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).