Interacting boson problems can be QMA-hard

Tzu-Chieh Wei,1, 2 Michele Mosca,1, 3, 4 and Ashwin Nayak3, 1, 4 1Institute for , University of Waterloo, Waterloo, Ontario, Canada 2Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada 3Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada 4Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada (Dated: Nov. 18, 2009) Computing the ground-state energy of interacting problems has recently been shown to be hard for QMA, a quantum analogue of the NP. Fermionic problems are usually hard, a phenomenon widely attributed to the so-called sign problem. The corresponding bosonic problems are, according to conventional wisdom, tractable. Here, we demonstrate that the complexity of interacting boson problems is also QMA-hard. Moreover, the bosonic version of N-representability problem is QMA-. Consequently, these problems are unlikely to have efficient quantum algorithms.

PACS numbers: 03.67.-a, 05.30.Jp, 89.70.Eg

Many important model Hamiltonians in physics, such The fact that interacting fermion problems are hard as the Hubbard model (both fermionic and bosonic) and motivates us to investigate the corresponding bosonic those for superconductivity and the quantum Hall effect, problems. Their complexity seems less explored. Could involve at most two-body interactions [1]. The ground- bosonic problems be so hard as to be intractable? We state wavefunction and energy of these Hamiltonians play show that generic nearest-neighbor two-body interact- a key role in understanding these fascinating phenom- ing boson problems are indeed QMA-hard. Inspired by ena. In some of these phenomena, are the major Ref. [5], we study the bosonic version of the (fermionic) players. Problems involving fermionic particles such as N-representability problem [18, 19], where one is given electrons often seem to be computationally more difficult a two-particle reduced density matrix ρ and needs to than those with bosonic counterparts. This intractabil- decide whether there exists a consistent global N-body ity is often attributed to the so-called “sign problem”, state σ. This problem has been vastly explored in quan- occurring in Quantum Monte Carlo simulations [2]. On tum chemistry [20], as its solution would enable efficient the other hand, bosonic problems do not suffer from the solution of ground-state energy for generic two-body in- sign problem [3] and are thus regarded as tractable. teracting fermionic systems. We show that the bosonic Schuch and Verstraete and Liu, Christandl and Ver- N-representability problem is also QMA-complete. Ad- straete have recently shown that computing the ground- ditionally, we show that the bosonic N-representability state energy of general interacting electrons is QMA- problem given only diagonal elements is NP-hard. hard [4, 5]. The complexity class QMA (Quantum QMA-hardness of interacting boson problems. Consider † Merlin-Arthur) is a generalization of the class NP (non- boson creation and annihilation operators aj, aj for the deterministic polynomial time) to the quantum realm. It j’th site or mode, obeying the commutations [22]: was introduced by Kitaev [6] in the so-called local Hamil- † † † tonian problem (LHP), where, roughly speaking, the goal [ai, aj] = 0 = [ai , aj], [ai, aj] = δij. (1) is to determine the ground-state energy of a Hamil- The use of these operators preserves the symmetry of tonian involving only few-body interaction terms. QMA- the bosonic wavefunctions under permutations, and any hard problems are regarded as difficult, unlikely to be N-boson wavefunction can be represented as follows: solved efficiently even by a quantum computer. However, X a quantum computer, if given the solution to any problem † i1 † im |ψi = ci1,...,im (a1) ...(am) |Ωi, (2) in QMA, along with a suitable “certificate” or “witness i1+...+im=N state”, can efficiently verify whether the solution is cor- where ik (0 ≤ ik ≤ N) denotes the number of bosons rect or not. In fact, as a result of a series of works [7–9], th even for nearest-neighbor two-body interactions among at the k site, m is the total number of sites, and |Ωi spin-1/2 particles on a two-dimensional lattice, the LHP denotes the vacuum state without any bosons. Note that is QMA-complete. With higher magnitude of spins in one we restrict ourselves to states with exactly N bosons [21]. dimension, the LHP can be QMA-complete as well [10]. We construct a bosonic Hamiltonian Hbose, whose Understanding and classifying the complexity of physi- ground-state energy determines the ground-state energy cal models and investigating hard problems using statis- of the following quantum spin glass model H: tical mechanical tools have become important research 3 X X µν (µ) (ν) endeavors [4–17], as a result of interplay between physics, H = cij σi ⊗ σj , (3) chemistry, mathematics and computer science. hi,ji µ,ν=0 2

(0) N N where i, j label the site, σ = 11 denotes the identity grows exponentially in N, i.e., Nm & (δ+1) /δ when and σ(1) = σx, σ(2) = σy, and σ(3) = σz are the three N is large. Given an N-boson state ρ(N), the two-boson µν . The coefficients cij are real but arbi- reduced state is calculated by tracing out but two (2) (N) (N) trary. Oliveira and Terhal [9] showed that determining bosons: ρ ≡ TrN−2ρ , where ρ is in general a the ground-state energy of H is QMA-hard, even if the in- mixture of states |ψi of the form (2). More precisely, teractions are restricted to nearest neighbor sites hi, ji in ρ(2) is given via its matrix elements: the two dimensional square lattice. By way of , (2) 1 † † solving the ground-state energy of Hbose is also QMA- ρ ≡ ha a a a i, (5) ijkl N(N − 1) k j i hard. To construct Hbose , we use the Schwinger boson correspondence between and boson states (see, e.g., where the bracket indicates the expectation value over Ref. [23]) given by the following map: the state ρ(N). Note that the one-boson reduced density (1) (N) (1) † x † † y † † z † † matrix ρ ≡ TrN−1ρ , defined via ρ ≡ ha aii/N, is σ ↔ a bi + b ai, σ ↔ i(b ai − a bi), σ ↔ a ai − b bi, ik k i i i i i i i i i completely determined once ρ(2) is known: where the operators a , b correspond to distinct sites. i i (1) X (2) It is easy to verify that the bosonic operators obey the ρik = ρilkl. (6) commutation relations of the corresponding Pauli opera- l tors. We can regard the qubit at site i as a single boson Informally, the bosonic N-representability problem that can be in one of two different degrees of freedom: (with m sites) asks whether there is an N-boson state † zi † 1−zi |zii ↔ (ai ) (bi ) |Ωi with zi ∈ {0, 1}, correspond- whose two-particle reduced density matrix equals a given ing to the dual-rail encoding of a photonic qubit in the state ρ. To deal with technical issues related to precision, Knill-Laflamme-Milburn linear-optics quantum compu- we are promised that when there is no N-boson state con- tation scheme [24]. Hence, N can be represented sistent with it, every two-particle reduced state is “far by N bosons in 2N sites (or N sites with each boson away” from ρ. Formally, we are given a two-boson den- possessing two distinct internal degrees of freedom): sity matrix ρ of size [m(m + 1)/2] × [m(m + 1)/2], and a real number β ≥ 1/poly(N), with all numbers specified † z1 † 1−z1 † zN † 1−zN |z1, . . . , zN i ↔ (a1) (b1) ··· (aN ) (bN ) |Ωi. with poly(N) bits of precision. We would like to decide if: (“YES” case) There exists an N-boson state σ such As any two-local qubit Hamiltonian can be written as a that Tr (σ) = ρ, or if (“NO” case) For all N-boson linear combination of terms with at most two Pauli opera- N−2 states σ, kTr (σ) − ρk ≥ β. tors, the corresponding bosonic Hamiltonian can be writ- N−2 1 We show that the bosonic N-representability is QMA- ten as a combination of products of at most two annihila- hard under Turing reductions [26]. In other words, we tion and two creation operators. To ensure that there be show that given an efficient algorithm for bosonic N- exactly one boson on the pair of sites corresponding to i, representability, we can efficiently determine the ground we add the following extra terms: P ≡ (a†a +b†b −11)2, i i i i i state energy of H , a QMA-hard problem as estab- which commute with other terms in the Hamiltonian. bose lished above. In the sequel, we refer to an algorithm for The total bosonic Hamiltonian is then N-representability as the “membership oracle”. † † X The first step is to write the two-particle interacting Hbose ≡ H(a , b , a, b) + c Pi, (4) terms in Hbose in terms of a complete orthonormal set, i P Q, of two-particle observables: Htwo-body ≡ Q∈Q γQ Q, which involves at most nearest-neighbor two-body inter- where the number of elements l ≡ |Q| ∼ O(m4). Note actions [25]. By making the weight c large enough, e.g., that poly(m) is poly(N), and so is poly(l). The observ- P µν i,j,µ,ν cij N(N − 1)/2, we guarantee that the ground ables Q are constructed as in the fermionic case [5]. We state of the full Hamiltonian has exactly one boson per define aI ≡ ai2 ai1 , for all pairs I = (i1, i2), i1 ≤ i2. (Note site. Thus Hbose may be represented with at most a that in the case of fermions i1 < i2.) We fix a total order polynomially larger number of bits as compared to H. (denoted by ≺) on pairs of indices I. The observables in Thus, if one can compute the ground-state energy of gen- Q are defined as follows: eral bosonic Hamiltonians with at most two-body inter- actions, one can solve general spin-1/2 two-local Hamil- 1 † † XIJ ≡ √ (aI aJ + aJ aI ), for I ≺ J, (7) tonian problems. As solving the latter is QMA-hard, nI nJ solving the former is QMA-hard as well. This shows that −i † † interacting boson problems are generally difficult. YIJ ≡ √ (aI aJ − aJ aI ), for I ≺ J, (8) nI nJ QMA-hardness of bosonic N-representability problem. 1 Z ≡ a† a , for I ≺ L, (9) We consider the number of bosons N to be fixed, and I n I I assume that the number of modes m that the bosons oc- I cupy is large enough, i.e., m ≥ δN for some constant where the factor nI = 1 if i1 6= i2, nI = 2 if i1 = i2, and L δ > 0. The number of different ways Nm that N identi- denotes the last pair in the ordering. These operators are N+m−1 cal bosons can occupy m sites is Nm = N , which Hermitian and the XIJ and YIJ have expectation values 3 in the interval [−1, 1] and the ZI have expectation values this, we construct a QMA proof system, i.e., describe a in [0, 1] for any two-particle state. In the two-particle witness state τ (over polynomially many qubits) for the Hilbert space, they are orthogonal to each other under N-representability of a given two-boson density matrix ρ, the trace operator, e.g., Tr(XIJ ZK ) = 0 for all I, J, K. and a polynomial-time V (the “veri- They also form a basis for any two-boson states ρ(2): fier”) that expects such a pair ρ, τ as input. The verifier determines probabilistically whether a given two-boson (2) X ρ = ZL + αZI (ZI − ZL) density matrix ρ is N-representable, or is far from being I≺L N-representable. In the “YES” case, V outputs “YES” 1 X with probability p1, and in the “NO” case, V outputs + α X + α Y  , (10) 2 (XIJ ) IJ (YIJ ) IJ “NO” with probability at least 1−p0. For the problem to I≺J be in QMA, the gap p1 −p0 should be at least 1/poly(N); −poly(N) (2) this gap can be amplified to 1 − e [6, 29]. where αQ = Tr(Qρ ) for all Q ∈ Q. Using Eq. (6) For the witness state, we represent an N-boson state † we see that the expectation value hai aki of the one- σ using m qudits (d-dimensional quantum systems), via body terms, can be expressed as linear combination of the following correspondence, a.k.a. Holstein-Primakoff (N) P 0 the αQ. Thus we have Tr(Hboseρ ) = Q∈Q γQ αQ, bosons (see, e.g., Ref. [23]): 0 where γQ includes the contribution from both one-body and two-body terms, and α are the coefficients of 1 + † † 1 − Q ai ↔ Ai ≡ S , a ↔ A ≡ S , (2) (N) p z i i i p z i ρ ≡ TrN−2 ρ . Finding the ground-state energy s + Si s + Si + 1 is equivalent to minimizing the linear function f(~α) = P 0 where S± are the raising/lowering operators for i’th γQ αQ subject to the constraint that ~α ∈ KN , i Q∈Q spin, and 2s ≥ N. The spin operators above satisfy where KN denotes the convex set of all ~α such that the corresponding state ρ(2) is N-representable. the bosonic commutation relations provided the total The above minimization of energy, subject to the con- spin magnitude is s. The boson number states |ni ∈ vex constraint that ρ is N-representable, belongs to {|0i, |1i, ..., |2si} at one site correspond to the spin states a class of convex optimization problems which can be |sni ∈ {|si, |s − 1i, ..., | − si}, and d = 2s + 1. A bosonic † † † † solved using the shallow-cut ellipsoid algorithm [27, 28] observable O = ai ajalak + akal ajai is transformed into with the aid of an N-representability membership oracle. ˜ † † † † O = Ai AjAlAk + AkAl AjAi, which is a tensor product If KN is contained in a ball of radius centred at the of at most four single-qudit observables, in contrast to origin, and it contains a ball of radius r, the run time the non-local string operators in the fermionic case [5].  is poly log(R/r) and the error in the solution due to The expectation value hO˜i can be estimated efficiently. computation with finite precision is 1/poly(R/r). The One method is to explicitly diagonalize the observable O˜ algorithm will be efficient and of polynomially bounded P as λi|θiihθi|, and measure the given qudit representa- error, if R/r is at most poly(l). i tionσ ˜ of the bosonic state σ in the basis {|θii}. Repeat- From the discussion leading to Eq. (10)√ it follows that ing the measurement on polynomially many copies ofσ ˜, KN is contained in a ball of radius R = l centered at we can estimate hO˜i. We note that a us- the origin. However, the method used by Liu et al. [5] ing qudits with d = 2s + 1 can be implemented efficiently of regarding aj as creation operator for a hole in site by an equivalent circuit using qubits. j cannot be employed to suitably bound r from below. The witness τ consists of polynomially many blocks, Instead, we explicitly construct a set of N-boson states where each block has m qudits that represent a stateσ ˜ such that the convex hull of the corresponding vectors that is claimed to be an N-boson state with TrN−2(˜σ) = {αQ} contains a ball of radius 1/poly(l). Relying on ρ. The verifier V measures, on each block, the observable the property that bosons can occupy the same site, for P † P z k AkAk = ms − k Sk to check whether the particle each Q we construct states so as to maximize or minimize number is N. If not, V outputs “NO”. This measure- αQ. The resulting points have the property that for any ment projects each block onto the space of fixed particle coordinate axis, there exist at least two points at constant number states. If the particle number is N, V contin- distance along that axis. As a consequence, we show that ues to perform measurements for a suitable set of ob- their convex hull (which is contained in KN ) contains a servables (e.g., those corresponding to Q) using the pro- ball with radius r ≥ 1/poly(l), centered at the center of jected states. It compares whether the outcomes match mass of the points. the expectation values specified by ρ, to check for con- An algorithm for bosonic N-representability thus sistency. It outputs “YES” if the errors are less than enables efficient calculation of the ground-state en- β/poly(N) (for a suitable polynomial), otherwise outputs ergy of the Hamiltonian Hbose. Consequently, N- “NO”. When ρ is N-representable, the prover supplies representability is QMA-hard. polynomially many copies of the correct state σ such that QMA-completeness. Is the bosonic N-representability in- TrN−2(σ) = ρ, and the verifier always answers “YES” side QMA or even harder? We show that the bosonic (i.e., p1 = 1), as the measurement outcome is always N-representability problem is indeed inside QMA, im- consistent with ρ. When ρ is not N-representable, the plying that the problem is QMA-complete. To establish prover can cheat by entangling different blocks of qudits. 4

Using a Markov argument, first employed by Aharonov Concluding remarks. We have shown that two families and Regev [30] and later by Liu [28], one can show of boson problems are QMA-complete, implying that in that the verifier will still output “NO” with probability the worst-case scenario they are unlikely to be solved effi- ≥ β/poly(N). Thus N-representability is in QMA [31], ciently even by quantum computers. However, this does and hence QMA-complete. This in turn implies that de- not preclude the possibility of efficient approximation al- termining the ground-state energy of interacting bosons gorithms. Approaches such as mean-field theory, path- with two-body interactions is also QMA-complete. integral quantum Monte Carlo [3], and more recently Further results. We follow the same argument as matrix product states [33] and Multiscale Entanglement in Ref. [5] and conclude that pure-state bosonic N- Renormalization Ansatz [34] are important endeavors to- representability is in QMA(k), as the essential point is wards classical approximation algorithms. It is possible to verify the purity of the certificate. Next, consider that many physical models fall into “easy” instances that the bosonic N-representability problem when only the can be solved efficiently by these schemes. A more spec- † † ulative direction is to develop quantum approximation diagonal elements Dij ≡ hai ajajaii are specified. If one considers the case m = 2N and the mapping by the algorithms, which have potential speedup. Schwinger representation, one finds that the solution en- ables one to solve the ground-state energy of local spin Acknowledgments. TCW thanks Zhengfeng Ji and Berni Hamiltonians which only contain σz operators. The lat- Alder for useful discussions. This work was supported by ter corresponds to a classical spin-glass problem, and is ARO/NSA, CIFAR, IQC, MITACS, NSERC, an Ontario known to be NP-hard [17]. Thus the problem of deciding ERA, ORF, QuantumWorks, Government of Canada, N-representability given {Dij} is also NP-hard. and Ontario-MRI.

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