Uniform Diagonalization Theorem for Complexity Classes of Promise Problems including Randomized and Quantum Classes

Diagonalization in the spirit of Cantor's diagonal arguments is a widely used tool in theoretical computer sciences to obtain structural results about computational problems and complexity classes by indirect proofs. The Uniform Diagonalization Theorem allows the construction of problems outside complexity classes while still being reducible to a specific decision problem. This paper provides a generalization of the Uniform Diagonalization Theorem by extending it to promise problems and the complexity classes they form, e.g. randomized and quantum complexity classes. The theorem requires from the underlying computing model not only the decidability of its acceptance and rejection behaviour but also of its promise-contradicting indifferent behaviour - a property that we will introduce as "total decidability" of promise problems. Implications of the Uniform Diagonalization Theorem are mainly of two kinds: 1. Existence of intermediate problems (e.g. between BQP and QMA) - also known as Ladner's Theorem - and 2. Undecidability if a problem of a complexity class is contained in a subclass (e.g. membership of a QMA-problem in BQP). Like the original Uniform Diagonalization Theorem the extension applies besides BQP and QMA to a large variety of complexity class pairs, including combinations from deterministic, randomized and quantum classes.Comment: 15 page

On Perfect Completeness for QMA

Whether the class QMA (Quantum Merlin Arthur) is equal to QMA1, or QMA with one-sided error, has been an open problem for years. This note helps to explain why the problem is difficult, by using ideas from real analysis to give a "quantum oracle" relative to which they are different. As a byproduct, we find that there are facts about quantum complexity classes that are classically relativizing but not quantumly relativizing, among them such "trivial" containments as BQP in ZQEXP.Comment: 9 pages. To appear in Quantum Information & Computatio

Quantum learning algorithms imply circuit lower bounds

We establish the first general connection between the design of quantum algorithms and circuit lower bounds. Specifically, let $\mathfrak{C}$ be a class of polynomial-size concepts, and suppose that $\mathfrak{C}$ can be PAC-learned with membership queries under the uniform distribution with error $1/2 - \gamma$ by a time $T$ quantum algorithm. We prove that if $\gamma^2 \cdot T \ll 2^n/n$, then $\mathsf{BQE} \nsubseteq \mathfrak{C}$, where $\mathsf{BQE} = \mathsf{BQTIME}[2^{O(n)}]$ is an exponential-time analogue of $\mathsf{BQP}$. This result is optimal in both $\gamma$ and $T$, since it is not hard to learn any class $\mathfrak{C}$ of functions in (classical) time $T = 2^n$ (with no error), or in quantum time $T = \mathsf{poly}(n)$ with error at most $1/2 - \Omega(2^{-n/2})$ via Fourier sampling. In other words, even a marginal improvement on these generic learning algorithms would lead to major consequences in complexity theory. Our proof builds on several works in learning theory, pseudorandomness, and computational complexity, and crucially, on a connection between non-trivial classical learning algorithms and circuit lower bounds established by Oliveira and Santhanam (CCC 2017). Extending their approach to quantum learning algorithms turns out to create significant challenges. To achieve that, we show among other results how pseudorandom generators imply learning-to-lower-bound connections in a generic fashion, construct the first conditional pseudorandom generator secure against uniform quantum computations, and extend the local list-decoding algorithm of Impagliazzo, Jaiswal, Kabanets and Wigderson (SICOMP 2010) to quantum circuits via a delicate analysis. We believe that these contributions are of independent interest and might find other applications

Quantum vs Classical Proofs and Subset Verification

We study the ability of efficient quantum verifiers to decide properties of exponentially large subsets given either a classical or quantum witness. We develop a general framework that can be used to prove that QCMA machines, with only classical witnesses, cannot verify certain properties of subsets given implicitly via an oracle. We use this framework to prove an oracle separation between QCMA and QMA using an "in-place" permutation oracle, making the first progress on this question since Aaronson and Kuperberg in 2007. We also use the framework to prove a particularly simple standard oracle separation between QCMA and AM.Comment: 23 pages, presentation and notation clarified, small errors fixe

AM with Multiple Merlins

We introduce and study a new model of interactive proofs: AM(k), or Arthur-Merlin with k non-communicating Merlins. Unlike with the better-known MIP, here the assumption is that each Merlin receives an independent random challenge from Arthur. One motivation for this model (which we explore in detail) comes from the close analogies between it and the quantum complexity class QMA(k), but the AM(k) model is also natural in its own right. We illustrate the power of multiple Merlins by giving an AM(2) protocol for 3SAT, in which the Merlins' challenges and responses consist of only n^{1/2+o(1)} bits each. Our protocol has the consequence that, assuming the Exponential Time Hypothesis (ETH), any algorithm for approximating a dense CSP with a polynomial-size alphabet must take n^{(log n)^{1-o(1)}} time. Algorithms nearly matching this lower bound are known, but their running times had never been previously explained. Brandao and Harrow have also recently used our 3SAT protocol to show quasipolynomial hardness for approximating the values of certain entangled games. In the other direction, we give a simple quasipolynomial-time approximation algorithm for free games, and use it to prove that, assuming the ETH, our 3SAT protocol is essentially optimal. More generally, we show that multiple Merlins never provide more than a polynomial advantage over one: that is, AM(k)=AM for all k=poly(n). The key to this result is a subsampling theorem for free games, which follows from powerful results by Alon et al. and Barak et al. on subsampling dense CSPs, and which says that the value of any free game can be closely approximated by the value of a logarithmic-sized random subgame.Comment: 48 page

A Quantum Time-Space Lower Bound for the Counting Hierarchy

We obtain the first nontrivial time-space lower bound for quantum algorithms solving problems related to satisfiability. Our bound applies to MajSAT and MajMajSAT, which are complete problems for the first and second levels of the counting hierarchy, respectively. We prove that for every real d and every positive real epsilon there exists a real c>1 such that either: MajMajSAT does not have a quantum algorithm with bounded two-sided error that runs in time n^c, or MajSAT does not have a quantum algorithm with bounded two-sided error that runs in time n^d and space n^{1-\epsilon}. In particular, MajMajSAT cannot be solved by a quantum algorithm with bounded two-sided error running in time n^{1+o(1)} and space n^{1-\epsilon} for any epsilon>0. The key technical novelty is a time- and space-efficient simulation of quantum computations with intermediate measurements by probabilistic machines with unbounded error. We also develop a model that is particularly suitable for the study of general quantum computations with simultaneous time and space bounds. However, our arguments hold for any reasonable uniform model of quantum computation.Comment: 25 page

Quantum Complexity, Relativized Worlds, and Oracle Separations

Η κλάση πολυπλοκότητας QMA, που ορίσθηκε από τον Watrous, το 2000, είναι το κβαντικό ανάλογο της MA, που ορίσθηκε από τον Babai, το 1985, και η οποία είναι μία γενίκευση της κλάσης NP. Η κλάση MA γενικεύει την NP με την εξής έννοια: η επαληθευτική διαδικασία στην κλάση MA είναι πιθανοκρατική, ενώ στην NP είναι πλήρως ντετερμινιστική. Το 2014, οι Grilo, Kerenidis και Sikora, απέδειξαν ότι η κβαντική απόδειξη που ανακύπτει στον ορισμό της QMA μπορεί, σε κάθε περίπτωση, να αντικατασταθεί από μία, κατάλληλα ορισμένη, κβαντική κατάσταση-υποσύνολο. Οι Grilo κ.ά. ονόμασαν την κλάση αυτή SQMA, για ‘subset-state quantum Merlin-Arthur.’ ΄Αρα QMA ⊆ SQMA, και κάποιος θα μπορούσε να γράψει ότι QMA = SQMA, μιά και ο εγκλεισμός SQMA ⊆ QMA ισχύει τετριμμένα. Μετά από αυτό το αποτέλεσμα, από τους Grilo κ.ά., οι Fefferman και Kimmel, το 2015, απέδει- ξαν ότι υπάρχει κάποιο κβαντικό μαντείο A—παρόμοιο με αυτό που εισήγαγαν οι Aaronson και Kuperberg, το 2006, για να δείξουν ότι υπάρχει μαντείο A τέτοιο ώστε QMAA 1 6⊆ QCMAA—το οποίο είναι τέτοιο ώστε QMAA = SQMAA 6⊆ QCMAA. Σημειώνουμε εδώ ότι η QCMA είναι αυτή η έκδοση της QMA, που ορίσθηκε από τους Aharonov και Naveh, το 2002, σύμφωνα με την οποία η προς επαλήθευση απόδειξη είναι πλήρως κλασσική, π.χ. μία συμβολοσειρά 0-1-χαρακτήρων, και η QMA1 είναι η έκδοση τέλειας πληρότητας της QMA, δηλαδή είναι η έκδοση της QMA κατά την οποία για κάθε ΝΑΙ απάντηση, στο εξεταζόμενο πρόβλημα απόφασης, υπάρχει μία απόδειξη που κά- νει τον επαληθευτή να απαντήσει ΝΑΙ με πιθανότητα ίση με ένα. Στον μαντειακό τους διαχωρισμό οι Fefferman και Kimmel, εισήγαγαν, και χρησιμοποίησαν, μία ενδιαφέρουσα διαδικασία κατά την οποία κάποιος μπορεί να αποδείξει ότι L ∈/ QCMA, για κάποια γλώσσα L που ικανοποιεί κάποιες συγκεκριμένες ιδιότητες. Χρησιμοποιώντας αυτό το αποτέλεσμα των Fefferman και Kimmel, αποδεικνύουμε ότι υπάρχει κάποιο κβαντικό μαντείο τέτοιο ώστε SQMAA 1 6⊆ QCMAA. Σημειώνουμε εδώ ότι η κλάση SQMA1 είναι η έκδοση τέλειας πληρότητας της SQMA. Στην απόδειξή μας χρησιμοποιήσαμε την εν λόγω διαδικασία των Fefferman και Kimmel, μία εκδοχή των βασικών μαντειακών τους κατασκευών, όπως και το πρόβλημα απόφασης που χρησιμοποίησαν για την απόδειξη του διαχωρισμού τους. Σημειώνουμε εδώ ότι το αποτέλεσμά μας συνεπάγεται αυτό των Fefferman και Kimmel, μιά και ισχύει ότι SQMA1 ⊆ SQMA. Αφού διατυπώσουμε και αποδείξουμε το αποτέλεσμά μας, κάνουμε μία παράκαμψη για να εξερευνή- σουμε τον κόσμο των μαντειακών διαχωρισμών τόσο στον κλασσικό όσο και τον κβαντικό κόσμο. Εξερευνούμε κάποια αποτελέσματα, και τις υποβόσκουσες μεθόδους τους, που είναι σχετικά με την χρήση κλασσικών ή κβαντικών μαντείων σε μαντειακούς διαχωρισμούς που αφορούν σε κλασσικές ή κβαντικές κλάσεις πολυπλοκότητας. ΄Αρα εξερευνούμε κάποιες πολύ ενδιαφέρουσες πτυχές των διαχωριστικών αποτελεσμάτων που είναι σχετικά με σχετικιστικούς κόσμους. Τελικά, επιστρέφουμε, στο ερευνητικό τοπίο, ώστε να προσεγγίσουμε την ερώτηση σχετικά με την υποτιθέμενη ύπαρξη, ή όχι, ενός μαντείου A που είναι τέτοιο ώστε QMAA 1 6⊆ SQMAA 1 . Καταγρά- φουμε τις πρώτες μας προσπάθειες, και ιδέες, μέχρι τώρα.The complexity class QMA, defined by Watrous, in 2000, is the quantum analogue of MA, defined by Babai, in 1985, which, in turn, is a generalization of the class NP. The class MA generalizes the class NP in the sense that the verification procedure of the purported proof, put forth by the prover, is carried out by a probabilistic machine, rather than a deterministic one—as the definition of the class NP demands. In 2014, Grilo, Kerenidis, and Sikora, proved that the quantum proof, in the setting of QMA, may always be replaced by, an appropriately defined, quantum subset state—without any conceptual loss. That is, QMA ⊆ SQMA. Grilo et al., named their new class SQMA, for subset-state quantum MerlinArthur. Thus, one could write that SQMA = QMA, as the inclusion SQMA ⊆ QMA holds trivially. After this result, by Grilo, Kerenidis, and Sikora, Fefferman and Kimmel, in 2015, used this new characterization of QMA, and further proved that there exists some quantum oracle A—similar to that Aaronson and Kuperberg introduced, and used, in 2006, to show that QMAA 1 6⊆ QCMAA—which is such that QMAA = SQMAA 6⊆ QCMAA. Here, QCMA is that version of QMA, defined by Aharonov, and Naveh, in 2002, in which the purported proof is purely-classical, that is, a bitstring, and QMA1 is the perfect completeness version of QMA. In their separation, Fefferman and Kimmel introduced, and used, an interesting template to obtain oracle separations against the class QCMA. Drawing upon this recent result, by Fefferman and Kimmel, we prove that there exists some quantum oracle A, such that SQMAA 1 6⊆ QCMAA. We note that the class SQMA1 is the perfect completeness version of the class SQMA. In our proof, we used the template of Fefferman and Kimmel, a modified version of their basic quantum oracle construction, as well as the basic decision problem, that they themselves used for their separation. Note that our result implies that of Fefferman and Kimmel, as the inclusion xiii SQMA1 ⊆ SQMA holds. After we state and prove our result, we take a detour to explore a bit the world of oracle separations, both in the classical and the quantum setting. That is, we explore some results, and their underlying methods, about classical and quantum oracles being employed for proving separations— about classical, or quantum, complexity classes. Hence, we investigate some gems pertaining to the, not few at all, nor uninteresting, privileged relativized worlds. Finally, we return, to the research setting, to approach the open question of whether there exists some classical, or quantum, oracle A, such that QMAA 1 6⊆ SQMAA 1 , or not. We record our efforts, and some of our first ideas, thus far

Quantum Meets the Minimum Circuit Size Problem

In this work, we initiate the study of the Minimum Circuit Size Problem (MCSP) in the quantum setting. MCSP is a problem to compute the circuit complexity of Boolean functions. It is a fascinating problem in complexity theory - its hardness is mysterious, and a better understanding of its hardness can have surprising implications to many fields in computer science. We first define and investigate the basic complexity-theoretic properties of minimum quantum circuit size problems for three natural objects: Boolean functions, unitaries, and quantum states. We show that these problems are not trivially in NP but in QCMA (or have QCMA protocols). Next, we explore the relations between the three quantum MCSPs and their variants. We discover that some reductions that are not known for classical MCSP exist for quantum MCSPs for unitaries and states, e.g., search-to-decision reductions and self-reductions. Finally, we systematically generalize results known for classical MCSP to the quantum setting (including quantum cryptography, quantum learning theory, quantum circuit lower bounds, and quantum fine-grained complexity) and also find new connections to tomography and quantum gravity. Due to the fundamental differences between classical and quantum circuits, most of our results require extra care and reveal properties and phenomena unique to the quantum setting. Our findings could be of interest for future studies, and we post several open problems for further exploration along this direction