Quantum Distinguishing Complexity, Zero-Error Algorithms, and Statistical Zero Knowledge

We define a new query measure we call quantum distinguishing complexity, denoted QD(f) for a Boolean function f. Unlike a quantum query algorithm, which must output a state close to |0> on a 0-input and a state close to |1> on a 1-input, a "quantum distinguishing algorithm" can output any state, as long as the output states for any 0-input and 1-input are distinguishable. Using this measure, we establish a new relationship in query complexity: For all total functions f, Q_0(f)=O~(Q(f)^5), where Q_0(f) and Q(f) denote the zero-error and bounded-error quantum query complexity of f respectively, improving on the previously known sixth power relationship. We also define a query measure based on quantum statistical zero-knowledge proofs, QSZK(f), which is at most Q(f). We show that QD(f) in fact lower bounds QSZK(f) and not just Q(f). QD(f) also upper bounds the (positive-weights) adversary bound, which yields the following relationships for all f: Q(f) >= QSZK(f) >= QD(f) = Omega(Adv(f)). This sheds some light on why the adversary bound proves suboptimal bounds for problems like Collision and Set Equality, which have low QSZK complexity. Lastly, we show implications for lifting theorems in communication complexity. We show that a general lifting theorem for either zero-error quantum query complexity or for QSZK would imply a general lifting theorem for bounded-error quantum query complexity

Complexity-Theoretic Foundations of Quantum Supremacy Experiments

In the near future, there will likely be special-purpose quantum computers with 40-50 high-quality qubits. This paper lays general theoretical foundations for how to use such devices to demonstrate "quantum supremacy": that is, a clear quantum speedup for some task, motivated by the goal of overturning the Extended Church-Turing Thesis as confidently as possible. First, we study the hardness of sampling the output distribution of a random quantum circuit, along the lines of a recent proposal by by the Quantum AI group at Google. We show that there\u27s a natural average-case hardness assumption, which has nothing to do with sampling, yet implies that no polynomial-time classical algorithm can pass a statistical test that the quantum sampling procedure\u27s outputs do pass. Compared to previous work - for example, on BosonSampling and IQP - the central advantage is that we can now talk directly about the observed outputs, rather than about the distribution being sampled. Second, in an attempt to refute our hardness assumption, we give a new algorithm, inspired by Savitch\u27s Theorem, for simulating a general quantum circuit with n qubits and m gates in polynomial space and m^O(n) time. We then discuss why this and other known algorithms fail to refute our assumption. Third, resolving an open problem of Aaronson and Arkhipov, we show that any strong quantum supremacy theorem - of the form "if approximate quantum sampling is classically easy, then the polynomial hierarchy collapses" - must be non-relativizing. This sharply contrasts with the situation for exact sampling. Fourth, refuting a conjecture by Aaronson and Ambainis, we show that the Fourier Sampling problem achieves a constant versus linear separation between quantum and randomized query complexities. Fifth, in search of a "happy medium" between black-box and non-black-box arguments, we study quantum supremacy relative to oracles in P/poly. Previous work implies that, if one-way functions exist, then quantum supremacy is possible relative to such oracles. We show, conversely, that some computational assumption is needed: if SampBPP=SampBQP and NP is in BPP, then quantum supremacy is impossible relative to oracles with small circuits

On Constructing One-Way Quantum State Generators, and More

As a quantum analogue of one-way function, the notion of one-way quantum state generator is recently proposed by Morimae and Yamakawa (CRYPTO\u2722), which is proved to be implied by the pseudorandom state and can be used to devise a construction of one-time secure digital signature. Due to Kretschmer\u27s result (TQC\u2720), it\u27s believed that pseudorandom state generator requires less than post-quantum secure one-way function. Unfortunately, it remains to be unknown how to achieve the one-way quantum state generator without the existence of post-quantum secure one-way function. In this paper, we mainly study that problem and obtain the following results: We propose two variants of one-way quantum state generator, which we call them the weak one-way quantum state generator and distributionally one-way quantum state generator, and show the existences among these three primitives are equivalent. The distributionally one-way quantum state generator from average-case hardness assumption of a promise problem belongs to $\textsf{QSZK}$ is obtained, and hence a construction of one-way quantum state generator is implied. We construct quantum bit commitment with statistical binding (sum-binding) and computational hiding directly from the average-case hardness of a complete problem of $\textsf{QSZK}$. To show the non-triviality of the constructions above, a quantum oracle $\mathcal{U}$ is devised relative to which such promise problem in $\textsf{QSZK}$ doesn\u27t belong to $\mathsf{QMA}^{\mathcal{U}}$. Our results present the first non-trivial construction of one-way quantum state generator from the hardness assumption of complexity class, and give another evidence that one-way quantum state generator probably requires less than post-quantum secure one-way function

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

Classical and quantum sublinear algorithms

This thesis investigates the capabilities of classical and quantum sublinear algorithms through the lens of complexity theory. The formal classification of problems between “tractable” (by constructing efficient algorithms that solve them) and “intractable” (by proving no efficient algorithm can) is among the most fruitful lines of work in theoretical computer science, which includes, amongst an abundance of fundamental results and open problems, the notorious P vs. NP question. This particular incarnation of the decision-versus-verification question stems from a choice of computational model: polynomial-time Turing machines. It is far from the only model worthy of investigation, however; indeed, measuring time up to polynomial factors is often too “coarse” for practical applications. We focus on quantum computation, a more complete model of physically realisable computation where quantum mechanical phenomena (such as interference and entanglement) may be used as computational resources; and sublinear algorithms, a formalisation of ultra-fast computation where merely reading or storing the entire input is impractical, e.g., when processing massive datasets such as social networks or large databases. We begin our investigation by studying structural properties of local algorithms, a large class of sublinear algorithms that includes property testers and is characterised by the inability to even see most of the input. We prove that, in this setting, queries – the main complexity measure – can be replaced with random samples. Applying this transformation yields, among other results, the state-of-the-art query lower bound for relaxed local decoders. Focusing our attention onto property testers, we begin to chart the complexity�theoretic landscape arising from the classical vs. quantum and decision vs. verification questions in testing. We show that quantum hardware and communication with a powerful but untrusted prover are “orthogonal” resources, so that one cannot be substituted for the other. This implies all of the possible separations among the analogues of QMA, MA and BQP in the property-testing setting. We conclude with a study of zero-knowledge for (classical) streaming algorithms, which receive one-pass access to the entirety of their input but only have sublinear space. Inspired by cryptographic tools, we construct commitment protocols that are unconditionally secure in the streaming model and can be leveraged to obtain zero-knowledge streaming interactive proofs – and, in particular, show that zero-knowledge is achievable in this model