When a local Hamiltonian must be frustration-free Or Sattatha, Siddhardh C. Morampudib,1, Chris R. Laumannc, and Roderich Moessnerb a Computer Science Division, University of California, Berkeley, CA 94704; bCondensed Matter Division, Max-Planck-Institut fur Physik Komplexer Systeme, 01187 Dresden, Germany; and cDepartment of Physics, University of Washington, Seattle, WA 98195
Edited by Peter W. Shor, Massachusetts Institute of Technology, Cambridge, MA, and approved April 1, 2016 (received for review October 7, 2015)
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quantum satisfiability local Hamiltonian critical exponents universality
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A
| hardcore lattice gas |
n overwhelming majority of systems of physical interest can be described via local Hamiltonians: H=
M X
Πi .
[1]
i=1
Here, the “k-local” operator Πi acts on a k-tuple of the microscopic degrees of freedom, best thought of as qudits for the computer scientists among our readers or spins for the physicists. The M operators define an interaction (hyper)graph G, displayed in Fig. 1. A surprisingly diverse and important class of such model Hamiltonians is defined by the additional property of being “frustrationfree”: The ground state jψi of H is a simultaneous ground state of each and every Πi. This class comprises both commuting Hamiltonians—for which ½Πi , Πj = 0 ∀i, j—such as the toric code, general quantum error correcting codes, and Levin–Wen models (1–3), and noncommuting ones, such as the Affleck-Kennedy-Lieb-Tasaki (AKLT) and Rokhsar–Kivelson models (4–6). Their particular usefulness is also related to the fact that many of these examples can be viewed as “local parent Hamiltonians” for generalized matrix product states (7). In general, frustration-free conditions provide analytic control of ground state properties in otherwise largely inaccessible quantum problems. Determining whether a given Hamiltonian H is frustration-free is well known in quantum complexity theory as the quantum satisfiability (QSAT) problem. QSAT is widely believed to be intractable, in the sense that no general purpose classical or quantum algorithm can efficiently determine whether a given Hamiltonian is frustration-free (“satisfiable”). The technical statement is that QSAT is www.pnas.org/cgi/doi/10.1073/pnas.1519833113
complete for the Quantum Merlin Arthur complexity class (QMA1complete) (8), even when restricted to qubits and k = 3 (9) or when the interaction is between neighboring qudits on a line (10). Fortunately, such hardness results apply only in the worst case. For instance, in contrast to the hardness result, it is immediately obvious that a fully disconnected interaction graph G can be analyzed efficiently by considering each term Πi individually. The central result reported here is a sufficient combinatorial criterion for a local Hamiltonian H to be frustration-free. In fact, we provide a lower bound for the dimension of the satisfying subspace. This amounts to a generalization of Shearer’s theorem (11) from classical probability theory to the quantum world. We first formulate the result in Theorem 1, followed by an intuitive explanation of its content, with a technical proof relegated to SI Appendix. We then turn to applying Theorem 1, for which we enlist the results available on statistical mechanics of hardcore objects with negative fugacity on the interaction graph to deduce statements regarding QSAT, producing new bounds on the satisfiability threshold for a large class of 1D, 2D, and 3D interaction graphs. With the help of the cavity method, we are able to conjecture improved lower bounds for the satisfiability of QSAT on regular and Erd} os–Rényi random graphs, canonical models for quantum constraint optimization problems. These statement hold just as well for classical satisfiability—but for some of these classical models better bounds are known (12, 13). We close with an outlook, including a discussion of the role of a universality that emerges in our analysis, as well as conjectures on when our results are exact or tight. Theorem 1. Given a local projector Hamiltonian H as in Eq. 1 with interaction graph G and relative projector rank p = RðIm Πi Þ for all i, then
Rðker HÞ ≥ ZðG, − pÞ > 0 if ZðG, − p′Þ > 0 for all 0 ≤ p′ ≤ p. Significance Quantum computers promise computational power qualitatively superior to that achievable classically. This power will not be unlimited: Beyond much-touted applications, such as breaking encryption schemes, entire classes of problems are known to be intractable even for quantum computers. This work addresses a question of great practical relevance: In between these two extremes of certain (in)tractability, how can one efficiently diagnose the nature and properties of a given problem instance? To achieve this, we adopt a strategy of transferring insights from statistical physics and classical computing into the quantum realm. This provides a new perspective on the complexity of quantum problems and allows us to analyze a canonical class of quantum optimization problems in unprecedented explicit detail. Author contributions: O.S., S.C.M., C.R.L., and R.M. designed research, performed research, contributed new reagents/analytic tools, analyzed data, and wrote the paper. The authors declare no conflict of interest. This article is a PNAS Direct Submission. 1
To whom correspondence should be addressed. Email: [email protected].
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. 1073/pnas.1519833113/-/DCSupplemental.
PNAS | June 7, 2016 | vol. 113 | no. 23 | 6433–6437
PHYSICS
A broad range of quantum optimization problems can be phrased as the question of whether a specific system has a ground state at zero energy, i.e., whether its Hamiltonian is frustration-free. Frustrationfree Hamiltonians, in turn, play a central role for constructing and understanding new phases of matter in quantum many-body physics. Unfortunately, determining whether this is the case is known to be a complexity-theoretically intractable problem. This makes it highly desirable to search for efficient heuristics and algorithms to, at least, partially answer this question. Here we prove a general criterion—a sufficient condition—under which a local Hamiltonian is guaranteed to be frustration-free by lifting Shearer’s theorem from classical probability theory to the quantum world. Remarkably, evaluating this condition proceeds via a fully classical analysis of a hardcore lattice gas at negative fugacity on the Hamiltonian’s interaction graph, which, as a statistical mechanics problem, is of interest in its own right. We concretely apply this criterion to local Hamiltonians on various regular lattices, while bringing to bear the tools of spin glass physics that permit us to obtain new bounds on the satisfiable to unsatisfiable transition in random quantum satisfiability. We are then led to natural conjectures for when such bounds will be tight, as well as to a novel notion of universality for these computer science problems. Besides providing concrete algorithms leading to detailed and quantitative insights, this work underscores the power of marrying classical statistical mechanics with quantum computation and complexity theory.
The area covered by the bubbles is largest if they do not overlap, corresponding to a large number of violating configurations; we thus expect a lower bound on RðkerHÞ when they are fully disjoint. To calculate this lower bound, we expand the product in Eq. 3 and introduce a collection of occupation variables ni = 0,1 that indicate the presence of Πi in each term of the expansion, P X ð−1Þ ni Πn11 ⋯ΠnMM . Rðker HÞ = [4] fni g
Fig. 1. (Left) The interaction graph of a k = 3-local Hamiltonian. The degrees of freedom, qudits, are denoted by circles; the squares, which indicate the operators Πi appearing in H, are joined to the qudits on which they act. In the hardcore lattice gas mapping, each square may be occupied by a particle covering also the adjacent circles (shaded blobs), on which it must not overlap with another particle. (Right) A k = 6 interaction graph forming a triangular lattice of the operators Πi . The corresponding hardcore lattice gas corresponds to the hard hexagon model, which is exactly solvable.
Without loss of generality, we have taken the terms Πi to be projectors so that the satisfiability condition reduces to the presence of a zero-energy ground state. (This may always be done by shifting and deforming the eigenvalues of the local Hamiltonian terms without influencing the frustration-free ground state space.) Here, the relative dimension of a subspace X of the full Hilbert space H is simply RðXÞ = dimX=dimH, and, ZðG, λÞ is the partition function for a hardcore lattice gas of fat particles living on the (hyper) edges of the interaction graph at fugacity λ (Fig. 1). More precisely, X P Y ZðG, λÞ = λ ni 1 − ni nj , [2] fni g
i↔j
where i ↔ j runs over all projectors that share qudits and the sum runs over occupations ni = 0,1 of the lattice gas. Intuition Let us first consider the case of classical projectors, Πi, diagonal in the standard computational basis, where we can provide a simple pictorial Q representation of the content of the Shearer bound. The operator i ð1 − Πi Þ projects onto the zero-energy space of the Hamiltonian. If it is nonzero, the Hamiltonian is frustration-free. Moreover, its expectation value in the maximally mixed (infinite temperature) state gives the relative dimension of the zero-energy space, Rðker HÞ =
M Y
ð1 − Πi Þ.
If two dependent projectors, Πi and Πj, are thus occupied, then that term is zero Pwhen their bubbles are made disjoint. Otherwise, it is given by p ni . Thus, P Y X Rðker HÞ ≥ ð−pÞ ni 1 − ni nj [5] i↔j fni g = ZðG, − pÞ, where i ↔ j runs over projectors that share qudits. We have derived inequality [5] under the assumption that it is possible to make the dependent bubbles disjoint. If p is small enough, this is always the case; in the Venn diagram, the bubbles can be made disjoint without covering more area than the total space contains. Shearer (11, 14) showed that the above intuitive lower bound is correct for classical projectors as long as p ≤ pc, where pc is the first zero of ZðG, − pÞ. This is the classical analog of our quantum generalization, Theorem 1. The classical sketch above makes little sense for noncommuting quantum projectors. In the language of probability, this reflects the failure of the inclusion–exclusion principle (Eq. 4) for the relative dimension of vector spaces. Nonetheless, the result holds; the proof— our fundamental technical advance—is provided in SI Appendix, Proof of Theorem 1. Statistical Mechanical Transcription A remarkable aspect of Theorem 1 is that it maps the quantum satisfiability problem onto the classical statistical mechanical problem of determining the position of the first negative fugacity zero λc ðGÞ of the partition function ZðG, λÞ. In general, evaluating Z is computationally hard, but for many infinite graphs it may nonetheless be accomplished in a satisfactory manner using statistical mechanical tools. [Technically, computing Z is #P-hard almost everywhere (15).] In the thermodynamic limit (number of qudits N → ∞), the first zero λc ðGÞ can be identified with a well-known critical point λc ðG∞ Þ of
[3]
i=1
If all of the projectors act on different qudits, then this expectation value simply factors Rðker HÞ =
M Y
ð1 − Πi Þ = ð1 − pÞM ,
i=1
where p = Πi is the relative dimension of Πi. This is nonzero as long as p < pc = 1. Now suppose that some of the projectors share qudits; we call such pairs of projectors dependent, as they introduce dependence into the expectation value in Eq. 3. When can we, nevertheless, guarantee that RðkerHÞ remains positive? The essence of the Shearer bound is captured by the Venn diagram in Fig. 2, where each bubble represents the fraction of configuration space “knocked out” by a projector, with overlapping bubbles representing configurations multiply penalized by the corresponding projectors. 6434 | www.pnas.org/cgi/doi/10.1073/pnas.1519833113
Fig. 2. Shearer bound for dependent projectors. Projectors are independent if they do not share any qudit (Top Left), and there will always exist configurations violating any combination of such projectors simultaneously. In the Venn diagram (Top Right), this is represented by the mutual intersections of the shaded circles, each of which denotes the fraction of configuration space penalized by the projector it is labeled with. By contrast, for dependent projectors (Bottom Left), such an overlap is not guaranteed. A lower bound for the relative dimension of the satisfying space can be obtained by assuming the projectors Πi and Πl do not share any violating configurations with the projector Πj with which they share a qudit (Bottom Right).
Sattath et al.
Examples As a first application of Theorem 1, we calculate the critical threshold λc for several infinite graphs. We do this using the cavity method, a well-known technique for studying statistical (31) and quantum (32– 34) models on infinite graphs that are locally tree-like. On trees and chains, the results are rigorously exact whereas on infinite random graphs they are less rigorous, but often just as exact. The heart of the cavity method is to introduce an auxiliary system of messages that propagate in both directions between the hyperedges and the nodes of the interaction graph G. For the model given by Eq. 2, the derivation is straightforward and quite intuitive at positive fugacity where the cavity messages correspond to marginal distributions for the gas particles (SI Appendix, The Cavity Derivations). The resulting equations continue smoothly to the negative fugacity regime. Adopting the convention that indexes i, j, . . . label hyperedges and a, b, . . . label sites, the belief propagation (BP) equations (for a recent pedagogical treatment, see ref. 35, chap. 14) are qa→i =
li→a =
λ+1+
λ P
lj→a je∂ani 1 − lj→a
λ
λ + Πbe∂ina
λ qb→i
−λ
,
where li→a and qa→i give the probability that hyperedge i is occupied in the absence of the connection to a and in the absence of everything except a, respectively. We note that a special case of these equations has been used previously to study the number of dimer coverings of random graphs (k = 2, positive fugacity) (36). Given a solution of the BP equations, the cavity estimate of the free energy F = log Z of the system is given by a sum of the free energies associated to the addition of individual elements of the interaction graph; i.e., Sattath et al.
Table 1. Summary of the critical threshold for various infinite interaction graphs Lattice 1D chain Triangular Square Square Checkerboard Hexagonal Simple cubic t-regular tree
pc
Qudits
Projectors
k
(38) (18, 20, 23) 0.1193 (17, 23) 0.0889 (23) 0.0688 (23) 0.1547 (23) 0.0744 (18) 1 4ðt − 1Þ (19, 39)
Vertices Edges Edges Vertices Black Edges Edges Vertices
Edges Vertices Vertices Edges Red Vertices Vertices Edges
2 6 4 2 4 3 6 2
1 4 pffiffi 5 5 − 11 2
F=
X a
Fa +
X i
Fi −
X
Fai ,
[6]
ðaiÞ
where Fa, Fi, and Fai correspond to the change in free energy due to the addition of sites a, hyperedges i, and the links ai between them, respectively. The full expressions may be found in SI Appendix, The Cavity Derivations. The Chain On the infinite chain, the BP equations can be solved by uniform messages qa→i = q, li→a = l. After some algebra and taking the root of the BP equations pffiffiffiffiffiffiffiffiffiffiffiffi corresponding to qðλ = 0Þ = 0, one finds l = q = 1 + ð1 − 1 + 4λ Þ=2λ. This expression suggests that λc = −1=4 as q becomes complex for λ < λc. Indeed, it is easy to check that the free energy density f = F=N (Eq. 6) has the expansion near λc: f = −log 2 + 2ðλ − λc Þ1=2 − 2ðλ − λc Þ + ⋯. The free energy goes complex for λ < λc, which reflects the accumulation of partition function zeros and indicates the failure of the lower bound. The free energy density is precisely −log 2 at the critical point. For qudits of local dimension q, this provides a lower bound on the dimension of the ground space dimðker HÞ = qN Rðker HÞ ≥ ðq=2ÞN . In fact, careful application of Theorem 1 for finite chains agrees with the exact result dimðker HÞ = ðq=2ÞN ðN + 1Þ for open chains derived using matrix-product techniques (37, 38). It is interesting to note that the entropy per site is zero for qubits at criticality, but has a finite value for larger q. Regular Trees The BP equations on infinite regular trees with site degree t and hyperedge degree k can be solved under the ansatz that all messages are the same. For algebraic details, see SI Appendix, The Cavity Derivations. The resulting critical threshold is 1 ðk − 1Þk−1 λc = − . kk t−1
[7]
For k = 2-local trees, this agrees with previous results obtained by matrix product states on trees (39). As far as we are aware, Eq. 7 provides a strictly better lower bound on satisfiability of infinite regular trees than any previous literature. The corresponding expansion of the free energy density is f = f0 ðt, kÞ + Aðt, kÞðλ − λc Þ + Bðt, kÞðλ − λc Þ3=2 + ⋯.
[8]
We thus recover the expected nonanalyticity (ϕ = 3=2) in the free energy density for the infinite-dimensional hardcore singularity (24). PNAS | June 7, 2016 | vol. 113 | no. 23 | 6435
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the hardcore lattice gas referred to as the hardcore singularity. The critical fugacity λc ðG∞ Þ upper bounds the λc of any finite subgraph by monotonicity (11). Thus, Theorem 1 implies that all subgraphs of G∞ are frustration-free as long as the relative rank of the projectors satisfies p < − λc ðG∞ Þ. The hardcore singularity has been studied extensively in the statistical mechanical literature (16–25) and its location is known for many infinite lattices. Table 1 summarizes some of these critical values and indicates their translation into QSAT interaction graphs. In addition, there exist rigorous bounds on λc that may be efficiently calculated on any sufficiently simple lattice (16). The singularity exhibits universal features just like standard phase transitions. In particular, the free energy density f ðλÞ = ð1=NÞlog ZðλÞ near the critical point λc has universal exponents (21, 22) due to its connection with the so-called Yang-Lee edge singularity (24–28). Technically, this means that the leading nonanalytic part of the free energy f ∼ ðλ − λc ÞϕD near λc, where ϕD is a noninteger critical exponent that depends only on the spatial dimension D of the lattice (21, 22, 24). It is known analytically that ϕ1 = 1=2, ϕ2 = 5=6 (20, 29, 30), and ϕD = 3=2 for D above the upper critical dimension DC = 6 (24). The critical exponents control the lower bounds on the dimension of the ground space of Hamiltonians close to the critical threshold. ϕD Indeed, by Theorem 1, Rðker HÞ ≥ enf ðλÞ ∝ eanðλ − λc Þ +Oðλ−λc Þ for λ > λc, so the critical exponents show up directly—and perhaps somewhat unexpectedly—as anomalous growth laws in D = 1,2 (SI Appendix, Improved Bounds Using the Critical Exponents). In the following, observation of the expected exponent in the random k-QSAT ensemble helps confirm the validity of the nonrigorous cavity analysis.
Random k-QSAT We now apply Theorem 1 to random k-QSAT. Random satisfiability has been a workhorse for the study of typical case complexity and heuristic algorithms. By tuning the density α = M=N of Erd} os–Rényitype random interaction graphs, a cornucopia of phases and phase transitions in the structure of the satisfying space has been discovered. In the quantum case, three phases for random k-QSAT on qubits and rank-1 projectors are believed to be (i) at low density, a product state satisfiable phase (PRODSAT); (ii) at intermediate densities for k sufficiently large, a satisfiable phase in which all ground states are entangled (ENTSAT); and (iii) at high density, an unsatisfiable (UNSAT) phase. The arguably most interesting of these phases, ENTSAT, has been shown to exist using the quantum Lovász local lemma (QLLL) (40) for k ≥ 13. The application of Theorem 1 to the Erd} os–Rényi (ER) ensemble is not straightforward, as the interaction graphs exhibit an unbounded degree distribution, so that a strict application of Shearer’s theorem provides no useful information. However, as explained in SI Appendix, Local Degree Fluctuations for Random k-QSAT, such local degree fluctuations do not appear to be the source of unsatisfiability in k-QSAT for qubits. Neglecting this local rare-region effect, we calculate the self-consistent solutions to the disorder-averaged cavity equations, using standard methods (population dynamics) (35). For λ > λc, this numerical technique converges to a population of messages that represents the distribution of q and l in the infinite random graph and from which we can directly estimate the disorder-averaged free energy f and occupation number density hni = ∂f =∂ðlog λÞ. The latter is particularly useful, because as λ → λc, we expect hni to exhibit a square-root singularity by universality; i.e., hni ∼ ðλ − λc Þ1=2. Observing this singularity allows us to estimate λc accurately and confirms the validity of the numerical approach. By fitting these singularities, we extract a new and improved (lower) but nonrigorous threshold for the existence of an ENTSAT phase, k ≥ 7 (Fig. 3). Conclusion and Open Questions By extending the classical Shearer theorem to quantum mechanical systems, we have provided a new statistical mechanical criterion for determining whether a local Hamiltonian is frustration-free. We have applied this criterion to a large class of regular and random Hamiltonians. These instances cover many of the geometries that have been studied in quantum complexity and as parent Hamiltonians for wavefunction-based many-body physics. In the context of random satisfiability problems, we have provided a set of new lower bounds on the existence of the ENTSAT phase. In particular, these bounds suggest that the ENTSAT phase is eminently more reachable in simulations than previously established. Theorem 1 depends on the dimension p of the projectors Πi relative to the Hilbert space dimension and not the absolute local dimension q of the qudits. At small q (e.g., qubits), there are many Hamiltonians where Rðker HÞ is strictly larger than the bound of Theorem 1 (compare the discussion of random k-QSAT). Nonetheless, for large q we conjecture that Theorem 1 becomes tight. This is in sharp contrast to the classical case where the length-4 cycle (periodic chain) already provides a counterexample to the analogous statement (41). However, it is easy to show numerically that this counterexample breaks down for quantum projectors. 1. Kitaev A (2003) Fault-tolerant quantum computation by anyons. Ann Phys 303(1): 2–30. 2. Gottesman D (2010) An introduction to quantum error correction and fault-tolerant quantum computation. Quantum Information Science and Its Contributions To Mathematics, Proceedings of Symposia in Applied Mathematics, ed Lomonaco SJ, Jr (Am Math Soc, Providence, RI), Vol 68, pp 13–58. 3. Levin MA, Wen XG (2005) String-net condensation: A physical mechanism for topological phases. Phys Rev B 71:045110. 4. Affleck I, Kennedy T, Lieb EH, Tasaki H (1987) Rigorous results on valence-bond ground states in antiferromagnets. Phys Rev Lett 59(7):799–802. 5. Rokhsar DS, Kivelson SA (1988) Superconductivity and the quantum hard-core dimer gas. Phys Rev Lett 61(20):2376–2379.
6436 | www.pnas.org/cgi/doi/10.1073/pnas.1519833113
Fig. 3. Our current understanding of the phase diagram of the random k-QSAT. For small α, the instances are PRODSAT, with the transition out of PRODSAT approaching αP → 1 from below as k grows (45). For large α, the instances are guaranteed to be UNSAT, with the best known upper bound (46) for the satisfiability transition at αuS = 0.573 · 2k at k ≥ 4. In between, there may be an ENTSAT phase if αS > αP . The best previous lower bound for the satisfiability transition, αlS, was given by the quantum Lovász local lemma (QLLL, up triangles) (40) and is roughly exponential at large k. This work obtains a better lower bound through the Shearer criterion (squares), lowering the threshold for the existence of an ENTSAT phase from k = 13 to k = 7.
If, indeed, Theorem 1 is tight, there are several striking consequences. The geometrization theorem (37) states that Rðker HÞ is minimized by almost all choices of the Πi s. Coupled with tightness, the lattice gas partition function Z then provides a complete characterization of quantum satisfiability for almost all Hamiltonians with large enough qudits. It also directly lifts the universality of the lattice gas critical exponents to the counting of the ground state entropy of almost all quantum Hamiltonians in the frustration-free regime. In this sense, the conjecture amounts to an even larger scope of transferring insights from classical statistical mechanics into the quantum complexity domain. Although Theorem 1 can guarantee the existence of zero energy states, it does not construct them. This is in contrast to classical SAT and commuting QSAT, where efficient constructive classical (under Lovász’s and Shearer’s assumptions) and quantum algorithms (under Lovász’s assumption) are known (41–44). Analogously constructing the wavefunctions corresponding to the solutions of the noncommuting quantum problem would represent a milestone for quantum complexity theory. ACKNOWLEDGMENTS. We thank Dorit Aharonov, David Gosset, Antonello Scardicchio, Shivaji Sondhi, Mario Szegedy, and Umesh Vazirani for valuable discussions. This work was supported by the ARO Grant W922NF-09-1-0440, NSF Grants CCF-0905626 and PHY-1520535, the DFG Grant SFB 1143, ERC Grant 0308301, and a Sloan Research Fellowship.
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PNAS | June 7, 2016 | vol. 113 | no. 23 | 6437
Edited by Peter W. Shor, Massachusetts Institute of Technology, Cambridge, MA, and approved April 1, 2016 (received for review October 7, 2015)
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quantum satisfiability local Hamiltonian critical exponents universality
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A
| hardcore lattice gas |
n overwhelming majority of systems of physical interest can be described via local Hamiltonians: H=
M X
Πi .
[1]
i=1
Here, the “k-local” operator Πi acts on a k-tuple of the microscopic degrees of freedom, best thought of as qudits for the computer scientists among our readers or spins for the physicists. The M operators define an interaction (hyper)graph G, displayed in Fig. 1. A surprisingly diverse and important class of such model Hamiltonians is defined by the additional property of being “frustrationfree”: The ground state jψi of H is a simultaneous ground state of each and every Πi. This class comprises both commuting Hamiltonians—for which ½Πi , Πj = 0 ∀i, j—such as the toric code, general quantum error correcting codes, and Levin–Wen models (1–3), and noncommuting ones, such as the Affleck-Kennedy-Lieb-Tasaki (AKLT) and Rokhsar–Kivelson models (4–6). Their particular usefulness is also related to the fact that many of these examples can be viewed as “local parent Hamiltonians” for generalized matrix product states (7). In general, frustration-free conditions provide analytic control of ground state properties in otherwise largely inaccessible quantum problems. Determining whether a given Hamiltonian H is frustration-free is well known in quantum complexity theory as the quantum satisfiability (QSAT) problem. QSAT is widely believed to be intractable, in the sense that no general purpose classical or quantum algorithm can efficiently determine whether a given Hamiltonian is frustration-free (“satisfiable”). The technical statement is that QSAT is www.pnas.org/cgi/doi/10.1073/pnas.1519833113
complete for the Quantum Merlin Arthur complexity class (QMA1complete) (8), even when restricted to qubits and k = 3 (9) or when the interaction is between neighboring qudits on a line (10). Fortunately, such hardness results apply only in the worst case. For instance, in contrast to the hardness result, it is immediately obvious that a fully disconnected interaction graph G can be analyzed efficiently by considering each term Πi individually. The central result reported here is a sufficient combinatorial criterion for a local Hamiltonian H to be frustration-free. In fact, we provide a lower bound for the dimension of the satisfying subspace. This amounts to a generalization of Shearer’s theorem (11) from classical probability theory to the quantum world. We first formulate the result in Theorem 1, followed by an intuitive explanation of its content, with a technical proof relegated to SI Appendix. We then turn to applying Theorem 1, for which we enlist the results available on statistical mechanics of hardcore objects with negative fugacity on the interaction graph to deduce statements regarding QSAT, producing new bounds on the satisfiability threshold for a large class of 1D, 2D, and 3D interaction graphs. With the help of the cavity method, we are able to conjecture improved lower bounds for the satisfiability of QSAT on regular and Erd} os–Rényi random graphs, canonical models for quantum constraint optimization problems. These statement hold just as well for classical satisfiability—but for some of these classical models better bounds are known (12, 13). We close with an outlook, including a discussion of the role of a universality that emerges in our analysis, as well as conjectures on when our results are exact or tight. Theorem 1. Given a local projector Hamiltonian H as in Eq. 1 with interaction graph G and relative projector rank p = RðIm Πi Þ for all i, then
Rðker HÞ ≥ ZðG, − pÞ > 0 if ZðG, − p′Þ > 0 for all 0 ≤ p′ ≤ p. Significance Quantum computers promise computational power qualitatively superior to that achievable classically. This power will not be unlimited: Beyond much-touted applications, such as breaking encryption schemes, entire classes of problems are known to be intractable even for quantum computers. This work addresses a question of great practical relevance: In between these two extremes of certain (in)tractability, how can one efficiently diagnose the nature and properties of a given problem instance? To achieve this, we adopt a strategy of transferring insights from statistical physics and classical computing into the quantum realm. This provides a new perspective on the complexity of quantum problems and allows us to analyze a canonical class of quantum optimization problems in unprecedented explicit detail. Author contributions: O.S., S.C.M., C.R.L., and R.M. designed research, performed research, contributed new reagents/analytic tools, analyzed data, and wrote the paper. The authors declare no conflict of interest. This article is a PNAS Direct Submission. 1
To whom correspondence should be addressed. Email: [email protected].
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. 1073/pnas.1519833113/-/DCSupplemental.
PNAS | June 7, 2016 | vol. 113 | no. 23 | 6433–6437
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A broad range of quantum optimization problems can be phrased as the question of whether a specific system has a ground state at zero energy, i.e., whether its Hamiltonian is frustration-free. Frustrationfree Hamiltonians, in turn, play a central role for constructing and understanding new phases of matter in quantum many-body physics. Unfortunately, determining whether this is the case is known to be a complexity-theoretically intractable problem. This makes it highly desirable to search for efficient heuristics and algorithms to, at least, partially answer this question. Here we prove a general criterion—a sufficient condition—under which a local Hamiltonian is guaranteed to be frustration-free by lifting Shearer’s theorem from classical probability theory to the quantum world. Remarkably, evaluating this condition proceeds via a fully classical analysis of a hardcore lattice gas at negative fugacity on the Hamiltonian’s interaction graph, which, as a statistical mechanics problem, is of interest in its own right. We concretely apply this criterion to local Hamiltonians on various regular lattices, while bringing to bear the tools of spin glass physics that permit us to obtain new bounds on the satisfiable to unsatisfiable transition in random quantum satisfiability. We are then led to natural conjectures for when such bounds will be tight, as well as to a novel notion of universality for these computer science problems. Besides providing concrete algorithms leading to detailed and quantitative insights, this work underscores the power of marrying classical statistical mechanics with quantum computation and complexity theory.
The area covered by the bubbles is largest if they do not overlap, corresponding to a large number of violating configurations; we thus expect a lower bound on RðkerHÞ when they are fully disjoint. To calculate this lower bound, we expand the product in Eq. 3 and introduce a collection of occupation variables ni = 0,1 that indicate the presence of Πi in each term of the expansion, P X ð−1Þ ni Πn11 ⋯ΠnMM . Rðker HÞ = [4] fni g
Fig. 1. (Left) The interaction graph of a k = 3-local Hamiltonian. The degrees of freedom, qudits, are denoted by circles; the squares, which indicate the operators Πi appearing in H, are joined to the qudits on which they act. In the hardcore lattice gas mapping, each square may be occupied by a particle covering also the adjacent circles (shaded blobs), on which it must not overlap with another particle. (Right) A k = 6 interaction graph forming a triangular lattice of the operators Πi . The corresponding hardcore lattice gas corresponds to the hard hexagon model, which is exactly solvable.
Without loss of generality, we have taken the terms Πi to be projectors so that the satisfiability condition reduces to the presence of a zero-energy ground state. (This may always be done by shifting and deforming the eigenvalues of the local Hamiltonian terms without influencing the frustration-free ground state space.) Here, the relative dimension of a subspace X of the full Hilbert space H is simply RðXÞ = dimX=dimH, and, ZðG, λÞ is the partition function for a hardcore lattice gas of fat particles living on the (hyper) edges of the interaction graph at fugacity λ (Fig. 1). More precisely, X P Y ZðG, λÞ = λ ni 1 − ni nj , [2] fni g
i↔j
where i ↔ j runs over all projectors that share qudits and the sum runs over occupations ni = 0,1 of the lattice gas. Intuition Let us first consider the case of classical projectors, Πi, diagonal in the standard computational basis, where we can provide a simple pictorial Q representation of the content of the Shearer bound. The operator i ð1 − Πi Þ projects onto the zero-energy space of the Hamiltonian. If it is nonzero, the Hamiltonian is frustration-free. Moreover, its expectation value in the maximally mixed (infinite temperature) state gives the relative dimension of the zero-energy space, Rðker HÞ =
M Y
ð1 − Πi Þ.
If two dependent projectors, Πi and Πj, are thus occupied, then that term is zero Pwhen their bubbles are made disjoint. Otherwise, it is given by p ni . Thus, P Y X Rðker HÞ ≥ ð−pÞ ni 1 − ni nj [5] i↔j fni g = ZðG, − pÞ, where i ↔ j runs over projectors that share qudits. We have derived inequality [5] under the assumption that it is possible to make the dependent bubbles disjoint. If p is small enough, this is always the case; in the Venn diagram, the bubbles can be made disjoint without covering more area than the total space contains. Shearer (11, 14) showed that the above intuitive lower bound is correct for classical projectors as long as p ≤ pc, where pc is the first zero of ZðG, − pÞ. This is the classical analog of our quantum generalization, Theorem 1. The classical sketch above makes little sense for noncommuting quantum projectors. In the language of probability, this reflects the failure of the inclusion–exclusion principle (Eq. 4) for the relative dimension of vector spaces. Nonetheless, the result holds; the proof— our fundamental technical advance—is provided in SI Appendix, Proof of Theorem 1. Statistical Mechanical Transcription A remarkable aspect of Theorem 1 is that it maps the quantum satisfiability problem onto the classical statistical mechanical problem of determining the position of the first negative fugacity zero λc ðGÞ of the partition function ZðG, λÞ. In general, evaluating Z is computationally hard, but for many infinite graphs it may nonetheless be accomplished in a satisfactory manner using statistical mechanical tools. [Technically, computing Z is #P-hard almost everywhere (15).] In the thermodynamic limit (number of qudits N → ∞), the first zero λc ðGÞ can be identified with a well-known critical point λc ðG∞ Þ of
[3]
i=1
If all of the projectors act on different qudits, then this expectation value simply factors Rðker HÞ =
M Y
ð1 − Πi Þ = ð1 − pÞM ,
i=1
where p = Πi is the relative dimension of Πi. This is nonzero as long as p < pc = 1. Now suppose that some of the projectors share qudits; we call such pairs of projectors dependent, as they introduce dependence into the expectation value in Eq. 3. When can we, nevertheless, guarantee that RðkerHÞ remains positive? The essence of the Shearer bound is captured by the Venn diagram in Fig. 2, where each bubble represents the fraction of configuration space “knocked out” by a projector, with overlapping bubbles representing configurations multiply penalized by the corresponding projectors. 6434 | www.pnas.org/cgi/doi/10.1073/pnas.1519833113
Fig. 2. Shearer bound for dependent projectors. Projectors are independent if they do not share any qudit (Top Left), and there will always exist configurations violating any combination of such projectors simultaneously. In the Venn diagram (Top Right), this is represented by the mutual intersections of the shaded circles, each of which denotes the fraction of configuration space penalized by the projector it is labeled with. By contrast, for dependent projectors (Bottom Left), such an overlap is not guaranteed. A lower bound for the relative dimension of the satisfying space can be obtained by assuming the projectors Πi and Πl do not share any violating configurations with the projector Πj with which they share a qudit (Bottom Right).
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Examples As a first application of Theorem 1, we calculate the critical threshold λc for several infinite graphs. We do this using the cavity method, a well-known technique for studying statistical (31) and quantum (32– 34) models on infinite graphs that are locally tree-like. On trees and chains, the results are rigorously exact whereas on infinite random graphs they are less rigorous, but often just as exact. The heart of the cavity method is to introduce an auxiliary system of messages that propagate in both directions between the hyperedges and the nodes of the interaction graph G. For the model given by Eq. 2, the derivation is straightforward and quite intuitive at positive fugacity where the cavity messages correspond to marginal distributions for the gas particles (SI Appendix, The Cavity Derivations). The resulting equations continue smoothly to the negative fugacity regime. Adopting the convention that indexes i, j, . . . label hyperedges and a, b, . . . label sites, the belief propagation (BP) equations (for a recent pedagogical treatment, see ref. 35, chap. 14) are qa→i =
li→a =
λ+1+
λ P
lj→a je∂ani 1 − lj→a
λ
λ + Πbe∂ina
λ qb→i
−λ
,
where li→a and qa→i give the probability that hyperedge i is occupied in the absence of the connection to a and in the absence of everything except a, respectively. We note that a special case of these equations has been used previously to study the number of dimer coverings of random graphs (k = 2, positive fugacity) (36). Given a solution of the BP equations, the cavity estimate of the free energy F = log Z of the system is given by a sum of the free energies associated to the addition of individual elements of the interaction graph; i.e., Sattath et al.
Table 1. Summary of the critical threshold for various infinite interaction graphs Lattice 1D chain Triangular Square Square Checkerboard Hexagonal Simple cubic t-regular tree
pc
Qudits
Projectors
k
(38) (18, 20, 23) 0.1193 (17, 23) 0.0889 (23) 0.0688 (23) 0.1547 (23) 0.0744 (18) 1 4ðt − 1Þ (19, 39)
Vertices Edges Edges Vertices Black Edges Edges Vertices
Edges Vertices Vertices Edges Red Vertices Vertices Edges
2 6 4 2 4 3 6 2
1 4 pffiffi 5 5 − 11 2
F=
X a
Fa +
X i
Fi −
X
Fai ,
[6]
ðaiÞ
where Fa, Fi, and Fai correspond to the change in free energy due to the addition of sites a, hyperedges i, and the links ai between them, respectively. The full expressions may be found in SI Appendix, The Cavity Derivations. The Chain On the infinite chain, the BP equations can be solved by uniform messages qa→i = q, li→a = l. After some algebra and taking the root of the BP equations pffiffiffiffiffiffiffiffiffiffiffiffi corresponding to qðλ = 0Þ = 0, one finds l = q = 1 + ð1 − 1 + 4λ Þ=2λ. This expression suggests that λc = −1=4 as q becomes complex for λ < λc. Indeed, it is easy to check that the free energy density f = F=N (Eq. 6) has the expansion near λc: f = −log 2 + 2ðλ − λc Þ1=2 − 2ðλ − λc Þ + ⋯. The free energy goes complex for λ < λc, which reflects the accumulation of partition function zeros and indicates the failure of the lower bound. The free energy density is precisely −log 2 at the critical point. For qudits of local dimension q, this provides a lower bound on the dimension of the ground space dimðker HÞ = qN Rðker HÞ ≥ ðq=2ÞN . In fact, careful application of Theorem 1 for finite chains agrees with the exact result dimðker HÞ = ðq=2ÞN ðN + 1Þ for open chains derived using matrix-product techniques (37, 38). It is interesting to note that the entropy per site is zero for qubits at criticality, but has a finite value for larger q. Regular Trees The BP equations on infinite regular trees with site degree t and hyperedge degree k can be solved under the ansatz that all messages are the same. For algebraic details, see SI Appendix, The Cavity Derivations. The resulting critical threshold is 1 ðk − 1Þk−1 λc = − . kk t−1
[7]
For k = 2-local trees, this agrees with previous results obtained by matrix product states on trees (39). As far as we are aware, Eq. 7 provides a strictly better lower bound on satisfiability of infinite regular trees than any previous literature. The corresponding expansion of the free energy density is f = f0 ðt, kÞ + Aðt, kÞðλ − λc Þ + Bðt, kÞðλ − λc Þ3=2 + ⋯.
[8]
We thus recover the expected nonanalyticity (ϕ = 3=2) in the free energy density for the infinite-dimensional hardcore singularity (24). PNAS | June 7, 2016 | vol. 113 | no. 23 | 6435
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the hardcore lattice gas referred to as the hardcore singularity. The critical fugacity λc ðG∞ Þ upper bounds the λc of any finite subgraph by monotonicity (11). Thus, Theorem 1 implies that all subgraphs of G∞ are frustration-free as long as the relative rank of the projectors satisfies p < − λc ðG∞ Þ. The hardcore singularity has been studied extensively in the statistical mechanical literature (16–25) and its location is known for many infinite lattices. Table 1 summarizes some of these critical values and indicates their translation into QSAT interaction graphs. In addition, there exist rigorous bounds on λc that may be efficiently calculated on any sufficiently simple lattice (16). The singularity exhibits universal features just like standard phase transitions. In particular, the free energy density f ðλÞ = ð1=NÞlog ZðλÞ near the critical point λc has universal exponents (21, 22) due to its connection with the so-called Yang-Lee edge singularity (24–28). Technically, this means that the leading nonanalytic part of the free energy f ∼ ðλ − λc ÞϕD near λc, where ϕD is a noninteger critical exponent that depends only on the spatial dimension D of the lattice (21, 22, 24). It is known analytically that ϕ1 = 1=2, ϕ2 = 5=6 (20, 29, 30), and ϕD = 3=2 for D above the upper critical dimension DC = 6 (24). The critical exponents control the lower bounds on the dimension of the ground space of Hamiltonians close to the critical threshold. ϕD Indeed, by Theorem 1, Rðker HÞ ≥ enf ðλÞ ∝ eanðλ − λc Þ +Oðλ−λc Þ for λ > λc, so the critical exponents show up directly—and perhaps somewhat unexpectedly—as anomalous growth laws in D = 1,2 (SI Appendix, Improved Bounds Using the Critical Exponents). In the following, observation of the expected exponent in the random k-QSAT ensemble helps confirm the validity of the nonrigorous cavity analysis.
Random k-QSAT We now apply Theorem 1 to random k-QSAT. Random satisfiability has been a workhorse for the study of typical case complexity and heuristic algorithms. By tuning the density α = M=N of Erd} os–Rényitype random interaction graphs, a cornucopia of phases and phase transitions in the structure of the satisfying space has been discovered. In the quantum case, three phases for random k-QSAT on qubits and rank-1 projectors are believed to be (i) at low density, a product state satisfiable phase (PRODSAT); (ii) at intermediate densities for k sufficiently large, a satisfiable phase in which all ground states are entangled (ENTSAT); and (iii) at high density, an unsatisfiable (UNSAT) phase. The arguably most interesting of these phases, ENTSAT, has been shown to exist using the quantum Lovász local lemma (QLLL) (40) for k ≥ 13. The application of Theorem 1 to the Erd} os–Rényi (ER) ensemble is not straightforward, as the interaction graphs exhibit an unbounded degree distribution, so that a strict application of Shearer’s theorem provides no useful information. However, as explained in SI Appendix, Local Degree Fluctuations for Random k-QSAT, such local degree fluctuations do not appear to be the source of unsatisfiability in k-QSAT for qubits. Neglecting this local rare-region effect, we calculate the self-consistent solutions to the disorder-averaged cavity equations, using standard methods (population dynamics) (35). For λ > λc, this numerical technique converges to a population of messages that represents the distribution of q and l in the infinite random graph and from which we can directly estimate the disorder-averaged free energy f and occupation number density hni = ∂f =∂ðlog λÞ. The latter is particularly useful, because as λ → λc, we expect hni to exhibit a square-root singularity by universality; i.e., hni ∼ ðλ − λc Þ1=2. Observing this singularity allows us to estimate λc accurately and confirms the validity of the numerical approach. By fitting these singularities, we extract a new and improved (lower) but nonrigorous threshold for the existence of an ENTSAT phase, k ≥ 7 (Fig. 3). Conclusion and Open Questions By extending the classical Shearer theorem to quantum mechanical systems, we have provided a new statistical mechanical criterion for determining whether a local Hamiltonian is frustration-free. We have applied this criterion to a large class of regular and random Hamiltonians. These instances cover many of the geometries that have been studied in quantum complexity and as parent Hamiltonians for wavefunction-based many-body physics. In the context of random satisfiability problems, we have provided a set of new lower bounds on the existence of the ENTSAT phase. In particular, these bounds suggest that the ENTSAT phase is eminently more reachable in simulations than previously established. Theorem 1 depends on the dimension p of the projectors Πi relative to the Hilbert space dimension and not the absolute local dimension q of the qudits. At small q (e.g., qubits), there are many Hamiltonians where Rðker HÞ is strictly larger than the bound of Theorem 1 (compare the discussion of random k-QSAT). Nonetheless, for large q we conjecture that Theorem 1 becomes tight. This is in sharp contrast to the classical case where the length-4 cycle (periodic chain) already provides a counterexample to the analogous statement (41). However, it is easy to show numerically that this counterexample breaks down for quantum projectors. 1. Kitaev A (2003) Fault-tolerant quantum computation by anyons. Ann Phys 303(1): 2–30. 2. Gottesman D (2010) An introduction to quantum error correction and fault-tolerant quantum computation. Quantum Information Science and Its Contributions To Mathematics, Proceedings of Symposia in Applied Mathematics, ed Lomonaco SJ, Jr (Am Math Soc, Providence, RI), Vol 68, pp 13–58. 3. Levin MA, Wen XG (2005) String-net condensation: A physical mechanism for topological phases. Phys Rev B 71:045110. 4. Affleck I, Kennedy T, Lieb EH, Tasaki H (1987) Rigorous results on valence-bond ground states in antiferromagnets. Phys Rev Lett 59(7):799–802. 5. Rokhsar DS, Kivelson SA (1988) Superconductivity and the quantum hard-core dimer gas. Phys Rev Lett 61(20):2376–2379.
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Fig. 3. Our current understanding of the phase diagram of the random k-QSAT. For small α, the instances are PRODSAT, with the transition out of PRODSAT approaching αP → 1 from below as k grows (45). For large α, the instances are guaranteed to be UNSAT, with the best known upper bound (46) for the satisfiability transition at αuS = 0.573 · 2k at k ≥ 4. In between, there may be an ENTSAT phase if αS > αP . The best previous lower bound for the satisfiability transition, αlS, was given by the quantum Lovász local lemma (QLLL, up triangles) (40) and is roughly exponential at large k. This work obtains a better lower bound through the Shearer criterion (squares), lowering the threshold for the existence of an ENTSAT phase from k = 13 to k = 7.
If, indeed, Theorem 1 is tight, there are several striking consequences. The geometrization theorem (37) states that Rðker HÞ is minimized by almost all choices of the Πi s. Coupled with tightness, the lattice gas partition function Z then provides a complete characterization of quantum satisfiability for almost all Hamiltonians with large enough qudits. It also directly lifts the universality of the lattice gas critical exponents to the counting of the ground state entropy of almost all quantum Hamiltonians in the frustration-free regime. In this sense, the conjecture amounts to an even larger scope of transferring insights from classical statistical mechanics into the quantum complexity domain. Although Theorem 1 can guarantee the existence of zero energy states, it does not construct them. This is in contrast to classical SAT and commuting QSAT, where efficient constructive classical (under Lovász’s and Shearer’s assumptions) and quantum algorithms (under Lovász’s assumption) are known (41–44). Analogously constructing the wavefunctions corresponding to the solutions of the noncommuting quantum problem would represent a milestone for quantum complexity theory. ACKNOWLEDGMENTS. We thank Dorit Aharonov, David Gosset, Antonello Scardicchio, Shivaji Sondhi, Mario Szegedy, and Umesh Vazirani for valuable discussions. This work was supported by the ARO Grant W922NF-09-1-0440, NSF Grants CCF-0905626 and PHY-1520535, the DFG Grant SFB 1143, ERC Grant 0308301, and a Sloan Research Fellowship.
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