Fundamental causal bounds of quantum random access memories

TL;DR

This work argues that fundamental causality and locality constraints in quantum many-body systems bound QRAM scalability, challenging the prospect of asymptotic $O(\log N)$ circuit depth for $O(N)$ qubits. It develops two complementary frameworks—Lieb–Robinson bounds on lattice models and a relativistic quantum field theory description—to translate information-propagation limits into concrete bounds on QRAM size across 1D, 2D, and 3D hardware, using a hybrid quantum acoustic QRAM as a concrete example. The authors derive explicit velocity and size bounds, e.g., $N \lesssim 8.9\times 10^{12}$ in 1D and up to $O(10^{24})$ in 3D, with 2D designs potentially reaching $O(10^{20})$–$O(10^{22})$ via teleportation-based routing, highlighting how fundamental physics constrains practical QRAM performance. These results inform hardware design strategies and future directions for achieving scalable quantum memory in data-intensive quantum computing workloads.

Abstract

Quantum devices should operate in adherence to quantum physics principles. Quantum random access memory (QRAM), a fundamental component of many essential quantum algorithms for tasks such as linear algebra, data search, and machine learning, is often proposed to offer $\mathcal{O}(\log N)$ circuit depth for $\mathcal{O}(N)$ data size, given $N$ qubits. However, this claim appears to breach the principle of relativity when dealing with a large number of qubits in quantum materials interacting locally. In our study we critically explore the intrinsic bounds of rapid quantum memories based on causality, employing the relativistic quantum field theory and Lieb-Robinson bounds in quantum many-body systems. In this paper, we consider a hardware-efficient QRAM design in hybrid quantum acoustic systems. Assuming clock cycle times of approximately $10^{-3}$ seconds and a lattice spacing of about 1 micrometer, we show that QRAM can accommodate up to $\mathcal{O}(10^7)$ logical qubits in 1 dimension, $\mathcal{O}(10^{15})$ to $\mathcal{O}(10^{20})$ in various 2D architectures, and $\mathcal{O}(10^{24})$ in 3 dimensions. We contend that this causality bound broadly applies to other quantum hardware systems. Our findings highlight the impact of fundamental quantum physics constraints on the long-term performance of quantum computing applications in data science and suggest potential quantum memory designs for performance enhancement.

Figures (5)

• Figure 1: During the $k$th step of initialization, the $k$th address qubit $\ket{k}$ follows the yellow branches in the figure, connecting the blue sites from left to right for each quantum router. $\ket{k}$ is then routed with a controlled-SWAP gate, which exchanges the qubit with the right channel, if the control qubit is in state $\ket{1}$, and exchanges with the left channel, if the control qubit is in state $\ket{0}$. The $\ket{k}$ qubit reaches the green site and a SWAP gate between $\ket{k}$, and the routing qubit will complete the initialization process.
• Figure 2: Correspondence of Feynman diagrams of Eqn. (\ref{['EffectiveLagrangianDensity']}) as quantum gates for realization of QRAM.
• Figure 3: Realization of the controlled-SWAP gate from basic gates: beam-splitter and CZ.
• Figure 4: Bounds of QRAM size $N$ for dimensions 1, 2, and 3. Bounds of QRAM size $N$ for dimensions 1, 2, and 3. Here we assume the lattice spacing of $10^{-6}~$m and the clock cycle time of $10^{-3}~$s. The horizontal axis is equal to the velocity limit determined by $\sum_{j=1}^\nu \sqrt{d} \left(\frac{\lambda_j}{m}\right)^{1/2}$ or $\sqrt{\frac{\lambda^{(d)}}{\rho}}$. This velocity is taken to be at most on the order of typical sound speed in solids: about $6000~$m/s.
• Figure 5: Heat plot for bounds of 1 dimension QRAM size $N$. We also assume that the lattice spacing is $10^{-3}~$m. The speed limit is again on the order of the sound speed in solids. For parameters in the vertical axes we indicate $\sum_{j=1}^\nu \left(\frac{\lambda_j}{m}\right)$ or $\frac{\lambda^{(1d)}}{\rho}$.

Theorems & Definitions (1)

• Proposition 3.1