Undecidability of the Spectral Gap in One Dimension

The spectral gap problem - determining whether the energy spectrum of a system has an energy gap above ground state, or if there is a continuous range of low-energy excitations - pervades quantum many-body physics. Recently, this important problem was shown to be undecidable for quantum systems in two (or more) spatial dimensions: it is provably impossible to determine in general whether a system is gapped or gapless, a result which has many unexpected consequences for the physics of such systems. However, there are many indications that one dimensional systems are simpler than their higher-dimensional counterparts: for example, they cannot have thermal phase transitions or topological order, and there exist highly-effective numerical algorithms such as DMRG for gapped 1D systems, exploiting the fact that such systems obey an entropy area-law. Furthermore, the spectral gap undecidability construction crucially relied on aperiodic tilings, which are easily seen to be impossible in 1D. So does the spectral gap problem become decidable in 1D? In this paper we prove this is not the case, by constructing a family of 1D spin chains with translationally-invariant nearest neighbour interactions with undecidable spectral gap. This not only proves that the spectral gap of 1D systems is just as intractable, but also predicts the existence of qualitatively new types of complex physics in 1D spin chains. In particular, it implies there are 1D systems with constant spectral gap and unique classical ground state for all systems sizes up to an uncomputably large size, whereupon they switch to a gapless behaviour with dense spectrum.


October 4, 2018
The spectral gap problem-determining whether the energy spectrum of a system has an energy gap above ground state, or if there is a continuous range of low-energy excitations-pervades quantum many-body physics. Recently, this important problem was shown to be undecidable for quantum systems in two (or more) spatial dimensions: it is provably impossible to determine in general whether a system is gapped or gapless, a result which has many unexpected consequences for the physics of such systems. However, there are many indications that one dimensional systems are simpler than their higher-dimensional counterparts: for example, they cannot have thermal phase transitions or topological order, and there exist highly-effective numerical algorithms such as DMRG for gapped 1D systems, exploiting the fact that such systems obey an entropy area-law. Furthermore, the spectral gap undecidability construction crucially relied on aperiodic tilings, which are easily seen to be impossible in 1D. So does the spectral gap problem become decidable in 1D? In this paper we prove this is not the case, by constructing a family of 1D spin chains with translationally-invariant nearest neighbour interactions with undecidable spectral gap. This not only proves that the spectral gap of 1D systems is just as intractable, but also predicts the existence of qualitatively new types of complex physics in 1D spin chains. In particular, it implies there are 1D systems with constant spectral gap and unique classical ground state for all systems sizes up to an uncomputably large size, whereupon they switch to a gapless behaviour with dense spectrum.
One-dimensional spin chains are an important and widely-studied class of quantum many-body systems. The quantum Ising model, for example, is a classic model of magnetism; the 1D Ising model with transverse fields is the textbook example of a quantum phase transition. It is also one of a handful of quantum many-body systems which can be completely solved analytically. Indeed, most known exactly solvable quantum many-body models are in 1D [1-3]. Even for 1D systems that are not exactly solvable, the density matrix renormalisation group (DMRG) algorithm [4] works extremely well in practice, and recent results have even yielded provably efficient classical algorithms for all 1D gapped systems [5].
There are several other indications that ground states of (finite) gapped 1D systems are qualitatively simpler than in higher dimensions. They obey an entanglement area-law, hence have an efficient classical descriptions in terms of matrix product states [6,7]. Furthermore, thermal phase transitions [8] and topological order [9] are both ruled out for 1D quantum systems. For classical 1D systems, satisfiability and tiling problems become tractable. For the simplest class of spin chains-qubit chains with translationally invariant nearest-neighbour interactions-the spectral gap problem has been completely solved when the system is frustration-free [10].
Contrast this with the situation in 2D and higher, where even simple theoretical models such as the 2D Fermi-Hubbard model (believed to underlie hightemperature superconductivity) cannot be reliably solved numerically even for moderately large system sizes [11,12]; the entropy area-law remains an unproven conjecture [13]; and the spectral gap problem is undecidable [14,15]. This latter result holds under strict definitions of gapped and gapless (see fig. 1), even if we impose the condition that the ground state is unique with a constant spectral gap above it in the gapped case, and the entire spectrum is continuous in the gapless case. For classical systems, satisfiability and tiling problems are NP-hard [16] and undecidable [17] (respectively) in two dimensions and higher.
On the other hand, not all questions concerning 1D quantum systems are easy. For example, Haldane conjectured in 1983 that a spin-1 antiferromagnetic Heisenberg model is gapped in the thermodynamic limit [18,19]. The Haldane conjecture remains open to this day [20]. Another example is the local Hamiltonian problem: approximating a quantum system's ground state energy to inverse polynomial precision is in general QMA hard [21,22], even with translationally-invariant nearest neighbour interactions [23,24].
Main Result. The many-body quantum systems we consider in this paper are one-dimensional spin chains of qudits on a line, i.e. with a Hilbert space (C d ) ⊗ N , where d is the local physical dimension, and N the length of the chain. The spins are coupled by translationally-invariant local interactions: a nearest-neighbour term h (2) , which is a d 2 × d 2 Hermitian matrix, and a d × d-sized local term h (1) which is also Hermitian. Both h (1) and h (2) are independent of the system size N. The overall Hamiltonian H N will be a sum of the local terms: (1) (Following standard notation, subscripts indicate the spin(s) on which the operator acts non-trivially, with the operator implicitly extended to the whole chain by tensoring with 1 on all other spins.) More precisely, H N defines a sequence of Hamiltonians on increasing chain lengths. The thermodynamic limit will be taken via N → ∞.
In order to be completely unambiguous about what we mean by the two terms gapped and gapless, we use a very strong definition. For {H N } to be gapless, we require that there exists a finite interval of size c above its ground state such that the spectrum of H N becomes dense therein as N goes to infinity. In contrast, {H N } is gapped if there exists γ > 0 such that for all N ∈ N, H N have a unique ground state and a spectral gap ∆(H N ) > γ where ∆(H N ) is the difference in energy between the (unique) ground state and the first excited state1 (see fig. 1). 1 Note that gapped is not defined as the negation of gapless; there are systems that fall into neither class. The reason for choosing such strong definitions is to deliberately avoid ambiguous cases (such as systems with degenerate ground states). Our constructions will allow us to use these strong definitions, because we are able to guarantee that each instance falls into one of the two classes.
For such one-dimensional spin chains, we give a constructive proof of the following theorem.
Theorem 1. There exist (explicitly constructible) d × d matrices a, a , a and d 2 × d 2 matrices b, b , b , b , b with the following properties: 1. a and b are diagonal with entries in Z.
4. For any n ∈ N and any rational number 0 < β ≤ 1 (which can be chosen arbitrary small), if one defines the Hamiltonian H N (n) Here, for n ∈ N with binary expansion n 1 n 2 . . . n |n | , we define φ(n) to be its binary fraction with interleaved 1s, i.e. φ(n) = 0.n 1 1n 2 1 . . . n |n |−1 1n |n | . the theorem proves that even an arbitrarily small perturbations of a classical Hamiltonian can have an undecidable spectral gap in the thermodynamic limit. There have been many previous results over the years relating undecidability to classical and quantum physics [17, . We refer to the introduction of [15] for a detailed historical account of these previous results.
So where is the difficulty in extending the two-dimensional result of Cubitt et al. to one-dimensional systems? One of the key ingredients in the 2D construction is a classical aperiodic tiling. The particular tiling used in [15], due to Robinson [51], exhibits a fractal structure, i.e. a fixed density of structures at all length scales. This ingredient is crucial if we are to translate the undecidability result to a one-dimensional system. Under the physical constraint of retaining a finite local dimension, however, this seems impossible in one dimension: all finite tile sets which can tile the line admit a periodic tiling. Even in the case of a degenerate ground state subspace, there is only a constant amount of information that can be passed from tile to tile (d bits for neighboring spins of dimension d)-it is therefore impossible to enforce a fractal structure classically.
Quantum mechanics can in principle circumvent this constraint, since entanglement can introduce long-range correlations. Yet even though it is known that one can obtain correlations between far-away sites that decay only polynomially, the resulting Hamiltonians are gapless [52,53].
The key new idea here is a 1D construction that creates-within the system's ground state-a periodic partition of the spin chain into segments, but whose length and period are related to the halting time of a Turing machine. This subtle interplay between the dynamics of a Turing machine, the periodic quantum ground state structure and the energy spectrum, plays the role of the classical aperiodic tilings of the 2D construction.
Hamiltonian Construction. A (classical) Turing machine is a simple model of computation consisting of an infinite "tape" divided into cells, and a "head" which steps left or right along the tape. The machine is always in one of a finite number of possible "internal states" There is one special internal state, denoted q f , which tells the machine to halt when it enters this state. Each cell can have one "symbol" written in it, from a finite set of possible symbols {σ Σ i=1 }. A finite table of "transition rules" determine how the machine should behave for each possible combination of symbol and internal state. At each time step, the machine reads the symbol in the cell currently under the head and looks up this symbol and the current internal state in the transition rule table. The transition rule specifies a symbol to overwrite in current cell, a new internal state to transition to, and whether to move the head left or right one step along the tape. The "input" to a Turing machine is whatever symbols are initially written on the tape, and the "output" is whatever is left written on the tape when it halts.
Despite its apparent simplicity, Turing machines can carry out any computation that it is possible to perform. Indeed, Turing constructed a universal Turing machine: a single set of transition rules that can perform any desired computation, 6 determined solely by the input. Given an input n to a universal Turing machine M, the Halting Problem asks whether M halts on input n.
We construct a Hamiltonian whose spectral gap encodes the Halting Problem. More precisely, we construct a 1D, translationally invariant, nearest-neighbour, spin chain Hamiltonian H N = H N (M, n) on the Hilbert space H = (C d ) ⊗ N , such that H N is gapped in the limit N → ∞ if M halts on input n, and gapless otherwise.
In the earlier 2D construction, this was accomplished by combining a trivial Hamiltonian with one that has a dense spectrum. The dense Hamiltonian is modified such that for a Halting instance, its lowest eigenvalue is pushed up by a constant, revealing the gap present due to the trivial part of the spectrum. In a non-Halting instance the Hamiltonian remains gapless (see fig. 1).
We know how to construct a Hamiltonian whose ground state energy is dependent on the outcome of a (quantum) computation: Feynman and Kitaev's history state construction, used ubiquitously throughout quantum complexity proofs [21-24, 54-58]. In brief, this construction allows one to take a circuit C with gates U 1 , . . . , U T acting on m qubits, and embed it into a Hamiltonian H C on n = m + poly log T qubits, such that the ground state is a superposition over histories of the computation, i.e. a state of the form |Ψ ∝ T t=0 |t |ψ t . Every "snapshot" of the computation |ψ t is entangled with a so-called clock register |t . For T computational steps, one can implement such a clock with a local Hamiltonian using poly log T qubits. The state |ψ 0 is thus input to the circuit, and |ψ t = U t · · · U 1 |ψ 0 is the state of the circuit after t gates. A later construction due to Gottesman and Irani [23] similarly encodes the evolution of a quantum Turing machine, instead of a quantum circuit. As the transition rules of a Turing machine do not depend on the head location, a benefit of encoding Turing machines rather than circuits is that the resulting Hamiltonians are naturally translationally invariant.
By adding a projector to "penalize" a subset of the possible outcomes of the computation, as encoded in |T |ψ T , the ground state in these cases is pushed up in energy by Θ(T −2 ). This energy shift can be exploited by combining the circuit Hamiltonian with a term that has a dense spectrum in the thermodynamic limit, i.e.
where H guard is a diagonal gapped Hamiltonian that ensures the ground state lies either completely in the dense sector of the Hilbert space-where H C and H dense are defined-or completely in the gapped trivial sector. If the computation output in in H C is penalized, the dense spectrum is pushed up, which in turn unveils the constant spectral gap of some trivial Hamiltonian H trivial . Yet even though we can easily penalize an embedded Turing machine reaching a halting state in this way (i.e. by adding a penalty term for the head being in any terminating state q f ), a history state Hamiltonian is insufficient for the undecidability proof. i) The energy penalty decreases as the embedded computation becomes longer [59]. However, we require a constant energy penalty density across the spin chain. ii) If we try to circumvent this problem by subdividing the tape to spawn multiple copies of the Turing machine, we need to know the space required beforehand in which the computation halts, if it halts-which is also undecidable.
Cubitt et al. circumvent this by spawning a fixed density of computations across an underlying Robinson lattice. Like this, within every area A, the halting case obtains an energy penalty ∝ A-the ground state energy density therefore differs by a constant for the Halting and non-Halting cases, allowing the ground state energy to diverge in the Halting case, which uncovers the spectral gap. The fractal properties of the Robinson tiling further ensure that that every possible tape length appears with a non-zero density in the large system size limit, so knowledge of the Turing machine's required runtime space is unnecessary.
We replace the fractal Robinson tiling with a 2-local "marker" Hamiltonian H on (C c ) ⊗ N , where the markers-a special spin state | -bound sections of tape used for the Turing machine. H is diagonal with respect to boundary markers-i.e. H commutes with | |. Thus any eigenstate |ψ has a welldefined signature with respect to these boundaries, where the signature sig |ψ is defined as the binary string with 1's where boundaries are located, and 0's everywhere else. We construct H in such a way that two consecutive markers bounding a segment will introduce an energy bonus that falls off quickly as the length of the segment increases: e.g. any eigenstate |ψ with a signature sig |ψ = (. . . , 0, 1, 0, . . . , 0, 1 length w , 0, . . .) will pick up a bonus of exp(−p(w)) for some fixed polynomial p. This bonus will be strictly smaller in magnitude than any potential penalty obtained from a computation running on the same segment of length w, i.e. when the TM head runs out of tape (see fig. 2).
To the marker Hamiltonian, we add a history state Hamiltonian H prop (φ, M). Here φ(n) = 0.n 1 1n 2 1 . . . n |n | 00 . . . encodes an input parameter n ∈ N with |n| binary digits as binary fraction, where the digits of n are interleaved by 1s.
The second parameter M is a classical universal Turing machine. We construct H prop to encode the following computation: 1. A Quantum Turing machine performs phase estimation on a single-qubit unitary that encodes the input φ.
2. The classical universal TM M uses the binary expansion of φ as input and performs a computation on it.
Up to a slight modification for 1. this is the same construction as in [15]. The Hamiltonian H prop is set up to spawn one instance of the computation per segment, and we penalize the TM M running out of available tape up to the next boundary marker. We finally add a trivial Hamiltonian H trivial with ground state energy −1 and constant spectral gap. The overall Hamiltonian is then where µ = 2 − |φ | , and |φ| = 2|n| is a small constant in the parameter n, and where β > 0 can be chosen arbitrarily small.
Phase Estimation. Quantum phase estimation (QPE) can be performed exactly when there is sufficient tape [60]. In case there is insufficient space for the full binary expansion of the input parameter φ, the output is truncated, and the resulting output state is not necessarily a product state in the computational basis anymore.
As in the 2D model, we have to allow for the possibility that the QPE truncates φ, possibly resulting in the universal TM dovetailed to the QPE switching its behavior to halting. In the 2D construction of [15], one could circumvent this by simply subtracting off the energy contribution from truncated phase-estimation outputs. However, this is not possible in the 1D construction, since we cannot à priori know the length of the segments on which the Turing machine runs. Instead, we augment the QPE algorithm by a short program which verifies that the expansion has been performed in full, and otherwise inflicts a large enough energy penalty to offset the case that the UTM now potentially halts on the perturbed QPE output.
To this end, we make use of the specific encoding of φ: the interleaved 1s are flags indicating how many digits to expand. Like this, before the inverse quantum Fourier transform, we know that the least-significant qubit is exactly in state |+ if the expansion was completed, and has overlap at least µ = 2 − |φ | with |− otherwise. By adding a penalty term to the Hamiltonian for said digit in state |− , we can penalize those segments with insufficient tape for a full expansion of the input, independently of whether the universal TM then halts or not on a faulty input. This result manifests as a kink of the lower energy bound for a too-short segment of length w in fig. 3. Yet since the marker Hamiltonian H is attenuated by µ as well, the energy remains nonnegative throughout for these segments. Therefore, the only segments left to be analyzed are those for which the input can be assumed un-truncated.
This results in two possibilities for H N . In case M(φ) does not halt, any instance of the TM running on any tape length will run out of tape space, incurring the penalty explained in fig. 2. This halting penalty will always dominate the bonus coming from the segment length, and we show the ground state energy to be λ min (H N ) ≥ 0. In case the TM does halt, there will be minimal segment length w halt above which segments will not pick up the penalty from exhausting the tape. Since the bonus given by the Marker Hamiltonian is decreasing with increasing segment length, the optimal energy configuration will therefore be achieved by partitioning the whole chain into segments of length w halt , each of which picks up a tiny-but finite-negative energy contribution.
where T halt is the number of computation steps till halting. As the system size N increases, the ground state energy will therefore diverge to −∞, and the claim of theorem 1 follows.
Discussion. In spite of indications that 1D spin chains are simpler systems than higher dimensional lattice models, we have shown that the spectral gap problem is undecidable even in dimension one. This resolves one of the main open questions left in [14]. At the same time, the construction we present has some distinguishing features from the 2D construction.
In the 2D case, the ground state behaves as a highly non-classical model, showing all features of criticality, for any system size where the Universal Turing machine embedded in the model does not halt. If the machine eventually halts, starting from the corresponding system size the ground state will suddenly transition to a classical, product state. The construction we have presented shows the opposite property: the ground state is a product classical state unless the machine halts, in which case it becomes critical. Both cases can be seen as an example of a size-driven phase transition [24], but since in the 1D construction we transition from trivial to gapless instead of vice-versa, any numerical study of the ground-state properties (for which we have algorithms which have provably polynomial running time in the system size, given that we are working in 1D [5]) will not reveal any exotic or unusual quantum phenomena. They will instead find a classical state, up to the point at which the universal Turing machine halts (if it does indeed halt), i.e. when the system size is increased above an uncomputable quantity. Therefore, not only is there no algorithm that can correctly predict whether the Hamiltonian is gapped or not, but also the known efficient algorithms for computing ground state properties will fail to predict the correct thermodynamic properties of the state, even properties as elementary as the decay of correlations.
Our findings extend to periodic boundary conditions, albeit in a limited fashion, for a number of spins promised to be coprime to some number P. This comes at the cost of a local dimension that grows linearly with P. The general periodic case with fixed local dimension remains open. As in 2D, the reduction also shows that the ground state energy density of 1D spin chains is, in general, uncomputable.
We conclude by commenting on some limitations of our result and on some open questions which are still to be addressed. First of all, as in the case of 2D systems, the model we present is extremely artificial, with a very large local dimension. It remains an interesting question whether it is possible to find more natural models showing this feature, or whether there is a local dimension threshold below which quantum systems necessarily behave in a predictable way [10]. We did not try to optimize the dimension of the local Hilbert space in our construction: while size-driven phase transitions can happen in 2D with very small local dimension [24], these low-dimensional constructions are decidable. It is certainly still possible that below some threshold on the local dimension the spectral gap problem becomes decidable. Determining if this threshold exists and if and how it depends on the lattice dimension remains a very interesting open question, and one which is now also interesting in 1D. [44] David Elkouss and David Pérez-García. "Inapproximability of the capacity of information stable finite state channels". In: arXiv:1601.06101 (2016).
[51] Raphael M. Robinson. "Undecidability and nonperiodicity for tilings of the plane". In: Inventiones mathematicae 12.  .1 In order to spawn a fixed density of computations in 1D without the aid of a fractal underlying structure, we need to know an optimal segment length to subdivide the spin chain into. In the halting case, this should be just enough tape for the computation to terminate. However, if we aim to construct a reduction from the Halting Problem, we cannot know the space required beforehandwhich, in particular, could be uncomputably large, or infinite! One way out is to spawn Turing machines on tapes of all possible lengths, and do this with a fixed density. In 2D this can be achieved using an underlying fractal tiling such as that due to Robinson [51], see fig. 4. The two-dimensional construction thus crucially depends on one's ability to create structures of all length scales, in order to define "lines" of all sizes,2 which are then used as a tape for running a Quantum Turing machine: the key property of the fractal which makes the construction work is that every possible tape length indeed appears with a non-zero density in the large system size limit.
As already mentioned, constructing a fractal tiling with a fixed density of structures of all length scales seems impossible in one dimension. We therefore replace the fractal Robinson tiling with a "marker" Hamiltonian, where the markers bound sections of tape used for the Turing machine (just like the lower boundaries of the squares in fig. 4). We will construct the Hamiltonian in such a way that two consecutive markers bounding a segment will introduce an energy bonus that falls off quickly as the length of the segment increases. This bonus will be weak enough to permit an executing QTM to "extend" the tape as needed, in the sense that the bonus due to the marker boundaries is strictly smaller in magnitude than the potential penalty introduced when the QTM head runs out of tape (see fig. 2).
In this section we will give an explicit construction of the Marker Hamiltonian we have discussed in the main text. It will be a local Hamiltonian H on a chain of qudits with a special spin state | , which we call a boundary, and  In the non-halting case a), there will never be any halting penalty, no matter how much tape there is available. In the halting case b), there is a threshold side length after which each rectangle of size × larger than the threshold contributes a penalty of magnitude Ω(exp(− ))-which yields a small but nonzero ground state energy density; the ground state energy diverges.
which will separate the different tape segments. For a product state |ψ , we define a signature with respect to these boundaries as the binary string with 1's where boundaries are located, and 0's everywhere else, which we will denote by sig |ψ . The Hamiltonian we construct will leave the signature invariant, i.e. sig |ψ = sig H |ψ for all |ψ . This property allows us to block-diagonalize H with respect to states of the same signature. For a given block signature, say  (1, 0, 0, 0, 1, 0, 0, 1), the Hamiltonian gives an energy bonus (i.e. a negative energy contribution) to each 1-bounded segment, which is large when the boundary markers are close, and becomes smaller the longer the segment. This introduces a notion of boundaries that are "attracted" to each other, and our goal is to have a falloff as ∼ −1/g(l) in the segment's length l, where g is a function we can choose. In brief, "attraction", in this context, simply means that the energy bonus given by H to pairs of boundary symbols grows the closer they are to each other.
For reasons of clarity, we start by constructing a Hamiltonian where the falloff is a fixed function g that is asymptotically bounded as Ω(2 l ) ≤ g ≤ O(4 l ).
In a second step, we allow the falloff to be tuned, replacing l by an arbitrary exponential in l, such that the falloff is doubly exponential in the segment length.

A.2. A Marker Hamiltonian with a Quick Falloff
We start with the following lemma.
is a 3-local Hamiltonian which is positive semi-definite, and block-diagonal with respect to the subspaces spanned by states with identical signature sig.
Proof. The first two claims are true by construction. The Hamiltonian H is further block-diagonal with respect to sig because sig H |ψ = sig |ψ ∀ |ψ ∈ H , as none of the local terms ever affect the subspaces spanned by the boundary symbol | .
As a second step, we employ a boundary trick by Gottesman and Irani [23] to ensure that blocks not terminated by a boundary marker have a ground state energy at least 2 higher than -terminated blocks. It is worth emphasizing that this is not achieved by a term that only acts on the boundary, but in a translationally-invariant way, i.e. by adding the same one-and two-local terms throughout the chain. In brief, it exploits the fact that while there are n spins in the chain, there is only n − 1 edges between them. We state this rigorously in the following remark.

Remark 3 (Gottesman and Irani [23]
). Give an energy bonus of strength 4 to | , and an energy penalty of 2 to | appearing next to any symbol (including itself. I.e. if | appears at the end of the chain there will be a net bonus of 2, otherwise a net penalty of zero). Collect these terms in a Hamiltonian P . Then, apart from positive semi-definiteness, H + P has the same properties claimed in lemma 2, but any block not terminated by a boundary will have energy ≥ −2, while all properly-terminated blocks will have a ground state energy −4.
Proof. The first claim is straightforward, as P does not change the interaction structure of H. The last claim follows from the fact that the only way of obtaining a net bonus is to place a boundary symbol at the end of the spin chain, where it picks up a net bonus of 2. The maximum possible bonus of any state is thus 4, which will be achieved by signatures that are properly bounded on either side.
From now on, when we talk of "properly bounded", we always mean a signature with boundary blocks at each end. Individual cases where only one side carries a boundary will be mentioned as such explicitly then. We want to point out that the same trick allows us to shift the overall energy of H + P up by 4, since it is possible to express 41 as a local term in a translationally-invariant fashion: achieves this, as it-analogously to remark 3-exploits the difference in the number of one-local compared to two-local couplings.
For now, and in order to keep things simple, we will keep the energy offset around, and lift it in one go towards the end. In the following, the "good" blocks will therefore be those that have ground space energy −4, all of which are properly bounded. Remark 3 allows us to analyze the blocks more closely, which we do in the following lemma.  Proof. If there are two neighbouring 1s in the signature s, the penalty term | | picks up an energy contribution of 2. Since H prop is already positive semi-definite and block-diagonal with respect to signatures, any state |ψ with support fully contained in the block corresponding to signature s must thus necessarily satisfy ψ| H |ψ ≥ ψ| P |ψ ≥ 2. The first claim follows. So let us assume that all 1s are spaced away from each other with at least one 0. Within the 2-dimensional 0 subspace spanned by the local basis states | and | . We note that the penalized substring | is also an invariant, meaning that no transition rule can create or destroy this configuration. Any state that, when expanded in the computational basis, has at least one expansion term with said substring will thus necessarily have all terms with this specific substring. The same arguments holds for the invariant substring | , and the second claim follows.

Lemma 4. Let H be as in lemma 2, but including the boundary trick terms
Since any eigenstate of R s picks up the full penalty contribution of 2, the third claim follows.
If neither of the invariant substrings | and | occur, we can assume that all 1-bounded segments of 0s lie within the span of the states | · · · , | · · · , . . . , | · · · , | · · · . (3) Since there is no penalty acting on any of those states, the ground state energy of G s equals −4. Each such segment of contiguous 0s thus defines a separate path graph, where the vertices are precisely these states, linked by the transition rules given in H prop . Denote the path graphs corresponding to these segments with G 1 , . . . , G n , where we assume that there are n 1-bounded segments of 0s in signature s. As each segment is independent of the others, the overall graph spanned by these individual paths is the Cartesian product of the individual paths, i.e. G = G 1 G 2 . . . G n . This is precisely a hyperlattice with side lengths uniquely determined by the lengths of the individual segments.
The transition rules in h prop therefore result in a block G s = ∆ G , i.e. the Hamiltonian is precisely the Laplacian of the graph of determined by the transition rules (see e.g. [24]). We further know that the Laplacian of a Cartesian product of graphs decomposes as and the last claim follows.
A more direct route to eq. (4) is to note that H prop is by definition the Laplacian of a graph with vertices given by strings of the alphabet { , , }, and edges by the transition rules in lemma 2. Those connected graph components that do not carry a penalty due to an invalid configuration (which either holds for all vertices, or none) are lattices in n dimensions-where n is the number of 1-bounded segments-and side lengths determined by the segments' lengths. Equation (4) is precisely the Laplacian of this grid graph.
For the sake of clarity, we will keep calling the segments of consecutive zeros bounded by on either side "1-bounded segments", and when talking about the entire string we use the term "properly bounded". We will henceforth re-label the states in eq. (3) as |1 , . . . , |w , where w denotes the length of the segment. Our next step will be to add a 2-local bonus term which gives an energy bonus to the arrow appearing to the left of the boundary, i.e. to | .
Lemma 5. Define H := H + P + P + B, where • P is taken from remark 3, gives a penalty of 1/2 to any boundary term, and 3. G s breaks up into sum of terms of the form 1 ⊗ ∆ w ⊗ 1, where ∆ w is a perturbed path graph Laplacian ∆ w := ∆ w − |w w| (where |w labels the last of the basis states given in eq. (3), as mentioned).
Proof. The first two claims follow immediately from lemma 4, since all of the newly-introduced terms leave signatures and penalized substrings invariant, and are at most 2-local.
Since the Cartesian graph product is associative and commutative, it is enough to show the decomposition for the case of two graphs G 1 and G 2 , and a single vertex v ∈ G 1 which we want to give a bonus of −1 to. Denote the bonus matrix for G 1 with B 1 . We have that the adjacency matrix Vertex v is thus mapped to a family of product vertices (v, v ) v ∈G 2 , which are precisely the corresponding bonus'ed vertices in G = G 1 G 2 that have to receive a bonus of −1. The bonus term for G is thus B = B 1 ⊗ 1, and the claim follows.
We know that any Laplacian eigenvalues µ, ν of two graphs G 1 , G 2 combine to a Laplacian eigenvalue µ + ν of G 1 G 2 (see e.g. [61, Ch. 1.4.6]). It is straightforward to extend this fact to the case of bonus'ed graphs, which will allow us to analyse the spectrum of each signature block H s .
The reader will have noticed that in contrast to lemma 4, lemma 5 does not make any claims about the ground state energy of the individual blocks. Naïvely, one could assume that the ground state energy of each block will diverge to +∞ with the number of boundaries present, as each of them carries a penalty of +1/2-but how does this balance with the bonus of −1, which we apply to only a single basis state in the graph Laplacian's ground space, and not on each vertex?
In order to answer this question, let us step back for a moment and develop a bound for the lowest eigenvalue of a modified path graph Laplacian ∆ w . We will do this in a series of technical lemmas. Lemma 6. ∆ w has precisely one negative eigenvalue.
As a next step, we will lower-bound the minimum eigenvalue of ∆ w .
Proof. We first observe that ∆ w is tridiagonal, e.g.
We can thus expand the determinant p w (λ) := det(∆ w − λ1) using the continuant recurrence relation (see [62, Ch. III]) f 0 := 1 As can be easily verified, a solution to this relation is given by the expression There is of course no hope to resolve p w (λ) = 0 for λ directly, so we go a different route. First note that p w (λ) is necessarily analytic, since it is the characteristic polynomial of ∆ w . We can calculate p w (−1/2) = (−1) 1+w 2 −w , and thus know that sign p w (−1/2) = 1 for w odd, and −1 for w even. If we can show that p w (−1/2 − 1/2 w ) has the opposite sign, then by the intermediate value theorem we know there has to exist a root on the interval [−1/2 − 1/2 w , −1/2], and the claim follows. First substitute p w (−1/2 − 1/2 w ) =: Then B w , a 1,w and a 2,w are real positive for all w. We distinguish two cases. 30 w even. If w is even, we need to show p w (−1/2 − 1/2 w ) ≥ 0, which is For w evew, y w ≥ x w , so it suffices to show which is true for all w ≥ 2.
w odd. Now y w ≤ x w , and it suffices to show which also holds true for all w ≥ 0. This finishes the proof.
And finally, using a similar approach, we will obtain an upper bound for the minimum eigenvalue of ∆ w . Lemma 8. The minimum eigenvalue of ∆ w satisfies λ ≤ −1/2 − 4 −w .
Proof. The idea is to extend the area around −1/2 for which p w is positive for w odd, and negative for w even, respectively. We start with p w from eq. (5), and substitute p w (−1/2 − 1/4 w ) =: A w /B w , where-almost as above, but replacing 2 −w by 4 −w -we have Then B w , a 1,w and a 2,w are real positive for all w. We distinguish even and odd cases.
w even. If w is even, we want to show that p w (−1/2 − 1/4 w ) ≤ 0, which is equivalent to where a := a 1,w a 2,w ∈ [1, 2] For w even, y w ≥ x w as before, so we cannot continue as before. Note that, for all w ≥ 0, and therefore It thus suffices to show It is straightforward to verify that this inequality holds for all w.
w odd. For odd w, y w ≤ x w . Analogously to before one can show Canceling the minus signs flips the inequality sign, and reduces the odd case to what we have shown for w even. The claim follows.
We summarize these findings in the following corollary.
Let us now analyse what this means for the spectrum of H . We are only interested in those blocks G s which correspond to modified grid Laplacians-all other cases are bounded away by a constant in lemma 5. In brief, the answer will be that the negative energy shift of −1/2 in corollary 9 will be precisely offset by the shift of 1/2 for any occurrence of the boundary state | .
Combining lemma 5 with corollary 9, we obtain the following theorem. 3. If s is bounded and without consecutive boundaries, H s = G s + R s as in lemma 5. In that case, the minimum eigenvalue λ of G s satisfies − i 1/2 w i ≤ λ + 7/2 ≤ − i 1/4 w i , where w i is the length of the i th contiguous 0-segments in the signature s. In that case, furthermore, G s has a spectral gap of size ≥ 1/2.
Proof. Claim 1 can be shown by explicitly considering an arbitrary signature, but with one missing boundary. We will only discuss the left boundary. The right then immediately follows from the fact that one could at most gain an extra bonus there from B in lemma 5. First consider the case that the left boundary looks like s = 01 · · · . By moving the boundary from the site to its right, we either break up a double boundary (in case s = 011 · · · ), or enlarge a segment (in case s = 010 · · · 01 · · · ). In the first case, we obtain i) a net bonus of 2 by remark 3, ii) a net bonus of 2 from breaking up a double boundary from lemma 2, iii) a bonus > 0 from creating a 1-bounded segment. In the second case, we also obtain i), but decrease the bonus from the segment to its right. This can at most be a penalty of 1/2, though, and the claim follows.
Claim 2 can be broken up in cases as well. Assume the double boundary is either on the left, or right (e.g. s = 110 · · · ). By deleting the second site boundary, one obtains a net bonus of at least 1. The same holds true for a site in the middle, as can be easily seen.
Claim 3 follows from corollary 9 and lemma 5. Every 1-bounded segment is terminated by a boundary, whose penalty of 1/2 from lemma 5 precisely offsets the −1/2 shift of the ground state of ∆ w . The leftover overall energy shift of −7/2 stems from the original −4 ground state from remark 3, and the single penalty of the left boundary of magnitude 1/2. The gap claim follows from lemma 5 (i.e. that R s ≥ G s + 2) and the spectral gap of ∆ w .
The transition rules in lemma 2 are those of a unary counter, as depicted in eq. (3). It is clear that if we allow for an increase in the local dimension we can use more complicated transition rules-and assume that they are 2local-to model the evolution of a more sophisticated calculation (e.g. the binary counter construction of [15], or the Quantum Thue System constructions of [24]). Instead of the linear exponential dependence on the segment length w in theorem 10, we then have the following theorem.
Theorem 11 (Marker Hamiltonian). Take a Hamiltonian H as H in theorem 10, but with 2-local transition rules describing a path graph evolution of length f (w) on a segment of length w. Furthermore, we add an energy shift of 7/2 by adding a term 7/2 N i=1 . Then H = s H s as before. We have H 0 ≥ 0, and either H s ≥ 1/2, or its minimum eigenvalue satisfies where w i is the i th segment length.
Proof. Precisely the same argument as in the proof of theorem 10, taking into account an energy shift of +7/2 due to the mismatch in the number of one-local and two-local couplings available in a system with open boundary conditions, see remark 3.
We conclude with the following two remarks.
Remark 12. On a spin chain with nearest neighbour interactions and local dimension d (including the boundary symbol ), one can obtain a path graph evolution length f (w) = (d − c 1 )w, or alternatively f (w) = (d − c 2 ) w , where c 1 and c 2 are constant. Each signature block H s of the corresponding Hamiltonian thus has a unique lowest-energy eigenvalue Proof. In the first case, we impose that each boundary term is followed by a sequence of states |0 , |1 , . . . , |d 1 , the latter of which we allow to be followed by |d 1 only. Now penalize a boundary term to the right of anything but |d 1 .
The second proof is similar, where instead of counting once we count modulo d 2 , and penalize the boundary state to appear to the right of anything but |d 2 .

B. Augmented Phase Estimation QTM
Just as in the two-dimensional case, we will use a phase estimation QTM to extract the input to a universal TM from the phase of a specific gate. In contrast to the original construction, we will need to be able to detect and penalize the case where the phase estimation does not terminate with the full binary expansion. This can be done with a slight modification to the original procedure from [15].
Theorem 14 (Phase-estimation QTM (Cubitt et al. [15])). There exists a family of QTMs P n indexed by n ∈ N, all with identical internal states and symbols but differing transition rules, with the property that on input N ≥ |n| written in unary, P n halts deterministically after O(poly(N)2 N ) steps, uses N + 3 tape, and outputs the binary expansion of n padded to N digits with leading zeros.
As the authors state, it is crucial that N does not determine the binary expansion that is written to the tape, only the number of digits in the output. The authors construct this family of QTMs explicitly, in three parts: 1. Apply the controlled U k -gates, where U is the phase gate encoding n (see fig. 5).
2. Detect the least significant bit.
The problem with using this series of steps unchanged is linked to the fact that we cannot apply the standard inverse quantum Fourier transform, for two reasons. First, we need the result of the QFT to be exact-so using approximate QFT is not an option. This in turn would imply we need an infinite local dimension, as we need a potentially infinite set of controlled phase gates. In the 2D construction, it suffices for the authors to provide a phase gate with minimum rotation α = 2 − |n | , since the case of too-short-segments can be independently detected there (see [15, sec. 5.3] for an extensive discussion).
However, in 1D, we cannot á priori know whether there is enough tape space for the full expansion, so finding the least significant bit is not always possible. A simple solution is as follows. By remark 13, we can always assume that the tape has length at least 10, and ≡ 0 (mod 2). We can then encode the input n as follows: n = n 1 n 2 · · · n |n | enc −→ φ := n 1 1n 2 1 · · · 1n |n | 0, i.e. we interleave the bits of n with 1s. In this way, by always reading pairs of bits, we know that once the second bit is 0, all digits of φ have been extracted.
In the following, we will assume that all inputs φ are always in the form eq. (6). The quantum phase estimation procedure can then be modified as follows.
1. Apply the controlled U k -gates, where U is the phase gate encoding n (see fig. 5).
2. Move the head to the least significant bit on the tape, and transition to a unique head symbol there.
3. Detect the least significant bit.
Steps 1, 3 and 4 are unchanged. In the next two sections we will rigorously show how this modification suffices to signal expansion success, and penalize all segments with insufficient space for the full expansion.

B.1. Expansion-Success-Signalling Quantum Phase Estimation
As a first step, we consider the requirement that the input N written in unary on the tape is longer than |φ| + 3. The tape is the space between two boundary symbols on a segment. As such, the segment length determines the maximum unary number N that we can write on the tape initially. Since we cannot á priori lower-bound the segment length to guarantee that N ≥ |φ| + 3, we have to consider the case N < |φ| + 3.
We will analyze the behaviour of this by going through the explicit construction of [15] step by step, and analyse how a too-small N affects the program flow. The phase estimation QTM is defined on the tape, but such that the tape has multiple tracks: a quantum track, where the quantum operations are performed, as well as classical tracks which are used for the control logic of the QTM-we refer the reader to [15, sec. 6.1.1&6.2] for details. The QTM follows five steps. Preparation Stage. The first cell of the quantum track is the ancilla qubit for the phase estimation, and the following N cells are the output qubits for the phase estimation.
1. Copy the quantum track's unary 1 · · · 1 to a separate input track, in binary.
This TM can work within a length N + 1 tape ([15, lem. 30]), so there is no issue with this step. We can thus assume that the separate input track contains the number N written in binary, and padded with 0s.
2. The N + 1 qubits in the quantum track are then initialized to |1 (|+ ) ⊗ N . Again, there is no issue.
Control-Phase Stage. This stage applies the first part of the phase estimation algorithm shown in fig. 5. It is crucial to note here that just because the input size N is not long enough to do the full phase estimation, the algorithm which is applied is still run as intended for N steps. If φ has binary expansion φ = 0.φ 1 · · · φ |n | , then the output on the first N qubits is |0 + e 2πi2 N −2 φ |1 · · · |0 + e 2πi2 0 φ |1 .

(7)
Signalling Expansion Success Since we only want to consider the full binary expansion of φ as a good input for the dovetailed universal TM, we need to have a way of signaling whether the full expansion has been delivered, or only a truncated version. We know that in eq. (7), the first qubit will be in state |+ if and only if the expansion happened in full. This is captured in the following lemma. Lemma 15. If we assume the phase φ in theorem 14 to be interleaved with 1s and terminating with a 0 as in eq. (6), and if N-the number of expansion bits-was even, the state post the controlled-U k stage, eq. (7), has the following properties: 2. Otherwise-if the phase estimation truncated φ-then | −| (|0 +e 2πi2 N −1 φ |1 )| 2 = Ω(2 − |φ | ).
Proof. The first claim follows since the least significant non-zero digit of φ is 1 by assumption, so 2π2 N −1 φ = 0 (mod 2π). For the second claim there are two extreme cases of φ to analyze; all others can easily be seen to be bounded by those. The first case is if there is only one more bit of 1 past where the expansion happened, i.e. a single 1 that is cut off: 2π2 N −1 φ = 0.1φ |n | 0 · · · (mod 2π), and φ |n | = 0. Then The other case is 2π2 N −1 φ = 0.1 · · · 10 · · · (mod 2π), with ≤ |φ| 1s. Then In order to temporarily transition to a specific head state q ? over the leftmost qubit which we just showed to have large overlap with |− in case of a truncated output, we dovetail the controlled phase stage with the following trivial machine. The head state q ? together with the underlying qubit will later allow us to discriminate between the two cases in lemma 15. to the output of the phase estimation. It is crucial to observe again that the control flow for the application of the Fourier transform TM does not change behaviour simply because the tape is too short to contain all |φ| digits of φ.
The trouble is that since we cannot necessarily locate the least significant bit if the expansion was truncated, we possibly apply the "wrong" inverse QFT. Thus, from hereon, we cannot guarantee that the output is related to the input in any way to keep the dovetailed UTM halting, if it were to halt on the fully-expanded φ, or likewise non-halting. As mentioned in the main text, we note that we do not need to care about this problem: we already have an independent state we can penalize (q ? over |− ) in case the QPE truncated the expansion.

B.2. On Proper QTM Behaviour
As in the two-dimensional construction, we have to ensure that one can write a valid history state Hamiltonian from the defined quantum Turing machine. One requirement is that when the QTM is specified by a partial isometry for the transition rules, they can be uniquely completed to a unitary transition function.
In Cubitt et al. [15]'s case, the authors ensured this by requiring that the QTM was proper, as defined in [15, Def. 20]-meaning that the QTM head moves deterministically on a subset of good inputs. This not only means that there should never be an explicit transition for a head state into a superposition, but also that any intermediate superposition on the quantum tape does not result in the head splitting up into distinct states. For TM tapes that were too short, the authors could not guarantee this property (just as we cannot here). This was not an issue in the 2D construction, since the energy contribution from these cases could be obtained by exact diagonalization (the binary length of φ is known, hence also an upper bound on the too-short-segment length) and subtracted from the final Hamiltonian.
The reason for proper behaviour in the good case-i.e. long enough tape-is more subtle. Assume for now we have a non-halting instance φ. If the QTM head were to move in some superposition, it could be that on some long but finite track, one head path reaches the boundary. Since there is no more tape, the clock moves this head to an idling tape. This head path is thus not able to interfere back with the other head paths. The other head paths could now think that one has a halting instance, skewing the result. It is therefore crucial that the QTM we design behaves properly for long enough tapes. Proof. The phase estimation terminates with success probability of 1 if the tape is long enough, and we refer the reader to [15] for a discussion of the proper QTMs they use, and whose existence we can thus assume.
We point out that for us it suffices that for too short tapes, we can inflict an independent penalty on the head state q ? in lemma 16. Whatever happens after that (since the tape is left in superposition) we do not care about, as we will discuss in the next section. So, as in the 2D case, we do not need to ensure that the QTM behaves properly in this case.
i.e. the product of internal states times the possible head positions times all possible tape configurations. Equation (8) allows us to choose a falloff exponent f such that 1 λ min = T 3 (w) < T 3 max (w) < 2 f (w) , The top panel shows the case for which the dovetailed universal TM will not halt. Depending on the segment length w, we have the following two cases: 1. For w < |n| + 5, there is not enough tape for the phase expansion. By lemma 16, we know that with probability ≥ µ, the phase estimation results in a string where the head symbol q ? is over a tape qubit |− , which shows the phase estimation truncated the output. Therefore, the head will be penalized by P with overlap ≥ µ. In order to account for the fact that the part of the computation following on from the garbage state coming out of the interrupted phase estimation could well halt, even if φ encodes a non-halting instance, we scale the lower bound in this area down by a factor µ-which is still non-negative, as µ is just a constant prefactor. Observe that it is not essential that we inflict the penalty term at the end of the history state (see e.g. [24, Cor. 44]).
2. For w ≥ |n| + 5 the phase estimation finishes exactly, and the universal TM retrieves the complete input on which it will not halt; the energy penalty P applies as well.
In either case, the history state evolution is of length T = T(w), i.e. the runtime of the computation until the head bumps into the right marker or the clock driving the computation runs out of time, both of which depends on the segment length w. In both cases, the last step of the computation will be completely penalized. This pushes the corresponding associated Hamiltonian's ground state energy up by Θ(1/T 2 ).
In case the dovetailed universal TM does halt, there is no further forward transition4. The TM head will not feel the penalty P, and the ground state energy is that of an unfrustrated history state Hamiltonian, i.e. zero. Observe that this happens at a point T halt which is obviously independent from w. The precise statement is that once there is enough tape such that the entire evolution of the (halting) TM can be contained, no halting penalty will be felt. This happens once w is such that T(w) ≥ T halt . Define this segment length to be w halt .
After including the Marker Hamiltonian H ( f ) s in H s , we obtain the ground state energy bounds shown in fig. 3. The dashed blue line shows an upper bound on the negative magnitude of the energy bonus E(w) induced by the

D.1. Periodic Boundary Conditions
Theorem 1 can, in a limited fashion, be extended to periodic boundary conditions, which we summarize in the following lemma and theorem.
Lemma 26. Theorem 1 holds, even on 1D spin chains with periodic boundary conditions, and under the assumption that the spin chain instances all have length coprime to P, at the cost of a local dimension that grows with P.
Proof. Take the Hamiltonian from Theorem 1. The only difference to the open boundary conditions case is that there is no mismatch between the number of 1and 2-local terms, so we will have to modify those parts of the proof carefully.
We first note that remark 3 relies on this boundary trick. In the periodic case, however, we cannot use it. The reason for remark 3 was to enforce all segments to have right-boundaries-otherwise a segment which is half-unbounded on the right would pick up the bonus from the marker Hamiltonian, but no penalty due to the TM running out of tape. This problem never occurs on a ring: if there is at least one marker present, it is automatically guaranteed that each segment is properly bounded. Therefore, if we drop the term P , lemma 4 goes through, but such that the resulting Hamiltonian has a ground state energy of 0, not −4.
The next step which needs amendment is in theorem 10, where we note that there is no leftover penalty of 1/2 from the leftmost boundary marker-bonus and penalty terms from lemma 5 precisely cancel. To this end, there is no energy shift necessary.
The last issue is with lemma 23: while one can straightforwardly create a Hamiltonian with constant negative ground state energy when there are open boundary conditions, this is not the case with periodic systems. To circumvent this, we assume we have a trivial Hamiltonian H triv with unique classical ground state with energy 0 and first excited state 1. We then shift everything else up by a constant. Under the stated assumption that the spin loop has a length coprime to P, the positive energy shift can be achieved by adding an ancilliary Hilbert space of dimension P, and adding local projectors that enforce a tiling á la 1, 2, 3, . . . , P. Since this tiling has to be broken at least at one site on the ring, there is a constant energy shift.
where H equals H from theorem 1, with the P-periodic tiling enforced. In the non-halting case, H tot will be gapped with ∆ ≥ 1, and unique ground state. In the halting case, H will have an energy that diverges to −∞ (despite the constant energy shift inflicted by the P-periodic tiling), and therefore pulls the dense spectrum of H dense with it. The claim of the theorem follows.