Publications

While quantum computing offers the promise of exponential advantages, limited quantum speedups are known, especially for practical applications. To open new avenues for quantum advantages, we propose Unconventional Quantum Advantages for Computation (U-QuACs), with respect to unconventional resources such as space (number of bits or quantum bits of memory required to solve a problem), accuracy of solution, communication, or energy consumption. We focus on space-efficient quantum algorithms, where we seek to design algorithms that solve a problem using much less space than the total size of the input. A natural setting in which space is critical is the streaming model of computation, where the input data arrives sequentially in pieces that must each be processed individually. Streaming is motivated by a variety of problems including analysis of internet traffic or social networks. We design the first exponential quantum space advantage for a natural streaming problem, which also constitutes the first quantum advantage for approximating a discrete optimization problem, albeit with respect to space.

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In this paper we introduce and study the STREAMINGCYCLES problem, a random order streaming version of the Boolean Hidden Hypermatching problem that has been instrumental in streaming lower bounds over the past decade. In this problem the edges of a graph G, comprising n/l disjoint length-l cycles on n vertices, are partitioned randomly among n players. Every edge is annotated with an independent uniformly random bit, and the players' task is to output, for some cycle in G, the sum (modulo 2) of the bits on its edges, after one round of sequential communication.Our main result is an lO(l) lower bound on the communication complexity of STREAMINGCYCLES, which is tight up to constant factors in the exponent. Applications of our lower bound for STREAMINGCYCLES include an essentially tight lower bound for component collection in (almost) random order graph streams, making progress towards a conjecture of Peng and Sohler [SODA'18] and the first exponential space lower bounds for random walk generation.

We investigate the space complexity of two graph streaming problems: MAX-CUT and its quantum analogue, QUANTUM MAX-CUT. Previous work by Kapralov and Krachun [STOC 19] resolved the classical complexity of the classical problem, showing that any (2 - ?)-approximation requires O(n) space (a 2-approximation is trivial with O(log n) space). We generalize both of these qualifiers, demonstrating O(n) space lower bounds for (2 - ?)-approximating MAX-CUT and QUANTUM MAX-CUT, even if the algorithm is allowed to maintain a quantum state. As the trivial approximation algorithm for QUANTUM MAX-CUT only gives a 4-approximation, we show tightness with an algorithm that returns a (2 + ?)-approximation to the QUANTUM MAX-CUT value of a graph in O(log n) space. Our work resolves the quantum and classical approximability of quantum and classical Max-Cut using o(n) space.We prove our lower bounds through the techniques of Boolean Fourier analysis. We give the first application of these methods to sequential one-way quantum communication, in which each player receives a quantum message from the previous player, and can then perform arbitrary quantum operations on it before sending it to the next. To this end, we show how Fourier-analytic techniques may be used to understand the application of a quantum channel.

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