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Register a new userQuantum Predictive Learning and Communication Complexity with Single Input
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Authors
Dmitry Gavinsky
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We define a new model of quantum learning that we call Predictive Quantum (PQ). This is a quantum analogue of PAC, where during the testing phase the student is only required to answer a polynomial number of testing queries. We demonstrate a relational concept class that is efficiently learnable in PQ, while in any reasonable classical model exponential amount of training data would be required. This is the first unconditional separation between quantum and classical learning. We show that our separation is the best possible in several ways; in particular, there is no analogous result for a functional class, as well as for several weak
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In this paper, we develop a theory of learning nonlinear input-output maps with fading memory by dissipative quantum systems, as a quantum counterpart of the theory of approximating such maps using classical dynamical systems. The theory identifies the properties required for a class of dissipative quantum systems to be {em universal}, in that any input-output map with fading memory can be approximated arbitrarily closely by an element of this class. We then introduce an example class of dissipative quantum systems that is provably universal. Numerical experiments illustrate that with a small number of qubits, this class can achieve comparable performance to classical learning schemes with a large number of tunable parameters. Further numerical analysis suggests that the exponentially increasing Hilbert space presents a potential resource for dissipative quantum systems to surpass classical learning schemes for input-output maps.
We study a new type of separation between quantum and classical communication complexity which is obtained using quantum protocols where all parties are efficient, in the sense that they can be implemented by small quantum circuits with oracle access to their inputs. More precisely, we give an explicit partial Boolean function that can be computed in the quantum-simultaneous-with-entanglement model of communication, however, every interactive randomized protocol is of exponentially larger cost. Furthermore, all the parties in the quantum protocol can be implemented by quantum circuits of small size with blackbox access to the inputs. Our result qualitatively matches the strongest known separation between quantum and classical communication complexity and is obtained using a quantum protocol where all parties are efficient.
Efficient distributed computing offers a scalable strategy for solving resource-demanding tasks such as parallel computation and circuit optimisation. Crucially, the communication overhead introduced by the allotment process should be minimised -- a key motivation behind the communication complexity problem (CCP). Quantum resources are well-suited to this task, offering clear strategies that can outperform classical counterparts. Furthermore, the connection between quantum CCPs and nonlocality provides an information-theoretic insights into fundamental quantum mechanics. Here we connect quantum CCPs with a generalised nonlocality framework -- beyond the paradigmatic Bells theorem -- by incorporating the underlying causal structure, which governs the distributed task, into a so-called nonlocal hidden variable model. We prove that a new class of communication complexity tasks can be associated to Bell-like inequalities, whose violation is both necessary and sufficient for a quantum gain. We experimentally implement a multipartite CCP akin to the guess-your-neighbour-input scenario, and demonstrate a quantum advantage when multipartite Greenberger-Horne-Zeilinger (GHZ) states are shared among three users.
The quantum Zeno effect (QZE) is the phenomenon where the unitary evolution of a quantum state is suppressed e.g. due to frequent measurements. Here, we investigate the use of the QZE in a class of communication complexity problems (CCPs). Quantum entanglement is known to solve certain CCPs beyond classical constraints. However, recent developments have yielded CCPs where super-classical results can be obtained using only communication of a single d-level quantum state (qudit) as a resource. In the class of CCPs considered here, we show quantum reduction of complexity in three ways: using i) entanglement and the QZE, ii) single qudit and the QZE, iii) single qudit. The final protocol is motivated by experimental feasibility, and we have performed a proof of concept experimental demonstration.
A major open problem in communication complexity is whether or not quantum protocols can be exponentially more efficient than classical protocols on _total_ Boolean functions in the two-party interactive model. The answer appears to be ``No. In 2002, Razborov proved this conjecture for so far the most general class of functions F(x, y) = f(x_1 * y_1, x_2 * y_2, ..., x_n * y_n), where f is a_symmetric_ Boolean function on n Boolean inputs, and x_i, y_i are the ith bit of x and y, respectively. His elegant proof critically depends on the symmetry of f. We develop a lower-bound method that does not require symmetry and prove the conjecture for a broader class of functions. Each of those functions F(x, y) is obtained by what we call the ``block-composition of a ``building block g : {0, 1}^k by {0, 1}^k --> {0, 1}, with an f : {0, 1}^n -->{0, 1}, such that F(x, y) = f(g(x_1, y_1), g(x_2, y_2), ..., g(x_n, y_n)), where x_i and y_i are the ith k-bit block of x and y, respectively. We show that as long as g itself is ``hard enough, its block-composition with an_arbitrary_ f has polynomially related quantum and classical communication complexities. Our approach gives an alternative proof for Razborovs result (albeit with a slightly weaker parameter), and establishes new quantum lower bounds. For example, when g is the Inner Product function for k=Omega(log n), the_deterministic_ communication complexity of its block-composition with_any_ f is asymptotically at most the quantum complexity to the power of 7.
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