Total Functions in QMA

Serge Massar1 and Miklos Santha2,3

1Laboratoire d’Information Quantique CP224, Université libre de Bruxelles, B-1050 Brussels, Belgium. 2CNRS, IRIF, Université Paris Diderot, 75205 Paris, France. 3Centre for Quantum Technologies & MajuLab, National University of Singapore, Singapore. December 15, 2020

The complexity class QMA is the quan- either output a circuit with length less than `, or tum analog of the classical complexity class output NO if such a circuit does not exist. The NP. The functional analogs of NP and functional analog of P, denoted FP, is the subset QMA, called functional NP (FNP) and of FNP for which the output can be computed in functional QMA (FQMA), consist in ei- polynomial time. ther outputting a (classical or quantum) Total functional NP (TFNP), introduced in witness, or outputting NO if there does [37] and which lies between FP and FNP, is the not exist a witness. The classical com- subset of FNP for which it can be shown that the plexity class Total Functional NP (TFNP) NO outcome never occurs. As an example, factor- is the subset of FNP for which it can be ing (given an integer n, output the prime factors shown that the NO outcome never occurs. of n) lies in TFNP since for all n a (unique) set TFNP includes many natural and impor- of prime factors exists, and it can be verified in tant problems. Here we introduce the polynomial time that the factorisation is correct. complexity class Total Functional QMA TFNP can also be defined as the functional ana- (TFQMA), the quantum analog of TFNP. log of NP ∩ coNP [37]. We show that FQMA and TFQMA can be TFNP contains many natural and impor- defined in such a way that they do not de- tant problems, including factoring, local search pend on the values of the completeness and problems[29, 42, 35], computational versions of soundness probabilities. We provide ex- Brouwer’s fixed point theorem[41] and finding amples of problems that lie in TFQMA, Nash equilibria[22, 17]. Although there proba- coming from areas such as the complex- bly do not exist complete problems for TFNP, ity of k-local Hamiltonians and public key there are many syntactically defined subclasses of quantum money. In the context of black- TFNP that contain complete problems, and for box groups, we note that Group Non- which some of the above natural problems can be Membership, which was known to belong shown to be complete. For recent work in this to QMA, in fact belongs to TFQMA. We direction, see [23]. also provide a simple oracle with respect to The quantum analog of NP is QMA [34]. which we have a separation between FBQP QMA has been extensively studied, and contains and TFQMA. a rich set of complete problems, see e.g. [12]. These complete problems are all promise prob- arXiv:1805.00670v4 [quant-ph] 12 Dec 2020 1 Introduction lems. For instance the most famous one, the k-local Hamiltonian problem, involves a promise Classical complexity classes are generally defined that the ground state energy of the input k-local as consisting of decision problems. But functional Hamiltonian is either less than b or greater than analogs of these classes can also be defined. The a, with a−b = 1/q(n), for some polynomial q(n), functional analog of NP is denoted FNP (Func- and the problem is to determine which is the case. tional NP). As a simple example, the functional Functional QMA, the problem of producing a analog of the travelling salesman problem is the quantum state that serves as witness for a QMA following: given a weighted graph and a length `, problem was first introduced in the unpublished manuscript [28]. For instance the functional ana- Serge Massar: [email protected] log of the k-local Hamiltonian problem is the fol-

1 lowing: given the classical description of a k-local 2 Definitions Hamiltonian, either output a state with energy less than b, or output NO if such a state does not 2.1 QMA exist. We denote by Hn the Hilbert space of n qubits. In [28] it was observed that there is no obvi- For pure states we use the Dirac ket notation ous reduction of FQMA problems to QMA prob- |ψi, whereas for density matrices we just use the lems. This should be opposed to the case of NP Greek letter ρ. We denote by In the identity ma- complete problems for which finding a witness re- trix acting on n qubits. duces to solving the decision problem. We denote by poly the set of all functions It is well known that the definition of QMA f : N → N, where N = {1, 2, ...}, for which does not depend on the values of the complete- there exists a polynomial time deterministic Tur- f(n) n ness and soundness probabilities, as they can be ing machine that outputs 1 on input 1 . Note brought exponentially close to 1 and 0 respec- that if f ∈ poly then there exists a polynomial q tively [34, 36, 40]. We discuss different definitions such that for all n ∈ N, f(n) < q(n). of Functional QMA. We show that with an ap- Computational processes that can be carried propriate definition, based on the notion of eigen- out in polynomial time are sometimes called effi- basis of a quantum verification procedure, one cient. can prove a similar amplification result. These theoretical considerations are the topic of Section Definition 1. Quantum Verification Proce- 2. dure. A quantum verification procedure is a family of polynomial time uniform quantum cir- In Section2 we also introduce the functional cuits Q = {Qn : n ∈ N} with Qn taking as in- class TFQMA (Total Functional QMA) as the put (x, |ψi ⊗ |0k(n)i) where x ∈ {0, 1}n is a bi- subset of FQMA such that only the YES an- nary string of length n, |ψi is a state of m(n) swer of the FQMA problem occurs, i.e. for all qubits, and both m = m(n) and k = k(n) belong classical inputs x there exists a witness. Simi- to poly. The last k qubits, initialized to the state larly to , the problems in are k TFNP TFQMA |0 i, form the ancilla Hilbert space Hk, and the not promise problems, rather they have a struc- m-qubit states |ψi form the witness Hilbert space ture such that one can prove that only the YES Hm. The outcome of the run of Qn is a random answer occurs. bit which is obtained by measuring the first qubit The main aim of the present paper is to show in the computational basis. We denote this out- that TFQMA is an interesting and rich complex- come by Qn(x, |ψi), and we interpret the outcome ity class. In Section3 we provide examples of 1 as accept and the outcome 0 as reject. problems that belong to TFQMA. These are related to problems previously studied in quan- Note that a quantum verification procedure can tum complexity, such as commuting quantum k- of course also take as input a mixed state ρ, rather SAT, commuting k-local Hamiltonian, the Quan- than a pure state |ψi. Mixed states can be writ- tum Lovász Local Lemma (QLLL) [8] and pub- ten as convex combinations of pure states. The lic key quantum money based on knots [20]. We acceptance (rejection) probability for the mixed show how these problems can be adapted to fit state is the convex combination of the acceptance (rejection) probabilities for the constituent pure into the TFQMA framework. Then in Section 4 we consider relativized problems. In the con- states. Abusing slightly the notation, we use text of black-box groups, we show that Group the same notation Qn(x, ρ) for the outcome of Non-Membership, which was known to belong to the quantum verification procedure on the mixed state ρ. QMA [50], in fact belongs to TFQMA. We also exhibit problems based on the Quantum Fourier Definition 2. (a,b)–Quantum Verification Transform (QFT) and provide a simple oracle Procedure. Let q ∈ poly, and let a, b : → with respect to which there is a separation be- N [0, 1] be polynomial time computable functions tween and . FBQP TFQMA which satisfy In the conclusion we present open questions raised by the present work. a(n) − b(n) ≥ 1/q(n) . (1)

2 We say that a quantum veriﬁcation procedure Q As a consequence the precise values of the is an (a, b)-quantum veriﬁcation procedure (or bounds a and b are irrelevant. Traditionally they shortly an (a, b)-procedure) if for every x of are taken to be 2/3 and 1/3. We will do here the length n, one of the following holds: same.

∃|ψi : Pr[Qn(x, |ψi) = 1] ≥ a, (2) Deﬁnition 4. We deﬁne the class QMA as (2/3, 1/3). ∀|ψi : Pr[Qn(x, |ψi) = 1] ≤ b. (3) QMA

We call a and b the completeness and soundness We will come back to the QMA amplification probabilities of the quantum verification proce- procedure below. dure. We now turn to a particular kind of (a, b)- procedure which will be our main topic of study: Definition 3. QMA and coQMA. Let a, b be functions as in Definition 2. The class Definition 5. a-Total Quantum Verification QMA(a, b) is the set of languages L ⊆ {0, 1}∗ Procedure. Let a : N → [0, 1] be a polynomially such that there exists an (a, b)-procedure Q, where time computable function. We say that a quan- for every x, we have x ∈ L if and only if Equa- tum verification procedure Q is an a-total quan- tion (2) holds (and consequently, x∈ / L if and tum verification procedure (or shortly an a-total only if Equation (3) holds). procedure) if for every x of length n, the follow- We call Q a quantum verification procedure ing holds: for L. For x ∈ L, we say that a |ψi satisfying Equation (2) is a witness for x. ∃|ψi : Pr[Qn(x, |ψi) = 1] ≥ a . (4) The class coQMA(a, b) is the set of languages L ⊆ {0, 1}∗ such that there exists an (a, b)- Note that an a-total procedure is also an (a, b)- quantum verification procedure Q0, where for ev- procedure for all b satisfying the conditions of ery x, we have x ∈ L if and only if Equation (3) Definition2. Note that the language associated ∗ holds (and consequently, x∈ / L if and only if to an a-total procedure is L = {0, 1} . That is Equation (2) holds). the decision problem for total procedures is trivial, since for all x ∈ {0, 1}∗ there exists a witness It is of course essential to understand to for x. Therefore for total procedures, the only what extent the above definitions depend on the interesting questions concern the witnesses. bounds a and b. Obviously we can decrease a and In order to prepare for a detailed study of to- increase b: QMA(a, b) ⊆ QMA(a0, b0) with a0 ≤ tal procedures, we therefore delve deeper into the a and b0 ≥ b, so long as a0(n) − b0(n) ≥ 1/q0(n), structure of the witness space. for some q0 ∈ poly. But can one increase a and decrease b? This 2.2 Structure of the witness space. was first addressed by Kitaev who showed that the separation a − b could be amplified to expo- The methods used in [36, 40] to obtain Theo- nentially close to 1 by using multiple copies of rem1 are based on Jordan’s lemma [30] (for a the input state and multiple copies of the verifi- short proof of Jordan’s lemma, see [43]). The cation circuit [34], that is by increasing both m use of Jordan’s lemma in this context provides and k. This was further improved in [36] (see also important insights into the structure of the wit- [40]) where it was shown that by running forwards ness space. A succinct proof of this structure was and backwards the original quantum verification given in [3]. We state here these result which will procedure, only one copy of the input state was play an important role in what follows. needed to obtain the same amplification, that is one needs only increase k. Theorem 2. Structure of witness space [36, 3]. Given a quantum verification procedure Q = n Theorem 1. QMA Amplification [34, 36, {Qn}, for all x ∈ {0, 1} , there exists a basis m 40]. For all a, b be functions as in Definition 2, BQ(x) = {|ψii : 1 ≤ i ≤ 2 } of the witness for all r ∈ poly, we have QMA(a, b) ⊆ QMA(1− space Hm such that the acceptance probability of 2−r, 2−r). linear combinations of the basis states does not

3 involve interferences, that is for all αi such that 2.3 Relations P |α |2 = 1, we have i i Consider a quantum verification procedure Q. In P this section we are interested in the set of states Pr[Qn(x, i αi|ψii) = 1] P 2 on which Q accepts with high probability. We = |αi| Pr[Qn(x, |ψii) = 1] . (5) i are also interested in the set of states on which Proof. This result follows from the spectral de- Q rejects with high probability. This leads us to composition of the POVM element correspond- the following definitions. ing to the quantum verification procedure giving outcome 1, see beginning of Section 6.1 of [3] for Definition 7. Accepting and rejecting den- details. sity matrices and subspaces. Let Q = {Qn} be a quantum verification procedure and fix a ∈ Definition 6. Eigenbasis, spectrum and [0, 1]. eigenspaces of a quantum verification pro- We define the following relations over binary cedure. Fix a quantum verification procedure strings and density matrices: n Q = {Qn}, x ∈ {0, 1} , and an eigenbasis ≥a RQ (x, ρ) = 1 if Pr[Qn(x, ρ) = 1] ≥ a , BQ(x) = {|ψii} of Q for x. ≤a Given |ψii ∈ BQ(x), we call RQ (x, ρ) = 1 if Pr[Qn(x, ρ) = 1] ≤ a . (9)

pi = Pr[Qn(x, |ψii) = 1] (6) Using the notion of eigenspace HQ(x, p) of Q introduced previously, we deﬁne the following bi- the acceptance probability of |ψii. nary relations over binary strings and quantum We call the set of acceptance probabilities the states: spectrum of Q for x: ≥a HQ (x, |ψi) = 1 if |ψi ∈ Span({HQ(x, p): p ≥ a}) , Spect(Q, x) = {p ∈ [0, 1] : ∃|ψii ∈ BQ(x) ≤a HQ (x, |ψi) = 1 if |ψi ∈ Span({HQ(x, p): p ≤ a}) . such that Pr[Qn(x, |ψii) = 1] = p} . (7) (10)

Given p ∈ Spect(Q, x), we call To simplify notation, we denote

HQ(x, p) = Span({|ψii ∈ BQ(x) ≥a n ≥a o RQ (x) = ρ : RQ (x, ρ) = 1 , (11) : Pr[Qn(x, |ψii) = 1] = p}) (8) ≤b n ≤b o RQ (x) = ρ : RQ (x, ρ) = 1 , (12) the eigenspace of Q for x with acceptance proba- ≥a n ≥a o bility p. HQ (x) = |ψi : HQ (x, |ψi) = 1 , (13) ≤b n ≤b o The eigenbasis is not necessarily unique: if two HQ (x) = |ψi : HQ (x, |ψi) = 1 , (14) states |ψ i, |ψ 0 i ∈ B (x) have the same accep- i i Q and we will generally express results in terms tance probability, than a unitary transformation ≥a ≤b of the sets RQ (x),RQ (x) and the subspaces acting on |ψii, |ψi0 i yields a new eigenbasis. How- ≥a ≤b ever, as the following result shows, this is the only HQ (x), HQ (x), rather then the corresponding freedom one has when choosing an eigenbasis. relations. The following result explains how these defini- Theorem 3. Uniqueness of the spectrum tions are related. and eigenspaces of Q. Given a quantum veri- ∗ Theorem 4. Partial equivalence between fication procedure Q = {Qn} and x ∈ {0, 1} , the spectrum Spect(Q, x) of Q and the eigenspaces accepting and rejecting density matrices and subspaces. Let a, b be functions as in HQ(x, p) of Q with acceptance probability p ∈ Spect(Q, x) are unique and do not depend on the Definition 2 and let Q be an (a, b)-procedure. Then, choice of eigenbasis BQ(x). 1. we have the inclusion Proof. Follows from the uniqueness of the spectral decomposition of the POVM element corre- H≥a(x) ⊆ R≥a(x) sponding to the quantum verification procedure Q Q ≤a ≤a giving outcome 1. HQ (x) ⊆ RQ (x) (15)

4 ≥a ≤a 0 (where we view HQ (x) and HQ (x) not as combination a|ψi + b|ψ i does not necessarily be- ≥a ≥a sets of pure states, but as the sets of den- long to RQ (x). On the other hand HQ (x) is a sity matrices associated to these pure states); subspace and hence closed under linear combinations. Second, R≥a(x) does not transform simply ≥a Q 2. and in the other direction, if RQ (x) is non under the amplification procedure described in ≥a ≥a empty, then HQ (x) is non empty, while if [36, 40], while HQ (x) does transform in a simple ≤a ≤a way, see Theorem5 below. RQ (x) is non empty, then HQ (x) is non empty We therefore adopt a definition of FQMA ≥a based on the subspace HQ (x). Proof. We consider the ≥ a case, the ≤ a case is For reasons that we discuss in the next para- similar. graphs, we define Functional QMA in terms of Denote by {|ψii} the basis of eiegnestates of Q two relations: for x. ≥a Definition 8. Functional QMA (FQMA). Point 1 is trivial: HQ (x) is constituted of all Let a, b be functions as in Definition 2. linear combinations of eigenstates |ψii whose acceptance probability p is greater or equal than The class FQMA(a, b) is the set i ≥a ≤b a. Hence, using Eq. (5), these states belong to {(HQ (x, |ψi), HQ (x, |ψi))} of pairs of re- ≥a lations, where Q is an (a, b)-procedure. RQ (x). ≥a Point 2 is also easy. Since RQ (x) is non empty, In order to motivate the above definition, it is there is at least one density matrix ρ whose ac- interesting to consider the following example. ceptance probability is greater or equal than a. We can write ρ as a convex combination of pure Example 1. Consider a function : N → states. At least one of these pure states must [0, 1/3) that decreases faster than 1/poly(n) for have acceptance probability greater or equal than any polynomial poly(n), for instance (n) = −n−2 a. We write this state in the basis of eigenstates 2 as The example consists of a quantum verification X procedure Q whose spectrum is the set |ψi = αi|ψii i 1 2 2 X X Spect(Q, x) = { , − (n), } . (17) = αi|ψii + αi|ψii . (16) 3 3 3 i:pi

5 are accepted with probability 1 and those which wish that the soundness and completeness thresh- are accepted with probability 0, hence Functional olds of Q be mapped onto the soundness and com- NP can be described by a single relation. In pleteness thresholds of Q0, where we recall that the case of Functional QMA, there are three the soundness and completeness thresholds must kinds of states in the witness Hilbert space, those be polynomial time computable, see Definition2. which are accepted with probability greater than That is, if Q is an (a, b)-procedure such that there a, those which are accepted with probability less exists an e–map from Q to Q0 via f, then Q0 than b, and the states which have intermediate is an a0, b0-procedure with a0(n) = f(a(n)) and acceptance probabilities. Hence it is natural to b0(n) = f(b(n)). describe Functional QMA by two relations. Note that eigenspace preserving maps are tran- sitive: if there exists an e-map from Q to Q0, and if there exists an e-map from Q0 to Q00, then there 2.5 FQMA amplification exists an e-map from Q to Q00. Note also that if We now show how the QMA Amplification results we require that the inverse f −1 of the strictly in- [36, 40] apply to Functional QMA. We will show creasing function f in Definition9 is also polyno- that, as for QMA, the bounds a and b that ap- mial time computable, then eigenspace preserving pear in the the definition of FQMA(a, b) can be maps are an equivalence relation. changed at will. The analysis is based on [36, 40], Theorem 5. QMA Amplification[36] is an but needs some new concepts, as we need to show eigenspace preserving map. Let Q be a quan- that the structure of the witness space does not tum verification procedure. Let a, b be functions change under amplification. as in Definition 2. For all r ∈ poly there exists 0 Definition 9. Eigenspace preserving map a quantum verification procedure Q , such that 0 of quantum verification procedures. Let Q there exists an e-map from Q to Q , and such that and Q0 be two quantum verification procedures. the polynomial time computable strictly increas- We say that there exists an eigenspace preserving function f that defines the e–map (see Defi- −r −r ing map from Q to Q0 if for all x ∈ {0, 1}∗: nition9) satisfies f(a) ≥ 1−2 and f(b) ≤ 2 . Proof. One checks that the QMA amplification 1. there exists a basis B (x) = {|ψ i} of the Q i procedure of [36] is an eigenspace preserving map witness Hilbert space H which is a joint m with the above properties. eigenbasis of Q and Q0 for x; Note that the amplification procedure of [36] 2. there exists a polynomial time computable does not allow us to choose f(a) and f(b) arbi- strictly increasing function f : [0, 1] → [0, 1] trarily. For this reason we introduce the following such that if pi = Pr[Qn(x, |ψii) = 1] is the deamplification procedure. acceptance probability of |ψii for Qn, and 0 0 Theorem 6. QMA Deamplification is an pi = Pr[Qn(x, |ψii) = 1] is the acceptance 0 0 eigenspace preserving map. Let Q be a quan- probability of |ψii for Qn, then pi = f(pi). tum verification procedure. Let a, b and a0, b0 be In what follows we will refer to an eigenspace pre- pairs of functions as in Definition 2 with a ≥ serving map simply as an e–map. a0 > b0 ≥ b. Then there exists a quantum verification procedure Q0, such that there exists an As a consequence, if there exists an e-map from e-map from Q to Q0, and such that the polyno- 0 Q to Q , then most questions about witnesses mial time computable strictly increasing function for Q can be reduced to questions about wit- f that defines the e–map (see Definition9) sat- 0 ≥a ≥f(a) 0 0 nesses for Q , in particular HQ (x) = HQ0 (x) isfies f(a) = a and f(b) = b . ≤b ≤f(b) ≥a and HQ (x) = HQ0 (x). However RQ (x) 6= Proof. We construct Q0 as follows. ≥f(a) ≤a ≤f(a) 0 0 RQ0 (x) and RQ (x) 6= RQ0 (x) which is one Let z, z : N → [0, 1], with z > z , be two of the reasons why we define functional QMA in polyomial time computable functions to be fixed ≥a ≤a terms of HQ (x) and HQ (x). below. The reason why we require that the function On any input (x, |ψi) run Q; if Q accepts, then f be polynomial time computable is because we accept with probability z ∈ [0, 1] and reject with

6 probability 1 − z; if Q rejects, then accept with Definition 11. Efficiently preparable probability z0 ∈ [0, 1] and reject with probability states. Let m ∈ poly. A family of density 0 n 1 − z . matrices {ρ(x): x ∈ {0, 1} , n ∈ N} is efficiently 0 It is immediate to check that Q e–maps to Q , preparable if ρ(x) acts on Hm(n) and if there with the the strictly increasing function f that exists a polynomial time uniform family of defines the e–map (see Definition9) given by quantum circuits Q = {Qn : n ∈ N} with Qn taking as input (x, |0ki) with x ∈ {0, 1}n and f(p) = (z − z0)p + z0 . (18) k ∈ poly with k ≥ m, and where ρ(x) is obtained 0 We now solve for z and z the equations f(a) = by tracing out the last k − m qubits of Qn(x). a0 and f(b) = b0. It is easy to check that z, z0 are rational functions of a, b, a0, b0 hence polynomial Definition 12. The language class BQP. time computable, that z, z0 ∈ [0, 1], and that z > BQP ⊆ QMA is the set of languages L ⊆ ∗ z0 since a ≥ a0 > b0 ≥ b. {0, 1} such that there exists:

Theorem 7. FQMA is independent of the 1. an (2/3, 1/3) quantum verification procedure bounds (a, b). Let Q be a quantum verification Q = {Qn : n ∈ N} with Qn taking as input procedure. Let a, b and a0, b0 be pairs of functions (x, |ψi ⊗ |0ki), where x ∈ {0, 1}n is a binary as in Definition 2 with a0 < 1−2−r and b0 > 2−r, string of length n, |ψi is a state of m qubits, for some r ∈ poly. Then there exists a quantum with m, k ∈ poly; verification procedure Q0, such that there exists an e-map from Q to Q0, and such that the strictly 2. an efficiently preparable set of density ma- increasing function f that defines the e–map (see trices {ρ(x)} where ρ(x) acts on Hm; Definition9) satisfies f(a) = a0 and f(b) = b0. and where for every x, we have x ∈ L if and only Proof. We first use the amplification procedure of if [36] to construct an intermediate quantum veri- 00 fication procedure Q , such that there exists an Pr[Qn(x, ρ(x)) = 1] ≥ 2/3, (19) e-map from Q to Q00 (as follows from Theorem 5). The parameters of the amplification proce- and x∈ / L if and only if dure are chosen such that the strictly increasing 00 ∀|ψi, Pr[Qn(x, |ψi) = 1] ≤ 1/3. (20) function f1 that defines the e–map from Q to Q (see Definition9) satisfies f (a) ≥ 1 − 2−r and 1 Definition 13. Functional BQP. The class f (b) ≤ 2−r. 1 FBQP is the subset of pairs of relations We then apply to Q00 deamplification as in The- {(H≥2/3(x, |ψi), H≤1/3(x, |ψi))} in , with orem6 to obtain the quantum verification proce- Q Q FQMA Q an (2/3, 1/3)-procedure, such that there ex- dure Q0. The parameters of the deamplification ists an efficiently preparable set of density ma- procedure are chosen such that the strictly in- trices {ρ(x)} and for all x, if H≥2/3(x, |ψi) is creasing function f2 that defines the e–map from Q 00 0 0 ≥2/3 Q to Q (see Definition9) satisfies f2(f1(a)) = a non empty then ρ(x) ∈ RQ (x, |ψi). 0 and f2(f1(b)) = b . As a consequence of Theorem7, the precise 2.7 Total Functional QMA values of the bounds a and b are irrelevant to the We now address the central topic of this study, definition of FQMA. Therefore, similarly to the the subset of FQMA for which there always ex- definition of QMA we make the following defini- ists a witness. We had previously introduced a– tion. total procedures in Definition5. We can now de- Definition 10. We define the class FQMA as fine the corresponding functional classes. FQMA(2/3, 1/3). Definition 14. Totality. A pair of relations (H≥a(x, |ψi), H≤b(x, |ψi)) in (a, b) 2.6 FBQP Q Q FQMA is called total if for all inputs x there exists ≥a The class BQP is the set of decision problems that at least one witness |ψi, i.e. if HQ (x) is non can be efficiently solved on a quantum computer. empty.

7 Definition 15. Total Functional QMA 2.9 1– and/or 0–Quantum Verification Proce- (TFQMA). Let a, b be functions as in dures Definition 2. The class TFQMA(a, b) is Because of the amplification theorem, in the set (H≥a(x, |ψi), H≤b(x, |ψi)) of pairs of FQMA Q Q all the above definitions we can replace the upper total relations, i.e. the set of pairs of relations bound a (by convention taken to be 2/3) in the in FQMA where Q is an a–total verification definition by 1 − 2−r, and the lower bound b (by procedure. convention taken to be 1/3) by 2−r, for any poly- The class = (1/3, 2/3) is TFQMA TFQMA nomial r. However sometimes one can show that the set of total relations in . FQMA one can take a = 1 and/or b = 0, which is po- tentially a stronger statement. In particular the We emphasize that problems in TFQMA are inclusions TFgapQMA(1, 0) ⊆ TFgapQMA ⊆ not promise problems: they satisfy that for all TFQMA might be strict. x there exists at least one witness. In analogy When a = 1 and/or b = 0 we are dealing with with and , we expect problems in FNP TFNP exact quantum computation. This means that the to be simpler than general problems in TFQMA quantum circuit is made out of a finite (or possi- . FQMA bly enumerable) set of quantum gates, and all operations (state preparation, gates, measurements 2.8 Gapped quantum verification procedures in the computational basis) are implemented with zero error. Exact quantum computation has been A sub-class of quantum verification procedures studied in several contexts. In particular QMA1 which will be of interest are those which have is the subset of QMA in which the accepting a gap in their spectrum. They are defined as probability in the case of YES instances is 1, follows. i.e. QMA1 = QMA(1, 1/3). Complete problems for QMA1 were described in [13, 24]. For Definition 16. Gapped (a,b)–Quantum arguments why it appears difficult to prove that Verification Procedure. An (a, b)-procedure QMA = QMA1, see [2]. Q is a gapped (a, b)- procedure if for every x To simplify notation when dealing with exact of length n, there are strictly no eigenstates with quantum computation, we write acceptance probability comprised between a and b. 1 ≥1 ≥a ≤b RQ(x) = RQ (x) , (22) As a consequence the spaces HQ (x) and HQ (x) 0 ≤0 generate the entire witness Hilbert space: RQ(x) = RQ (x) , (23) 1 ≥1 HQ(x) = H (x) , (24) H = Span(H≥a(x) ∪ H≤b(x)) . (21) Q m Q Q 0 ≤0 HQ(x) = HQ (x) . (25) Definition 17. gapQMA. Let a, b be functions as in Definition 2. The class gapQMA(a, b) is 3 Problems in TFQMA the set of languages L ⊆ {0, 1}∗ such that there exists a gapped (a, b)-procedure Q, where for ev- 3.1 Preliminary Considerations ery x, we have x ∈ L if and only if Equation (2) 3.1.1 Introduction holds (and consequently, x∈ / L if and only if Equation (3) holds). In this section and the next one we provide examples of problems in TFQMA which are not Definition 18. Total Functional gap obviously in FBQP. QMA (TFgapQMA). Let a, b be func- At the heart of any such example is a quantum tions as in Definition 2. The class verification procedure Q. The minimum require- TFgapQMA(a, b) is the set of pairs of to- ments for an example to be included in our list tal relations {(H≥a(x, |ψi), H≤b(x, |ψi)} for is to be able to show that Q is an a-total quan- Q Q ≥a which Q is a gapped (a, b)-quantum verification tum verification procedure, i.e. that RQ (x) is ≥a procedure. non empty for all x (and consequently HQ (x) We define the class TFgapQMA as is non empty). However in some cases we can TFgapQMA(2/3, 1/3). also determine the eigenbasis BQ(x) = {|ψii}

8 of Q for x, and (at least partially) characterise that the SWAP test has outputted "Symmetric", ≥a ≤b the pair of relations (HQ (x, |ψi), HQ (x, |ψi)) in while in the second case that it has outputted TFQMA(a, b). The precise formulation of the "Anti-symmetric". examples will depend on how much we can say about them. 3.2 Eigenstates of commuting k-local Hamil- When possible we shall formulate the prob- tonian lems in such a way that they belong to TFgapQMA(1, 0), as this is the strongest state- 3.2.1 Background ment. For convenience in what follows, the input size Definition 19. k-local Hamiltonian. Fix will not necessarily be denoted by n, which can k, d ∈ N. A qudit is a quantum system of di- be reserved for some parameter in the input. mension d. Denote by H the Hilbert space of n qudits. Let A ∈ poly.A k-local Hamiltonian is a Hermitian matrix acting on H which can be 3.1.2 Measurements PA(n) written as H = a=1 Ha, where each term Ha It is often convenient to describe some of the steps (sometimes called constraint) is a Hermitian op- of a quantum verification procedures as a mea- erator that acts non trivially on at most k qudits surement. By this we mean that we carry out and whose matrix elements in the computational an ideal quantum measurement (also known as basis have an efficient classical description. a quantum non demolition measurement) which is realised as a unitary transformation. Consider The k-local Hamiltonian problem is to deter- P the operator A = a aΠa, where the a’s are the mine whether the ground state of H has energy eigenvalues of A and Πa are orthogonal projec- ≤ b or ≥ a, with a − b ≥ 1/poly(n), for some tors. Measuring A on state |ψi corresponds to polynomial poly(n), with the promise that only carrying out the unitary evolution one of these cases occurs. The k-local Hamilto- nian problem is QMA complete [34, 31] even when X |ψi|0i → Πa|ψi|ai (26) k = 2 [33]. a The commuting k-local Hamiltonian is the where the second register is an ancilla that reg- case where the operators Ha commute. It was isters the outcome of the measurement. Subse- shown by Bravyi and Vyalyi that the commuting quent operations can then be carried out condi- 2-local Hamiltonian problem is in NP [14]. Some tional on the state of the ancilla. additional cases of commuting k-local Hamilto- A specific measurement we will use several nian problems also in NP are: the 3-local Hamil- times is the projection on the antisymmetric and tonian where the systems are qubits [5], the 3- symmetric spaces. This can be realised by imple- local Hamiltonian where the systems are qutrits menting the SWAP test [15] as follows: one car- and the interaction graph is planar or more gener- ries out a Hadamard transform on an ancilla, then ally nearly Euclidean [5], the planar square lattice a conditional SWAP, and finally a Hadamard on of qubits with plaquette-wise interactions [47]; the ancilla. approximating the ground state energy when the interaction graph is a locally expanding graph [6]. 1 |φi|ψi|0i → √ |φi|ψi (|0i + |1i) The complexity of the commuting k-local Hamil- 2 tonian problem in the general case is unknown. 1 → √ (|φi|ψi|0i + |ψi|φi|1i) A particularly interesting case is when each Ha 2 is a projector, that is has only 0, 1 eigenvalues. 1 → (|φi|ψi + |ψi|φi) |0i The k-local Hamiltonian problem in this case re- 2 duces to the question whether H has a frustration 1 + (|φi|ψi − |ψi|φi) |1i (27) free eigenstate, an eigenstate with eigenvalue 0. 2 This is known as quantum k-SAT ( denoted k- After these operations, if the ancilla is in the |0i QSAT), and was introduced in [13] where it was state one has projected onto the symmetric space, shown that 2-QSAT is in P and k-QSAT for k ≥ 4 while if it is in the |1i state one has projected onto is QMA1 complete (where QMA1 is the subset the anti–symmetric space. In the first case we say of QMA in which the accepting probability in the

9 case of YES instances is 1). It was later shown A = n constraints with 0, 1 eigenvalues (projec- that 3-QSAT is also QMA1 complete [24]. tors), and where by hypothesis each constraint can be measured with zero error in polynomial 3.2.2 Notation time using a quantum computer. Denote by H1(x) the subspace of the space of Consider a commuting k-local Hamiltonian H = 2n + 1 qubits H spanned by states satisfying PA 2n+1 a=1 Ha, [Ha,Ha0 ] = 0, acting on n qubits. Since one of the two conditions: the Ha’s are hermitian and commute, they pos- sess a common eigenbasis. That is, there exists a 1. States with the ﬁrst qubit set to 0, the next basis B = {|ψhji} of the Hilbert space Hn, where n qubits a frustration free state, i.e. a state A each basis state |ψhji is also an eigenstate of all such that its eigenvalue sequence is h = 0 , the constraints Ha: and the last n qubits are in an arbitrary state; Ha|ψhji = ha|ψhji , (28)

hψh0j0 |ψhji = δh0hδj0j . (29) 2. States with the ﬁrst qubit set to 1 and the remaining 2n qubits the antisymmetric lin- A We denote by h = (h1, ..., hA) ∈ R the string ear combination of two orthogonal eigen- of eigenvalues of Ha. The same symbol h is also states with the same eigenvalue sequence h = used as the ﬁrst index in labelling the basis states (h1, ..., hn): |ψhji. The second index j ∈ Jh labels orthogonal states within the subspaces with the same eigen- 1 value h. H (x) = Span( 0 0 We denote by E the energy of the eigenstate: {|0i|ψ0Aji|ψh0j0 i : j ∈ J0A , (h , j ) ∈ F } 1 A X ∪ {√ |1i |ψhji|ψhj0 i − |ψhj0 i|ψhji E = ha . (30) 2 0 0 a=1 : h ∈ G, j, j ∈ Jh, j 6= j } We denote by ). (33)

F = {(h, j)} (31) Denote by H0(x) the orthogonal subspace: the sets of indices of the basis B. Since B is a H0(x)= Span( n 0 0 A basis we have |F | = 2 . We denote by {|0i|ψhji|ψh0j0 i :(h, j), (h , j ) ∈ F, h 6= 0 } 0 0 0 G = {h : ∃j (h, j) ∈ F } (32) ∪{|1i|ψhji|ψh0j0 i :(h, j), (h , j ) ∈ F, h 6= h } ∪{|1i|ψhji|ψhji :(h, j) ∈ F } the set of possible strings of eigenvalues of H . a 1 Since there may exist orthogonal eigenstates with ∪{√ |1i |ψhji|ψhj0 i + |ψhj0 i|ψhji 2 the same string of eigenvalues, |G| ≤ 2n with : h ∈ G, j, j0 ∈ J , j 6= j0} equality not necessarily attained. h ) Note that given an eigenstate |ψhji, one can efficiently determine the string h = (h1, ..., hA) (34) by measuring each H in succession, where the a Theorem 8. The pair of subspaces order is immaterial since the H commute. a (H1(x), H0(x)) defined in Definition 20 be- In the case where each H is a projector, h ∈ a a long to (1, 0)-total functional gap : {0, 1}, and h ∈ {0, 1}A. QMA (H1(x), H0(x)) ∈ TFgapQMA(1, 0) . (35) 3.2.3 Frustration-Free or Degenerate Eigenspace of commuting quantum k-SAT Definition 20. Frustration free or degener- Proof. We will show that there is a quantum ver- ate eigenspace of commuting quantum k- ification procedure Q that accepts with probabil- SAT with n constraints. Denote by x the clas- ity 1 on the states in H1(x) and accepts with sical description of a commuting k-local Hamil- probability 0 on the states in H0(x). Further- 1 tonian acting on the space of n qubits Hn, with more we will show that H (x) is non empty

10 for all x. This then implies that Q is a (1, 0) form gapped total quantum verification procedure, |1i 1 0 √ |ψhji|ψhj0 i − |ψhj0 i|ψhji , (37) and (H (x), H (x)) ∈ TFgapQMA(1, 0). 2 Quantum verification procedure. where |ψ i and |ψ 0 i are different basis vectors We first describe Q. hj hj with the same eigenvalue sequence. This state is Measure the first qubit, obtaining outcome b. also accepted by probability 1 by Q. If b = 0, measure all the Ha on the first n Hence H1(x) is non empty for all x. qubits, let the result be h. Accept if h = 0A, and otherwise reject. Note that if one or more of the constraints If b = 1, measure all the Ha both on the first n Ha = In is the identity operator, then there is qubits and on the second n qubits, yielding two no frustration free state, and the witnesses are n bit eigenvalue sequences h = (h1, ..., hn) and necessarily of the form given in Equation (37). 0 0 0 0 Note that the existence argument in the proof h = (h1, ..., hn). Reject if h 6= h . If h = h0 carry out a SWAP test. Reject if the of Theorem8 is based on the pigeonhole princi- SWAP test outputs "Symmetric", accept if the ple, and therefore the problem frustration free or SWAP test outputs "Antisymmetric". degenerate eigenspace of commuting quantum k- Q is a gapped (1, 0)-procedure. SAT with n constraints has a form very similar to Note that the states enumerated in Equa- the problems in the Polynomial Pigeonhole Prin- tions (33) and (34) form a basis of the space of ciple (PPP) class introduced in [41]. It has the following classical analog: given a k-SAT formula 2n + 1 qubits. Consequently we have H2n+1 = Span(H1(x) ∪ H0(x)). with n variables and n clauses, either find a sat- It is straightforward to check that Q leaves all isfying assignment, or find two assignments such the states enumerated in Equations (33) and (34) that the clauses all have the same value. invariant, and accepts with probability 1 on the states in Equation (33), and rejects with proba- 3.2.4 Almost degenerate states of commuting k- bility 1 on the states in Equation (34). Therefore Hamiltonian. Q is a gapped (1, 0)-procedure, and the states Definition 21. Almost degenerate enumerated in Equations (33) and (34) form an eigenspace of commuting k-local Hamil- eigenbasis of Q for x. tonian. Denote by x the classical description Q a total procedure. of a commuting k-local Hamiltonian acting on 1 PA To prove that Q is total, we show that H (x) the space of n qubits Hn, H = a=1 Ha, with is non empty for all x. the local terms bounded by 0 ≤ Ha ≤ In/A, To this end, for every basis vector |ψhji ∈ B and where by hypothesis each k-local term can we consider its associated eigenvalue sequence be measured with zero error in polynomial h = (h1, ..., hn) ∈ G. The basis B comprises time using a quantum computer, and where by 2n states. The number of associated eigenvalue hypothesis the eigenvalues of the k-local terms sequences |G| is less or equal than 2n. There- can be efficiently computed, and efficiently added fore, by the pigeonhole principle, either |G| = 2n and subtracted with zero error. and there is a one to one mapping between basis Denote by H1(x) the subspace of the space of states and bits strings, in which case there is a ba- 2n qubits H2n spanned by the antisymmetric lin- A sis state |ψhji with eigenvalue sequence h = 0 ; ear combination of two orthogonal eigenstates, n 1 1 2 2 or |G| < 2 and there is at least one collision, i.e. |ψh1j1 i and |ψh2j2i, (h , j ) 6= (h , j ), with al- at least two basis states with the same eigenvalue most identical energies |E1 − E2| ≤ 2−n, where 1 P 1 2 P 2 sequence. E = a ha and E = a ha: In the first case a witness is provided by the H1(x) = Span( frustration free state 1 {√ |ψh1j1 i|ψh2j2 i − |ψh2j2 i|ψh1j1 i 0 2 |0i|ψ A i|ψ i , (36) 0 j :(h1,2, j1,2) ∈ F, (h1, j1) 6= (h2, j2), 1 2 −n where |ψ0i is any state of n qubits. |E − E | ≤ 2 } In the second case there exists a state of the ) (38)

11 0 0 0 and denote by H (x) the orthogonal subspace: |ψhji and |ψh0j0 i with (h, j) 6= (h , j ), such that the corresponding energies differ by at most 2−n. H0(x) = Span( The quantum verification procedure therefore ac- {|ψhji|ψhji|(h, j) ∈ F } cepts with probability 1 on the antisymmetric lin- 1 ear combination of these states. ∪ {√ |ψ 1 1 i|ψ 2 2 i + |ψ 2 2 i|ψ 1 1 i 2 h j h j h j h j :(h1,2, j1,2) ∈ F, (h1, j1) 6= (h2, j2)} 1 3.2.5 Multiple copies of eigenstates of commuting ∪ {√ |ψh1j1 i|ψh2j2 i − |ψh2j2 i|ψh1j1 i 2 k-local Hamiltonian. :(h1,2, j1,2) ∈ F, (h1, j1) 6= (h2, j2), The quantum no–cloning principle suggests an- 1 2 −n |E − E | > 2 } other type of problem, namely producing several ) (39) copies of a state that has certain properties. In order to translate this requirement into a quan- Theorem 9. The pair of subspaces tum verification procedure it must be possible to (H1(x), H0(x)) defined in Definition 21 be- verify these properties efficiently. We illustrate long to (1, 0)-total functional gap : QMA this in the case of a commuting k-local Hamil- 1 0 PA (H (x), H (x)) ∈ TFgapQMA(1, 0) . (40) tonian H = a=1 Ha. The required property is that the states be joint eigenstates of all the Ha’s Proof. Quantum verification procedure. with the same eigenvalues. We first describe Q. Note that creating a single joint eigenstate Carry out a SWAP test. Reject if the SWAP of the Ha’s (with random eigenvalues) is easy: test outputs "Symmetric". take the completely mixed state (half of a max- If the SWAP test outputs "Antisymmetric", imally entangled state) and measure all the Ha measure all the Ha on the first n qubits and operators on the state. To create two iden- on the second n qubits to obtain the eigenval- tical copies, we can try the following proce- 1 1 1 1 2 2 2 2 ues h = (h1, h2, ..., hA) and h = (h1, h2, ..., hA). dure: start with the maximally entangled state 1 P 1 2 + −n/2 P2n−1 Compute the energies E = a ha and E = |φ i = 2 i=0 |ii1|ii2 (which can be effi- P 2 1 2 −n a ha. Reject if |E − E | > 2 and accept ciently produced). Now measure the Ha’s on 1 2 −n if |E − E | ≤ 2 . (Recall that according to the first system. Denote by h = (h1, ..., hn) our hypothesis, the difference of energies can be the measured eigenvalues. If the corresponding computed exactly using an efficient classical al- eigenspace is one-dimensional, the state after the ∗ ∗ gorithm.) measurement is |ψhi1|ψhi2, where |ψ i denotes Q is a gapped (1, 0)-procedure. the complex conjugate of the state |ψi in the Note that the states enumerated in Equa- standard basis. (If the corresponding eigenspace tions (38) and (39) form a basis of the space is degenerate with degeneracy Jh, the state af- of 2n qubits. Consequently we have H = −1/2 PJh ∗ 2n ter the measurement is J j=1 |ψhji1|ψ i2 1 0 h hj Span(H (x) ∪ H (x)). where {|ψhji; j = 1, ..., Jh} is an orthonormal ba- It is straighforward to check that Q leaves all sis of the eigenspace with eigenvalues h). Thus if the states enumerated in Equations (38) and (39) the Ha’s are real in the standard basis, we can ef- invariant, and accepts with probability 1 on the ficiently create two identical eigenstates. But we states in Equation (38), and rejects with proba- do not know an efficient procedure to create two bility 1 on the states in Equation (39). Therefore identical eigenstates when the Ha’s are complex, Q is a gapped (1, 0)-procedure, and the states nor do we know of an efficient procedure to create enumerated in Equations (38) and (39) form an three identical eigenstates when the Ha are real. eigenbasis of Q for x. These remarks lead to the following problem: Q is a total procedure. To prove that Q is total, we show that H1(x) Definition 22. Multiple copies of eigen- is non empty for all x. states of commuting k-local Hamiltonian. n Since there are 2 states |ψhji and their en- Denote by x the classical description of a com- ergies lie in the interval [0, 1], by the pigeonhole muting k-local Hamiltonian acting on the space PA principle, there are at least two different states, of n qubits Hn, H = a=1 Ha, and where by hy-

12 pothesis each k-local term can be measured with 3.3 Quantum Lovász Local Lemma zero error in polynomial time using a quantum The Quantum Lovász Local Lemma (QLLL) in- computer. troduced in [8] provides conditions under which Denote by H1(x) the subspace of the space of the quantum k-SAT problem is satisﬁable. The 3n qubits H spanned by the products of states 3n satisﬁability conditions were extended in [45] and with the same eigenvalues h: [27]. H1(x) = Span( As an example we give the following result

{|ψhji|ψhj0 i|ψhj00 i taken from [8]: Let {Π1, ..., Πm} be a k-QSAT in- 0 00 stance where all projectors have rank at most r. If : h ∈ G, j, j , j ∈ Jh} every qubit appears in at most D = 2k/(e · r · k) ) (41) projectors, then the problem is satisfiable. For and denote by H0(x) the orthogonal subspace: our purposes we will call the hypothesis of this statement the QLLL condition. H0(x) = Span( A Constructive Quantum Lovász Local Lemma {|ψhji|ψh0j0 i|ψh000j00 i provides conditions under which the frustration :(h, j), (h0, j0), (h00, j00) ∈ F, free state can be efficiently constructed by a quan- h 6= h0 OR h0 6= h00 OR h00 6= h} tum algorithm, i.e. is in FBQP. Initial results ) . (42) used commutativity of the constraints [48, 44]. This condition was dropped in [21] which pro- (For definiteness we have considered the case vides a constructive algorithm under a uniform where we request 3 copies of the eigenstates. The gap constraint defined as follows: let = 1/q(n) case where the Ha are complex and we request 2 for some polynomial q(n), then for any subset copies can be treated in the same way). P S of the constraints the gap of HS = i∈S Πi is Theorem 10. The pair of subspaces greater than , where the gap is the difference be- (H1(x), H0(x)) defined in Definition 22 between the two smallest eigenvalues of HS. Note long to (1, 0)-total functional gap QMA: that there is no known efficient quantum algorithm that can check whether the uniform gap 1 0 (H (x), H (x)) ∈ TFgapQMA(1, 0) . (43) constraint is satisfied. Proof. Quantum verification procedure. It is not known how the constructive algorithm We first describe Q. of [21] works when the uniform gap condition does not hold. It may be that it always outputs a state Measure all the Ha on qubits 1, ..., n, on qubits n + 1, ..., 2n, and on qubits 2n + 1, ..., 3n. Accept close to the ground state. It may also be that it sometimes outputs a state far from the ground if the outcomes h = (h1, ..., hA) are equal. Oth- erwise reject. state. If the latter is true, then this gives rise to Q is a gapped (1, 0)-procedure. an interesting problem in TFQMA. Note that the states enumerated in Equa- Definition 23. Quantum verification proce- tions (41) and (42) form a basis of the space dure for ground state energy under QLLL of 3n qubits. Consequently we have H3n = conditions. Denote by x the classical descrip- 1 0 Span(H (x) ∪ H (x)). tion {Π1, ..., Πm} of a k-QSAT instance satis- It is straighforward to check that Q leaves all fying the QLLL condition. Denote by H = −1 Pm the states enumerated in Equations (41) and (42) m i=1 Πi the Hamiltonian obtained by sum- invariant, and accepts with probability 1 on the ming all the projectors, rescaled to have eigenstates in Equation (41), and rejects with proba- values in the interval [0, 1]. bility 1 on the states in Equation (42). Therefore Denote by QQLLL the following quantum veri- Q is a gapped (1, 0)-procedure, and the states fication procedure: enumerated in Equations (38) and (39) form an On input (x, |ψi), apply to |ψi the phase es- eigenbasis of Q for x. timation algorithm [46, 32, 19] for the unitary Q a total procedure. operator U = exp(iπH) to `(n) bits of precision, 1 Since {|ψhji} is a basis of Hn, H (x) is non where ` ∈ poly. empty for all x. In fact Dim H1(x) ≥ 2n. In order to implement the unitary operator U = exp(iπH) and its powers in the phase esti-