Realizable Hamiltonians for Universal Adiabatic Quantum Computers

Haverford College Haverford Scholarship

Faculty Publications Physics

2008

Jacob D. Biamonte

Peter John Love Haverford College

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Repository Citation "Realizable Hamiltonians for Universal Adiabatic Quantum Computers" Jacob D. Biamonte, Peter J. Love, Phys. Rev. A , 78 , 012352 (2008), http://www.arxiv.org/abs/0704.1287

This Journal Article is brought to you for free and open access by the Physics at Haverford Scholarship. It has been accepted for inclusion in Faculty Publications by an authorized administrator of Haverford Scholarship. For more information, please contact [email protected]. PHYSICAL REVIEW A 78, 012352 ͑2008͒

Realizable Hamiltonians for universal adiabatic quantum computers

Jacob D. Biamonte1,* and Peter J. Love2,† 1Oxford University Computing Laboratory, Wolfson Building, Parks Road, Oxford, OX1 3QD, United Kingdom 2Department of Physics, Haverford College, 370 Lancaster Avenue, Haverford, Pennsylvania 19041, USA ͑Received 27 April 2007; revised manuscript received 20 May 2008; published 28 July 2008͒ It has been established that local lattice spin Hamiltonians can be used for universal adiabatic quantum computation. However, the two-local model Hamiltonians used in these proofs are general and hence do not limit the types of interactions required between spins. To address this concern, the present paper provides two simple model Hamiltonians that are of practical interest to experimentalists working toward the realization of a universal adiabatic quantum computer. The model Hamiltonians presented are the simplest known quantum- Merlin-Arthur-complete ͑QMA-complete͒ two-local Hamiltonians. The two-local Ising model with one-local transverse ﬁeld which has been realized using an array of technologies, is perhaps the simplest quantum spin model but is unlikely to be universal for adiabatic quantum computation. We demonstrate that this model can be rendered universal and QMA-complete by adding a tunable two-local transverse ␴x␴x coupling. We also show the universality and QMA-completeness of spin models with only one-local ␴z and ␴x ﬁelds and two- local ␴z␴x interactions.

DOI: 10.1103/PhysRevA.78.012352 PACS number͑s͒: 03.67.Lx

I. INTRODUCTION system has at most k-local interactions. A “yes” instance is shown by providing a witness eigenstate with energy below What are the minimal physical resources required for uni- the lowest promised value. versal quantum computation? This question is of interest in The problem 5-LOCAL HAMILTONIAN was shown to be understanding the connections between physical and compu- QMA-complete by Kitaev ͓8͔. To accomplish this, Kitaev tational complexity, and for any practical implementation of modified the autonomous quantum computer proposed by ͓ ͔ quantum computation. In 1982, Barahona 1 showed that Feynman ͓9͔. This modification later inspired a proof of the finding the ground state of the random field Ising model is polynomial equivalence between quantum circuits and adia- NP-hard. Such observations fostered approaches to solving batic evolutions by Aharonov et al. ͓10͔͑see also Refs. ͓ ͔ problems based on classical 2 and later quantum annealing ͓11,12͔͒. Kempe, Kitaev, and Regev subsequently proved ͓ ͔ 3 . The idea of using the ground-state properties of a quan- QMA-completeness of 2-LOCAL HAMILTONIAN ͓14͔. Oliveira tum system for computation found its full expression in the and Terhal then showed that universality remains even when ͓ ͔ adiabatic model of quantum computation 4 . This model the two-local Hamiltonians act on particles in a subgraph of works by evolving a system from the accessible ground state the two-dimensional ͑2D͒ square lattice ͓15͔. Any QMA- of an initial Hamiltonian Hi to the ground state of a final complete Hamiltonian may realize universal adiabatic quan- Hamiltonian Hf, which encodes a problem’s solution. The tum computation, and so these results are also of interest for ෈͓ ͔ ͑ ͒ evolution takes place over parameters s 0,1 as H s the implementation of quantum computation. ͑ ͒ = 1−s Hi +sHf, where s changes slowly enough that transi- Since the 1-LOCAL HAMILTONIAN is efficiently solvable, ͓ ͔ tions out of the ground state are suppressed 5 . The simplest an open question is to determine which combinations of two- adiabatic algorithms can be realized by adding noncommut- local interactions allow one to build QMA-complete Hamil- ͚ ␴z ing transverse field terms to the Ising Hamiltonian ihi i tonians. Furthermore, the problem of finding the minimum ͚ ⌬ ␴x ͚ ␴z␴z ͑ ͓ ͔͒ + i i i + i,jJij i j, see Ref. 6 . However, it is unlikely set of interactions required to build a universal adiabatic that the Ising model with transverse field can be used to quantum computer is of practical, as well as theoretical, in- ͓ ͔ construct a universal adiabatic quantum computer 7 . terest: every type of two-local interaction requires a separate What then are the simplest Hamiltonians that allow uni- type of physical interaction. To address this question we versal adiabatic quantum computation? For this we turn to prove the following theorems: ͑ ͒ the complexity class quantum-Merlin-Arthur QMA , the Theorem 1. The problem: the 2-LOCAL ZZXX HAMIL- quantum analog of NP, and consider the QMA-complete TONIAN is QMA-complete, with the ZZXX Hamiltonian 1 ͓ ͔ problem k-LOCAL HAMILTONIAN 8 . One solves k-LOCAL given as HAMILTONIAN by determining if there exists an eigenstate with energy above a given value or below another—with a ␴z ⌬ ␴x ␴z␴z ␴x␴x HZZXX = ͚ hi i + ͚ i i + ͚ Jij i j + ͚ Kij i j . promise that one of these situations is the case—when the i i i,j i,j ͑1͒ Theorem 2. The problem: 2-LOCAL ZX HAMILTONIAN is *[email protected].ac.uk QMA-complete, with the ZX Hamiltonian given as †[email protected] 1 ␴z ⌬ ␴x ␴z␴x ␴x␴z ͑ ͒ In computer science parlance, “problems” such as SAT, and HZX = ͚ hi i + ͚ i i + ͚ Jij i j + ͚ Kij i j . 2 LOCAL HAMILTONIAN are capitalized. i i iϽj iϽj

Structure. In the present paper we briefly review the stan- spanned by all states with a single domain wall. The term ͉ ͗͘ ͉ dard circuit to adiabatic construction to show that 2-LOCAL Hclockint applies a penalty 1 1 t=1 to the first qubit to ensure HAMILTONIAN is QMA-complete when restricted to real- that the clock is in state ͉0͘ T at time t=0. valued Hamiltonians. We then show how to approximate the Hprop acts both on logical and clock qubits. It ensures that ground states of such two-local real Hamiltonians by the ZX the ground state is the history state corresponding to the ͚T and ZZXX Hamiltonians. We conclude this work by provid- given circuit. Hprop is a sum of T terms, Hprop= t=1Hprop,t, ing references confirming our claim that the Hamiltonians in where each term checks that the propagation from time t ͑ ͒ ͑ ͒ Յ Յ Eqs. 1 and 2 are highly relevant to experimentalists at- −1 to t is correct. For 2 t T−1, Hprop,t is defined as tempting to build a universal adiabatic quantum computer. def ͉ ͗͘ ͉ ͉ ͗͘ ͉ Hprop,t = 1 t −1 t −1 − Ut t t −1 II. THE PROBLEM † ͉ ͗͘ ͉ ͉ ͗͘ ͉ ͑ ͒ − Ut t −1 t + 1 t t , 7 The translation from quantum circuits to adiabatic evolu- ͉ ͗͘ ͉ ͉ ͗͘ ͉ where operators t t−1 = 110 100 ͑t−1,t,t+1͒, etc., act on tions began when Kitaev ͓8͔ replaced the time dependence of clock qubits t−1, t, and t+1 and where the operator Ut is the gate model quantum algorithms with spatial degrees of free- tth gate in the circuit. For the boundary cases ͑t=1,T͒, one dom using the nondegenerate ground state of a positive ͑ writes Hprop,t by omitting a clock qubit t−1 and t+1, respec- semidefinite Hamiltonian: tively͒. 0=H͉␺ ͘ = ͑H + H + H + H ͉͒␺ ͘. ͑3͒ We have now explained all the terms in the Hamiltonian hist in clock clockinit prop hist from Eq. ͑3͒—a key building block used to prove the QMA- To describe this, let T be the number of gates in the quantum completeness of 5-LOCAL HAMILTONIAN ͓8͔. The construc- circuit with gate sequence UT ¯U2U1 and let n be the num- tion reviewed in the present section was also used in a proof ber of logical qubits acted on by the circuit. Denote the cir- of the polynomial equivalence between quantum circuits and ͉ ͘ ͉␺ ͘ cuit’s classical input by x and its output by out . The his- adiabatic evolutions ͓10͔. Which physical systems can imple- tory state representing the circuit’s entire time evolution is ment the Hamiltonian model of computation from Eq. ͑3͒? Ideally, we wish to find a simple Hamiltonian that is in prin- 1 ͉␺ ͘ ͓͉ ͘ ͉ ͘T ͉ ͘ ͉ ͉͘ ͘T−1 ciple realizable using current, or near-future technology. The hist = x 0 + U1 x 1 0 ͱT +1 ground states of many physical systems are real-valued, such as the ground states of the Hamiltonians from Eqs. ͑1͒ and + U U ͉x͘ ͉11͉͘0͘ T−2 + 2 1 ¯ ͑2͒. So a logical first step in our program is to show the ͉ ͘ ͉ ͘T͔ ͑ ͒ + UT ¯ U2U1 x 1 , 4 QMA-completeness of general real-valued local Hamilto- nians. where we have indexed distinct time steps by a T qubit unary clock. In the following, tensor product symbols separate operators acting on logical qubits ͑left͒ and clock qubits ͑right͒. A. The QMA-completeness of real-valued Hamiltonians Hin acts on all n logical qubits and the first clock qubit. By Bernstein and Vazirani showed that arbitrary quantum cir- annihilating time-zero clock states coupled with classical in- cuits may be represented using real-valued gates operating ͉͑ ͘ ͉ ͒͘ ͓ ͔ put x, Hin ensures that valid input state x 0¯0 is in on real-valued wave functions 16 . Using this idea, one can the low-energy eigenspace: show that 5-LOCAL REAL HAMILTONIAN is already QMA- ͓ ͔ n complete—leaving the proofs in Ref. 8 otherwise intact and ͑ ͒ ͑ ͉ ͗͘ ͉͒ ͉ ͗͘ ͉ changing only the gates used in the circuits. Hin from Eq. 5 Hin = ͚ 1 − xi xi 0 0 1 ͑ ͒ i=1 and Hclock from Eq. 6 are already real-valued and at most ͑ ͒ two-local. Now consider the terms in Hprop from Eq. 7 for n † 1 the case of self-inverse elementary gates Ut =U ͩ ͪ ͓ ͑ ͒xi␴z͔ ͑ ␴z ͒ ͑ ͒ t + ͚ 1 − −1 i 1 + 1 . 5 4 i=1 U 1 ͑ ␴z ͒͑ ␴z ͒ ͑ ␴z ͒␴x͑ ␴z ͒ Hprop,t = 1 − ͑t−1͒ 1 + ͑t+1͒ − 1 − ͑t−1͒ t 1 + ͑t+1͒ . Hclock is an operator on clock qubits ensuring that valid unary 4 4 ͉ ͘ ͉ ͘ ͉ ͘ clock states 00¯0 , 10¯0 , 110¯0 , etc., span the low- ͑8͒ energy eigenspace For the boundary cases ͑t=1,T͒, we define T−1

H = ͚ ͉01͗͘01͉͑ ͒ 1 1 clock t,t+1 H = ͑1 + ␴z ͒ − U ͑␴x + ␴x␴z ͒, t=1 prop,1 2 2 1 2 1 1 2 T−1 1 1 1 ͫ͑ ͒ ␴z ␴z ͚ ␴z␴z ͬ ͑ ͒ z x z x = T −1 + − − ͑ ͒ , 6 ͑ ␴ ͒ ͑␴ ␴ ␴ ͒ ͑ ͒ 1 1 T t t+1 Hprop,T = 1 − ͑T−1͒ − UT T − ͑T−1͒ T . 9 4 t=1 2 2 where the superscript ͑t,t+1͒ indicates the clock qubits acted The terms from Eqs. ͑8͒ and ͑9͒ acting on the clock space are on by the projection. This Hamiltonian has a simple physical already real-valued and at most three-local. As an explicit interpretation as a line of ferromagnetically coupled spins example of the gates Ut, let us deﬁne a universal real-valued with twisted boundary conditions, so that the ground state is and self-inverse two-qubit gate

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1 1 R ͑␾͒ = ͑1 + ␴z͒ + ͑1 − ␴z͒ ͓sin͑␾͒␴x + cos͑␾͒␴z͔. ij 2 i 2 i i j ͑␾͒ ͑␲/ ͒ The gate sequence Rij Rij 2 recovers the universal gate from Refs. ͓17͔. This is a continuous set of elementary gates parametrized by the angle ␾. Discrete sets of self-inverse gates which are universal are also readily constructed. For example, Shi showed that a set comprising the controlled- NOT ͑CNOT͒ plus any one-qubit gate whose square does not ͓ ͔ ␴z␴x preserve the computational basis is universal 13 . We imme- FIG. 1. The ZZXX gadget used to approximate the operator i j diately see that a universal set of self-inverse gates cannot using only ␴x␴x and ␴z␴z interactions. The present figure presents a contain only the CNOT and a single one-qubit gate. However, diagrammatic representation of the perturbation Hamiltonian V ͑ ͒ the set ͕CNOT,X,cos ␺X+sin ␺Z͖ is universal for any single =V1 +V2 +V3 from Eq. 12 applied to qubits i, j and k. Not shown value of ␺ which is not a multiple of ␲/4. in the present figure is an overall constant energy shift of D. A reduction from 5-LOCAL to 2-LOCAL HAMILTONIAN was accomplished by the use of gadgets that reduced three-local V = ͓Y + D͑␴x + 1͔͒ 1 − A␴z ͉0͗͘0͉ , Hamiltonian terms to two-local terms ͓14͔. From the results 1 j k i k in Ref. ͓14͔͑see also Ref. ͓15͔͒ and the QMA-completeness V = B͑␴x + ͒ ␴x, V = C␴z ͉1͗͘1͉ . ͑12͒ of 5-LOCAL REAL HAMILTONIAN, it now follows that 2-LOCAL 2 j 1 k 3 i k REAL HAMILTONIAN is QMA-complete and universal for The term V2 above allows the mediator qubit k to undergo adiabatic quantum computation. We note that the real prod- virtual excitations and applies an ␴x term to qubit j during uct ␴y ␴y, or tensor powers thereof, are not necessary in L L i j transitions between the − and + subspaces. During excita- any part of our construction, and so Hamiltonians composed L ␴z tion into +, the term V3 applies a term to qubit i. This of the following pairwise products of real-valued Pauli ma- perturbation is illustrated in Fig. 1 trices are QMA-complete and universal for adiabatic quan- Let ⌸Ϯ be projectors on LϮ; for arbitrary operator O we ͓ ͔ tum computation 25 define OϮϯ=⌸ϮO⌸ϯ ͑OϮϮ=⌸ϮO⌸Ϯ͒ and let ␭͑O͒ denote the lowest eigenvalue of O. One approximates ␭͑H ͒ of the ͕1,1 ␴x,1 ␴z,␴x 1, targ desired low-energy effective two-local Hamiltonian by a re- ␴z ,␴x ␴z,␴z ␴x,␴x ␴x,␴z ␴z͖. ͑10͒ ˜ 1 alizable two-local physical Hamiltonian H=Hp +V, where To prove our theorems ͑1͒ and ͑2͒, we will next show that ␭͑H˜ ͒ is calculated using perturbation theory. The spectrum of ͑ ͒ ˜ one can approximate all the terms from Eq. 10 using either H−− is approximated by the projection of the self-energy op- the ZX or ZZXX Hamiltonians—the Hamiltonians from Eqs. erator ⌺͑z͒ for real-valued z which has the following series ͑1͒ and ͑2͒, respectively. We do this using perturbation expansion ͓ ͔ theory 14,15 to construct gadget Hamiltonians that ap- 0th 1st 2nd ␴z␴x ␴x␴z proximate the operators i j and i i with terms from the ␴z␴z ␴x␴x ⌺ ͑z͒ = H + V + V G ͑z͒V ZZXX Hamiltonian as well as the operators i i and i j −− p− −− −+ ++ +− with terms from the ZX Hamiltonian. V G ͑z͒V G ͑z͒V + −+ ++ + ++ +−

B. The ZZXX gadget 3rd O͑ʈVʈ4␦−3͒ ¯ ͑13͒ We use the ZZXX Hamiltonian from Eq. ͑1͒ to construct + + . ␴z␴x ␴x␴x ␴z␴z the interaction i j from and interactions. Let Note that with our penalty Hamiltonian H =0, and for the ␣ ␴z␴x ͉ ͗͘ ͉ −− Heff= ij i j 0 0 k, where qubit k is an ancillary qubit perturbing Hamiltonian V=V1 +V2 +V3 only V1 is nonzero in and deﬁne the penalty Hamiltonian Hp and corresponding the low energy subspace, V and V are nonzero in the high- Green’s function G͑z͒ as follows: 1 3 energy subspace, and only V2 induces transitions between the ␦ def two subspaces. The nonzero projections are ␦͉ ͗͘ ͉ ͑ ␴z͒ ͑ ͒ ͑ ͒−1 ͑ ͒ Hp = 1 1 k = 1 − and G z = z1 − Hp . 11 k ͓ ␴z ͑␴x ͔͒ ͉ ͗͘ ͉ 2 V1−− = Y + A i + D j + 1 0 0 k,

Hp splits the Hilbert space into a degenerate low-energy L ͕͉͉ ͉͘ ͉͘∀ ෈͕ ͖͖ V = B͑␴x + ͒ ͉0͗͘1͉ , eigenspace − =span sisj 0 si ,sj 0,1 , in which qu- 2−+ j 1 k ͉ ͘ ␦ L bit k is 0 , and a energy eigenspace + ͕͉͉ ͉͘ ͉͘∀ ෈͕ ͖͖ ͉ ͘ ͑␴x ͒ ͉ ͗͘ ͉ =span sisj 1 si ,sj 0,1 , in which qubit k is 1 . V2+− = B j + 1 1 0 k, First, we give the ZZXX Hamiltonian which produces an effective ␴z␴z interaction in the low-energy subspace. Let Y V3++ = V3, be an arbitrary ZZXX Hamiltonian acting on qubits i and j and consider a perturbation V=V +V +V that breaks the L z x 1 2 3 − V = ͓Y + C␴ + D͑␴ + 1͔͒ ͉1͗͘1͉ . ͑14͒ zero eigenspace degeneracy by creating an operator O͑⑀͒ + i j k close to Heff in this space: The series expansion of the self-energy follows directly:

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z ͓ ␴z ͑␴x ͔͒ ͉ ͗͘ ͉ ␴ Y 1st: Y − A i + D j + 1 0 0 k, coupling which depends on the strength of the i term in . If Y is regarded as the net uncontrolled physical Hamiltonian ͑ ͒ ␴z B2 coupling i and j a source of error it is only the local i field 2nd: ͑␴x + ͒2 ͉0͗͘0͉ , which contributes to an error in the ␴z␴z coupling strength. ␦ j 1 k z − We make the following choices for our gadget parameters A, B, C, and D: B2C ͑␴x ͒␴z͑␴x ͒ ͉ ͗͘ ͉ 2/3 2/3 3rd: 2 j + 1 i j + 1 0 0 k ␦ ␣ ␦ ͑z − ␦͒ A = ␣ , B = ͩ ͪ ¯E, C = ijͩ ͪ , D =2␦1/3¯E2/3, ij ¯ 2 ¯ B2 E E + ͑␴x + 1͒Y͑␴x + 1͒ ͉0͗͘0͉ ͑ ͒ ͑z − ␦͒2 j j k 21 4DB2 where ¯E is an energy scale parameter to be fixed later. We + ͑␴x + 1͒3 ͉0͗͘0͉ . ͑15͒ ͑ ͒ ͑z − ␦͒2 j k expand the self-energy 16 in the limit where z is constant ͓ ͑ ͒Ӷ␦͔ ͑ ␦͒−1Ӎ 1 ͑ 1 ͒ z=O 1 . Writing z− − ␦ +O ␦2 gives The self-energy in the low-energy subspace ͑where qubit k is in state ͉0͒͘ is therefore ¯E4/3 ⌺͑ ͒ ˜ ␣ ␴z␴x ͑␴x ͒ ͑ʈ ʈ4␦−3͒ 0 −− = Y + ij i j +8 1/3 j + 1 + O V + ¯ . 2B2C ␦ ⌺ ͑z͒Ӎ˜Y + ͩ − Aͪ␴z −− ͑z − ␦͒2 i ͑22͒ 2B2 4DB2 2B2C For the self-energy to become O͑⑀͒ close to Y +␣ ␴z␴x ͩ ͪ͑␴x ͒ ␴z␴x ij i j + + D + + 1 + ͉ ͗͘ ͉ ͑ ͒ z − ␦ ͑z − ␦͒2 j ͑z − ␦͒2 i j 0 0 k, the error terms in Eq. 22 must be bounded above by ⑀ through an appropriate choice of ␦. Define a lower ͑ʈ ʈ4␦−3͒ ͑ ͒ + O V + ¯ . 16 bound on the spectral gap ␦ as an inverse polynomial in ⑀: ␦Ն¯E⑀−r, where ¯E is a constant and integer rՆ1. Now bound ˜Y is the interaction between qubits i and j which is the origi- r by considering the ͑weak͒ upper bound on ʈVʈ: nal physical interaction dressed by the effect of virtual excitations into the high-energy subspace ␦ 2/3 ͉␣ ͉ ␦ 2/3 ʈVʈ Յ ʈYʈ + ͉␣ ͉ +4␦1/3¯E2/3 +2¯Eͩ ͪ + ij ͩ ͪ . 2 ij B ¯E 2 ¯E ˜Y = Y + ͑␴x + ͒Y͑␴x + ͒. ͑17͒ ͑ ␦͒2 j 1 j 1 z − ͑23͒ In practice there will always be some interaction between The largest term in ␦−3ʈVʈ4 is O͓¯E͑¯E/␦͒1/3͔, and so in order qubits i and j. We assume Y is a ZZXX Hamiltonian and that ␦−3ʈVʈ4 Ͻ⑀ we require rՆ3. This also bounds the term express the dressed Hamiltonian ˜Y in terms of modified cou- below fourth order, ¯E4/3␦−1/3 =O͑¯E⑀͒ and so for zӶ␦ we ob- pling coefficients. Writing the physical Hamiltonian ʈ⌺ ͑ ͒ ʈ ͑⑀͒ ⌺͑ ͒ ¯ ⑀͑␴x ͒ tain −− z −Heff =O . In fact, 0 −−=Heff+E +1 . ␴z ␴z ⌬ ␴x ⌬ ␴x ␴z␴z ␴x␴x ͑ ͒ j Y = hi i + hj j + i i + j j + Jij i j + Kij i j . 18 Now apply theorem ͑3͒ from Ref. ͓14͔ and it follows that ͉␭͑ ͒ ␭͑˜ ͉͒ ͑⑀͒ ͑ ͒ The new dressed coupling strengths are Heff − H =O . It also follows from lemma 11 of Ref. ͓14͔ that the ground-state wave function of H is also 2 eff 2B close to the ground state of our gadget. ,ͪ h ͩ1+ ۋ h i i ͑z − ␦͒2 The ZZXX Hamiltonian ͑1͒ allows for the direct realization of all terms in Eq. ͑10͒ except for ␴z␴x and ␴x␴z inter- ͑⑀͒ 2B2 2B2 actions. These terms can be approximated with only O -K , error using the gadget in the present section—thereby show + ͪ +1ͩ ⌬ ۋ ⌬ i i ͑z − ␦͒2 ͑z − ␦͒2 ij ing that the ZZXX Hamiltonian can efficiently approximate all terms from Eq. ͑10͒. Similarly, the ZX Hamiltonian allows 2B2 ͑ ͒ for the direct realization of all terms in Eq. 10 except for ,ͪ +1ͩ ⌬ ۋ ⌬ j j ͑z − ␦͒2 ␴z␴z and ␴x␴x interactions. These terms will be approximated with only O͑⑀͒ error by defining gadgets in the com- 2B2 2B2 ing sections—showing that the ZX Hamiltonian can also be ͒ ͑ K ͩ1+ ͪ + ⌬ ͑19͒ ۋ K ij ij ͑z − ␦͒2 ͑z − ␦͒2 i used to efficiently approximate all terms from Eq. 10 . with additional couplings C. The ZZ from ZX gadget 2B2 2B2 We approximate the operator ␤ ␴z␴z using the ZX Hamil- ⌬ 1 + h ␴z␴x. ͑20͒ ij i j ͑z − ␦͒2 j ͑z − ␦͒2 i i j tonian in Eq. ͑2͒ by defining a penalty Hamiltonian as in Eq. ͑ ͒ 11 . The required perturbation is a sum of terms V=V1 We see that the effect of the gadget on any existing physical +V2: interaction is to modify the coupling constants, add an over- ␴z␴z ͉ ͗͘ ͉ ͑␴z ␴z͒ ␴x ͑ ͒ all shift in energy, and to add a small correction to the V1 = Y + A 0 0 k, V2 = B i − j k. 24

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The nonzero projections are terms in Eq. ͑10͒ except for ␴x␴x interactions. These interactions can also be approximated with only O͑⑀͒ error by V = Y ͉1͗͘1͉ , V = ͑Y + A ͒ ͉0͗͘0͉ , 1++ k 1−− 1 k deﬁning an additional gadget in the next section.

V = B͑␴z − ␴z͒ ͉1͗͘0͉ , V = B͑␴z − ␴z͒ ͉0͗͘1͉ . 2+− i j k 2−+ i j k D. The XX from ZX gadget ͑25͒ An ␴x␴x coupling may be produced from the ␴z␴x cou- V1 does not couple the low- and high-energy subspaces and pling as follows. We deﬁne a penalty Hamiltonian and cor- V2 couples the subspaces but is zero in each subspace. The responding Green’s function series expansion of the self-energy follows directly: ␦ 1 ͑ ␴x͒ ␦͉ ͗͘ ͉ ͉ ͗͘ ͉ ͑ ͒ 1st:͑Y + A ͒ ͉0͗͘0͉ , Hp = 1 − k = − − , G++ = − − k. 33 1 k 2 z − ␦

B2͑␴z − ␴z͒2 This penalty Hamiltonian splits the Hilbert space into a low- i j ͉ ͗͘ ͉ 2nd: 0 0 k, k ͉ ͘ ͑z − ␦͒ energy subspace in which the ancilla qubit is in state + =͉͑0͘+͉1͒͘/ͱ2 and a high-energy subspace in which the an- ͱ 2 cilla qubit k is in state ͉−͘=͉͑0͘−͉1͒͘/ 2. B z z z z 3rd: ͑␴ − ␴ ͒Y͑␴ − ␴ ͒ ͉0͗͘0͉ . ͑26͒ The perturbation is a sum of two terms V=V1 +V2, where ͑z − ␦͒2 i j i j k V1 and V2 are given by Note that in this case the desired terms appear at second x x z V = Y 1 + A͉ + ͗͘+ ͉ , V = B͑␴ − ␴ ͒␴ . ͑34͒ order in the expansion, rather than at third order as was the 1 k k 2 i j k case for the ZX from ZZXX gadget. The terms which dress The nonzero projections are the physical Hamiltonian Y coupling qubits i and j appear at ͉ ͗͘ ͉ ͉ ͗͘ ͉ third order. The series expansion of the self-energy in the V1++ = − − V1 − − k low-energy subspace is ͉ ͗͘ ͉ = Y − − k, 2B2͑1−␴z␴z͒ ⌺͑ ͒ ͑˜ ͒ i j ͑ʈ ʈ4␦−3͒ ͑ ͒ z −− = Y + A1 + + O V , 27 ͉ ͗͘ ͉ ͉ ͗͘ ͉ ͑z − ␦͒ V1−− = Y + + k + A + + k, where the dressed interaction ˜Y is defined as ͉ ͗͘ ͉ ͉ ͗͘ ͉ V2+− = − − V2 + + k B2 ͑␴x ␴x͉͒ ͗͘ ͉ ˜Y = Y + ͑␴z − ␴z͒Y͑␴z − ␴z͒. ͑28͒ = B i − j − + k, ͑z − ␦͒2 i j i j ͉ ͗͘ ͉ ͉ ͗͘ ͉ We assume that the physical interaction Y between i and j V2−+ = + + V2 − − k qubits is a ZX Hamiltonian and express the dressed Hamil- = B͑␴x − ␴x͉͒ + ͗͘− ͉ . ͑35͒ tonian in terms of modified coupling constants. Writing the i j k physical Hamiltonian Once more we see that the perturbation V1 does not couple ␴z ␴z ⌬ ␴x ⌬ ␴x ␴z␴x ␴x␴z ͑ ͒ the subspaces, whereas V2 couples the subspaces but is zero Y = hi i + hj j + i i + j j + Jij i j + Kij i j . 29 in each subspace. This perturbation is illustrated in Fig. 3. We obtain modified coupling strengths The series expansion of the self-energy follows: 2B2͑h − h ͒ 2B2͑h − h ͒ 1st:͑Y + A1͒ ͉ + ͗͘+ ͉ , h + j i . ͑30͒ k ۋ h + i j , h ۋ h i i ͑z − ␦͒2 j i ͑z − ␦͒2 B2͑␴x − ␴x͒2 In this case only the local Z field strengths are modified. i j ͉ ͗͘ ͉ 2nd: + + k, We choose values for the perturbation interaction ͑z − ␦͒ ␤ ␦ ͱ ij ␤ strengths as follows: B= 2 and A= ij and expand the self-energy in the limit where z is constant ͓z=O͑1͒Ӷ␦͔: B2 3rd: ͑␴x − ␴x͒Y͑␴x − ␴x͒. ͑36͒ ͑z − ␦͒2 i j i j ⌺͑ ͒ ˜ ␤ ␴z␴z ͑ʈ ʈ4␦−3͒ ͑ ͒ 0 −− = Y + ij i j + O V . 31 ␦ Again we see that the desired term appears at second order, We again choose to be an inverse power in a small param- while the third-order term is due to the dressing of the physi- ¯ −s eter ⑀ so that ␦ՆE⑀ , and again use the ͑weak͒ upper bound cal interaction Y between qubits i and j. In the low-energy on ʈVʈ: subspace the series expansion of the self-energy to third or- ʈ ʈ Յ ʈ ʈ ␤ ͱ ␤ ␦ ͑ ͒ der is V Y + ij + 2 ij . 32 2 2͑ ␴x␴x͒ The largest term in ʈVʈ4␦−3 is 4␤ ␦−1, and so in order that 2B 1 − i j ij ⌺͑z͒ = ͑˜Y + A1͒ + + O͑ʈVʈ4␦−3͒, ͑37͒ ʈVʈ4␦−3Ͻ⑀ we require rՆ1. −− ͑z − ␦͒ Using the gadget defined in the present section, the ZX Hamiltonian can now be used to efficiently approximate all where the dressed interaction ˜Y is defined as

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FIG. 2. The ZZ from ZX gadget: The present figure presents a FIG. 3. The XX from ZX gadget: The present figure presents a diagrammatic representation of the perturbative Hamiltonian V diagrammatic representation of the perturbative Hamiltonian V ͑ ͒ ͑ ͒ =V1 +V2 from Eq. 34 applied to qubits i, j and k. In addition to the =V1 +V2 from Eq. 24 applied to qubits i, j and k. In addition to these terms shown in the present figure, there is an overall energy terms shown in the present figure, there is an overall energy shift of / ␴x shift of A/2. A 2. The penalty term applied to qubit k is the basis. tonian terms from Eq. ͑3͒ using either the ZZXX or ZX B2 Hamiltonians. It also follows from Lemma ͑11͒ of Ref. ͓14͔ ˜ ͑␴x ␴x͒ ͑␴x ␴x͒ ͑ ͒ Y = Y + i − j Y i − j . 38 H ͑z − ␦͒2 that the ground-state wave function of eff is also close to the ground state of our gadget. So to complete our proof, it is Once more we assume the physical Hamiltonian Y is a ZX enough to show that each gadget satisfies the criteria given in Hamiltonian ͓25͔ and we describe the effects of dressing to theorem ͑3͒ from Ref. ͓14͔. low order in terms of the new dressed coupling strengths III. CONCLUSION 2B2͑⌬ − ⌬ ͒ 2B2͑⌬ − ⌬ ͒ j i The objective of this work was to provide simple model ⌬ ۋ ⌬ i j ⌬ ۋ ⌬ i i + , j i + , ͑z − ␦͒2 ͑z − ␦͒2 Hamiltonians that are of practical interest for experimental- ͑39͒ ists working toward the realization of a universal adiabatic quantum computer. Accomplishing such a task also enabled and in this case only the local X field strengths are modified. us to find the simplest known QMA-complete two-local ␥ x x Choosing values for our gadget parameters A= ij and B Hamiltonians. The ␴ ␴ coupler is realizable using systems ␥ ␦ ͱ ij ͓ ͔ = 2 and expanding the self-energy in the limit where z is including capacitive coupling of flux qubits 18 and spin constant ͓z=O͑1͒Ӷ␦͔ gives models implemented with polar molecules ͓19͔. In addition, ␴ ␴ ͓ ͔ a z x coupler for flux qubits is given in Ref. 20 . The ZX ⌺͑ ͒ ˜ ͉ ͗͘ ͉ ␥ ␴x␴x ͉ ͗͘ ͉ ͑ʈ ʈ4␦−3͒ 0 −− = Y + + k + ij i j + + k + O V . and ZZXX Hamiltonians enable gate model ͓21͔, autonomous ͓ ͔ ͓ ͔ ͑40͒ 22 , measurement-based 23 , and universal adiabatic quantum computation ͓10,14,15͔, and may also be useful for As before, this self-energy may be made O͑⑀͒ close to the quantum annealing ͓24͔. For these reasons, the reported target Hamiltonian by a bound ␦Ն¯E⑀−1. Hamiltonians are of interest to those concerned with the Summary. The proof of theorem ͑1͒ follows from the si- practical construction of a universal adiabatic quantum com- multaneous application of the ZZXX gadget illustrated in Fig. puter. 1 to realize all ␴z␴x terms in the target Hamiltonian using a ZZXX Hamiltonian. Similarly, application of the two gadgets ACKNOWLEDGMENTS illustrated in Figs. 2 and 3 to realize ␴x␴x and ␴z␴z terms in We thank C. J. S. Truncik, R. G. Harris, W. G. Macready, the target Hamiltonian proves the first part of theorem ͑2͒. M. H. S. Amin, A. J. Berkley, P. Bunyk, J. Lamothe, T. Our result is based on theorem ͑3͒ from Ref. ͓14͔ which Mahon, and G. Rose. J.D.B. and P.J.L. completed part of this allowed us to approximate ͓with O͑⑀͒ error͔ all the Hamil- work while on staff at D-Wave Systems Inc.

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͓15͔ R. Oliveira and B. Terhal, e-print arXiv:quant-ph/0504050. ͑1999͒. ͓16͔ E. Bernstein and U. Vazirani, SIAM J. Comput. 26, 1411 ͓21͔ A. Barenco, D. Deutsch, A. Ekert, and R. Jozsa, Phys. Rev. ͑1997͒. Lett. 74, 4083 ͑1995͒. ͓17͔ T. Rudolph and L. Grover, e-print arXiv:quant-ph/0210187. ͓22͔ D. Janzing, Phys. Rev. A 75, 012307 ͑2007͒. ͓18͔ D. V. Averin and C. Bruder, Phys. Rev. Lett. 91, 057003 ͓23͔ S. D. Bartlett and T. Rudolph, Phys. Rev. A 74, 040302͑R͒ ͑2003͒. ͑2006͒. ͓19͔ A. Micheli, G. Brennen, and P. Zoller, Nat. Phys. 2, 341 ͓24͔ S. Suzuki, H. Nishimori, and M. Suzuki, Phys. Rev. E 75, ͑2006͒. 051112 ͑2007͒. ͓20͔ T. P. Orlando, J. E. Mooij, L. Tian, C. H. van der Wal, L. S. ͓25͔ The QMA completeness of this subset of Hamiltonians was Levitov, S. Lloyd, and J. J. Mazo, Phys. Rev. B 60, 15398 found independently by D. Bacon ͑unpublished͒.

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