The complexity of quantum spin systems on a two-dimensional square lattice

arXiv:quant-ph/0504050v1 7 Apr 2005

Roberto Oliveira

∗

Barbara M. Terhal

†

June 11, 2008

Abstract The problem 2-LOCAL HAMILTONIAN has been shown to be complete for the quantum computational class QMA [1]. In this paper we show that this important problem remains QMAcomplete when the interactions of the 2-local Hamiltonian are restricted to a two-dimensional (2-D) square lattice. Our results are partially derived with novel perturbation gadgets that employ mediator qubits which allow us to manipulate 2-local interactions. As a side result, we obtain that quantum adiabatic computation using 2-local interactions restricted to a 2-D square lattice is equivalent to the circuit model of quantum computation.

1

Introduction

The novel possibilities that quantum mechanics brings to information processing have been the subject of intense study in recent years. In particular, much interest has been devoted to understanding the strengths and weaknesses of quantum computing as it pertains to important problems in computer science and physics. An important part of this research program consists of understanding which families of quantum systems are computationally complex. This complexity can manifest itself in two ways. On the one hand, a positive result shows that a given family of systems is “complicated enough” to efficiently implement universal quantum computation. On the other hand, a negative result shows that certain questions about such systems are unlikely to be efficiently answerable. A proof of QMAcompleteness offers compelling evidence of the negative kind while also locating the given problem in the complexity hierarchy, since QMA, –the class of decision problems that can be efficiently solved on a quantum computer with access to a quantum witness–, is analogous to the classical complexity classes NP and MA. More precisely, the class QMA is defined as Definition 1 (QMA) A language L = Lyes ∪ Lno ⊆ {0, 1}∗ is in QMA if there is an efficient quantum circuit family {Cx }x∈{0,1}∗ such that (i) for all x ∈ Lyes there exists a quantum witness state | ψx i in the input space of Cx such that Cx (| ψx ihψx |) outputs 1 with probability ≥ 2/3; and (ii) for all x ∈ Lno and all quantum states | ξi in the input space of Cx , Cx (| ξihξ |) outputs 1 with probability ≤ 1/3. The work on finding QMA-complete problems was jump-started by a ‘quantum Cook-Levin Theorem’ proved by Kitaev [2] (see also the survey [3]). Kitaev showed that the promise problem ∗ †

IBM Watson Research Center, Yorktown Heights, NY, USA 02139. [email protected] IBM Watson Research Center, Yorktown Heights, NY, USA 02139. [email protected]

1

k-LOCAL HAMILTONIAN for k = 5 is QMA-complete. The problem k-LOCAL HAMILTONIAN is

defined as: Definition Pr 2 (k-LOCAL HAMILTONIAN) Given is a k-local Hamiltonian, –a self-adjoint operator– , H = j=1 Hj on n qubits where each term Hj acts on at most k qubits. Here r ≤ poly(n), ||Hj || ≤ poly(n) and the entries of Hj are specified by poly(n) bits. Furthermore, we have two constants α, β with α < β. The problem is to decide whether the smallest eigenvalue of H, also called the ‘ground state energy’ of H, denoted as λ(H), is at most α. The promise is that either λ(H) ≤ α or λ(H) > β. Kitaev’s result was strengthened in Ref. [4] which proved QMA-completeness of 3-LOCAL HAMILTONIAN. The strongest result to date, found by Kempe, Kitaev and Regev [1], states that the problem 2-LOCAL HAMILTONIAN is still QMA-complete. In the positive direction it was first shown by Aharonov et al. [5] that adiabatic quantum computation using 3-local Hamiltonians is computationally equivalent to quantum computation in the circuit model. In the adiabatic computation paradigm one starts the computation in the groundstate, i.e. the eigenstate with smallest eigenvalue, of some Hamiltonian H(t = 0). The computation proceeds by slowly (at a rate at most poly(n)) changing the parameters of the Hamiltonian H(t). The adiabatic theorem (see Ref. [6] for an accessible proof thereof) states that if the instantaneous Hamiltonian H(t) has a sufficiently large spectral gap, – i.e. the difference between the second smallest eigenvalue and the smallest eigenvalue is Ω(1/poly(n))–, then the state at time t during the evolution is the ground-state of the instantaneous Hamiltonian H(t). At the end of the computation (t = T ), one measures the qubits in the ground-state of the final Hamiltonian H(T ). Ref. [1] improved on the result by Aharonov et al. by showing that any efficient quantum computation can be efficiently simulated by an adiabatic computation employing only 2-local Hamiltonians. These results on the complexity of Hamiltonians can be viewed as the first (see also Ref. [7]) in a field that is still largely unexplored as compared to the classical case The class of Hamiltonian problems is likely to be a very important class of problems in QMA. Hamiltonians govern the dynamics of quantum systems and as such contain all the physically important information about a quantum system. The problem of determining properties of the spectrum, in particular the ground state (energy) or the low-lying excitations, is a well-known problem for which a variety of methods, both numerical and analytical, (see e.g. [8, 9]) have been developed. Let us briefly review the classical situation. In some sense the LOCAL HAMILTONIAN problem is similar to MAX2SAT [10]. But perhaps a better analog is the set of problems defined with ‘classical’ Hamiltonians such as ISING SPIN GLASS, inspired by physics: Definition 3 (ISING SPIN GLASS) Given is an interaction graph G = (V, E) with Hamiltonian X X HG = Jij Zi ⊗ Zj + Γi Zi . (1) i,j∈E

i∈V

Here the couplings Jij and Γi are at most poly(n) and Z = | 0ih0 | − | 1ih1 | is the Pauli Z operator. Problem: is the smallest eigenvalue of HG smaller than some constant α? It is known that ISING SPIN GLASS is NP-complete in the case that (1) Γi = 0 and the graph is a two-level square lattice (2 connected levels of a 2-D square lattice) and (2) Γi = Γ 6= 0, Jij = 1 and the graph is planar [11]. Interestingly, ISING SPIN GLASS ∈ P for a planar graph with Γi = 0. 2

In this paper we prove some results on the complexity of a quantum version of this model, a quantum spin glass, restricted to a two-dimensional (2-D) square lattice or a planar graph with degree at most 4. Our results are based on two ideas. The first one is an important modification to the ‘quantum Cook-Levin’ circuit-to-3-local Hamiltonian construction that will prove QMAcompleteness of a 3-local Hamiltonian on a ‘spatially sparse’ graph. Secondly, we introduce a set of quantum ‘mediator qubit’ gadgets to manipulate 2-local interactions and perform reductions inside QMA1 . The general technique is based on the idea of perturbation gadgets introduced in Ref. [1]. Before we state the results, let us give a few more useful definitions. With a 2-local Hamiltonian HG defined on n qubits we can associate a (simple) interaction graph G = (V, E) where |V | = n. For every edge in e ∈ E between vertices a and b there is a nonzero 2-local term He on qubits a and P b such thatPHe is not 1-local nor proportional to the identity operator I. We can write HG = e∈E He + v∈V Hv where Hv is a potential 1-local term on the vertex v. A Pauli edge of an interaction graph G is an edge between vertices a and b associated with an operator αab Pa ⊗ Pb where Pa , Pb are Pauli matrices X = | 0ih1 | + | 1ih0 |, Y = −i| 0ih1 | + i| 1ih0 |, Z = | 0ih0 | − | 1ih1 | and αab is some real number. For an interaction graph in which every edge is a Pauli edge, the degree of a vertex is called its Pauli degree. For such a graph, the X- (resp. Y-, resp. Z-) degree of a vertex a is the number of edges with endpoint a for which Pa = X (resp. Pa = Y , resp. Pa = Z). We will sketch proofs of the following results. First we show that Theorem 4 2-LOCAL HAMILTONIAN on a planar graph with maximum Pauli degree equal to 4 is QMA-complete. With only a little more work, we prove that Theorem 5 2-LOCAL HAMILTONIAN with Pauli interactions on a subgraph of the 2-D square lattice is QMA-complete. Lastly, we answer an open problem in Ref. [5] as we show that Theorem 6 (Loosely) Universal quantum computation can be efficiently simulated by quantum adiabatic evolution of qubits interacting on a 2-D square lattice. We believe that our Theorem 5 is in some sense the strongest result that one can expect, since we consider it unlikely that 2-LOCAL HAMILTONIAN restricted to a linear chain is QMA-complete 2 . With regards to Theorem 6, one should note that Aharonov et al. has already proven that interactions of six-dimensional particles on a two-dimensional square lattice suffice for universal quantum adiabatic computation. Our improvement to qubits on a two-dimensional lattice is relevant with respect to practical feasibility. This paper is organized as follows. In Section 2 we show how to modify the completeness construction of a 2-local Hamiltonian so that the 2-local Hamiltonian is restricted to a spatially sparse graph. In Section 3 we introduce our perturbation gadgets. In Section 4 we use these gadgets to reduce the problem of a 2-local Hamiltonian on a spatially sparse graph to one on a planar graph of Pauli degree at most 4, Theorem 4. With a bit more work we further reduce it to a 2-local Hamiltonian on a 2-D square lattice, Theorem 5. Finally, Section 5 shows that the argument for proving Theorem 4 can be adapted to prove Theorem 6. 1 These gadgets are inspired by the idea of superexchange between particles with spin. Loosely speaking, superexchange is the creation of an effective spin ‘exchange’ interaction due to a mediating particle, first calculated by H.A. Kramers in 1934 [12]. 2 In physics, one-dimensional quantum problems are often known to be ‘easy’. However, it is an interesting open question whether 2-LOCAL HAMILTONIAN on a one-dimensional lattice is indeed in BQP (or even BPP).

3

2

A Spatially Sparse 2-local Hamiltonian Problem

We start by modifying the proof that 3-LOCAL HAMILTONIAN is QMA-complete in Ref. [1]. After that we will use the 3-local to 2-local reduction in Ref. [1]. The essential insight in is (1) to modify any quantum circuit to one in which any qubit is used a constant number of times and (2) make sure that the program to execute the gates in the correct time sequence is spatially local. First consider the following spatial layout of a quantum circuit, see Fig. 1(a). Assume that the quantum circuit is implemented in a one-dimensional array with N qubits where n qubits are input qubits and the other N − n qubits are ancilla qubits. Now, with only polynomial overhead, we modify this circuit such that the gate sequence is executed in R = poly(n) ‘rounds’ such that in every round only 1 gate is performed. In between the rounds all qubits are swapped forward to the next round, starting with the qubits at the bottom in the Figure. This implies that each qubit enters a gate at most 2 times. Let us label the gates with a time-index depending on when they are executed. Then it is clear that in this model time changes locally in the manner indicated in Fig. 1(b). With this circuit we will construct a corresponding 3-local Hamiltonian H such that if on some input | ξ, 0i Arthur’s ‘verifying’ circuit Vx for instance x accepts with probability more than 1 − ǫ, H has an eigenvalue less than ǫ. If Vx accepts with probability less than ǫ then all eigenvalues are larger than 1/2 − ǫ. Since it is known that 3-LOCAL HAMILTONIAN is in QMA [2], this will prove QMA-completeness of 3-LOCAL HAMILTONIAN on a restricted hypergraph. Let the number of computational qubits M = N R. We add T = (2R − 1)N clock-qubits c1 . . . cT such that time t is represented as | 1t 0T −t ic1 ...cT (as in Ref. [1]). These clock-qubits can be added in the ‘third’ dimension, that is, we put the clock-qubits in the plane above the computational qubits, see Fig. 3. The idea is that our 3-local Hamiltonian only has interactions that are spatially local in this two-level structure. The ‘swap’ (R − 1)N clock-qubits are placed in the plane above the computational qubits, in between the rows of qubits to be swapped. Every swap clock-qubit will interact with the two computational qubits to be swapped and with its nearest neighbor clock-qubits. The RN ‘gate’ clock-qubits are placed in the rows right above the computational qubits, half-way between two computational qubits. For every two-qubit gate we will have a 3-local interaction in H between the 2 adjacent computational qubits in the row and the gate clock-qubit above. For every one-qubit gate we will have a 2-local interaction in H between a computational qubit and a nearest neighbor gate clock-qubit. For every I on a computational qubit, there is a 1-local interaction of the nearest neigbor clock-qubit which ‘advances’ the time from t to t+1. Next to these interactions we need include terms in H that (1) initialize the ancillary qubits including all qubits in rows R = 2, 3 etc., (2) penalize a non-acceptance state | 0i at the end of the computation and (3) penalize clock states that are not allowed, i.e. not of the form | 1t 0T −t i. Let U1 . . . UT be the sequence of operations on the computational qubits of the quantum circuit Vx , one operation for every clock-qubit c1 , . . . , cT . The set of operations includes the actual gates, the I operations when only time advances and the swap gates. Let Qin be the set of n qubits that contain the input | ξi. Let qout be the final qubit that is measured in the quantum circuit. For every computational qubit q there is one gate clock-qubit ctq , as indicated in Fig. 3. The 3-local Hamiltonian is as follows: H = Jclock

T −1 X t=1

Hclock,t + Jevolv

T X

Hevol,t + Jstart

t=1

X

q|q ∈Q / in

4

Hstart,q + Hend ,

(2)

where the J-couplings will be chosen at most poly(n). Here Hclock,t = | 01ih01 |ct ct+1 ,  †  t = 1,  | 0ih0 |c1 + | 10ih10 |c1 c2 − U1 ⊗ | 1ih0 |c1 − U1 ⊗ | 0ih1 |c1 , Hevol,t = | 10ih10 |ct−1 ct + | 10ih10 |ct ct+1 − Ut ⊗ | 1ih0 |ct − Ut† ⊗ | 0ih1 |ct , 1 < t < T,  †  | 10ih10 | t = T, cT −1 cT + | 1ih1 |cT − UT ⊗ | 1ih0 |cT − UT ⊗ | 0ih1 |cT , Hstart,q = | 1ih1 |q ⊗ | 0ih0 |ctq ,

Hend = (T + 1)| 0ih0 |qout ⊗ | 1ih1 |cT .

Now the following lemma can be proved: Lemma 1 Let | ψi =

q

| ξi| 0M −n i

1 T +1

PT

t T −t i c1 ...cT t=0 | ξt iq1 ...qM | 1 0

where | ξt i = Ut | ξt−1 i for all 1 ≤ t ≤ T

and | ξ0 i = for some state | ξi of the input qubits. There exist sufficiently large (but poly(n)) couplings Jclock ≫ Jevolv ≫ Jstart = poly(n) and an ǫ > 0 such that the following holds. If the circuit Vx accepts with probability more than 1 − ǫ on some input | ξ, 0i then hψ |H| ψi < ǫ. If Vx accepts with probability less than ǫ on all inputs | ξ, 0i then all eigenvalues of H are larger than 1/2 − ǫ.

Proof [Sketch] There are 2 main differences between our Hamiltonian and the one in Ref. [1]. First, the term Hclock which penalizes the wrong clock states is in our case only between nearest neighbor clock-qubits; this is sufficient to make any state orthogonal to the proper clock states have eigenvalue at least Jclock . The initialization of all qubits 6∈ Qin to | 0i is done in a local fashion unlike what was done in Ref. [1]. The term Hstart,q penalizes qubit q for being in the state | 1i when the adjacent ‘gate’ clock-qubit is in the state | 0i. The gate clock-qubit will be in the state | 0i (enforced by a large Jclock ) as long as the execution of that gate is an event in the future. The technical details of the proof are derived by several applications of the projection Lemma in Ref. [1]. The interaction hypergraph of this 3-local Hamiltonian has a planar embedding, as suggested in Fig. 3. Note that all interactions between clock-qubits in H occur between adjacent clock-qubits. A computational qubit q interacts with at most two computational qubits (in a 3-local ‘gate’ and ‘swap’ interactions) and it interacts at most with two gate clock-qubits and one swap clock-qubit. ˜ Now we can use the result in Ref. [1] to approximate λ(H) of a 3-local Hamiltonian to λ(H) ˜ by a 2-local Hamiltonian H using a perturbation gadget (see also the next Section). Each 3-local ˜ which acts on the same qubits as H, plus at most term in H gets replaced by a new Hamiltonian H 3 × 27 new qubits. In Ref. [1] the three-local interaction is expanded as a sum over at most 27 terms each of which can be written as B1 ⊗ B2 ⊗ B3 with B1 , B2 , B3 ≥ 0 plus some 2-local terms. A single such interaction term is replaced by a 2-local interaction triangle involving 3 new qubits, as shown in Fig. 2. From this it follows that we have a 2-local Hamiltonian problem restricted to an interaction graph which is spatially sparse, that is, we can show that Proposition 1 The problem 2-LOCAL HAMILTONIAN on a spatially sparse interaction graph is QMA-complete. A spatially sparse interaction graph G is defined as a graph which (1) has maximal degree O(1), has a straight-line drawing in the plane such that (2) every edge crosses at most O(1) other edges, (3) every pair of edges that are either adjacent or crossing form a minimum angle of Ω(1) with each other and (4) any edge is of length O(1).

5

3

Perturbation Gadgets

In Ref. [1] the authors reduce the problem 3-LOCAL HAMILTONIAN to 2-LOCAL HAMILTONIAN by introducing a perturbation gadget. The idea is approximate λ(Heff ) of an effective (3-local) ˜ of a 2-local Hamiltonian H ˜ where λ(H) ˜ is calculated using perturbation Hamiltonian Heff by λ(H) ˜ theory. One sets H = H + V where H is the ‘unperturbed’ Hamiltonian which has a large spectral gap ∆ and V is a small perturbation operator. We will choose H such that it has a degenerate ground-space associated with eigenvalue 0 and the eigenvalues of the ‘excited’ eigenstates are at least ∆. The effect of the perturbation V is to lift the degeneracy in the ground-space and create the effective Hamiltonian in this space. More accurately, we have a Hilbert space H = H+ ⊕ H− where H− is the ground-space of H. Let Π± be the projectors on H± and we can define for some operator X, X±∓ = Π± XΠ∓ and X+ ≡ X++ . In order to calculate the perturbed eigenvalues, one introduces the self-energy operator Σ− (z) for real-valued z Σ− (z) = H− + V−− + V−+ G+ (I − V++ G+ )−1 V+− ,

(3)

where we can perturbatively expand (I − V++ G+ )−1 = I + V++ G+ + V++ G+ V++ G+ + . . . .

(4)

Here G+ , called the unperturbed Green’s function (or resolvent) in physics literature, is defined by G−1 + = zI+ − H+ .

(5)

In Ref. [1] the following theorem is proved ˜ − be the space of Theorem 7 ([1]) Let ||V || ≤ ∆/2 where ∆ is the spectral gap of H. Let H ˜ ˜ eigenstates of H with eigenvalues less than ∆/2 and let H|H˜ − be the restriction to this space. Let there be an effective Hamiltonian Heff with Spec(Heff ) ⊆ [a, b]. If the self-energy Σ− (z) for all z ∈ [a − ǫ, b + ǫ] where a < b < ∆/2 − ǫ for some ǫ > 0, has the property that ||Σ− (z) − Heff || ≤ ǫ,

(6)

˜ j of H| ˜ ˜ is ǫ-close to the jth eigenvalue of Heff . In particular then each eigenvalue λ H− ˜ ≤ ǫ. |λ(Heff ) − λ(H)|

3.1

(7)

Mediator Qubit Gadgets

The gadgets that we introduce below to accomplish the reduction are what we call mediator qubit gadgets and seem to be useful in general to manipulate 2-local interactions. The idea is that we replace a direct 2-local interaction between qubits with indirect interactions through a mediator qubit. In the ground-state of the unperturbed Hamiltonian H the mediator qubit is in state | 0i. The perturbation V is chosen such that interaction with the other qubits can flip the mediator qubit. The perturbative corrections to the self-energy, up to second order in perturbation theory, involve the process of flipping the mediator qubit by interaction with a qubit i and flipping the mediator qubit back to | 0i by a second interaction with a qubit j. If i = j we potentially obtain some 1-local terms. For i 6= j we obtain an effective 2-local interaction between i and j. 6

We first explain the action of the basic gadgets depicted in Figs. 4,5 and 6. We will assume that we have a simple graph (not a multigraph) of which the edges are Pauli edges (we will show in Section 3.3 how to reduce any 2-local term to terms involving only Pauli edges). We will illustrate the action in detail with the edge subdivision gadget; the other two gadgets are based on the same idea.

3.2

The Basic Gadgets

The Edge Subdivision Gadget. Assume that the 2-local operator associated with edge ab is of the form αab Pa ⊗ Pb where αab is constant or at most polynomial in n. The edge ab is part of a larger graph and we can always write its Hamiltonian as Heff = Helse − (−αab Pa + Pb )2 /2,

(8)

where Helse contains all other 1 or 2-local operators in the graph, that is, everything except the ˜ = H + V are the operator associated with edge ab. The terms in the gadget Hamiltonian H following p (9) H = ∆ΠZ+ [w], V = Helse + ∆/2 (−αab Pa + Pb ) ⊗ X[w]. Here ΠZ+ [w] = (I − Z[w])/2 is the projector onto the state | 1i of the mediator qubit w. The operator X[w] is the Pauli X operator acting on qubit w. The degenerate ground-space H− of |w H has the mediator qubit in the state | 0i. We have the following: H− = 0, G+ (z) = | 1ih1 z−∆ , V−− = Helse ⊗ | 0ih0 |w and p V+− = ∆/2(−αab Pa + Pb ) ⊗ | 1ih0 |w . (10) Thus the self-energy Σ− (z) equals     ||V ||3 ∆ 2 (−αab Pa + Pb ) ⊗ | 0ih0 |w + O . Σ− (z) = Helse + 2(z − ∆) (z − ∆)2

(11)

In order for Theorem 7 to apply the following must hold: (1) for z ∈ Spec(Heff ) ± ǫ, Σ− (z) should be ǫ-close to Heff for some small ǫ and (2) ||V || ≤ ∆/2. We have ||Heff || ≤ ||Helse || + C1 |αab |2 2 for some √ constant C1 . Thus we consider the interval |z| ≤ ||Helse || + C1 |αab | + ǫ. We also have ||V || ≤ ∆ (||Helse || + C2 |αab |) for some constant C2 . If we choose ∆ = (||Helse || + C3 |αab |)6 /ǫ2 ,

(12)

3

for some constant C3 , then we have ||V∆||2 ≤ O(ǫ) and z/∆ ≤ O(ǫ2 ). Thus for this choice we have ˜ = O(ǫ). Σ− (z) = Heff ⊗ | 0ih0 |w + O(ǫ). From Theorem 7 it follows that |λ(Heff ) − λ(H)| When ||Helse ||, |αab | and 1/ε are polynomial in n, it is clear from Eq. (12), that the norm of the gadget Hamiltonian which uses ∆ is polynomially larger than the norm of the effective Hamiltonian. This implies that the gadget can only be used a constant number of times in series if norms have to remain polynomial. The Cross Gadget. For the Cross Gadget we assume that we have a graph G which when embedded in the plane contains two crossing edges such as in Fig. 5. Assume that the operator on edge ad is αad Pa ⊗ Pd and on edge bc we have αbc Pb ⊗ Pc . Our desired Hamiltonian is Heff = Helse − (−αad Pa − αbc Pb + Pc + Pd )2 /2, 7

(13)

where where Helse contains all other edges in the graph G plus some additional edges ab, bd, cd and ac. Thus there will be terms in Helse which cancel, for example, the edge operator αad Pa ⊗ Pc ˜ = H + V with generated by the last term. As before we set H p (14) H = ∆ΠZ+ [w], V = Helse + ∆/2 (−αad Pa − αbc Pb + Pc + Pd ) ⊗ X[w],

and the analysis follows as for the edge subdivision gadget. Note that if there are no edges ab, bd, ˜ as indicated in Fig. 5. cd, or ac in Heff , there will be such edges in H, The Y Gadget. For the Y gadget we have a subgraph as in Fig. 6 where the operator on edge ab is αab Pa ⊗ Pb and on edge ac it is αac Pa ⊗ Pc . The Y gadget merges the 2 edges coming from vertex a at the cost of creating an additional edge between b and c. Our desired Hamiltonian is Heff = Helse − (Pa − αab Pb − αac Pc )2 /2, where Helse contains all other terms not involving edge ab and ac. As before we set p H = ∆ΠZ+ [w], V = Helse + ∆/2 (Pa − αab Pb − αac Pc ) ⊗ X[w],

(15)

(16)

and the analysis follows as before.

3.3

Two Composite Gadgets

We first present a gadget in which the edge subdivision gadget is used in parallel. The Pauli Reduction Gadget An arbitrary 2-local Hamiltonian on two qubits can be expanded in terms of products of Pauli-matrices, which gives at most 9 parallel Pauli edges. By applying the edge subdivision gadget for each Pauli term in parallel we can create a graph where m m each edge is associated with a single Pauli product of the form αm ab Pa ⊗ Pb , see Fig. 7. Thus we have X m 2 m (17) Heff = Helse − (−αm ab Pa + Pb ) /2, m

and choose

H=∆

X

ΠZ+ [wm ], V = Helse +

m

p

∆/2

X m

m m m (−αm ab Pa + Pb ) ⊗ X[w ].

(18)

The degenerate ground-space H− of H0 has all additional vertices w1 . . . w9 in the state | 0i. Let h(x) be the Hamming weight of a bitstring x of the qubits w1 . . . w9 . We have the following P | xihx | , V−− = Helse ⊗ | 00 . . . 0ih00 . . . 0 | and G+ = x z−h(x)∆ V+− =

p

∆/2

X m m (−αm ab Pa + Pb )| 00 . . . 1m . . . 0ih00 . . . 0 |,

(19)

m

where | 00 . . . 1m . . . 0i has qubit wm in the state | 1i. To second order in perturbation theory, there are no cross-gadget terms in Σ− (z). Thus the self-energy Σ− (z) to second order equals !   X ||V ||3 ∆ 2 m m m Σ− (z) = Helse + (−αab Pa + Pb ) ⊗ | 00 . . . 0ih00 . . . 0 | + O , (20) 2(z − ∆) m (z − ∆)2 which for ∆ = poly(n)/ǫ2 for some sufficiently large poly(n) gives Σ− (z) = Heff ⊗| 00 . . . 0ih00 . . . 0 |+ O(ǫ). 8

This gadget can of course also be applied to all edges e in the graph in parallel as we will do in our reduction. The important thing to note is that if we use any mediator qubit gadget, –Cross, Y or Edge Subdivision– , in parallel in a variety of places in a graph, we will not have any crossgadget terms to second order in perturbation theory. The higher order terms may be polynomially larger, but by choosing a sufficiently large (but polynomially-sized) gap ∆ we can always make the ˜ be arbitrarily close to λ(Heff ). approximation λ(H) The Triangle Gadget We will now show how the Y gadget can be used in order to reduce the degree of a vertex, see Fig. 8, by applying it 3 times in series. Our desired effective Hamiltonian is Heff = Helse + αab Pa ⊗ Pb + αac Pa ⊗ Pc , (21) where Helse contains everything except the operators on edges ab and ac. The Y gadget has gadget Hamiltonian p ′ (22) H1 = ∆1 ΠZ+ [w1 ] + Helse + αab αbc Pb ⊗ Pc + ∆1 /2 (Pa − αab Pb − αac Pc ) ⊗ X[w1 ],

′ is Helse + cI where c depends on αab and αac . We apply the Y gadget again in order where Helse to get rid of the additional bc edge. Note that we can pair the Pauli edges with endvertex b since they both have operator Pb on b. Thus we apply the Y gadget on vertex b and obtain Hamiltonian H2 . Lastly we can reduce the degree of the c vertex to what it was before by pairing the edges cw1 and cw2 in the Y gadget and obtain some Hamiltonian H3 . In the final Hamiltonian H3 vertex a has reduced its degree by 1, whereas vertices b and c have the same degree as in the original Heff in Eq. (21). For every step we can choose ∆ such that the lowest eigenvalue changes by at most Cǫ for some constant C. This implies that |λ(Heff ) − λ(H3 )| ≤ 3Cǫ.

4

2-LOCAL HAMILTONIAN

on a 2-D Square Lattice

Now we are ready to state our main reduction which we obtain by applying the gadgets in the previous section. Together with Proposition 1, this Lemma implies Theorem 4. Lemma be a 2-local Hamiltonian related to a spatially sparse graph G = (V, E) where P 2 Let HGP HG = e∈E He + v∈V Hv such that kHe k ≤ poly(n) and kHv k ≤ poly(n). Then there exists a graph Gsim which is planar with maximum Pauli degree at most 4 and a (polynomially bounded) 2-local Hamiltonian HGsim such that |λ(HG ) − λ(HGsim )| = O(ǫ),

(23)

for some constant ǫ > 0. Moreover, there is a planar straight-line drawing of Gsim such that all edges in Gsim have length O(1), and all angles between adjacent edges are Ω(1). Proof 1. Our first step is to construct a new graph and Hamiltonian in which every edge is a Pauli edge. This we will accomplish by parallel application of the Pauli reduction gadget on every edge of the graph, see Fig. 7. If in the planar straight-line drawing of G every edge crosses at Ccross ′ = 9Ccross . other edges then for this new graph every edge has crossing number at most Ccross ′ ⌉ 2. Next, we reduce the number of crossings per edge, by subdividing each edge at most ⌈log2 Ccross times, that is we apply the edge subdivision gadget everywhere in the graph this number of times, see Fig. 9. Every subdivision is done in parallel on all edges of the graph that need subdividing. 3. Then we use the edge subdivision gadget to localize each crossing, see Fig. 4. We apply the edge subdivision gadget in parallel on every crossing in the graph and we repeat the process 4 times so 9

that for all crossing edges ab, cd, the quadrilateral acbd contains only these points and the crossing edges. 4. We apply the Cross gadget, see Fig. 5, in parallel to every localized crossing in order to remove the crossing. This will also generate 2-local terms around the square. 5. We use the edge subdivision gadget in order to localize each vertex with degree more than 4, see Fig. 11. Then we are ready to reduce the degree of these vertices. 6. Note that the mediator vertices that are created with the previous procedures have (X-)degree 2 or 4. Consider the set of vertices with degree more than 4 and note that there is no direct edge between vertices of this kind due to the previous steps. For a vertex with X-degree dx , Y-degree dy , Z-degree dz we do the following. We pair the X-edges and apply to each pairing a Y gadget. This means we have reduced the X-degree to ⌈dx /2⌉. In parallel we pair the Y-edges and the Z-edges using the Y gadget, halving their degrees. We do this single perturbative step in parallel for all high-degree vertices in the graph. Then, in order to remove the additional edges that are created in the Y gadget, we apply the Y gadget twice more for every pairing, as in the Triangle gadget, illustrated in Fig. 8. We repeat this process O(1) number of times (since the maximum degree initially was O(1)) until the total degree of every vertex is at most 4. The pairing of the edges will lead to O(1) additional crossings which can be kept inside a small ‘ball’ around the vertex. The removal of these crossings will be our last step. 7. For every vertex whose degree reduction process has led to crossings in the ball around it we carry out steps 2, 3 and 4 again to remove these crossings. Note that due to the localization of each crossing, the additional edges created by the Cross gadget, give rise to mediator qubits with degree at most 4. Since the initial degree of every vertex was O(1), the number of crossings is O(1), and so will the number of steps be to remove the crossings. We obtain a graph is planar and which has degree at most 4, Theorem 7 then gives the result, Eq. (23).

4.1

Representation on a 2-D Square Lattice

Any planar graph G = (V, E) with maximal degree 4 in which the (straight-line) edges have length O(1) and adjacent edges form an angle of Ω(1) can be represented on a planar square lattice in the following sense: each vertex a of G is mapped to some lattice site φ(a) inside the square [−O(|V |), O(|V |)]2 , and each edge ab of G is mapped to a lattice path φ(ab) of length O(1) from φ(a) to φ(b) that does not cross any other vertices or any other path. To see this, one can look at Fig. 12 or follow these steps: draw a fine square grid on the plane. If the spacing between points on the grid is small enough, moving each vertex a of G to a vertex in the lattice (and redrawing the edges) still leaves the graph planar, with O(1)-length edges and Ω(1) angles. Now for each edge, draw a lattice path that stays close to the edge. If the grid is fine enough, these paths can never cross outside an O(1)-size square (indicated in grey in Fig. 12) around the vertices of the graph, because of the angle condition. By further refining the grid if necessary, one can reroute each of the paths stemming out of a vertex a inside of a’s square, so that no two different paths collide. It is easy to see that we only need the grid to have spacing Ω(1), and that all the other conditions above are satisfied. Clearly, this embedding can be found efficiently, given the adequate embedding of G. If H is a Hamiltonian that has G as (Pauli) interaction graph, one can use the edge subdivision gadget O(1) ˜ thus times in parallel to map each edge ab to a path of the same length as φ(ab). The Hamiltonian H ˜ obtained has interaction graph φ(G) and λ(H) is O(ǫ)-close to λ(H). Together with Proposition 1 and Lemma 2 this is the proof of Theorem 5. 10

5

Universal Quantum Adiabatic Computation

In Ref. [1] the authors show that their perturbation-theoretic reduction of 3-LOCAL HAMILTONIAN to 2-LOCAL HAMILTONIAN also reduces 3-local adiabatic computations to 2-local ones. There are two main ingredients in this proof. First, the authors note that the perturbation gadgets preserve both the ground-state energy of the Hamiltonian and its gap up to a small error. Secondly, if the gap of the 3-local Hamiltonian is Ω(1/poly(n)) to begin with, the ground-state of the final Hamiltonian obtained in the reduction is close to | ψi ⊗ | ξi, where | ψi is the same ground-state of the original 3-local Hamiltonian and | ξi is some other state of the additional vertices. Thus the result of the 3local computation can be read off that of the 2-local one, and the time necessary to reach the result is still polynomial. The same argument applies, mutatis mutandis, to Theorem 6. The construction of the 3-local Hamiltonian in Ref. [5] needs to be slightly modified as in Section 2 and be shown to have a large gap. From then on, the proof of Theorem 5 applies almost without modification, and we deduce that the final Hamiltonian has large gap and the appropriate ground-state.

6

Acknowledgements

We thank David DiVincenzo for an inspiring discussion about superexchange. We acknowledge support by the NSA and the ARDA through ARO contract number W911NF-04-C-0098.

References [1] J. Kempe, A. Kitaev, and O. Regev. The Complexity of the Local Hamiltonian Problem. In Proc. of 24th FSTTCS, pages 372–383, 2004, http://arxiv.org/abs/quant-ph/0406180. [2] A. Yu. Kitaev, A.H. Shen, and M.N. Vyalyi. Classical and Quantum Computation. Vol. 47 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2002. [3] D. Aharonov and T. Naveh. http://arxiv.org/abs/quant-ph/0210077.

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NP—A

Survey.

2002,

[4] J. Kempe and O. Regev. 3-Local Hamiltonian is QMA-complete. Quantum Information and Computation, 3(3):258–264, 2003, http://arxiv.org/abs/quant-ph/0302079. [5] D. Aharonov, W. van Dam, Z. Landau, S. Lloyd, J. Kempe, and O. Regev. Universality of Adiabatic Quantum Computation. In Proceedings of 45th FOCS, 2004, http://arxiv.org/abs/quant-ph/0405098. [6] A. Ambainis and O. Regev. An Elementary Proof of the Quantum Adiabatic Theorem. 2004, http://arxiv.org/abs/quant-ph/0411152. [7] D. Janzing, P. Wocjan, and T. Beth. Identity Check is QMA-complete. http://arxiv.org/abs/quant-ph/9610012.

1996,

[8] B. Nachtergaele. Quantum Spin Systems. To appear in the Encyclopedia of Mathematical Physics (Elsevier), 2004, http://arxiv.org/abs/math-ph/0409006. [9] M. Plischke and B. Bergersen. Equilibrium Statistical Physics. World Scientific, Singapore, 1994. 11

[10] C. H. Papadimitriou. Computational Complexity. Addison-Wesley, 1994. [11] F. Barahona. On the Computational Complexity of Ising Spin Glass Models. Jour. of Phys. A: Math. and Gen., 15:3241–3253, 1982. [12] H.A. Kramers. L’ interaction entre les atomes magn´etog`enes dans un cristal paramagn´etique. 1:182, 1934.

12

R=1

R=2

Time

R=3

(a)

(b)

Figure 1: (a) Two-dimensional layout of the quantum circuit in which qubits are indicated by circles. One and two-qubit gates are indicated by boxes. Swap operations between qubits are indicated with arrows. (b) The ‘time sweep’ indicated on top of the same two-dimensional lay-out.

Figure 2: In Ref. [1] a 3-local interaction indicated by the grey triangle gets replaced by 2-local interactions with new vertices.

13

=

Swap

=

Gate

q ct

q

Figure 3: The interaction hypergraph of the 3-local Hamiltonian for R = 3 and N = 6. The • qubits are computational qubits and ◦ qubits are clock-qubits. The interactions that are at most 3-local are indicated by triangles and interactions that are at most 2-local are indicated by line segments.

Pa

Pb

a

b

Pa a

X X w

Pb b

Figure 4: Edge subdivision gadget. An edge is subdivided into two edges by placing a mediator qubit vertex w in the middle. The operators Pa , Pb , X next to the edges indicate which Pauli operators correspond to those edges.

14

a

b

Pa

Pb

Pd

Pc

c

Pa

a

Pb

Pa P a

X X

Pd P d

Pc

c

b

Pb

Pb

X X

Pc

d

d

Pd

Pc

Figure 5: Cross gadget. A crossing between two edges is removed by placing a mediator qubit in the middle. Additional edges ab, ac, bd and cd are created.

b

Pb

c

Pa

a

Pa

Pb Pb

Pc

Pc

b

c

P

c

w

X

X X Pa

a

Figure 6: Y gadget. Two edges of the same type at vertex a are merged by the placement of a mediator qubit w. The additional edge bc is created.

15

Figure 7: Pauli reduction gadget. For each (pure) Pauli interaction between vertices a and b we insert a mediator qubit.

Pb

b

c

a

X

Pc

X

X

Pc

Pb w2

w3 w1

X

Figure 8: Triangle Gadget. Three applications of the Y gadget give rise to a ‘mediator triangle’ such that vertices b and c have the same degree as before and vertex a has reduced its degree by 1. 16

Figure 9: An edge that crosses C other edges is subdivided ⌈log C⌉ times by inserting a mediator qubit.

a

b

c

d

a

b

c

d

Figure 10: Localizing a crossing by applying the edge subdivision gadget four times.

17

Figure 11: Localizing a vertex.

18

Figure 12: A planar graph of maximal degree ≤ 4 and its representation in the lattice. In the gray squares, the paths are rerouted to avoid crossings.

19