Quantum Time-Space Tradeoffs for Matrix Problems

TL;DR

This work establishes tight quantum time-space tradeoffs for a broad set of matrix problems, showing that for many linear-algebra tasks, quantum algorithms do not outperform classical lower bounds under memory constraints. By combining matrix rigidity with a refined recording-query framework and extending the Borodin–Cook method to quantum circuits, the authors derive strong lower bounds for matrix-vector products, matrix multiplication, and Boolean variants, including $T=\, ext{Ω}igl(n^{2}/Sigr)$ for Ax and $T=\, ext{Ω}igl(n^{3}/\,\sqrt{S}\bigr)$ for $n imes n$ binary matrix multiplication, alongside improved Boolean bounds $T= ext{Ω}(n^{2.5}/S^{1/4})$. The results imply no asymptotic quantum advantage for these problems across a broad space regime, with matching deterministic quantum upper bounds in many cases, and also yield cumulative-memory lower bounds. The findings sharpen our understanding of quantum limits on fundamental matrix computations and provide a rigorous baseline for assessing potential quantum improvements in linear-algebraic workloads.

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. For example, for almost all 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 $x \in \{0,1\}^n$. We similarly prove that matrix multiplication for $n\times n$ binary matrices requires $T=Ω(n^3 / \sqrt{S})$. Because many of our lower bounds match deterministic algorithms with the same time and space complexity, we 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 $n\times n$ 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 used in prior work. Our tight lower bounds for linear algebra problems require adding a new bucketing method to the recording-query technique of Zhandry that lets us apply classical arguments to upper bound the success probability of quantum circuits.

Figures (5)

• Figure 1: A general quantum circuit with $T$ queries.
• Figure 2: An example of a valid 3-coloring (as in \ref{['dfn:L-coloring']}), where the pink and green squares on the right matrix correspond to the colored outputs. For the left two matrices, the black squares are fixed to the input $1$ while the white square are fixed to the input $0$. The pink and green squares in the left two matrices encode an input to $OR_4^4$ whose outputs are the colored entries of the right matrix.
• Figure 3: Visualization of a single iteration of \ref{['alg:finding-colorable-rectangle']}.
• Figure 4: Comparison of our lower bounds for Boolean matrix multiplication with those of prior work for both quantum and classical computation. The shaded region comes from the fact that the time must always be $\Omega(n^2)$. The endpoints mark choices of parameters where the upper and lower bounds match.
• Figure 5: The FFT graph with the space-efficient evaluations on one pass highlighted.

Theorems & Definitions (104)

• Proposition 2.1: Shannon
• Definition 2.2
• Proposition 2.3: Lemma 3.2 in Yes84
• Proposition 2.4: Lemma 4.3 in Abr91
• proof : Proof sketch
• Proposition 2.5: Aar05
• proof
• Definition 2.6: adapted from HM21
• Proposition 2.7: Theorem 3.3 in HM21
• Definition 2.8