We consider the time and space required for quantum computers to solve a wide variety of problems involving matrices, many of which have only been analyzed classically in prior work. Our main results show that for a range of linear algebra problems-including matrix-vector product, matrix inversion, matrix multiplication and powering-existing classical time-space tradeoffs, several of which are tight for every space bound, also apply to quantum algorithms with at most a constant factor loss. For example, for almost all fixed matrices A, including the discrete Fourier transform (DFT) matrix, we prove that quantum circuits with at most T input queries and S qubits of memory require T = Ω(n 2 /S) to compute matrix-vector product Ax for x ∈ {0, 1} n . We similarly prove that matrix multiplication for n × n binary matrices requires T = Ω(n 3 / √ S). Because many of our lower bounds are matched by deterministic algorithms with the same time and space complexity, our results show that quantum computers cannot provide any asymptotic advantage for these problems with any space bound.We obtain matching lower bounds for the stronger notion of quantum cumulative memory complexity-the sum of the space per layer of a circuit.We also consider Boolean (i.e. AND-OR) matrix multiplication and matrix-vector products, improving the previous quantum time-space tradeoff lower bounds for n × n Boolean matrix multiplication to T = Ω(n 2.5 /S 1/4 ) from T = Ω(n 2.5 /S 1/2 ).Our improved lower bound for Boolean matrix multiplication is based on a new coloring argument that extracts more from the strong direct product theorem that was the basis for prior work. To obtain our tight lower bounds for linear algebra problems, we require much stronger bounds than strong direct product theorems. We obtain these bounds by adding a new bucketing method to the quantum recording-query technique of Zhandry that lets us apply classical arguments to upper bound the success probability of quantum circuits.

WC obtain an exponential separation between consecutive levels in the hierarchy of classes of functions computable by syntactic read-k-times branching programs of polynomial size, for all k > 0, as conjectured by various authors [24, 22, 161, For every k, we exhibit two explicit functions that can be computed by linear-sized read-(k+l)-times branching programs but require size to be computed by any read-k-times branching program. The result actually gives the strongest possible separation -the exponential lower bound applies to both nondeterministic read-k-times branching programs and randomized read-k-times branching programs with Zsided error E, for some E > 0. The only previously known results are the separation between k = 1 and k = 2 [6] and a aopnration of non-deterministic read-k from deterministic road-(kink/ In2 + C), where C is some appropriate constant, for each k [IS]. A simple corollary of our results is that randomization is not more powerful than non-determinism for read-k-times branching programs, A combinatorial result that we prove along the way is a "hash-mixing lemma" [12] for fnmilics of hash functions that are almost universal, which may be of independent interest.

We demonstrate the limitations of various ordered representations that have been considered in the literature for symbolic model checking including BDDs [3], *-BMDs [6], HDDs [15], MTBDDs [13] and EVBDDs [25]. We introduce a lower bound technique that applies to a broad spectrum of such functional representations. Using an abstraction that encompasses all these representations, we apply this technique to show exponential size bounds for a wide range of integer and boolean functions that arise in symbolic model checking in the definition and implicit exploration of the state spaces. We give the first examples of integer functions including integer division, remainder, high/low-order words of multiplication, square root and reciprocal that require exponential size in all these representations. Finally, we show that there is a simple regular language that requires exponential size to be represented by any *-BMD, even though BDDs can represent any regular language in linear size.

In his survey paper on branching programs, Razborov asked the following question: Does every rectifier-switching network computing the majority of n bits have size n 1+0(1) ? We answer this question in the negative by constructing a simple oblivious branching program of size O[n log 3 nÂlog log n log log log n] for computing any threshold function. This improves the previously best known upper bound of O(n 3Â2 ) due to Lupanov. We also construct oblivious branching programs of size o(n log 4 n) for computing general mod functions. All previously known constructions for computing general mod functions have size 0(n 3Â2 ). ] 1997 Academic Press

In his survey paper on branching programs, Razborov [RazSl] asked the following question: Does every rectifier-switching network computing the majority of n bits have size n l + n ( l ) ? We answer this question in the negative by constructing a simple oblivious branching programof size n log3 n log log n log log log n for computing any threshold function. This improves the previously best known upper bound of O(n3I2) due to Lupanov [Lup65].