
Span programs and quantum time complexity
Span programs are an important model of quantum computation due to their...
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Span Program for Nonbinary Functions
Span programs characterize the quantum query complexity of binary functi...
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Quantum Algorithms for Connectivity and Related Problems
An important family of span programs, stconnectivity span programs, hav...
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Monotone Circuit Lower Bounds from Robust Sunflowers
Robust sunflowers are a generalization of combinatorial sunflowers that ...
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Quantum learning algorithms imply circuit lower bounds
We establish the first general connection between the design of quantum ...
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Leveraging Unknown Structure in Quantum Query Algorithms
Quantum span program algorithms for function evaluation commonly have re...
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Inferring Lower Runtime Bounds for Integer Programs
We present a technique to infer lower bounds on the worstcase runtime c...
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Span Programs and Quantum Space Complexity
While quantum computers hold the promise of significant computational speedups, the limited size of early quantum machines motivates the study of spacebounded quantum computation. We relate the quantum space complexity of computing a function f with onesided error to the logarithm of its span program size, a classical quantity that is wellstudied in attempts to prove formula size lower bounds. In the more natural bounded error model, we show that the amount of space needed for a unitary quantum algorithm to compute f with bounded (twosided) error is lower bounded by the logarithm of its approximate span program size. Approximate span programs were introduced in the field of quantum algorithms but not studied classically. However, the approximate span program size of a function is a natural generalization of its span program size. While no nontrivial lower bound is known on the span program size (or approximate span program size) of any concrete function, a number of lower bounds are known on the monotone span program size. We show that the approximate monotone span program size of f is a lower bound on the space needed by quantum algorithms of a particular form, called monotone phase estimation algorithms, to compute f. We then give the first nontrivial lower bound on the approximate span program size of an explicit function.
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