with We analyze the spectrum of K. If H (j) has eigenvalues E (j) n with corresponding eigenvectors |φ n /3, respectively. Recall that eigenvectors of a Hamiltonian span the whole Hilbert space over which the Hamiltonian is defined. Therefore, the eigenvectors of K listed above span the entire sectors of the Hilbert space associated with the ancillas set to 0 N A or 1 N A . Suppose |φ is another eigenvector of K. Since it is orthogonal to all of the previously listed eigenvectors, |φ can be written for some set of complex coefficients α x with x |α x | 2 = 1 and some set of normalized states |η x . The sum explicitly leaves out the x = 0 = 0 N and x = 2 N − 1 = 1 N binary strings because these states lie in the subspace spanned by eigenstates already listed. We wish to lower bound the energy of the state |φ .
We would like to show a lower bound for this quantity. We make the following claim: Claim 3. For any state |η , and any i,i+1 , which has energy E r . Then Here we invoke the variational principle twice. First, note that the expectation value of (the exact ground state of the whole chain) is exactly (r − 1)E (j) 0 /(N − 1). Since |ν is the exact ground state of that Hamiltonian, E r must be smaller than this quantity. Second, as before, Combining these observations yields Now we use this claim to complete the proof of Lemma 7. For any binary string x we can associate a sequence of indices 1 = i 0 < i 1 < . . . < i m < i m+1 = N + 1 such that for all k = 1, . . . , m and all i in the interval In other words, x can be decomposed into substrings of consecutive 0s and consecutive 1s, with i j representing the index of the "domain wall" that separates a substring of 0s from a substring of 1s. The parameter m is the number of domain walls. Using this notation, and letting For any x other than 0 N and 1 N , there is at least one domain wall and m ≥ 1. Thus we can say φ| K |φ ≥ E K 0 + 1/3 (89) We have shown that any state orthogonal to the states 0 N A ⊗|φ (0) n and 1 N A ⊗|φ (1) n will have energy at least 1/3 larger than the lowest energy state of the system. Without loss of generality, suppose E , or lies outside the sector associated with ancillas 0 N and 1 N , whichever has lowest energy. The three cases lead to spectral gaps of ∆ (0) /3, E 0 − E (0) 0 , and something larger than 1/3 (due to Eq. (89)), respectively. This proves all three items of the Lemma.

V. DISCUSSION
Our results paint an interesting landscape of the complexity of approximating ground states of gapped nearest-neighbor 1D Hamiltonians locally. On the one hand, we show all k-local properties of the ground state can be captured by an MPS with only constant bond dimension, an improvement over the poly(N ) bond dimension required to represent the global approximation. This constant scales like a polynomial in k and 1/ǫ, when parameters like ∆, ξ, and d are taken as constants. On the other hand, we give evidence that, at least for the case where the Hamiltonian is translationally invariant, finding the local approximation may not offer a significant speedup over finding the global approximation: we have shown that the ability to find even a constant-precision estimate of local properties would allow one to learn a constant precision estimate of the ground state energy with only O(log(N )) overhead. This reduction does not allow one to learn any global information about the state besides the ground state energy, so it falls short of giving a concrete relationship between the complexity of the global and local approximations. Nonetheless, the reduction has concrete consequences. In particular, at least one of the following must be true: (1) The ground state energy of a translationally invariant Hamiltonian on a chain of length N can be estimated to O(1) precision in O(log(N )) time.
(2) Local properties of the ground state of a translationally invariant Hamiltonian on a chain of length N cannot be estimated to O(1) precision in time independent of N .
In particular, the second item, if true, would seem to imply that, in the translationally invariant case when N → ∞, local properties cannot be estimated at all. Indeed, it is when the chain is very long, or when we are considering the thermodynamic limit directly that our our results are most relevant. In the translationally invariant case as N → ∞, our first proof method yields a local approximation that is a translationally invariant MPS. However, the MPS is non-injective 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 at infinity is also required [12,14]. Our second proof method, on the other hand, yields a periodic MPS (with period O(log(k/ǫ 2 ))) that is injective and can be constructed by a constantdepth quantum circuit made from spatially local gates. If we allow the locality of the gates to be O(log(k/ǫ 2 )), then the circuit can have depth 2. If we require the locality of the gates be only a constant, say 2, then the circuit can have depth poly(k, 1/ǫ).
The fact that the approximation is injective perhaps makes the latter method more powerful. Injective MPS are the exact ground states of some local gapped Hamiltonian [19,32]. Additionally, non-injective MPS form a set of measure zero among the entire MPS manifold, so variational algorithms that explore the whole manifold are most compatible with an injective approximation. In fact, since the approximation can be generated from a constant-depth circuit, the result justifies a more restricted variational ansatz using states of that form. This ansatz could provide several advantages over MPS in terms of number of parameters needed and ability to quickly calculate local observables, like the energy density. However, algorithms that perform variational optimization of the energy density generally suffer from two issues, regardless of the ansatz they use. First, they do not guarantee convergence to the global minimum within the ansatz set, and second, even when they do find the global minimum, the output does not necessarily correspond to a good local approximation. This stems from the fact that a state that is ǫ-close to the ground state energy density may actually be far, even orthogonal, to the actual ground state. Therefore, even a brute-force optimization over the ansatz set cannot be guaranteed to give any information about the ground state, other than its energy density.
This leaves open many questions regarding the algorithmic complexity of gapped local 1D Hamiltonians. For the general case on a finite chain, can one find a local approximation to the ground state faster than the global approximation? For translationally invariant chains, can one learn the ground state energy to O(1) precision in O(log(N )) time, and can one learn local properties in time independent of the chain length? Relatedly, in the thermodynamic limit, can one learn an ǫ-approximation to the ground state energy density in O(log(1/ǫ)) time, and can one learn local properties at all? These are interesting questions to consider in future work.
We would like to conclude by drawing the reader's attention to independent work studying the same problem by Huang [33], to appear simultaneously with our own.