Sign-curing local Hamiltonians: termwise versus global stoquasticity and the use of Clifford transformations

07/23/2020
by   Marios Ioannou, et al.
Freie Universität Berlin
0

We elucidate the distinction between global and termwise stoquasticity for local Hamiltonians and prove several complexity results. We prove coNP-hardness of deciding global stoquasticity in a fixed basis and Σ_2^p-hardness of deciding global stoquasticity under single-qubit transformations. We expand the class of sign-curing transformations by showing how Clifford transformations can sign-cure a class of disordered 1D XYZ Hamiltonians.

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1 Introduction

Computational quantum physics has distinguished so-called sign-free Hamiltonians from generic quantum systems with respect to their classical simulatability via stochastic Monte Carlo methods. Many heuristic computational Monte Carlo strategies exist, but these techniques do not shed full light on the question regarding the computational power of sign-free Hamiltonians as compared to general quantum systems. This motivated the introduction of the notion of so-called

stoquastic Hamiltonians with the aim of formally understanding the computational power, in an algorithmic and complexity sense, of sign-free Hamiltonians [BDT:stoq] .

Although a comprehensive understanding remains elusive to this point, significant progress has been made in this direction. Of particular interest is the power of stoquastic adiabatic computation, with quantum annealing being a prime example. While it has been shown that adiabatic computing using stoquastic frustration-free Hamiltonians can be efficiently classically simulated [BravyiTerhalMA], a recent breakthrough result shows that stoquastic adiabatic computation can be superpolynomially more powerful than classical computation with respect to the number of queries to an oracle [hastings:stoq-adia]. On the other hand, as it is strongly believed that general Hamiltonians are computationally more powerful than stoquastic Hamiltonians, there has been an increased interest in engineering non-stoquastic Hamiltonians for adiabatic computing [Ozfidan2019]. Interestingly, recent evidence [CAHY:design] suggests that the run-time for quantum adiabatic optimization algorithms might not benefit from using non-stoquastic over stoquastic Hamiltonians.

In [BDT:stoq] a Hamiltonian has been called stoquastic with respect to the computational basis when is real and additionally . We will refer to such Hamiltonians as being globally stoquastic. Stoquasticity of ensures that the thermal state is a non-negative matrix in the computational basis for any , and that the ground state has non-negative amplitudes in the computational basis. Global stoquasticity is a sufficient condition for the applicability of the path integral Monte Carlo method (see a brief review in Appendix A), but it is understood that it is not a necessary condition in various ways. For example, beyond stoquasticity, one can also examine conditions under which the partition function is efficiently expressible as a sum over non-negative weights [Suzuki, GAH:QMC, GH:sign].

In this light Refs. [Marvian2018, KT:stoq] and [Klassen2019] examined the question whether a Hamiltonian can be computationally-stoquastic i.e. whether the sign problem can be cured by a computationally-efficient basis changing transformation. Examples of such transformations are single-qubit (product) unitary gates or single-qubit Clifford transformations. In a similar vein, one can ask whether the sign problem can approximately be cured [Hangleiter2019] or, yet differently, whether low-depth circuits can transform away the sign problem in the ground state [torlai]. In these previous works it was found that deciding whether a local Hamiltonian is stoquastic or not, is computationally-hard. In particular, in Ref. [Klassen2019] it was shown that deciding if a two-local Hamiltonian is stoquastic under single-qubit unitary transformations, is NP-hard, while Ref. [Klassen2019] also gave an efficient algorithm for deciding stoquasticity under single-qubit unitary transformations for two-local Hamiltonians without any 1-local terms.

Naturally, these hardness results do not preclude the existence of heuristic or approximate strategies which find basis changes which can reduce the severity of the sign problem as explored in [Hangleiter2019], but this has so far not been explored systematically.

In this work we first clarify the distinction between the notion of global stoquasticity versus termwise stoquasticity for a local Hamiltonian (Section 2). In addition, we show that while it is computationally tractable to decide whether a -qubit Hamiltonian is termwise stoquastic [Marvian2018], it is coNP-complete to decide whether it is globally stoquastic (Theorem 2 in Section 3).

Next, we combine these insights with the previous NP-hardness result in [Klassen2019] to show that deciding global stoquasticity under single-qubit unitary transformations is -hard (Section 4). These results fit into the existing picture as shown in Table 1.

Last but not least, we give a family of disordered 1D XYZ models for which sign-curing by global Clifford transformations is provably more powerful than sign-curing by local unitary transformation (Theorem 4) in Section 5. Although this is a minor result, it shows that there is still much to explore about the class of possible sign-curing transformations.

We expect that our results will contribute to delineating the boundary betweem stoquastic and fully-quantum Hamiltonians, and the origin and mitigation of the sign problem.

Stoquastic -Termwise Global
Fixed Basis [Marvian2018] coNP-complete [*]
1-Local Unitary NP-hard [Klassen2019] -hard [*]
Table 1: Hardness results for deciding membership of a Hamiltonian to a particular stoquasticity class. [*] indicates proof given in this work.

2 Global versus Termwise Stoquasticity

We will be concerned with -qubit -local Hamiltonians : such Hamiltonians can be written as a sum of Hermitian terms each of which acts on at most qubits nontrivially and is usually 111For sparse Hamiltonians without locality structure, for example the Laplacian on a graph [CAHY:design], the notion of termwise stoquasticity is not meaningful.. We assume familiarity with the complexity classes NP, P and coNP. We denote a general polynomial in as when we don’t care about its degree or prefactors specifically.

A seemingly innocuous variation on the definition of global stoquasticity is to ask whether the local Hamiltonian is decomposable as a sum of local terms, each of which itself is stoquastic. More precisely, one can define the class of local Hamiltonians which are termwise stoquastic:

Definition 1 (Termwise Stoquastic ).

A -qubit -local Hamiltonian is -termwise stoquastic with respect to a basis , if it admits a decomposition into -local terms such that each -local Hermitian term is real and obeys .

Remark: For the set of real -local Hamiltonians, we can choose a real Pauli basis with each basis element acting on at most qubits nontrivially. We can then view a -local Hamiltonian with as a point in with . On the other hand, the set of -local stoquastic matrices generate a convex cone . The question whether a -local Hamiltonian is -termwise stoquastic is thus the question whether lies in . Caratheodory’s theorem then tells us that if lies in the cone, it is supported on points .

Note that the definition for global stoquasticity is equivalent to being -termwise stoquastic and being -termwise stoquastic implies being -termwise stoquastic. One can trivially see that any termwise stoquastic Hamiltonian is also globally stoquastic. As was stated in [BDT:stoq] and later shown explicitly in [Klassen2019] the converse also holds for two-local Hamiltonians, i.e. every two-local globally stoquastic Hamiltonian is -termwise stoquastic. Such equivalence can be generalized:

Lemma 1.

If each qubit of a -local Hamiltonian interacts with at most other qubits and , then global stoquasticity implies -termwise stoquasticity, where .

Proof.

Assume global stoquasticity of the Hamiltonian. Write where are all terms in which flip the qubits in the subset with . When , collects all the diagonal terms. Global stoquasticity implies that each term

is stoquastic as each such term gives rise to distinct non-zero matrix elements

. Terms in for involve qubits in the subset and at most qubits outside of (where they act Z-like). Hence is at most -local and is -termwise stoquastic with . ∎

The remaining question is thus whether the notion of global stoquasticity and -termwise stoquasticity is the same for arbitrary -local interactions. One can in fact construct the follow 3-local counterexample, due to Bravyi.

Proposition 1.

There exists a class of -local globally-stoquastic Hamiltonians which are not 3-termwise stoquastic.

Proof.

Consider a graph with vertices and define an Ising Hamiltonian on it:

(1)

where . Now consider the Hamiltonian where

is the lowest eigenvalue of

. We show that is globally-stoquastic but cannot be written as a sum of stoquastic 3-local terms when the Ising Hamiltonian is frustrated (and therefore is more than the sum of the minimal values of each term). The only non-zero, off-diagonal matrix elements of are where and . Since is the smallest eigenvalue of , it follows immediately that is globally stoquastic.

Now consider whether can be termwise stoquastic. The decomposition of must be of the form where each is stoquastic and acts on qubits at vertices and . We note that has to be diagonal in the Z-basis, since adding and subtracting bit-flipping terms in does not help in making termwise stoquastic. Each term is stoquastic iff and the most general form of is

(2)

with the restrictions , and .

The condition implies that

(3)

If is frustrated, as not all terms contribute negatively: this is then in contradiction with the requirement that and . ∎

Note that the result does not preclude the possibility that Hamiltonians constructed in this proposition are -termwise stoquastic. In fact, it is not hard to prove the following generalization

Proposition 2.

Let be a traceless -local Hamiltonian on qubits, with terms diagonal in the computational basis and let be its lowest eigenvalue. Let . is -termwise stoquastic if and only if there exists a -local decomposition of which is frustration-free. The frustration-free -local decomposition is a decomposition where each Hermitian classical term acts on at most qubits non-trivially and with the lowest eigenvalue of the term .

Proof.

Assume we have a frustration-free -local decomposition . A term is then stoquastic and -local and hence the sum of these terms is a -termwise stoquastic decomposition of . Assume is -termwise stoquastic, this then induces a decomposition of of the form where is traceless, diagonal and -local, and stoquasticity of each term implies that . In addition, we must have and implying that which is only possible when the form a frustration-free decomposition of . ∎

Determining a frustration-free -local decomposition of solves the problem of finding the ground state energy which is generically NP-complete, see e.g. Theorem 1. This suggests that determining whether or not a Hamiltonian is globally stoquastic, can in fact be a difficult problem. We will indeed prove in Theorem 2 that it is coNP-complete by showing that a yes instance of an NP-complete classical lowest eigenvalue problem can be 1-1 mapped onto a no instance of the problem of determining global stoquasticity.

Given that the notion of global versus termwise stoquasticity can be distinct, we can ask which is the right notion used in existing complexity and algorithmic results in the literature. In Appendix A we review the path integral quantum Monte Carlo method and observe that a notion of global stoquasticity suffices for there not to be a sign problem. On the other hand, the result that determining the lowest eigenvalue of a -local stoquastic Hamiltonian is StoqMA-complete [BBT:stoq] uses the fact that these Hamiltonians are -termwise stoquastic. It is not clear whether the lowest eigenvalue problem for only globally-stoquastic Hamiltonians would in fact be contained in StoqMA. The proof of containment of the lowest-eigenvalue problem in AM in [BDT:stoq] again only uses the global stoquasticity of the Hamiltonian as it only uses the sparsity of . The problem of exactly deciding whether a -termwise stoquastic Hamiltonian is frustration-free is contained in MA in [BravyiTerhalMA], and, assuming a constant spectral gap, in NP in [AG:NP].

3 Determining whether a local Hamiltonian is globally or termwise stoquastic

Given that the notion of global versus termwise stoquasticity can be distinct, we define GlobStoq and TermStoqas the problems of deciding whether a local Hamiltonian is globally stoquastic or whether it is -termwise stoquastic and ask how hard or easy it is to decide these problems.

First we ask about the efficiency with which one can determine whether a -local Hamiltonian is -termwise stoquastic with . In [Marvian2018] it was argued that , while [KT:stoq] gave a simple explicit strategy for two-local Hamiltonians. We provide the proof for completeness:

Proposition 3.

One can efficiently determine whether a -local Hamiltonian acting on qubits is -termwise stoquastic where , or .

Proof.

As in the proof of Lemma 1, we write with . We can ignore the purely diagonal terms in which flip no qubits, . Termwise stoquasticity of implies termwise stoquasticity of each as these correspond to different matrix elements. Now we can consider the set of -local stoquastic matrices which only flip the qubits in : there are such matrices, hence there will be extremal points, call them . One thus needs to solve the problem whether there are such that which is a linear feasibility problem. If there exists a (feasible) solution to this program, we move to the next term etc. until we have found solutions for all terms or for at least one value of there exists no feasible solution, in which case we output ‘not -termwise stoquastic’. ∎

Proposition 4.

When , for any constant .

Proof.

Unlike the case where is a constant, for one can only prove that the problem is contained in NP: for a yes instance the prover gives the stoquastic matrices in the decomposition: as these act nontrivially on at most qubits, they are given as matrices and there are at most of them. The verifier can check that the poly(n) entries of the -local Hamiltonian can be constructed from . ∎

If we bound the degree of each qubit in the interaction hypergraph, we have shown that global stoquasticity and -stoquasticity are equivalent in Lemma 1. In fact in such cases, instead of finding a convex decomposition as in Proposition 3, it is simpler to test for global stoquasticity directly:

Proposition 5.

If each qubit of a -local Hamiltonian interacts with at most other qubits and , then the problem of deciding where the Hamiltonian is globally stoquastic can be solved efficiently, that is, GlobalStoq P.

Proof.

Consider all matrix entries where and differ only in a subset . We need to consider such subsets. For each subset , it is easy to check all matrix entries , where and differ only on , since at most terms in the Hamiltonian can contribute to these entries. ∎

It is clear that geometrically local Hamiltonians, where the qubits are distributed apart in and only interact with qubits away, have the property that each qubit interacts with at most other qubits and hence for these classes, global stoquasticity is equivalent to O(1)-termwise stoquasticity by Lemma 1 and we can efficiently test for both local and global stoquasticity. This is good news as these are commonly-studied models in physics.

3.1 Hardness of

In this section we will argue that is coNP-complete. To show this, we will consider the well-known NP-complete problem of determining the lowest energy of an Ising Hamiltonian with local fields, and show that the complement of this problem reduces to .

Theorem 1 (Planar Spin Glass (Ps) [Barahona_1982]).

Given a planar graph and an integer . Deciding whether there exists a configuration of such that

is an NP-complete problem.

Theorem 2.

The problem of deciding whether a -local Hamiltonian is globally stoquastic (GlobalStoq) is coNP-complete.

Proof.

It is straightforward to see that GlobStoq coNP, as there exists an efficiently verifiable witness for its no instances in the form of bitstrings for which . Since is a local Hamiltonian, we can efficiently evaluate such matrix elements and thus verify the witness.

We proceed by showing that GlobStoq is coNP-hard and hence coNP-complete by reducing PS (Theorem 1) to it. Given an instance of PS with graph such that , define the qubit Hamiltonian

(4)

where . As we can factor from every term in , only when bitstrings differ in the bit. Let us denote the first bits of as and the as respectively. We have

For a no instance of there exist and such that

where . Since the left hand side is an integer and ,

Therefore, a no instance of implies a yes instance of PS. Similarly for a yes instance of we have that . Again using the fact that , we have

and thus, a yes instance of corresponds to a no instance of PS which completes the reduction. ∎

Remark: Although deciding GlobStoq is coNP-complete for as in Eq. (3.1) and thus is coNP-complete in general, such Hamiltonians can be sign-cured by means of a single-qubit Hadamard transformation on the qubit.

We observe that under the assumption that , deciding - cannot be equivalent to GlobStoq.

4 Global stoquasticity via local basis changes

In the previous section we considered the difficulty of determining if a Hamiltonian is globally stoquastic in a fixed basis. Previous work [Marvian2018, KT:stoq, Klassen2019] has shown that it is NP-hard to decide if there is a local change of basis such that a Hamiltonian is termwise stoquastic. Now we combine both questions, and consider the problem of determining if there exists a local change of basis such that a Hamiltonian is globally stoquastic. We start with a definition:

Definition 2.

Let be a family of unitaries. Then -GlobalStoq is the following problem:

Input: Local Hamiltonian

Problem: Decide if there is a unitary such that is globally stoquastic.

Observe that GlobalStoq is the special case of -GlobalStoq where

only contains the identity matrix. Our result is the following.

Theorem 3.

Let be the set of 1-local product unitaries. Then, restricted to a certain class of 3-local Hamiltonians, -GlobalStoq is -complete.

To prove this theorem, we will need three different ingredients:

  • a convenient -complete problem to reduce from

  • a "gadget" Hamiltonian terms to restrict the possible basis change unitaries

  • a Hamiltonian which encodes our -complete problem.

4.1 Complete problem for

The complexity class

sits in the second level of the polynomial hierarchy, and is the class of problems that can be solved by a polynomial time non-deterministic Turing machine (an

NP machine), which can make queries to an oracle to NP.

The complement of is . A canonical complete problem for is 3-SAT [Schaefer2002]:

Definition 3 (3-Sat).

Input: Boolean formula in conjunctive normal form with 3 literals per clause.

Problem: Decide if for all , there exists a such that is true.

The complement of 3-SAT is complete for (since ). We could reduce directly from this problem to -GlobalStoq, using the same construction as in Section 4.3, to obtain -hardness, but the resulting Hamiltonian would be 4-local. In order to achieve a hardness result for a 3-local Hamiltonian, we first find a more convenient complete problem for .

Definition 4 (Minmax--Sat).

Input: Boolean formula in conjunctive normal form with at most 2 literals per clause; and an integer .

Problem: Decide if for all , there exists a , such that satisfies at least clauses.

The problem MINMAX--SAT is related to 3-SAT, in the same way that MAX-2-SAT is related to 3-SAT. In fact we can prove MINMAX--SAT is -complete, by reducing from -SAT, with exactly the same proof method as the reduction from 3-SAT to MAX-2-SAT.

Lemma 2.

MINMAX--SAT is -complete.

Proof.

Let be a formula specifying an instance of 3-SAT of clauses on on bits. We construct a formula of clauses on bits, with at most 2 literals per clause, such that if and satisfies , then there exists such that satisfies clauses of ; and if does not satisfy then satisfies strictly less than clauses, for any .

For each clause containing three literals , replace it with 10 clauses

Then if the original clause is satisfied, there is a choice of such that 7 of these clauses are satisfied, but there is no choice of such that more than 7 are satisfied. And if the clause is not satisfied (all 3 of are false), then any choice of will satisfy at most six of these clauses.

Then 3-SAT for reduces to MINMAX--SAT for and . Since if for all , there exists such that is true, then for all there exists such that clauses of are satisfied. And if there exists such that for all is unsatisfied, then there exists such that for all , strictly less than clauses of are satisfied. ∎

An immediate consequence of Lemma 2 is that the complement of MINMAX--SAT, which we call MINMAX--SAT, is -complete.

Definition 5 (Minmax--Sat).

Input: Boolean formula in conjunctive normal form with at most 2 literals per clause; and an integer .

Problem: Decide if there exists , such that for all , violates at least clauses.

4.2 Gadgets

In [Klassen2019] two-local Hamiltonian gadget terms were constructed such that a local change of basis could make the total Hamiltonian stoquastic if and only if the original Hamiltonian could be made stoquastic by a restricted type of basis of change.

In the following, we use to denote the single-qubit Hadamard matrix and for we define

Lemma 3 (Lemma 5.1 of [Klassen2019]).

Let H be a two-local Hamiltonian on qubits. For each qubit , add three ancilla qubits and define the two-local gadget Hamiltonian

Then the following are equivalent:

  1. there exists a product of single qubit unitaries such that is a globally stoquastic Hamiltonian.

  2. there exists such that is globally stoquastic, where .

Now we present a small extension to these gadgets, in the form of an additional gadget Hamiltonian , to further restrict the type of possible basis changes on certain qubits.

Here denotes the concatenation of two strings and , so .

Lemma 4 (Extension of Lemma 3).

Let be a Hamiltonian on qubits. First add an ancilla qubit for each and define the two-local gadget Hamiltonian

(5)

Then add three more ancilla qubits for each of the qubits. Defining the total Hamiltonian on qubits as

where is as defined in Lemma 3, the following are equivalent:

  1. there exists a product of single qubit unitaries such that is globally stoquastic.

  2. there exists such that is globally stoquastic, where .

Proof.

We first prove that 2 1. Let . If there exists such that is globally stoquastic, it is easy to check that is also globally stoquastic. Therefore by Lemma 3, there exists a product unitary such that is globally stoquastic.

For the converse direction, Lemma 3 implies that if 1. holds then there exists and such that is globally stoquastic, where

Suppose for a contradiction that or is 1 for some . Then there is a positive off-diagonal matrix entry where differ at location . This cannot be cancelled out by (since this acts trivially on qubit ), implying that is not globally stoquastic. But this is a contradiction and we can conclude that .

Finally if is globally stoquastic then is globally stoquastic.

4.3 Hardness construction

For a 2-SAT formula in conjunctive normal form, with and ,

we define a corresponding Hamiltonian on qubits:

where

Recalling that , it can be seen that for all , has the following properties:

  1. All off-diagonal elements of are non-negative.

  2. The diagonal matrix entries satisfy:

    where and .

  3. is the number of clauses of violated by .

Properties 1. and 2. can be easily checked. To see point 3., observe that

and so

Therefore, is equal to the number of clauses in that are not satisfied by , as claimed.

We are now ready to prove Theorem 3.

Proof. (of Theorem 3).

We consider Hamiltonians of the form

where is as defined above, and , are the gadget Hamiltonians of Lemma 4. These are chosen so that there exists a product unitary such that is globally stoquastic if and only if there exists such that

is globally stoquastic.

This happens if and only if all the off-diagonal terms of are non-negative and all the diagonal matrix entries are greater than or equal to .

As observed above, all the off-diagonal elements of are positive and the smallest diagonal matrix entry is of the form for some . Furthermore is equal to the number of unsatisfied clauses of . Therefore -GlobalStoq is a yes instance if and only if there exists such that for all at least clauses of are unsatisfied. That is, is a yes instance of -GlobalStoq if and only if is a yes instance of MINMAX--SAT. MINMAX--SATis -complete, and so -GlobalStoq is also -complete. ∎

5 Sign-Curing by Clifford Transformations

In the previous section, as well as in previous work, we considered the problem of sign-curing a Hamiltonian by single-qubit unitaries. Here we go beyond these single-qubit transformations and ask what -local Hamiltonians can be mapped onto stoquastic Hamiltonians by means of general Clifford transformations. The Clifford group is defined as

Definition 6 (Clifford group).

where is the Pauli group on n qubits, that is,

A simple example is a Hamiltonian which is a sum of commuting Pauli terms with Paulis such as a stabilizer Hamiltonians. is then obviously computationally-stoquastic as it can be transformed by the adjoint action of an element of the Clifford group (i.e. a Clifford transformation ) with where

is a tensor product of Z gates, so the transformed Hamiltonian

is classical. The existence of a Clifford transformation which performs this mapping is attributed to the fact that, modulo and signs, a Clifford exists when the product relations among the Paulis are those of .

More precisely and more generally, for a Hamiltonian , one specifies a set of independent Paulis among the such that other Pauli terms in are obtained by taking products of these independent elements (modulo prefactors). If we have a Hermiticity-preserving injective map such that each is mapped to a , i.e. so that for , then there is always a realization for which is a Clifford transformation. Hence specifying the action of such map acting on the independent set means that a Clifford transformation will exist (and can be found explicitly as a symplectic transformation [KS:sym]).

Assume that with Clifford and let

be globally stoquastic. We observe that if we wish to estimate thermal or ground state properties of

via a path integral quantum Monte Carlo method, see Appendix A, then we can simply use in this simulation: the Pauli terms in Hamiltonian may have high-weight, but this does not make the MC method inefficient (although it may require different Metropolis update rules), since the number of such Pauli terms is the same as in and thus matrix elements of can still be determined efficiently.

We now will provide an example of a class of 1D disordered Heisenberg Hamiltonians for which one can prove that sign-curing using single-qubit unitary transformations is not possible, while Clifford transformations do sign-cure the Hamiltonians in this class.

Consider the disordered Heisenberg XYZ model on an open boundary 1D chain of qubits with arbitrary coupling coefficients:

(6)

In what follows, we will refer to pairs as edges of . It is known that a 1D translationally-invariant Heisenberg XYZ model has no sign problem [Suzuki] and we review the argument in Appendix A.2. It was shown in [KT:stoq] that if we seek to sign-cure an arbitrary XYZ Heisenberg model by single-qubit (local) basis changes, then single-qubit Clifford transformations suffice. However, not all 1D XYZ models can be sign-cured by single-qubit Clifford transformations and we give some explicit examples where such single-qubit Cliffords fall short in Appendix B. Given these examples, the following theorem then shows that global Clifford transformation are more powerful than single-qubit Clifford transformations.

Theorem 4.

The (open boundary) 1D disordered XYZ Heisenberg model in Eq. (6) is -termwise stoquastic by a Clifford transformation if, either for all even edges we have

(7)

or, for all odd edges we have

(8)
Proof.

We can see the commutation structure of the XX, YY and ZZ terms in the Heisenberg model in Fig. 1. The XX and YY terms can be chosen as the independent set . If Eq. (7) holds, we map the commuting XX, YY, ZZ terms between qubits and (odd edges of ) to terms which are tensor products of Z gates, which all mutually commute as required. Since , we just need to make sure that we choose the 3 terms with the right sign, so as to preserve all product relations. However, terms which are tensor products of Z gates are diagonal, their signs do not matter for stoquasticity so any consistent choice is valid. If instead Eq. (8) holds, we would have done the same for the terms between qubits and (even edges of ).

Now consider wlog that Eq. (7) holds and we need to map the commuting XX, YY, ZZ terms between qubits on even edges of to Pauli terms respecting all product relations. One can easily check that we can map these terms to tensor products of X gates, acting on at most 4 qubits, as shown in Fig. 1. In contrast to the purely diagonal terms mentioned above, we are now constraint in choosing the signs so that these off-diagonal terms have non-positive entries. As a consequence of Eq. (7) one can choose , , individually for each edge between qubits and , so all X-like terms are stoquastic. Note that in case a term which is mapped to a X-like term occurs at the boundary we replace its action on non-existing qubits by . ∎

Figure 1: Top: Graph summarising the (anti-) commutation relations between the Pauli terms in the XYZ Heisenberg model. An edge between two vertices representing Paulis indicates anticommutation between them while the absence of an edge indicates that they commute. Bottom: Representation of these terms when transformed under a Clifford transformation which preserves the commutation structure. , are chosen for each edge of independently.

Remark: For a disordered Heisenberg chain with periodic boundary this construction would not work as the XX and YY terms do not form an independent set.

We thank Itay Hen for interesting discussions. BMT and JK acknowledge funding from ERC grant EQEC No. 682726. MI acknowledges support by the DFG (CRC183, EI 519/14-1) and EU FET Flagship project PASQuanS. MM is supported by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA) and the Defense Advanced Research Projects Agency (DARPA), via the U.S. Army Research Office contract W911NF-17-C-0050. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the ODNI, IARPA, DARPA, or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon.

References

Appendix A Stoquastic Hamiltonians and The Sign Problem in the path integral Quantum Monte Carlo Method

In this Appendix we prove that sparse Hamiltonians which are globally stoquastic by an efficiently-computable curing transformation avoid the sign problem in the path integral Monte Carlo method. As target for the path integral Monte Carlo method we focus on estimating where