Quantum Query-Space Lower Bounds Using Branching Programs

TL;DR

This work introduces the restricted generalized quantum branching program (r-GQBP), a phase-only variant of GQBP, and proves its equivalence to quantum query circuits. Leveraging this equivalence, it derives explicit query-space lower bounds: for Promise-OR, any $(s,L)$-r-GQBP must satisfy $L = \Omega(\frac{n}{\sqrt{s}})$, and for the $(k,k+\delta)$-Hamming decision problem, $L = \Omega(\frac{n}{\delta\sqrt{s}})$; these results extend to non-constant symmetric functions, giving the bound $L^2 s = \Omega(n^2)$ and, via the circuit equivalence, $Q(f) = \Omega(\sqrt{n})$ with $O(\log n)$ qubits. The approaches rely on a hybrid-argument technique and provide an avenue to transfer lower bounds between r-GQBP and quantum-query circuits, contributing to our understanding of quantum query-space resources and potential time-space tradeoffs for quantum computation. The findings suggest that, under sub-logarithmic space, quadratic speedups are unlikely for broad symmetric-function classes, motivating further exploration of width–length tradeoffs and their implications for quantum algorithms and Turing machines.

Abstract

Branching programs are quite popular for studying time-space lower bounds. Bera et al. recently introduced the model of generalized quantum branching program aka. GQBP that generalized two earlier models of quantum branching programs. In this work we study a restricted version of GQBP with the motivation of proving bounds on the query-space requirement of quantum-query circuits. We show the first explicit query-space lower bound for our restricted version. We prove that the well-studied OR$_n$ decision problem, given a promise that at most one position of an $n$-sized Boolean array is a 1, satisfies the bound $Q^2 s = Ω(n^2)$, where $Q$ denotes the number of queries and $s$ denotes the width of the GQBP. We then generalize the problem to show that the same bound holds for deciding between two strings with a constant Hamming distance; this gives us query-space lower bounds on problems such as Parity and Majority. Our results produce an alternative proof of the $Ω(\sqrt{n})$-lower bound on the query complexity of any non-constant symmetric Boolean function.

Figures (3)

• Figure 1: The GQBP presented in bera2023generalized for the $Parity_n$ problem (reproduced here with the authors' kind permission) was found to be an r-GQBP. $+$ denotes $\tfrac{1}{\sqrt{2}}$ and $-$ denotes $-\tfrac{1}{\sqrt{2}}$. The green (dotted) and the red (dashed) lines correspond to the $0$ and the $1$ transitions.
• Figure 2: Figure (a) shows a two-layered one-transition r-GQBP $R$. Figure (b) is a three-layered two-transition r-GQBP $R'$ that is equivalent to $R$. Note that the first transition of $R'$ is query-dependent, whereas the second transition of $R'$ is query-independent.
• Figure 3: The state transitions using the hybrid argument.

Theorems & Definitions (15)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Definition 5: Generalized quantum branching program (GQBP)
• Definition 6
• Lemma 7
• Corollary 8
• Theorem 9
• Theorem 10