QMA-complete problems for stoquastic Hamiltonians and Markov matrices

QMA-complete problems for stoquastic Hamiltonians and Markov matrices

Stephen P. Jordan Institute for Quantum Information, California Institute of Technology, Pasadena, CA 91125    David Gosset Center for Theoretical Physics, Massachusetts Institute of Technology, 77 Massachusetts Ave, 6-304, Cambridge, MA 02139    Peter J. Love Department of Physics, Haverford College, 370 Lancaster Avenue, Haverford, PA 19041 Institute for Quantum Information, California Institute of Technology, Pasadena, CA 91125
Abstract

We show that finding the lowest eigenvalue of a 3-local symmetric stochastic matrix is 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 new QMA-complete problem arising in the classical setting of Markov chains, and new adiabatically universal Hamiltonians that arise in many physical systems.

preprint: MIT-CTP 4134

I Introduction

Quantum complexity theory is the study of the capabilities and limitations of computational devices operating according to the principles of quantum mechanics [7]. Because many of the classical constructs of computer science (e.g. circuits and clauses) are replaced by matrices, quantum complexity theory is sometimes referred to as matrix-valued complexity theory [10, 12]. In addition to its intrinsic interest this subject has many connections to issues of practical relevance to physical science, such as the difficulty of computing properties of quantum systems using either quantum or classical devices [24, 19, 11].

Perhaps the most basic classical complexity classes are P - the class of problems solved by a deterministic Turing machine in polynomial time, and NP - the class of problems whose verification lies in P. It is widely believed, but not proven, that NP is strictly larger than P [26].

Because quantum mechanics only predicts probabilities of events, the classical deterministic classes are not the most natural place to start if one seeks their quantum generalizations. The probabilistic generalization of P is BPP (Bounded-error Probabilistic Polynomial-time) - those problems solvable by a probabilistic Turing machine in polynomial time with bounded error [1]. The quantum generalization of this class is BQP (Bounded-error Quantum Polynomial-time) - the class of problems solvable in polynomial time with bounded error on a quantum computer [7].

The classical probabilistic generalization of NP is the class MA [5]. This generalizes NP to problems whose verification is in BPP. MA stands for Merlin-Arthur. Merlin, who is computationally unbounded but untrustworthy, provides a proof that Arthur can verify using his BPP machine. The class MA possesses a quantum generalization to QMA (Quantum Merlin Arthur) [27, 18, 2]. QMA may be intuitively understood as the class of decision problems that can be efficiently verified by a quantum computer.

Given a classical description of a decision problem of length , the prover, Merlin, provides a witness state to the verifier, Arthur. Arthur then peforms a -time quantum computation on the witness and either accepts or rejects. A problem is contained in QMA if, for all YES instances, there exists a witness causing Arthur to accept with probability greater than and for NO instances, there does not exist any witness that causes Arthur to accept with probability greater than . A problem is said to be QMA-complete if it is contained in QMA and every problem in QMA can be converted to an instance of in classical polynomial time.

Let us consider the following question: What is the ground state energy of a quantum system? This question lies at the core of many areas of physical science, including electronic structure theory and condensed matter physics. In quantum complexity theory this problem has been formalized (originally by Kitaev [18], see also, for example, [16]) as the -local Hamiltonian problem. For some systems, complexity-theoretic arguments suggest that efficient computation of the ground state energy is likely to remain beyond reach [6, 18].

A Hamiltonian , acting on qubits, is said to be -local if it is of the form

where each acts on at most -qubits. Thus, for example, 1-local Hamiltonians consist only of external fields acting on individual qubits, and 2-local Hamiltonians consist of 1-local terms and pairwise couplings between qubits. Physically realistic Hamiltonians are usually -local with small , often or , and each local term has bounded norm. Note that this notion of locality has nothing to do with spatial locality; a 2-local Hamiltonian may have long-range couplings but they must be pairwise.

Problem: -local Hamiltonian
Input: We are given a classical description of a -local Hamiltonian on qubits with . Each acts on at most qubits and has operator norm. In addition we are given two constants and such that , and .
Output: If has an eigenvalue answer YES. If all eigenvalues of are answer NO.
Promise: the Hamiltonian is such that it will produce either YES or NO.

Perhaps a more obvious formulation of this problem is to ask for an approximate ground state energy to within of the correct answer. However, if one can decide the answer to -local Hamiltonian in polynomial time, then one can solve the approximation version in polynomial time by a binary search. Thus, the approximation problem is of equivalent difficulty to the “decision” version, to within a polynomial factor.

The problem -local Hamiltonian is QMA-complete for  [16]. The -local Hamiltonian problem is specified by the matrix elements of the local terms of . YES instances possess the ground state as a witness. The verification circuit is the phase estimation algorithm - a suitably formalized version of the notion of energy measurement [18]. If the lowest eigenvalue of is less than (a YES instance) then Arthur will accept the ground state as a witness. However, if the lowest eigenvalue of is greater than (a NO instance) then Merlin cannot supply any eigenstate or superposition of eigenstates that will result in a measurement of energy less than .

It is considered unlikely that and therefore it is probably impossible to construct a general quantum (or classical) algorithm that finds ground state energies in polynomial time. However, many Hamiltonians studied in practice have additional restrictions beyond -locality. In particular, many physical systems are stoquastic, meaning that all of their off-diagonal matrix elements are nonpositive in the standard basis. This includes the ferromagnetic Heisenberg model, the quantum transverse Ising model, and most Hamiltonians achievable with Josephson-junction flux qubits [11]. In [11] it was shown that for any fixed , stoquastic -local Hamiltonian is contained in the complexity class AM. Thus, unless (which is believed to be unlikely), stoquastic -local Hamiltonian is not QMA-complete111Like MA, the class AM is a probabilistic generalization of NP, see [4].. It was also shown in [11] that, for any fixed , adiabatic quantum computation in the ground state of a -local stoquastic Hamiltonian can be simulated in . Thus, unless (which is also believed to be unlikely), such quantum computation is not universal. The work of [10] also defines a random stoquastic local Hamiltonian problem which is complete for the class AM.

These results were tightened further for stoquastic frustration free (SFF) Hamiltonians in [12]. A local Hamiltonian is frustration free if it can be written as a sum of terms

(1)

such that

  1. Each local operator is positive semidefinite

  2. The ground state of H satisfies for each

The work of  [12] showed that an adiabatic evolution along a path composed entirely of SFF Hamiltonians may be simulated by a sequence of classical random walks - that is, the adiabatic evolution may be simulated in the complexity class BPP.

These results were extended to the quantum -satisfiability problem in [10, 12]. The quantum -satisfiability problem was defined in [9] and we reproduce the definition here:

Problem: Quantum -SAT
Input: A set of -local projectors for where and a parameter
Output: If there is a state such that for each , then this is a YES instance. If every state satisfies

then it is a NO instance.
Promise: The instance is either YES or NO.

In [10] the stoquastic restriction of quantum -SAT was shown to be contained in MA for any constant , and MA-complete for – the first nontrivial example of an MA-complete problem. In [12] these results were extended to a simplified form of stoquastic quantum -SAT in which projectors all have matrix elements taken from the set , and the stoquastic constraints which appear as terms in the Hamiltonian are of the form .

The main intuition behind these results is that, by the Perron-Frobenius theorem, the ground state of a stoquastic Hamiltonian consists entirely of real positive amplitudes (given the appropriate choice of global phase). Thus the ground state is proportional to a classical probability distribution. For this reason, ground state properties are amenable to classical random walk algorithms and certain problems such as stoquastic -local Hamiltonian fall into classical probabilistic complexity classes such as AM. Diffusion Quantum Monte Carlo calculations for stoquastic Hamiltonians do not suffer from the sign problem because the negativity of the nonzero off-diagonal matrix elements guarantees that the transition probabilities in the associated random walk are all positive.

In this paper we first demonstrate that stoquastic Hamiltonians may be constructed which allow universal adiabatic quantum computation in a subspace. Then we show that the -local Hamiltonian problem is QMA-complete when restricted to stochastic Hamiltonians. These are Hamiltonians in which all matrix elements are real and nonnegative, and the sum of matrix elements in any row or column is one. Hence determining the lowest eigenstate of a symmetric stochastic matrix is QMA-hard. If is a stochastic Hamiltonian, then is stoquastic. Thus, our result also shows that determination of the highest lying eigenstate of a stoquastic matrix is QMA-hard, sharpening the intuition that it is the positivity of the ground state which causes its local Hamiltonian problem to fall in a classical class. We then show that universal adiabatic quantum computation is possible in the ground state of a stochastic frustration free Hamiltonian. Defining the computational problem stochastic -SAT in analogy to the definition of stoquastic -SAT given in [12], we show that this problem is QMA-complete for . (QMA is a slight variant of QMA such that in YES instances, Arthur can be made to accept with probability one[9].)

Ii QMA-completeness and adiabatic universality of stoquastic Hamiltonians

We start with the result of [8], which shows that for a Hamiltonian of the form

(2)

the 2-local Hamiltonian problem is QMA-complete if the coefficients , , , are allowed to have both signs. Furthermore, time-dependent Hamilonians that take the form at all times can perform universal adiabatic quantum computation [8].

Starting with a Hamiltonian of the form on qubits we can eliminate the negative matrix elements in each term using a technique from [14]. Essentially, the idea is that instead of representing the group by we use its regular representation:

can be rewritten as

(3)

Where each coefficient is positive and for each , is one of

(4)

with identity acting on the remaining qubits. For any , is a matrix in which each entry is either +1,-1, or 0. From we construct a matrix by making the following replacements

(5)

We can interpret as acting on qubits. The matrices of (5) act on the ancilla qubit that has been added. Each is 2-local or 1-local, thus each corresponding is 3-local or 2-local. Furthermore, each is a permutation matrix. Let

(6)

This is a linear combination of permutation matrices with negative coefficients. By construction, is therefore a 3-local stoquastic Hamiltonian. We can rewrite as

(7)

where

(8)

is the entry-wise absolute value of , and

The projectors and act on the ancilla qubit.

Equation 7 makes the relationship between the spectra of and clear. Let denote the eigenstates of with corresponding eigenvalues , and let denote the eigenstates of with corresponding eigenvalues . ( acts on qubits, so .) , which acts on a -dimensional Hilbert space, has two -dimensional invariant subspaces. The first is spanned by with eigenvalues . The second is spanned by eigenvectors with eigenvalues .

We can perform universal adiabatic quantum computation in such an eigenstate of a stoquastic Hamiltonian. To prove this, we make use of the universal adiabatic Hamiltonian from [8], which at all takes the form shown in equation 2. One can use the construction described above to obtain a stoquastic Hamiltonian corresponding to each instantaneous Hamiltonian . In this way we obtain a time varying Hamiltonian whose spectrum in the subspace exactly matches the spectrum of , the only difference being the addition of an ancilla qubit in the state. Because has no coupling between the subspace and the subspace, the adiabatic theorem may be applied within the subspace. The relevant eigenvalue gap is thus the same as that of , and so is the runtime.

In standard adiabatic quantum computation, the qubits are in the ground state of the instantaneous Hamiltonian. Thus, any disturbance to the state costs energy. This is thought to offer some protection against thermal noise [13]. When performing universal adiabatic quantum computation with , the qubits are not in the ground state. Thus, it is possible for the system to thermally relax out of the computational state. However, this can only occur by disturbing the ancilla qubit out of the state . By protecting the ancilla qubit, one can to a large degree protect the entire computation. Note that an energy penalty against the ancilla qubit leaving the state would be non-stoquastic. This is why the above construction fails to prove QMA-completeness and universal adiabatic quantum computation using the ground state of a stoquastic Hamiltonian, as we expect it must, based on the complexity-theoretic results of [11, 10, 20, 12].

Iii QMA-Complete problems for Markov Matrices

The second main result of our paper provides an example of a QMA-complete classical problem: finding the lowest eigenvalue of a symmetric Markov matrix. A matrix with all nonnegative entries, such that the entries in any given column sum to one is called a stochastic or Markov matrix. These matrices are named after Markov chains, which are stochastic processes such that given the present state, the future state is independent of the past states. Suppose a system has possible states. Then, its probability distribution at time is described by the -dimensional vector whose entries are nonnegative and sum to one. If the system is evolving according to a Markov process then its dynamics are completely specified by the equation where is a stochastic matrix. Note that, like quantum Hamiltonians, Markov matrices often have tensor product structure. For example, suppose we have two independent simultaneous Markov chains governed by and . Then their joint probability distribution is governed by .

Markov processes for which the Markov matrix is symmetric correspond to random walks on undirected weighted graphs. (Self-loops are allowed and correspond to diagonal matrix elements.) These matrices are doubly-stochastic: the sum of the entries in any row or column is one. By the Perron-Frobenius theorem, the highest eigenvalue of a symmetric stochastic matrix is one, and the corresponding eigenvector is the uniform distribution. The eigenvalue with next largest magnitude controls the rate of convergence of the process to its fixed point. A symmetric stochastic matrix is Hermitian and therefore one can also think of these matrices as Hamiltonians.

To prove that finding the lowest eigenvalue of a 3-local symmetric stochastic matrix is QMA-complete, we again use a reduction from the QMA-complete Hamiltonian of [8]. We must take the opposite sign convention from equation 3:

(9)

where the coefficients are the same as before (all positive) and . Now define:

(10)

where

and is the permutation matrix obtained by applying the replacement rules (5) to . By construction, is a 3-local, symmetric, doubly stochastic matrix. We can rewrite as

(11)

where . Thus, to determine an eigenvalue of to within we must find the corresponding eigenvalue of to within . Because is a two-local Hamiltonian on qubits with coupling strengths of order unity, is at most . Thus the problem of determining the eigenvalue of corresponding to the ground state of to polynomial precision is QMA-hard.

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

Here acts on the ancilla qubit, thereby giving it an energy penalty of size against leaving the state . For , is a stochastic Hamiltonian. For the energy penalty is large enough that the highest eigenvalue in the subspace lies below the lowest eigenvalue in the subspace. In this case the lower half of the spectrum of is the spectrum of scaled by , and the upper half of the spectrum of is the spectrum of scaled by and shifted up by .

Thus, we can obtain the ground energy of to polynomial precision by computing the lowest eigenvalue of to a higher but still polynomial precision. This reduction proves that finding the lowest eigenvalue of to polynomial precision is QMA-hard. Using the quantum algorithm for phase estimation, one easily shows that the problem of estimating the lowest eigenvalue of is contained in QMA (see [18]). Thus this problem is QMA-complete.

Iv Frustration Free Adiabatic Computation

It was stated in [12] that universal adiabatic quantum computation can be performed in the ground state of a 5-local frustration-free Hamiltonian. Let be a quantum circuit acting on qubits with gates. Let

(12)

be a state of qubits corresponding to the state of the time evolution of a quantum circuit specified by gates and

(13)

be a state of clock qubits. Bravyi and Terhal construct a parametrized 5-local Hamiltonian such that the ground state satisfies

We can think of first register in as consisting of “work” qubits on which the computation happens and the second register in as being a clock containing a time written in unary.

For , the minimal eigenvalue gap between the ground state and first excited state of is . By the adiabatic theorem 222Many versions of the adiabatic therem have been proven. For one example see appendix F of [15]., eigenvalue gap ensures that given , one obtains by applying and varying from zero to one over time. By measuring the clock register of , one obtains the result with probability . If this result is obtained, one finds the output of the circuit by measuring the first register of qubits in the computational basis. By repeating this process with copies of one succeeds with high probability. Alternatively, one can pad the underlying circuit with identity gates, in which case each trial succeeds with probability .

The construction from [12] invokes the fact that the spectrum of is independent of the form of the gates . By choosing a gate set which is composed of elements of simply connected unitary groups such as and one may construct a continuous path connecting each gate to the identity, and use a single parameter to transform all gates from the identity to the final circuit at once. The Hamiltonian at corresponds to the identity circuit, and its ground state is the uniform superposition of the clock states tensored with the initial data on the work qubits. In this ground state, the qubits of the clock register are entangled. It is standard to design adiabatic computations such that the initial Hamiltonian has a product state as its ground state, because such states should be easily produced by cooling or single-qubit measurements. In this section we construct a modified version of the construction from [12] that satisfies this condition and is still frustration free.

Let indicate the clock qubit and let indicate the work qubit. Let

For define

and let

It can be directly verified that each , , and is a projector. Here, for convenience, we define so that it varies from zero to one half rather than from zero to one as is done in [12]. Our frustration-free Hamiltonian is the following sum of projectors:

If are chosen from a universal set of two-qubit gates then is an efficient 5-local frustration-free adiabatic quantum computer. To see how this Hamiltonian achieves universal adiabatic computation, we examine the various terms one by one. The ground state of is the simultaneous zero eigenspace of , , and . commutes with and provides an energy penalty of at least unit size if the clock register is not in one of the unary states . Thus the low lying spectrum of is strictly contained in the ground space of .

For any bit string and integer , let

where is as defined in equation 13. (We also define .) There are such states and they form an orthonormal basis for the ground space of . In this basis, takes the block-diagonal form

where

is an by matrix and . Here denotes the Hamming weight of the bit string . The appearance of is the sole manifestation of . The rest of the matrix elements all come from the “hopping” action of .

has the unique ground state

(14)

where and are as defined in equations 12 and 13, , and is a normalization factor. This constitutes the ground state of . The first excited state of has energy . Because of the direct sum structure of , we can apply the adiabatic theorem directly to . The runtime of the adiabatic algorithm is thus determined by the gap between the ground and first excited states of . This takes its minumum at , where it is equal to . For questions of fault tolerance it is also useful to know the eigenvalue gap between the ground and first excited states of the full Hamiltonian . The first excited energy of is equal to the ground energy of , which is . Thus the minumum eigenvalue gap of occurs at and is equal to .

By equation 14, the ground state of is , and the ground state of is the same state produced by the scheme of [12].

V Stochastic Frustration Free Computation

In [12], Bravyi and Terhal showed that adiabatic quantum computation in the ground state of a stoquastic frustration free Hamiltonian can be efficiently simulated by a classical computer. In this section we show that in contrast, one can perform universal adiabatic quantum computation in the ground state of a stochastic frustration free (StochFF) Hamiltonian . (Alternatively, we can view this as computation in the highest energy state of the stoquastic Hamiltonian .)

It has been shown that the two-qubit CNOT gate, together with any one-qubit rotation whose square is not basis preserving, are sufficient to perform universal quantum computation [25]. All matrix elements in these gates are real numbers. If we choose from this gate set then is a 5-local real frustration free Hamiltonian. Examining the construction of section III one sees that it can be applied to any Hamiltonian with real matrix elements, and it increases the locality by one. This construction also preserves frustration-freeness, as we will show in the next paragraph. We can thus use this construction on to obtain a 6-local stochastic frustration free Hamiltonian whose ground state is universal for adiabatic quantum computation

To show that the mapping of section III preserves frustration-freeness, consider applying this mapping to a frustration free local Hamiltonian , where (where each is, up to an overall sign, a tensor product of Pauli operators and each is positive). We obtain the Hamiltonian

(15)

where and . When , is stochastic and has a zero energy ground state with an eigenvalue gap which is times the gap of . Furthermore we see from (15) (and the fact that each is positive semidefinite) that is a sum of positive semidefinite operators. Hence the Hamiltonian is frustration free.

Vi Generalizations

The constructions of sections III and II replace Hamiltonians with real matrix elements of both signs by computationally equivalent Hamiltonians with real positive matrix elements. In this section we show that this technique can be generalized to directly replace Hamiltonians with complex matrix elements by computationally equivalent Hamiltonians with only real positive matrix elements. However, in the process we necessarily introduce a two-fold degeneracy of the ground state.

Let be an arbitrary -local Hamiltonian. We may expand as

(16)

where each is a tensor product of or fewer Pauli matrices and each is positive. Each entry in each is or . We can replace the group with its left-regular representation

(17)

where

(18)

The eigenvectors of are where

The corresponding eigenvalues are

(19)

Let and be the real and imaginary parts of . That is, and are the unique real symmetric and anti-symmetric matrices such that

Further let and and similarly for , where denotes the entrywise absolute value. Applying the replacement (17) to and dividing by yields the stochastic Hamiltonian with the decomposition

(20)

where

, thus the spectrum of in the subspace matches that of up to a normalization factor of and a pair of extra ancilla qubits. If we write each projector in terms of the Pauli basis we obtain

where is the projector onto the eigenvalue eigenstate of the Pauli matrix . Thus an penalty on the first ancilla qubit will separate the subspace from the subspace. So, taking, , the stochastic Hamiltonian

has ground space spanned by and , where is the ground state of and is the ground state of . , thus is the ground state of . , thus is the complex conjugate of the ground state of .

A simple argument shows that the doubling in the spectrum of is a necessary property for any construction which maps an arbitrary Hamiltonian onto a real Hamiltonian , where is equal to within a fixed 1D subspace of the ancillas. Suppose that we have such a map which sends an arbitrary Hamiltonian which acts on a Hilbert space to a real Hamiltonian on a larger Hilbert space with the property that

(21)

where the state does not depend on the particular Hamiltonian but the operator may depend on . Then for any eigenvector of with energy we have:

(22)

Since is real, complex conjugating this equation gives:

(23)

To show that doubling exists in the spectrum it is sufficient to show that . To prove this, first use equation (23) to obtain

(24)

Then use equation (21) to obtain

(25)

Equating these expressions gives

(26)

This must hold for all Hamiltonians and eigenstates and therefore it must be the case that . So we have shown that the doubling in the spectrum of is a necessary feature of the type of maps we consider. For constructing universal adiabatic quantum computers the degeneracy induced by this construction may be problematic. However, for proving complexity-theoretic completeness results it is often irrelevant, as we see in the next section.

Vii Stochastic -Sat

The methods of the previous section can be used to show, roughly speaking, that deciding whether or not a Hamiltonian which is a sum of positive semidefinite stochastic operators is frustration free is as difficult as the general problem of deciding whether a Hamiltonian is frustration free. In this section we formalize this by defining a problem called stochastic -SAT, which we show to be -complete for .

We first recall the definition of stoquastic -SAT which is given in  [12].

Problem: Stoquastic -SAT
Input: Input: A set of -local Hermitian operators for where and a parameter

  1. Each is positive semidefinite

  2. Each has norm which is bounded by a polynomial in .

  3. Every is stoquastic.

Output: If has a zero energy ground state, then this is a YES instance. Otherwise if every eigenstate of has energy then it is a NO instance.
Promise: Either the ground state of H has energy 0, or else it has energy .

The stoquastic -SAT problem is therefore the problem of deciding if a given stoquastic Hamiltonian that is a sum of positive definite operators is frustration free, given that either this is the case or else its ground energy exceeds  [12]. Note that this definition of stoquastic -SAT looks somewhat different from the definition of quantum -SAT which was given in section I, which was stated entirely in terms of projectors. Given an instance of stoquastic -SAT, we can define operators which project onto the zero eigenspaces of the . When the Hamiltonians are stoquastic, these projectors are guaranteed to have nonnegative matrix elements in the computational basis  [12]. So given an instance of stoquastic -SAT with Hermitian positive semidefinite operators , it is possible to construct another instance of stoquastic -SAT with operators that are all projectors.

We now define a problem called stochastic -SAT, which is identical to stoquastic -SAT except that condition 3 is replaced by

. Every is a stochastic matrix.

We note that there does not appear to be an equivalence between this definition of stochastic -SAT and the corresponding definition where all the are (in addition) required to be projectors.

Given these two definitions and the foregoing map from an arbitrary Hamiltonian to a stochastic Hamiltonian, we now show how to reduce any instance of quantum 4-SAT to an instance of stochastic 6-SAT. Starting with an instance of quantum 4-SAT specified by a set of projectors (for ) we use the map of the previous section (with for concreteness) on each projector to obtain a set of 6-local positive semidefinite stochastic Hamiltonians where

(27)

(Note that refers to the operator obtained by applying the mapping from equation (20).) If the 4-SAT instance is satisfiable, then the stochastic 6-SAT instance will also be satisfiable. Define to be the maximum value of obtained for one of the terms when using the mapping of equation (20). If the 4-SAT instance is not satisfiable then for any state there is some projector such that . If we take the parameter of the stochastic 6-SAT instance to be related to the parameter of the quantum 4-SAT instance by then the stochastic 6-SAT instance will also be unsatisfiable. Therefore stochastic 6-SAT is hard. Stochastic 6-SAT is contained in since every instance of stochastic 6-SAT can be mapped to an instance of quantum 6-SAT by taking projectors which project onto everything but the zero eigenspaces of the .

So completeness of stochastic 6-SAT follows from the results of Bravyi  [9] on quantum -SAT. This is in contrast to stoquastic -SAT, which is contained in MA for every constant  [12].

Viii QMA-completeness for excited states

The local Hamiltonian problem refers specifically to ground state energies. Similarly, we have formulated a computational problem based on the highest energy of a given Hamiltonian. It is natural to ask about the complexity of estimating the excited state. We can formulate this as follows. Let be a -local Hamiltonian on the Hilbert space of qubits. Let denote the eigenvalues of , with corresponding eigenvectors . The -energy problem is as follows.

Problem: -energy
Input: We are given a classical description of , an integer , and a pair of parameters such that .
Output: If answer YES. If output NO.
Promise: is such that the answer is YES or NO.

In this section we will show that the -energy problem is QMA-complete for any . Showing QMA-hardness is the easier of the two proofs. This can be achieved as follows. Let and be a pair of -local Hamiltonians on qubits, with spectra , and , , respectively. Then

is a -local Hamiltonian on qubits. Its complete set of eigenvalues is , with corresponding eigenvectors . To prove QMA-hardness of a low-lying excited state let to be a Hamiltonian such that determining whether the ground energy is close to zero is QMA-hard. Given an integer , let , , and

has exactly states with negative energy, and its lowest nonnegative eigenvalue is . Thus determining the excited energy of is QMA-hard. In particular, it is interesting to note that by choosing we construct a Hamiltonian whose eigenvalue gap between the ground state and first excited state is QMA-hard to compute.

Next we show containment in QMA. The naive protocol would be for Merlin to provide Arthur with the state and for Arthur to use phase estimation to check that the registers each contain a state of energy at most . The problem is that for “no” instances there are many ways for Merlin to cheat. For example if but , the answer is “no” but Merlin can provide the state as a supposed witness. To prevent this, Arthur needs to somehow check that he has been given a set of orthogonal states that each have energy at most . Thus we propose the following protocol.

Arthur demands that Merlin give him the state

(28)

Arthur performs the projective measurement to see that the state given to him by Merlin lies in the antisymmetric subspace of . If this fails he rejects. He then throws away all but the first register and performs phase estimation of to precision better than . If the state has energy above he rejects. Otherwise he accepts.

It is clear that for YES instances, Arthur will accept the state with high probability. (The only source of error is imprecision in phase estimation.) We will next prove that for NO instances the acceptance probability is at most . Using standard methods[18, 21, 22] we can amplify this protocol to obtain polynomially small acceptance probability for NO instances.

Lemma 1

For any state in the antisymmetric subspace of and any state , where is the identity operator on .

Proof: Extend to an orthonormal basis for . Let be the set of functions such that . Thus . For any we have the corresponding Slater determinant state.