Haverford College Haverford Scholarship
Faculty Publications Physics
2010
Quantum-Merlin-Arthur-complete problems for stoquastic Hamiltonians and Markov matrices
Stephen P. Jordan
Peter John Love Haverford College
David Gosset
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Repository Citation Jordan, Stephen P., David Gosset, and Peter J. Love. "Quantum-Merlin-Arthur–complete problems for stoquastic Hamiltonians and Markov matrices." Physical Review A 81.3 (2010): 032331.
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 81, 032331 (2010)
Quantum-Merlin-Arthur–complete problems for stoquastic Hamiltonians and Markov matrices
Stephen P. Jordan Institute for Quantum Information, California Institute of Technology, Pasadena, California 91125, USA
David Gosset Center for Theoretical Physics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, 6-304, Cambridge, Massachusetts 02139, USA
Peter J. Love Department of Physics, Haverford College, 370 Lancaster Avenue, Haverford, Pennsylvania 19041, USA, and Institute for Quantum Information, California Institute of Technology, Pasadena, California 91125, USA (Received 1 June 2009; revised manuscript received 6 January 2010; published 29 March 2010) We show that finding the lowest eigenvalue of a 3-local symmetric stochastic matrix is Quantum-Merlin- Arthur-complete (QMA-complete). We also show that finding the highest energy of a stoquastic Hamiltonian is QMA-complete and that adiabatic quantum computation using certain excited states of a stoquastic Hamiltonian is universal. We also show that adiabatic evolution in the ground state of a stochastic frustration-free Hamiltonian is universal. Our results give a QMA-complete problem arising in the classical setting of Markov chains and adiabatically universal Hamiltonians that arise in many physical systems.
DOI: 10.1103/PhysRevA.81.032331 PACS number(s): 03.67.Ac, 75.10.Jm, 89.70.Eg
I. INTRODUCTION Given a classical description of a decision problem x of length n, the prover, Merlin, provides a witness state |ψ to the Quantum complexity theory is the study of the capabilities verifier, Arthur. Arthur then peforms a poly(n)-time quantum and limitations of computational devices operating according computation on the witness |ψ and either accepts or rejects. to the principles of quantum mechanics [1]. Because many of A problem is contained in QMA if, for all YES instances, the classical constructs of computer science (e.g., circuits and there exists a witness causing Arthur to accept with probability clauses) are replaced by matrices, quantum complexity theory greater than 2/3 and, for NO instances, there does not exist any is sometimes referred to as matrix-valued complexity theory witness that causes Arthur to accept with probability greater [2,3]. In addition to its intrinsic interest, this subject has many than 1/3. A problem X is said to be QMA-complete if it connections to issues of practical relevance to physical science, is contained in QMA and every problem in QMA can be such as the difficulty with computing properties of quantum converted to an instance of X in classical polynomial time. systems using either quantum or classical devices [4–6]. Let us consider the following question: What is the ground- Perhaps the most basic classical complexity classes are P, state energy of a quantum system? This question lies at the the class of problems solved by a deterministic Turing machine core of many areas of physical science, including electronic- in polynomial time, and NP (nondeterministic polynomial), structure theory and condensed-matter physics. In quantum the class of problems whose verification lies in P. It is complexity theory, this has been formalized (originally by widely believed, but not proven, that NP is strictly larger Kitaev [11]; see also, for example, [13]) as the k-local than P [7]. Hamiltonian problem. For some systems, complexity-theoretic Because quantum mechanics only predicts probabilities of arguments suggest that efficient computation of the ground- events, the classical deterministic classes are not the most state energy is likely to remain beyond reach [11,14]. natural place to start if one seeks their quantum generaliza- A Hamiltonian H , acting on n qubits, is said to be k-local tions. The probabilistic generalization of P is bounded-error if it is of the form probabilistic polynomial time (BPP), those problems solvable by a probabilistic Turing machine in polynomial time with bounded error [8]. The quantum generalization of this class is H = Hs , bounded-error quantum polynomial time (BQP), the class of s problems solvable in polynomial time with bounded error on a quantum computer [1]. The classical probabilistic generalization of NP is the class where each Hs acts on at most k qubits. Thus, for example, MA [9]. This generalizes NP to problems whose verification 1-local Hamiltonians consist only of external fields acting on is in BPP. MA stands for Merlin-Arthur. Merlin, who is individual qubits, and 2-local Hamiltonians consist of 1-local computationally unbounded but untrustworthy, provides a terms and pairwise couplings between qubits. Physically proof that Arthur can verify using his BPP machine. The class realistic Hamiltonians are usually k-local with small k,often MA possesses a quantum generalization to quantum-Merlin- 2 or 1, and each local term has bounded norm. Note that this Arthur (QMA) [10–12]. QMA may be intuitively understood notion of locality has nothing to do with spatial locality; a as the class of decision problems that can be efficiently verified 2-local Hamiltonian may have long-range couplings, but they by a quantum computer. must be pairwise.
1050-2947/2010/81(3)/032331(10) 032331-1 ©2010 The American Physical Society STEPHEN P. JORDAN, DAVID GOSSET, AND PETER J. LOVE PHYSICAL REVIEW A 81, 032331 (2010)
Problem: k-local Hamiltonian such that
Input. We are given a classical description of a k-local (i) each local operator Hs is positive semidefinite; = r = Hamiltonian H on n qubits H j=1 Hj with r poly(n). (ii) the ground state |ψ of H satisfies Hs |ψ =0 for each Each Hj acts on at most k qubits and has O(1) operator norm. s ∈{1,...,m}. In addition, we are given two constants a and b such that 0 a b, and b − a = >1/poly(n). The work of [3] showed that an adiabatic evolution along a Output. If H has an eigenvalue a, answer YES. If all path composed entirely of SFF Hamiltonians may be simulated eigenvalues of H are >b, answer NO. by a sequence of classical random walks—that is, the adiabatic Promise. The Hamiltonian is such that it will produce either evolution may be simulated in the complexity class BPP. YES or NO. These results were extended to the quantum k-satisfiability Perhaps a more obvious formulation of this problem is to (k-SAT) problem in [2,3]. The quantum k-SAT problem was ask for an approximate ground-state energy to within ± of defined in [16] and we reproduce the definition here. the correct answer. However, if one can decide the answer to k-local Hamiltonian in polynomial time, then one can solve Problem: Quantum k-SAT the approximation version in polynomial time with a binary Input. A set of k-local projectors {q } for q ∈{1,...,m}, search. Thus, the approximation problem is of equivalent where m = poly(n) and a parameter> ˜ 1/poly(n). difficulty to the “decision” version to within a polynomial Output. If there is a state |φ such that 1|φ =0 for each factor. q ∈{1,...,M}, then this is a YES instance. If every state |φ The problem k-local Hamiltonian is QMA-complete for satisfies k 2[13]. The k-local Hamiltonian problem is specified by M the matrix elements of the local terms of H. YES instances φ|q |φ ,˜ possess the ground state as a witness. The verification circuit is q=1 the phase estimation algorithm—a suitably formalized version of the notion of energy measurement [11]. If the lowest then it is a NO instance. eigenvalue of H is less than a (a YES instance), then Arthur Promise. The instance is either YES or NO. will accept the ground state as a witness. However, if the lowest In [2] the stoquastic restriction of quantum k-SAT was eigenvalue of H is greater than b (a NO instance), then Merlin shown to be contained in MA for any constant k, and = cannot supply any eigenstate or superposition of eigenstates MA-complete for k 6—the first nontrivial example of an that will result in a measurement of energy less than b. MA-complete problem. In [3] these results were extended It is considered unlikely that QMA ⊆ BQP, and therefore to a simplified form of stoquastic quantum k-SAT in which it is probably impossible to construct a general quantum (or projectors a all have matrix elements taken from the set { } classical) algorithm that finds ground-state energies in poly- 0,1/2,1 , and the stoquastic constraints which appear as terms = − nomial time. However, many Hamiltonians studied in practice in the Hamiltonian are of the form Ha 11 a. have additional restrictions beyond k-locality. In particular, The main intuition behind these results is that, by the many physical systems are stoquastic, meaning that all of their Perron-Frobenius theorem, the ground state of a stoquastic off-diagonal matrix elements are nonpositive in the standard Hamiltonian consists entirely of real positive amplitudes basis. This includes the ferromagnetic Heisenberg model, (given the appropriate choice of global phase). Thus, the the quantum transverse Ising model, and most Hamiltonians ground state is proportional to a classical probability distribu- achievable with Josephson-junction flux qubits [4]. In [4]it tion. For this reason, ground-state properties are amenable to was shown that for any fixed k, stoquastic k-local Hamiltonian classical random-walk algorithms and certain problems such as is contained in the complexity class AM. Thus, unless QMA ⊆ stoquastic k-local Hamiltonian fall into classical probabilistic AM (which is believed to be unlikely), stoquastic k-local complexity classes such as AM. Diffusion quantum Monte Hamiltonian is not QMA-complete.1 It was also shown in Carlo calculations for stoquastic Hamiltonians do not suffer [4] that, for any fixed k, adiabatic quantum computation in from the sign problem because the negativity of the nonzero the ground state of a k-local stoquastic Hamiltonian can be off-diagonal matrix elements guarantees that the transition probabilities in the associated random walk are all positive. simulated in BPPpath. Thus, unless BQP ⊆ BPPpath (which is also believed to be unlikely), such quantum computation is In this article we first demonstrate that stoquastic Hamil- not universal. The work of [2] also defines a random stoquastic tonians may be constructed which allow universal adiabatic local Hamiltonian problem which is complete for the class MA. quantum computation in a subspace. Then we show that the These results were tightened further for stoquastic 3-local Hamiltonian problem is QMA-complete when re- frustration-free (SFF) Hamiltonians in [3]. A local Hamil- stricted to stochastic Hamiltonians. These are Hamiltonians tonian is frustration free if it can be written as a sum of in which all matrix elements are real and non-negative, and terms the sum of matrix elements in any row or column is 1. Hence m determining the lowest eigenstate of a symmetric stochastic H = H , (1) matrix is QMA-hard. If H is a stochastic Hamiltonian, then s −H is stoquastic. Thus, our result also shows that determina- s tion of the highest-lying eigenstate of a stoquastic matrix is QMA-hard, sharpening the intuition that it is the positivity of 1Like MA, the class AM is a probabilistic generalization of NP, see the ground state which causes its local Hamiltonian problem for example [15]. to fall into a classical class. We then show that universal
032331-2 QUANTUM-MERLIN-ARTHUR–COMPLETE PROBLEMS FOR ... PHYSICAL REVIEW A 81, 032331 (2010) adiabatic quantum computation is possible in the ground where state of a stochastic frustration-free Hamiltonian. Defining ¯ = | | the computational problem stochastic k-SAT in analogy to HXZ αk Tk , (8) the definition of stoquastic k-SAT given in [3], we show that k this problem is QMA1-complete for k = 6. (QMA1 is a slight |Tk| is the entrywise absolute value of Tk, and variant of QMA such that in YES instances, Arthur can be 1 made to accept with probability one [16].) |+ = √ (|0 +|1 ) 2 1 II. QMA-COMPLETENESS AND ADIABATIC |− = √ (|0 −|1 ). UNIVERSALITY OF STOQUASTIC HAMILTONIANS 2 |− −| |+ +| We start with the result of [17], which shows that for a The projectors and act on the ancilla qubit. Equation (7) makes the relationship between the spectra Hamiltonian of the form of HXZ and HXZ clear. Let |ψ0 , |ψ1 ,...,|ψN−1 denote = + + + HXZ di Xi hi Zi Kij Xi Xj Jij Zi Zj , the eigenstates of HXZ with corresponding eigenvalues λ0 i i i,j i,j λ1 ··· λN−1, and let |ψ¯ 0 , |ψ¯ 1 ,...,|ψ¯ N−1 denote the (2) eigenstates of H¯XZ with corresponding eigenvalues λ¯ 0 n λ¯ 1 ··· λ¯ N−1.(HXZ acts on n qubits, so N = 2 .) HXZ, the 2-local Hamiltonian problem is QMA-complete if the which acts on a 2N-dimensional Hilbert space, has two coefficients di , hi , Kij , and Jij are allowed to have both signs. N-dimensional invariant subspaces. The first is spanned by Furthermore, time-dependent Hamilonians that take the form |ψj |− with eigenvalues λj . The second is spanned by HXZ at all times can perform universal adiabatic quantum eigenvectors |ψ¯ j |+ with eigenvalues −λ¯ j . computation [17]. We can perform universal adiabatic quantum computation Starting with a Hamiltonian of the form HXZ on n qubits, we in such an eigenstate of a stoquastic Hamiltonian. To prove can eliminate the negative matrix elements in each term using this, we make use of the universal adiabatic Hamiltonian a technique from [18]. Essentially, the idea is that instead H (t) from [17], which at all t takes the form shown in { − } XZ of representing the group Z2 by 1, 1 we use its regular Eq. (2). One can use the construction described previously representation: to obtain a stoquastic Hamiltonian HXZ(t) corresponding to each instantaneous Hamiltonian H (t). In this way we obtain 10 01 XZ , . a time-varying Hamiltonian HXZ(t) whose spectrum in the 01 10 |− subspace exactly matches the spectrum of HXZ(t), the H can be rewritten as only difference being the addition of an ancilla qubit in the XZ |− state. Because HXZ(t) has no coupling between the |− HXZ =− αkTk, (3) subspace and the |+ subspace, the adiabatic theorem may be k applied within the |− subspace. The relevant eigenvalue gap is thus the same as that of HXZ(t), and so is the run time. where each coefficient αk is positive and for each k, Tk is one of In standard adiabatic quantum computation, the qubits are in the ground state of the instantaneous Hamiltonian. Thus, any ±X, ±Z, ±Xi Xj , ±Zi Zj , (4) disturbance to the state costs energy. This is thought to offer some protection against thermal noise [19]. When performing with identity acting on the remaining qubits. For any k, Tk is a2n × 2n matrix in which each entry is either +1, −1, or 0. universal adiabatic quantum computation with HXZ(t), the n+1 n+1 qubits are not in the ground state. Thus, it is possible for From Tk we construct a 2 × 2 matrix Tk by making the following replacements: the system to thermally relax out of the computational state. However, this can only occur by disturbing the ancilla qubit 10 01 00 out of the state |− . By protecting the ancilla qubit, one 1 → , −1 → , 0 → . (5) 01 10 00 can to a large degree protect the entire computation. Note that an energy penalty against the ancilla qubit leaving the |− We can interpret Tk as acting on n + 1 qubits. The 2 × 2 state would be nonstoquastic. This is why the preceding matrices of (5) act on the ancilla qubit that has been added. construction fails to prove QMA-completeness and universal Each T is 2-local or 1-local; thus, each corresponding T adiabatic quantum computation using the ground state of a k k is 3-local or 2-local. Furthermore, each Tk is a permutation stoquastic Hamiltonian, as we expect it must, based on the matrix. Let complexity-theoretic results of [2–4,20]. HXZ =− αkTk. (6) III. QMA-COMPLETE PROBLEMS FOR k MARKOV MATRICES This is a linear combination of permutation matrices with The second main result of our article provides an example negative coefficients. By construction, HXZ is therefore a 3- of a QMA-complete classical problem: finding the lowest local stoquastic Hamiltonian. We can rewrite HXZ as eigenvalue of a symmetric Markov matrix. A matrix with HXZ = HXZ ⊗ |− −| − H¯XZ ⊗ |+ +|, (7) all non-negative entries, such that the entries in any given
032331-3 STEPHEN P. JORDAN, DAVID GOSSET, AND PETER J. LOVE PHYSICAL REVIEW A 81, 032331 (2010)
+ 1 column sum to 1 is called a stochastic or Markov matrix. These = |+ +| = + + Here σn+1 2 (11 Xn 1) acts on the ancilla qubit, matrices are named after Markov chains, which are stochastic thereby giving it an energy penalty of size (1 − p)against processes such that, given the present state, the future state is leaving the state |− .For0 p 1, Hp is a stochastic independent of the past states. Suppose a system has d possible Hamiltonian. For p<1/3, the energy penalty is large enough states. Then, its probability distribution at time t is described that the highest eigenvalue in the |− subspace lies below the by the d-dimensional vector xt whose entries are non-negative lowest eigenvalue in the |+ subspace. In this case, the lower and sum to 1. If the system is evolving according to a Markov half of the spectrum of Hp is the spectrum of HXZ scaled by process, then its dynamics are completely specified by the p/N, and the upper half of the spectrum of Hp is the spectrum = × equation xt+1 Mxt , where M is a d d stochastic matrix. of H¯XZ scaled by p/N and shifted up by 1 − p. Note that, like quantum Hamiltonians, Markov matrices often Thus, we can obtain the ground energy of HXZ to have a tensor product structure. For example, suppose we polynomial precision by computing the lowest eigenvalue of have two independent simultaneous Markov chains governed Hp to a higher but still polynomial precision. This reduction = = by xt+1 Mxt and yt+1 Nyt . Then their joint probability proves that finding the lowest eigenvalue of Hp to polynomial distribution z is governed by zt+1 = (M ⊗ N)zt . precision is QMA-hard. Using the quantum algorithm for Markov processes for which the Markov matrix is sym- phase estimation, one easily shows that the problem of metric correspond to random walks on undirected weighted estimating the lowest eigenvalue of Hp is contained in QMA graphs. (Self-loops are allowed and correspond to diagonal (see [11]). Thus, this problem is QMA-complete. matrix elements.) These matrices are doubly stochastic: The sum of the entries in any row or column is 1. By the Perron- IV. FRUSTRATION-FREE ADIABATIC COMPUTATION Frobenius theorem, the highest eigenvalue of a symmetric stochastic matrix is 1, and the corresponding eigenvector is It was stated in [3] that universal adiabatic quantum the uniform distribution. The eigenvalue with next largest computation can be performed in the ground state of a magnitude controls the rate of convergence of the process to 5-local frustration-free Hamiltonian. Let U = UL ···U2U1 be its fixed point. A symmetric stochastic matrix is Hermitian and a quantum circuit acting on n qubits with L = poly(n) gates. therefore one can also think of these matrices as Hamiltonians. Let To prove that finding the lowest eigenvalue of a 3-local ⊗ |ψ =U ···U |0 n (12) symmetric stochastic matrix is QMA-complete, we again use j j 1 a reduction from the QMA-complete HXZ Hamiltonian of [17]. be a state of n qubits corresponding to the jth state of the time We must take the opposite sign convention from Eq. (3), evolution of a quantum circuit specified by gates Uj and | =| t+1 L−t HXZ = αkSk, (9) ct 1 0 . (13) k be a state of L + 1 clock qubits. Bravyi and Terhal construct where the coefficients αk are the same as before (all positive) a parametrized 5-local Hamiltonian H (s) such that the ground and Sk =−Tk. Now define state |ψ(s) satisfies 1 1 L HˆXZ = αkSk, (10) | =√ | | N ψ(1) ψj cj , k + L 1 j=0 where L 1 n |ψ(0) =√ |0 |cj . N = αk + L 1 j=0 k | We can think of the first register in ψ(s) as consisting of and Sk is the permutation matrix obtained by applying the “work” qubits on which the computation happens and the ˆ replacement rules (5)toSk. By construction, HXZ is a 3-local, second register in |ψ(s) as being a clock containing a time ˆ symmetric, doubly stochastic matrix. We can rewrite HXZ as written in unary. 1 For s ∈ [0, 1], the minimal eigenvalue gap between the ˆ = ⊗|− −|+ ¯ ⊗ |+ +| 2 HXZ (HXZ HXZ ), (11) H s O /L N ground state and first excited state of ( )is (1 ). By the adiabatic theorem,2 1/poly(L) eigenvalue gap ensures ¯ = | | where HXZ k αk Sk . Thus, to determine an eigenvalue of that given |ψ0 , one obtains |ψ1 by applying H (s) and HXZ to within ± we must find the corresponding eigenvalue varying s from 0 to 1 over poly(L) time. By measuring L+1 of HˆXZ to within ±/N. Because HXZ is a 2-local Hamiltonian the clock register of |ψ(1) , one obtains the result |1 on n qubits with coupling strengths of order unity, N is at most with probability 1/(L + 1). If this result is obtained, one O(n2). Thus, the problem of determining the eigenvalue of finds the output of the circuit U by measuring the first HˆXZ corresponding to the ground state of HXZ to polynomial register of qubits in the computational basis. By repeating this precision is QMA-hard. process with O(L) copies of |ψ(1) , one succeeds with high To obtain a cleaner QMA-hard problem, we would like to construct a stochastic matrix whose lowest eigenvalue is QMA-hard to find. To do this, let 2Many versions of the adiabatic theorem have been proven. For one = − + + ˆ Hp (1 p)σn+1 pHXZ. example, see Appendix F of [21].
032331-4 QUANTUM-MERLIN-ARTHUR–COMPLETE PROBLEMS FOR ... PHYSICAL REVIEW A 81, 032331 (2010) probability. Alternatively, one can pad the underlying circuit the simultaneous zero eigenspace of H clock, H init, and H prop. with L identity gates, in which case each trial succeeds with H clock commutes with H prop(s) + H init and provides an energy probability 1/2. penalty of at least unit size if the clock register is not in t+1 L−t The construction from [3] invokes the fact that the spectrum one of the unary states |ct =|1 0 . Thus, the low-lying FF of H (1) is independent of the form of the gates Uj . By choosing spectrum of H (s) is strictly contained in the ground space a gate set which is composed of elements of simply connected of H clock. unitary groups such as SU(2) and SU(4), one may construct a For any bit string x ∈{0, 1}n and integer j ∈{1, 2,...,L}, continuous path connecting each gate to the identity and use a let single parameter s to transform all gates from the identity to = j = ··· | ⊗| the final circuit at once. The Hamiltonian at s 0 corresponds χx (Uj Uj−1 U1 x ) cj , to the identity circuit, and its ground state is the uniform | | 0 = superposition of the clock states tensored with the initial data where cj is as defined in Eq. (13). (We also define χx n on the work qubits. In this ground state, the qubits of the |x ⊗|c0 .) There are 2 (L + 1) such states and they form an clock register are entangled. It is standard to design adiabatic orthonormal basis for the ground space of H clock. In this basis, computations such that the initial Hamiltonian has a product H prop(s) + H init takes the block-diagonal form state as its ground state, because such states should be easily prop init produced by cooling or single-qubit measurements. In this H (s) + H = Mx , section we construct a modified version of the construction x∈{0,1}n from [3] that satisfies this condition and is still frustration free. Let c(j) indicate the jth clock qubit and let w(j) indicate where ⎡ ⎤ the jth work qubit. Let s +|x|−b ⎢ ⎥ H init =|1 1| ⊗|10 10| , ⎢ − − ⎥ j w(j) c(1),c(2) ⎢ b 1 b ⎥ clock ⎢ ⎥ H =|01 01| − . ⎢ −b 1 −b ⎥ j c(j 1),c(j) ⎢ ⎥ Mx = ⎢ ⎥ For j ∈{1,...,L− 1}, define ⎢ ...... ⎥ ⎢ . . . ⎥ prop = | | ⎢ ⎥ Hj (s) s 100 100 c(j),c(j+1),c(j+2) ⎣ −b 1 −b ⎦ + (1 − s)|110 110| + + − − c(j),c(j 1),c(j 2) b 1 s − s(1 − s)[Uj ⊗|110 100|c(j),c(j+1),c(j+2) √ is an L + 1byL + 1 matrix and b = s(1 − s). Here + † ⊗| | Uj 100 110 c(j),c(j+1),c(j+2)] |x| denotes the Hamming weight of the bit string x.The | | init and let appearance of x is the sole manifestation of H . The rest of the matrix elements all come from the “hopping” action of prop H (s) = s|10 10| + + (1 − s)|11 11| + H prop(s). L c(L),c(L 1) c(L),c(L 1) M00...0 has the unique ground state − s(1 − s)[UL ⊗|11 10|c(L),c(L+1) † L + U ⊗|10 11|c L ,c L+ ]. L ( ) ( 1) j N r |ψj |cj , (14) clock init prop It can be directly verified that each Hj , Hj , and Hj (s) j=0 is a projector. Here, for convenience, we define s so that it | | varies from 0 to 1/2 rather than from 0 to 1 as is done in where ψj and cj are as defined in Eqs. (12) and (13), [3]. Our frustration-free Hamiltonian is the following sum of r = s , and N is a normalization factor. This constitutes projectors: 1−s FF the ground state of√ H (s). The first excited state of M00...0 L has energy 1 − 2 s(1 − s) cos( π ). Because of the direct H clock =|0 0| + H clock, L+1 c(0) j sum structure of H FF(s), we can apply the adiabatic theorem j=1 directly to M00...0. The run time of the adiabatic algorithm is n thus determined by the gap between the ground and first excited H init = H init, j states of M00...0. This takes its minumum at s = 1/2, where it j=1 − π = 2 is equal to 1 cos[ (L+1) ] O(1/L ). For questions of fault L tolerance, it is also useful to know the eigenvalue gap between prop = prop H (s) Hj (s), the ground and first excited states of the full Hamiltonian j=1 H FF(s). The first excited energy of H√FF(s) is equal to the FF clock init prop − − π H (s) = H + H + H (s). ground energy of M10...0, which is 1 2 s(1 s) cos[ 2(L+1) ]. Thus the minumum eigenvalue gap of H FF(s) occurs at s = If U ···U are chosen from a universal set of two-qubit 1 L 1/2 and is equal to 1 − cos[ π ] = O(1/L2). gates, then H FF(s) is an efficient 5-local frustration-free 2(L+1) FF | ··· ⊗ adiabatic quantum computer. To see how this Hamiltonian By Eq. (14), the ground state of H (0) is 000 | ··· FF 1000 , and the ground state of H (1/2) is the same state achieves universal adiabatic computation, we examine the L FF √ 1 |ψ |c produced by the scheme of [3]. various terms one by one. The ground state of H (s)is L+1 j=0 j j
032331-5 STEPHEN P. JORDAN, DAVID GOSSET, AND PETER J. LOVE PHYSICAL REVIEW A 81, 032331 (2010)
V. STOCHASTIC FRUSTRATION-FREE COMPUTATION We can replace the group {1,i,−1, −i} with its left-regular representation In [3], Bravyi and Terhal showed that adiabatic quantum computation in the ground state of a SFF Hamiltonian can i → F be efficiently simulated by a classical computer. In this −1 → F 2 section we show that, in contrast, one can perform universal (17) − → 3 adiabatic quantum computation in the ground state of a i F stochastic frustration-free (StochFF) Hamiltonian H StochFF(s). 1 → F 4, [Alternatively, we can view this as computation in the highest where energy state of the stoquastic Hamiltonian −H StochFF(s).] ⎛ ⎞ It has been shown that the two-qubit controlled-NOT (CNOT) 0100 ⎜ ⎟ gate, together with any one-qubit rotation whose square is not ⎜ 0010⎟ basis preserving, are sufficient to perform universal quantum F = ⎜ ⎟ . (18) ⎝ 0001⎠ computation [22]. All matrix elements in these gates are real numbers. If we choose U1,...,UL from this gate set, 1000 then H FF(s) is a 5-local real frustration-free Hamiltonian. The eigenvectors of F are |v0 , |v1 , |v2 , |v3 , where Examining the construction of Sec. III, one sees that it can be applied to any Hamiltonian with real matrix elements, and it 1 3 |v = ilj |l . increases the locality by one. This construction also preserves j 2 frustration freeness, as we will show in the next paragraph. We l=0 can thus use this construction on H FF(s) to obtain a 6-local The corresponding eigenvalues are stochastic frustration-free Hamiltonian whose ground state is j F |vj =i |vj . (19) universal for adiabatic quantum computation. To show that the mapping of Sec. III preserves frustration Let Sj and Aj be the real and imaginary parts of αj Oj . That is, S and A are the unique real symmetric and antisymmetric freeness, consider applying this mapping to a frustration- j j = m = j j matrices such that free local Hamiltonian H j=1 Hj , where Hj k αk Sk j α O = S + iA . (where each Sk is, up to an overall sign, a tensor product j j j j of Pauli operators and each αj is positive). We obtain the + = | |+ − = | |− k Further, let Sj ( Sj Sj )/2 and Sj ( Sj Sj )/2, and Hamiltonian ± |·| similarly for Aj , where denotes the entrywise absolute + = ˆ + − 11 Xn+1 value. Applying the replacement (17)toH and dividing by Hp pH (1 p) = ˜ 2 N j αj yields the stochastic Hamiltonian H with the decomposition Nj 11 + Xn+1 = pHˆj + (1 − p) , (15) N 2 1 (0) (1) j H˜ = (H ⊗|v0 v0|+H ⊗|v1 v1| N = j = 1 + (2) ⊗| |+ (3) ⊗| | where Nj k αk and N j Nj . When p<3 , Hp H v2 v2 H v3 v3 ), (20) is stochastic and has a zero-energy ground state with an where eigenvalue gap which is p times the gap of H. Furthermore, we N (0) = + + − + + + − see from (15) (and the fact that each Hj is positive semidefinite) H Sj Sj Aj Aj , that Hp is a sum of positive semidefinite operators. Hence, the j Hamiltonian H is frustration free. p (1) = + − − + + − − H Sj Sj iAj iAj , j VI. GENERALIZATIONS (2) = + + − − + − − The constructions of Secs. III and II replace Hamiltonians H Sj Sj Aj Aj , j with real matrix elements of both signs by computationally equivalent Hamiltonians with real positive matrix elements. In H (3) = S+ − S− − iA+ + iA−. this section we show that this technique can be generalized to j j j j j directly replace Hamiltonians with complex matrix elements (1) by computationally equivalent Hamiltonians with only real H = H ; thus, the spectrum of H˜ in the |v1 subspace positive matrix elements. However, in the process we neces- matches that of H up to a normalization factor of N and a pair sarily introduce a twofold degeneracy of the ground state. of extra ancilla qubits. If we write each projector |vj vj | in Let H be an arbitrary k-local Hamiltonian. We may expand terms of the Pauli basis we obtain H as X X |v0 v0|=++, H = αj Oj , (16) X Y |v1 v1|=−+, j X X |v2 v2|=+−, where each Oj is a tensor product of k or fewer Pauli matrices |v v |=XY , and each αj is positive. Each entry in each Oj is ±1or±i. 3 3 − −
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a where ± is the projector onto the eigenvalue ±1 eigenstate We first recall the definition of stoquastic k-SAT which is of the Pauli matrix a.Thus,anX penalty on the first ancilla given in [3]. qubit will separate the |v1 , |v3 subspace from the |v0 , |v2 1 Problem: Stoquastic k-SAT subspace. So, taking, 0
HR|ψ |φ =E|ψ |φ . (23) is possible to construct another instance of stoquastic k-SAT with operators H˜j ={1 − j } that are all projectors. To show that doubling exists in the spectrum, it is sufficient We now define a problem called stochastic k-SAT, which | = to show that φ φ 0. To prove this, first use Eq. (23)to is identical to stoquastic k-SAT except that condition (iii) is obtain replaced by
(iii )EveryH is a stochastic matrix. (11 ⊗|φ φ|)HR|ψ |φ =E|ψ |φ φ|φ . (24) j We note that there does not appear to be an equivalence Then use Eq. (21) to obtain between this definition of stochastic k-SAT and the corre- (11 ⊗|φ φ|)H |ψ |φ =(H|ψ )|φ φ|φ . (25) sponding definition where all the Hj are (in addition) required R to be projectors. Equating these expressions gives Given these two definitions and the foregoing map from an arbitrary Hamiltonian to a stochastic Hamiltonian, we now H |ψ φ|φ =E|ψ φ|φ . (26) show how to reduce any instance of quantum 4-SAT to an This must hold for all Hamiltonians H and eigenstates |ψ , instance of stochastic 6-SAT. Starting with an instance of and therefore it must be the case that φ|φ =0. So we have quantum 4-SAT specified by a set of projectors {j } (for shown that the doubling in the spectrum of H is a necessary j ∈ 1,...,m), we use the map of the previous section (with p = 1 feature of the type of maps we consider. For constructing p 3 for concreteness) on each projector to obtain a set of universal adiabatic quantum computers, the degeneracy in- 6-local positive semidefinite stochastic Hamiltonians {Hj }, duced by this construction may be problematic. However, for where proving complexity-theoretic completeness results, it is often 2 11 + Xn+1 1 H = + ˜ . (27) irrelevant, as we see in the next section. j 3 2 3 j [Note that ˜ refers to the operator obtained by applying the VII. STOCHASTIC k-SAT j mapping from Eq. (20).] If the 4-SAT instance is satisfiable, The methods of the previous section can be used to show, then the stochastic 6-SAT instance will also be satisfiable. roughly speaking, that deciding whether or not a Hamiltonian Define Nmax to be the maximum value of N obtained for which is a sum of positive semidefinite stochastic operators is one of the terms j when using the mapping of Eq. (20). If frustration free is as difficult as the general problem of deciding the 4-SAT instance is not satisfiable, then for any state |φ | | whether a Hamiltonian is frustration free. In this section we there is some projector k such that φ k φ m .Ifwe formalize this by defining a problem called stochastic k-SAT, take the parameter ˜ of the stochastic 6-SAT instance to be whichweshowtobeQMA1-complete for k = 6. related to the parameter of the quantum 4-SAT instance
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by ˜ = , then the stochastic 6-SAT instance will also |ψ1 |ψ2 ...|ψc and for Arthur to use phase estimation to 3mNmax be unsatisfiable. Therefore, stochastic 6-SAT is QMA1-hard. check that the c registers each contain a state of energy at most Stochastic 6-SAT is contained in QMA1 since every instance a. The problem is that for NO instances there are many ways of stochastic 6-SAT can be mapped to an instance of quantum for Merlin to cheat. For example if λ1 a but λc b,the ⊗c 6-SAT by taking projectors j which project onto everything answer is NO, but Merlin can provide the state |ψ1 as a but the zero eigenspaces of the Hj . supposed witness. To prevent this, Arthur needs to somehow So QMA1 completeness of stochastic 6-SAT follows from check that he has been given a set of c orthogonal states that the results of Bravyi [16] on quantum k-SAT. This is in contrast each have energy at most a. Thus, we propose the following to stoquastic k-SAT, which is contained in MA for every protocol. constant k [3]. Arthur demands that Merlin give him the state 1 |W =√ sgn(π)|ψπ(1) |ψπ(2) ···|ψπ(c) . (28) VIII. QMA-COMPLETENESS FOR EXCITED STATES c! π∈Sc The local Hamiltonian problem refers specifically to ground Arthur performs the projective measurement to see that the state energies. Similarly, we have formulated a computational state given to him by Merlin lies in the antisymmetric subspace problem based on the highest energy of a given Hamiltonian. of H⊗c. If this fails, he rejects. He then throws away all It is natural to ask about the complexity of estimating the but the first register and performs phase estimation of H to cth excited state. We can formulate this as follows. Let precision better than . If the state has energy above b,he H be a k-local Hamiltonian on the Hilbert space H of n rejects. Otherwise, he accepts. qubits. Let λ λ ··· λ n denote the eigenvalues of H , 1 2 2 It is clear that for YES instances, Arthur will accept the with corresponding eigenvectors |ψ , |ψ ,...,|ψ n .The 1 2 2 state |W with high probability. (The only source of error (k,c,)-energy problem is as follows. is imprecision in phase estimation.) We will next prove that for NO instances the acceptance probability is at most 1 − Problem: (k,c, )-energy 1 c . Using standard methods [11,23,24], we can amplify this Input. We are given a classical description of H, an integer protocol to obtain polynomially small acceptance probability c 1, and a pair of parameters a,b such that b − a = > for NO instances. 1/poly(n). Lemma 1. For any state |φ in the antisymmetric subspace of ⊗c 1 Output. If λc a, answer YES. If λc b, output NO. H | ∈H | | |⊗ | and any state α1 , φ ( α1 α1 11) φ c where Promise. H is such that the answer is YES or NO. 11 is the identity operator on H⊗(c−1). In this section we will show that the (k,c,)-energy Proof. Extend |α1 to an orthonormal basis problem is QMA-complete for any c = O(1). Showing QMA- |α1 , |α2 ,...,|α2n for H.LetF be the set of hardness is the easier of the two proofs. This can be achieved functions f : {1, 2,...,c}→{1, 2,...,2n} such that (a) (b) n as follows. Let H and H be a pair of k-local Hamiltonians f (1) 032331-8 QUANTUM-MERLIN-ARTHUR–COMPLETE PROBLEMS FOR ... PHYSICAL REVIEW A 81, 032331 (2010) The quantity various techniques while retaining universality [13,26,27]. The universal Hamiltonian H of [17] is one outcome p (φ) =φ|(|ψ ψ |⊗11)|φ XZ j j j of this chain of reductions. Here we add one more step is the probability of obtaining |ψj if we measure the first to this chain, obtaining universal stochastic and stoquastic register of a state |φ in the eigenbasis of H. By Lemma 1 , Hamiltonians which resemble those arising in some systems of c−1 superconducting qubits [4]. The constructions given here are 1 at least 3-local, and so would require the use of perturbative pj (φ) 1 − . c gadgets to implement in terms of physical 2-local interactions. j=1 Finally, our results are of interest from a purely complexity- 1 Thus, with probability at least c , such a measurement would theoretic point of view. Stochastic matrices arise outside yield |ψj , with j>c− 1. Thus, if λc >b, then with the context of quantum mechanics, in Markov chains. Our 1 probability at least c a measurement of the observable H would reduction shows that finding the lowest eigenvalue of a certain yield energy of at least b. The phase estimation algorithm can class of exponentially large but efficiently describable doubly in poly(1/) time perform such an energy measurement with stochastic matrices is QMA-complete. (These stochastic matri- an exponentially smaller chance of making an error as large as ces correspond to Markov chains in which the “update rule” is a . Thus, the protocol is sound, which completes the proof that probabilistic selection over some set of updates which are local the (c, k, )-energy problem is QMA-complete for constant c in the tensor product sense.) In general, the problem of finding and k and polynomially small . eigenvalues of stochastic matrices is of interest because the eigenvalue of second-largest magnitude determines the mixing time of the corresponding Markov chain. There exist Markov IX. DISCUSSION AND CONCLUSIONS chains in which the eigenvalue of second-largest magnitude is negative and is the lowest-lying eigenvalue. We hope that The results presented in this article have several applica- the demonstration of a QMA-complete problem arising in a tions. Although calculating the ground-state energy of stoquas- classical setting will help shed further light on the class QMA tic Hamiltonians appears easier than calculating the ground- itself. state energy of generic Hamiltonians, our results suggest that calculating other eigenstates of stoquastic Hamiltonians remains hard. Because the wave functions of these states ACKNOWLEDGMENTS have amplitudes which are both positive and negative, the hardness of determining their energy supports the intuition We thank Sergey Bravyi for a helpful discussion. S.J. that it is the positivity of the amplitudes which makes thanks MIT’s Center for Theoretical Physics, RIKEN’s Digital the ground-state problem for stoquastic Hamiltonians easier. Materials Laboratory, and Caltech’s Institute for Quantum In- An extreme distinction between stochastic and stoquastic formation, Franco Nori, Sahel Ashhab, ARO/DTO’s QuaCGR Hamiltonians arises when the Hamiltonians are also frustration program, the US Department of Energy, the Sherman Fairchild free. Although adiabatic evolution with SFF Hamiltonians is Foundation, and the NSF for Grant No. PHY-0803371. D.G. simulable in BPP [3], we have shown that adiabatic evolution thanks Eddie Farhi, the W. M. Keck Foundation Center in the ground state of a stochastic frustration-free Hamiltonian for Extreme Quantum Information Theory, and the Natural is universal. Sciences and Engineering Research Council of Canada. P.J.L. Second, these results may be relevant for the physical thanks John Preskill and the Institute of Quantum Information implementation of quantum computers. The first proof of at Caltech for hosting an extended visit, during which part universality of adiabatic quantum computation used 5-local of this work was completed. This research was supported interactions [25]. Since then, the Hamiltonians have been in part by the National Science Foundation under Grant No. brought into incrementally more physically feasible form by PHY05-51164. [1] E. Bernstein and U. Vazirani, SIAM J. Comput. 26, 1411 [8] L. Adleman, in FOCS ’78: Proceedings of the 19th IEEE Sym- (1997). posium on Foundations of Computer Science (IEEE Computer [2] S. Bravyi, A. J. Bessen, and B. M. Terhal, e-print arXiv:quant- Society, 1978), pp. 75–83. ph/0611021. [9] L. Babai, in STOC ’85: Proceedings of the 17th Annual ACM [3] S. Bravyi and B. Terhal, SIAM J. Comput. 39, 1462 Symposium on Theory of Computing (ACM, 1985), pp. 421–429. (2009). [10] D. Aharonov and T. Naveh (2002), e-print arXiv:quant- [4] S. Bravyi, D. DiVincenzo, R. Oliveira, and B. Terhal, Quant. 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