Cumulative memory—the sum of space used per step over the duration of a computation—is a fine-grained measure of time-space complexity that was introduced to analyze cryptographic applications like password hashing. It is a more accurate cost measure for algorithms that have infrequent spikes in memory usage and are run in environments such as cloud computing that allow dynamic allocation and de-allocation of resources during execution, or when many instances of an algorithm are interleaved in parallel. We prove the first lower bounds on cumulative memory complexity for both sequential classical computation and quantum circuits. Moreover, we develop general paradigms for bounding cumulative memory complexity inspired by the standard paradigms for proving time-space tradeoff lower bounds that can only lower bound the maximum space used during an execution. The resulting lower bounds on cumulative memory that we obtain are just as strong as the best time-space tradeoff lower bounds, which are very often known to be tight. Although previous results for pebbling and random oracle models have yielded time-space tradeoff lower bounds larger than the cumulative memory complexity, our results show that in general computational models such separations cannot follow from known lower bound techniques and are not true for many functions. Among many possible applications of our general methods, we show that any classical sorting algorithm with success probability at least 1/poly( n ) requires cumulative memory \(\tilde{\Omega }(n^2) \) , any classical matrix multiplication algorithm requires cumulative memory Ω ( n 6 / T ), any quantum sorting circuit requires cumulative memory Ω ( n 3 / T ), and any quantum circuit that finds k disjoint collisions in a random function requires cumulative memory Ω ( k 3 n / T 2 ).
Cumulative memory—the sum of space used per step over the duration of a computation—is a fine-grained measure of time-space complexity that was introduced to analyze cryptographic applications like password hashing. It is a more accurate cost measure for algorithms that have infrequent spikes in memory usage and are run in environments such as cloud computing that allow dynamic allocation and de-allocation of resources during execution, or when many instances of an algorithm are interleaved in parallel. We prove the first lower bounds on cumulative memory complexity for both sequential classical computation and quantum circuits. Moreover, we develop general paradigms for bounding cumulative memory complexity inspired by the standard paradigms for proving time-space tradeoff lower bounds that can only lower bound the maximum space used during an execution. The resulting lower bounds on cumulative memory that we obtain are just as strong as the best time-space tradeoff lower bounds, which are very often known to be tight. Although previous results for pebbling and random oracle models have yielded time-space tradeoff lower bounds larger than the cumulative memory complexity, our results show that in general computational models such separations cannot follow from known lower bound techniques and are not true for many functions. Among many possible applications of our general methods, we show that any classical sorting algorithm with success probability at least 1/poly( n ) requires cumulative memory (\tilde{\Omega }(n^2) ) , any classical matrix multiplication algorithm requires cumulative memory Ω ( n 6 / T ), any quantum sorting circuit requires cumulative memory Ω ( n 3 / T ), and any quantum circuit that finds k disjoint collisions in a random function requires cumulative memory Ω ( k 3 n / T 2 ).
Time and space (memory) are two of the most important measures of cost in computation, even more so for quantum computation. Yet, our tools for proving unconditional quantum tradeoffs between time and space are surprisingly limited.
The first quantum time-space tradeoff lower bounds were proven for sorting by Klauck, Špalek and de Wolf. Unfortunately, their method is limited to proving output-oblivious lower bounds (i.e. the lower bounds only apply to algorithms with a non-adaptive output schedule) and other methods have yielded nothing beyond output-oblivious lower bounds for sorting. Here, we prove the first fully general quantum time-space tradeoff lower bound for sorting.
We do so by introducing a novel method based on the noise operator to add to the analysis toolkit for proving quantum query and time-space tradeoff lower bounds. By combining our resulting quantum noise stability bound with quantum recording query methods, we prove an Ω(n4/3(loglogn)/(S1/3logn)) lower bound on the number of queries that a fully general quantum algorithm with at most S qubits of memory requires to sort n numbers from [n2]. Applying our noise operator argument involves purely classical reasoning, which makes it particularly simple to use.
We also use it to prove that, for any strongly universal (pairwise independent) hash function family H from n bits to m bits, almost all hash functions in H require any algorithm with at most S qubits of memory to make Ω(nm/S) quantum queries to input x in order to compute h(x), even with very small success probability. Previously, Mansour, Nisan, and Tiwari had shown a matching classical lower bound for computing h(x) with both h and x as inputs using their hash mixing lemma. Our noise operator method allows us to use a related property of hash functions to prove our quantum lower bounds.
Abstract. 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 [Formula: see text], including the discrete Fourier transform matrix, we prove that quantum circuits with at most [Formula: see text] input queries and [Formula: see text] qubits of memory require [Formula: see text] to compute matrix-vector product [Formula: see text] for [Formula: see text]. We similarly prove that matrix multiplication for [Formula: see text] binary matrices requires [Formula: see text]. 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 [Formula: see text] Boolean matrix multiplication to [Formula: see text] from [Formula: see text]. 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.
We prove lower bounds on 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. Using a novel way of applying recording query methods we show that for many linear algebra problems—including matrix-vector product, matrix inversion, matrix multiplication and powering—existing classical time-space tradeoffs 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(n2/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(n3 / √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 at any space bound. We also improve the previous quantum time-space tradeoff lower bounds for n× n Boolean (i.e. AND-OR) matrix multiplication from T=Ω(n2.5/S1/2) to T=Ω(n2.5/S1/4) which has optimal exponents for the powerful query algorithms to which it applies. Our method also yields improved lower bounds for classical algorithms.
We investigate the quantum algorithms for dynamic programming by Ambainis et al. (SODA'19). While giving provable complexity speedups and applicable to a variety of NP-hard problems, these algorithms have a notable drawback: they require a large amount of Quantum Random Access Memory (QRAM), which potentially could be very challenging to implement in a physical quantum computer. In this work, we study how we can improve the space complexity by trading it for time, while still retaining a speedup over the classical algorithms. We show novel quantum time-space tradeoffs by combining different classical approaches with quantum techniques. For instance, we show that the Travelling Salesman Problem can be solved quantumly in O(1.859n) time and O(1.315n) QRAM space.
We study the problem of finding K collision pairs in a random function f : [N] → [N] by using a quantum computer. We prove that the number of queries to the function in the quantum random oracle model must increase significantly when the size of the available memory is limited. Namely, we demonstrate that any algorithm using S qubits of memory must perform a number T of queries that satisfies the tradeoff T 3 S ≥ Ω (K 3 N) . Classically, the same question has only been settled recently by Dinur [ 22 ], who showed that the Parallel Collision Search algorithm of van Oorschot and Wiener [ 36 ] achieves the optimal time–space tradeoff of T 2 S = Θ (K 2 N) . Our result limits the extent to which quantum computing may decrease this tradeoff. Our method is based on a novel application of Zhandry’s recording query technique [ 42 ] for proving lower bounds in the exponentially small success probability regime. As a second application, we give a simpler proof of the time–space tradeoff T 2 S ≥ Ω (N 3 ) for sorting N numbers on a quantum computer, which was first obtained by Klauck, Špalek, and de Wolf [ 30 ].
In a work by Raz (J. ACM and FOCS 16), it was proved that any algorithm for parity learning on n bits requires either Ω(n2) bits of classical memory or an exponential number (in n) of random samples. A line of recent works continued that research direction and showed that for a large collection of classical learning tasks, either super-linear classical memory size or super-polynomially many samples are needed. All these works consider learning algorithms as classical branching programs, which perform classical computation within bounded memory. However, these results do not capture all physical computational models, remarkably, quantum computers and the use of quantum memory. It leaves the possibility that a small piece of quantum memory could significantly reduce the need for classical memory or samples and thus completely change the nature of the classical learning task. Despite the recent research on the necessity of quantum memory for intrinsic quantum learning problems like shadow tomography and purity testing, the role of quantum memory in classical learning tasks remains obscure. In this work, we study classical learning tasks in the presence of quantum memory. We prove that any quantum algorithm with both, classical memory and quantum memory, for parity learning on n bits, requires either Ω(n2) bits of classical memory or Ω(n) bits of quantum memory or an exponential number of samples. In other words, the memory-sample lower bound for parity learning remains qualitatively the same, even if the learning algorithm can use, in addition to the classical memory, a quantum memory of size c n (for some constant c>0). Our result is more general and applies to many other classical learning tasks. Following previous works, we represent by the matrix M: A × X → {−1,1} the following learning task. An unknown x is sampled uniformly at random from a concept class X, and a learning algorithm tries to uncover x by seeing streaming of random samples (ai, bi = M(ai, x)) where for every i, ai∈ A is chosen uniformly at random. Assume that k,ℓ,r are integers such that any submatrix of M of at least 2−k·|A| rows and at least 2−ℓ·|X| columns, has a bias of at most 2−r. We prove that any algorithm with classical and quantum hybrid memory for the learning problem corresponding to M needs either (1) Ω(k · ℓ) bits of classical memory, or (2) Ω(r) qubits of quantum memory, or (3) 2Ω(r) random samples, to achieve a success probability at least 2−O(r). Our results refute the possibility that a small amount of quantum memory significantly reduces the size of classical memory needed for efficient learning on these problems. Our results also imply improved security of several existing cryptographical protocols in the bounded-storage model (protocols that are based on parity learning on n bits), proving that security holds even in the presence of a quantum adversary with at most c n2 bits of classical memory and c n bits of quantum memory (for some constant c>0).