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arXiv:2111.02296 · Leibniz international proceedings in informatics

On Polynomially Many Queries to NP or QMA Oracles

Sevag Gharibian, Dorian Rudolph

Published Nov 3, 2021• 0 citations
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Abstract

We study the complexity of problems solvable in deterministic polynomial time with access to an NP or Quantum Merlin-Arthur (QMA)-oracle, such as $P^{NP}$ and $P^{QMA}$, respectively. The former allows one to classify problems more finely than the Polynomial-Time Hierarchy (PH), whereas the latter characterizes physically motivated problems such as Approximate Simulation (APX-SIM) [Ambainis, CCC 2014]. In this area, a central role has been played by the classes $P^{NP[\log]}$ and $P^{QMA[\log]}$, defined identically to $P^{NP}$ and $P^{QMA}$, except that only logarithmically many oracle queries are allowed. Here, [Gottlob, FOCS 1993] showed that if the adaptive queries made by a $P^{NP}$ machine have a "query graph" which is a tree, then this computation can be simulated in $P^{NP[\log]}$. In this work, we first show that for any verification class $C\in\{NP,MA,QCMA,QMA,QMA(2),NEXP,QMA_{\exp}\}$, any $P^C$ machine with a query graph of "separator number" $s$ can be simulated using deterministic time $\exp(s\log n)$ and $s\log n$ queries to a $C$-oracle. When $s\in O(1)$ (which includes the case of $O(1)$-treewidth, and thus also of trees), this gives an upper bound of $P^{C[\log]}$, and when $s\in O(\log^k(n))$, this yields bound $QP^{C[\log^{k+1}]}$ (QP meaning quasi-polynomial time). We next show how to combine Gottlob's "admissible-weighting function" framework with the "flag-qubit" framework of [Watson, Bausch, Gharibian, 2020], obtaining a unified approach for embedding $P^C$ computations directly into APX-SIM instances in a black-box fashion. Finally, we formalize a simple no-go statement about polynomials (c.f. [Krentel, STOC 1986]): Given a multi-linear polynomial $p$ specified via an arithmetic circuit, if one can "weakly compress" $p$ so that its optimal value requires $m$ bits to represent, then $P^{NP}$ can be decided with only $m$ queries to an NP-oracle.

DOI
10.4230/lipics.itcs.2022.75

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We study the complexity of problems solvable in deterministic polynomial time with access to an NP or Quantum Merlin-Arthur (QMA)-oracle, such as PNPP^{NP}PNP and PQMAP^{QMA}PQMA, respectively. The former allows one to classify problems more finely than the Polynomial-Time Hierarchy (PH), whereas the latter characterizes physically motivated problems such as Approximate Simulation (APX-SIM) [Ambainis, CCC 2014]. In this area, a central role has been played by the classes PNP[log⁡]P^{NP[\log]}PNP[log] and PQMA[log⁡]P^{QMA[\log]}PQMA[log], defined identically to PNPP^{NP}PNP and PQMAP^{QMA}PQMA, except that only logarithmically many oracle queries are allowed. Here, [Gottlob, FOCS 1993] showed that if the adaptive queries made by a PNPP^{NP}PNP machine have a "query graph" which is a tree, then this computation can be simulated in PNP[log⁡]P^{NP[\log]}PNP[log]. In this work, we first show that for any verification class C∈{NP,MA,QCMA,QMA,QMA(2),NEXP,QMAexp⁡}C\in\{NP,MA,QCMA,QMA,QMA(2),NEXP,QMA_{\exp}\}C∈{NP,MA,QCMA,QMA,QMA(2),NEXP,QMAexp​}, any PCP^CPC machine with a query graph of "separator number" sss can be simulated using deterministic time exp⁡(slog⁡n)\exp(s\log n)exp(slogn) and slog⁡ns\log nslogn queries to a CCC-oracle. When s∈O(1)s\in O(1)s∈O(1) (which includes the case of O(1)O(1)O(1)-treewidth, and thus also of trees), this gives an upper bound of PC[log⁡]P^{C[\log]}PC[log], and when s∈O(log⁡k(n))s\in O(\log^k(n))s∈O(logk(n)), this yields bound QPC[log⁡k+1]QP^{C[\log^{k+1}]}QPC[logk+1] (QP meaning quasi-polynomial time). We next show how to combine Gottlob's "admissible-weighting function" framework with the "flag-qubit" framework of [Watson, Bausch, Gharibian, 2020], obtaining a unified approach for embedding PCP^CPC computations directly into APX-SIM instances in a black-box fashion. Finally, we formalize a simple no-go statement about polynomials (c.f. [Krentel, STOC 1986]): Given a multi-linear polynomial ppp specified via an arithmetic circuit, if one can "weakly compress" ppp so that its optimal value requires mmm bits to represent, then PNPP^{NP}PNP can be decided with only mmm queries to an NP-oracle.
The canonical hard problems for NP and its quantum analogue, Quantum Merlin-Arthur (QMA), are MAX-k-SAT and the k-local Hamiltonian problem (k-LH), the quantum generalization of MAX-k-SAT, respectively. In recent years, however, an arguably even more physically motivated problem than k-LH has been formalized - the problem of simulating local measurements on ground states of local Hamiltonians (APX-SIM). Perhaps surprisingly, [Ambainis, CCC 2014] showed that APX-SIM is likely harder than QMA. Indeed, [Ambainis, CCC 2014] showed that APX-SIM is P^{QMA[log]}-complete, for P^{QMA[log]} the class of languages decidable by a P machine making a logarithmic number of adaptive queries to a QMA oracle. In this work, we show that APX-SIM is P^{QMA[log]}-complete even when restricted to physically motivated Hamiltonians, obtaining as intermediate steps a variety of related complexity-theoretic results. Specifically, we first give a sequence of results which together yield P^{QMA[log]}-hardness for APX-SIM on well-motivated Hamiltonians such as the 2D Heisenberg model: - We show that for NP, StoqMA, and QMA oracles, a logarithmic number of adaptive queries is equivalent to polynomially many parallel queries. Formally, P^{NP[log]}=P^{||NP}, P^{StoqMA[log]}=P^{||StoqMA}, and P^{QMA[log]}=P^{||QMA}. (The result for NP was previously shown using a different proof technique.) These equalities simplify the proofs of our subsequent results. - Next, we show that the hardness of APX-SIM is preserved under Hamiltonian simulations (à la [Cubitt, Montanaro, Piddock, 2017]) by studying a seemingly weaker problem, ∀-APX-SIM. As a byproduct, we obtain a full complexity classification of APX-SIM, showing it is complete for P, P^{||NP},P^{||StoqMA}, or P^{||QMA} depending on the Hamiltonians employed. - Leveraging the above, we show that APX-SIM is P^{QMA[log]}-complete for any family of Hamiltonians which can efficiently simulate spatially sparse Hamiltonians. This implies APX-SIM is P^{QMA[log]}-complete even on physically motivated models such as the 2D Heisenberg model. Our second focus considers 1D systems: We show that APX-SIM remains P^{QMA[log]}-complete even for local Hamiltonians on a 1D line of 8-dimensional qudits. This uses a number of ideas from above, along with replacing the "query Hamiltonian" of [Ambainis, CCC 2014] with a new "sifter" construction.
The canonical problem for the class Quantum Merlin-Arthur (QMA) is that of estimating ground state energies of local Hamiltonians. Perhaps surprisingly, [Ambainis, CCC 2014] showed that the related, but arguably more natural, problem of simulating local measurements on ground states of local Hamiltonians (APX-SIM) is likely harder than QMA. Indeed, [Ambainis, CCC 2014] showed that APX-SIM is P^QMA[log]-complete, for P^QMA[log] the class of languages decidable by a P machine making a logarithmic number of adaptive queries to a QMA oracle. In this work, we show that APX-SIM is P^QMA[log]-complete even when restricted to more physical Hamiltonians, obtaining as intermediate steps a variety of related complexity-theoretic results. We first give a sequence of results which together yield P^QMA[log]-hardness for APX-SIM on well-motivated Hamiltonians: (1) We show that for NP, StoqMA, and QMA oracles, a logarithmic number of adaptive queries is equivalent to polynomially many parallel queries. These equalities simplify the proofs of our subsequent results. (2) Next, we show that the hardness of APX-SIM is preserved under Hamiltonian simulations (a la [Cubitt, Montanaro, Piddock, 2017]). As a byproduct, we obtain a full complexity classification of APX-SIM, showing it is complete for P, P^||NP, P^||StoqMA, or P^||QMA depending on the Hamiltonians employed. (3) Leveraging the above, we show that APX-SIM is P^QMA[log]-complete for any family of Hamiltonians which can efficiently simulate spatially sparse Hamiltonians, including physically motivated models such as the 2D Heisenberg model. Our second focus considers 1D systems: We show that APX-SIM remains P^QMA[log]-complete even for local Hamiltonians on a 1D line of 8-dimensional qudits. This uses a number of ideas from above, along with replacing the "query Hamiltonian" of [Ambainis, CCC 2014] with a new "sifter" construction.
What is the power of polynomial-time quantum computation with access to an NP oracle? In this work, we focus on two fundamental tasks from the study of Boolean satisfiability (SAT) problems: search-to-decision reductions, and approximate counting. We first show that, in strong contrast to the classical setting where a poly-time Turing machine requires Θ(n)Θ(n)Θ(n) queries to an NP oracle to compute a witness to a given SAT formula, quantumly Θ(log⁡n)Θ(\log n)Θ(logn) queries suffice. We then show this is tight in the black-box model - any quantum algorithm with "NP-like" query access to a formula requires Ω(log⁡n)Ω(\log n)Ω(logn) queries to extract a solution with constant probability. Moving to approximate counting of SAT solutions, by exploiting a quantum link between search-to-decision reductions and approximate counting, we show that existing classical approximate counting algorithms are likely optimal. First, we give a lower bound in the "NP-like" black-box query setting: Approximate counting requires Ω(log⁡n)Ω(\log n)Ω(logn) queries, even on a quantum computer. We then give a "white-box" lower bound (i.e. where the input formula is not hidden in the oracle) - if there exists a randomized poly-time classical or quantum algorithm for approximate counting making o(logn)o(log n)o(logn) NP queries, then BPPNP[o(n)]\text{BPP}^{\text{NP}[o(n)]}BPPNP[o(n)] contains a PNP\text{P}^{\text{NP}}PNP-complete problem if the algorithm is classical and FBQPNP[o(n)]\text{FBQP}^{\text{NP}[o(n)]}FBQPNP[o(n)] contains an FPNP\text{FP}^{\text{NP}}FPNP-complete problem if the algorithm is quantum.
We study the quantum-classical polynomial hierarchy, QCPH, which is the class of languages solvable by a constant number of alternating classical quantifiers followed by a quantum verifier. Our main result is that QCPH is infinite relative to a random oracle (previously, this was not even known relative to any oracle). We further prove that higher levels of PH are not contained in lower levels of QCPH relative to a random oracle; this is a strengthening of the somewhat recent result that PH is infinite relative to a random oracle (Rossman, Servedio, and Tan 2016). The oracle separation requires lower bounding a certain type of low-depth alternating circuit with some quantum gates. To establish this, we give a new switching lemma for quantum algorithms which may be of independent interest. Our lemma says that for any ddd, if we apply a random restriction to a function fff with quantum query complexity Q(f)≤n1/3\mathrm{Q}(f)\le n^{1/3}Q(f)≤n1/3, the restricted function becomes exponentially close (in terms of ddd) to a depth-ddd decision tree. Our switching lemma works even in a "worst-case" sense, in that only the indices to be restricted are random; the values they are restricted to are chosen adversarially. Moreover, the switching lemma also works for polynomial degree in place of quantum query complexity.
Classical approximation complexity asks what solution quality can be guaranteed with polynomial-time computation. The classes APX, PTAS, and FPTAS distinguish a fixed approximation ratio, approximation to any fixed accuracy, and approximation schemes whose running time is also polynomial in inverse accuracy. Their randomized counterparts are R-APX, R-PTAS, and R-FPTAS. We define bounded-error quantum counterparts BQ-APX, BQ-PTAS, and BQ-FPTAS. Membership requires a uniform quantum algorithm that, on every input, returns a feasible classical solution achieving at least the claimed approximation ratio with probability at least 2/3. Scores (objective values) must be efficiently classically computable. Running time includes parameter selection, preparation, measurement, decoding, and repetition. Many quantum optimization methods are used heuristically, and high benchmark scores alone do not establish these guarantees. We further establish a conditional hierarchy for logarithmic, polynomial, and exponential approximation factors. Assuming NP ⊈\nsubseteq⊈ BQP, the quantum classes form a strict hierarchy. Problems based on prime factorization and discrete logarithms give conditional quantum-classical separations. Certified Maximum Order has an exact quantum algorithm, while any randomized polynomial-time algorithm guaranteeing at least an inverse-polynomial approximation ratio on every input would yield efficient factoring. Discrete-Logarithm Fitting has an exact quantum algorithm and a deterministic one-half approximation, but any fixed improvement over one half would give a randomized polynomial-time algorithm for the safe-prime discrete logarithm problem. Our results show, under explicit complexity assumptions, that quantum computation can improve worst-case approximation guarantees. A quantum-classical gap for common problems such as MaxCut or MaxSAT remains open.
The polynomial-time hierarchy (PH) has proven to be a powerful tool for providing separations in computational complexity theory (modulo standard conjectures such as PH do not collapse). Here, we study whether two quantum generalizations of PH can similarly prove separations in the quantum setting. The first generalization, QCPH\rm{QCPH}QCPH , uses classical proofs, and the second, QPH\rm{QPH}QPH , uses quantum proofs. For the former, we show quantum variants of the Karp-Lipton theorem and Toda's theorem. For the latter, we place its third level, QΣ3\rm{Q\Sigma_3}QΣ3​ , into NEXP using the ellipsoid method for efficiently solving semidefinite programs. These results yield two implications for QMA(2)\rm{QMA(2)}QMA(2) , the variant of Quantum Merlin-Arthur ( QMA\rm{QMA}QMA ) with two unentangled proofs, a complexity class whose characterization has proven difficult. First, if QCPH=QPH\rm{QCPH = QPH}QCPH=QPH (i.e., alternating quantifiers are sufficiently powerful so as to make classical and quantum proofs ``equivalent''), then QMA(2) is in the counting hierarchy (specifically, in Ppppp{\rm P}^{{\rm pp}^{{\rm pp}}}Ppppp ). Second, because QMA(2)⊆QΣ3\rm{QMA(2)}\subseteq \rm{Q\Sigma_3}QMA(2)⊆QΣ3​ , QMA(2)\rm{QMA(2)}QMA(2) is strictly contained in NEXP unless QMA(2)=QΣ3\rm{QMA(2)}=\rm{Q\Sigma_3}QMA(2)=QΣ3​ (i.e., alternating quantifiers do not help in the presence of ``unentanglement'').