Introduction

Quantum algorithms executed on quantum computers have the potential to provide significant benefits over their classical equivalents1,2,3. However, a large portion of these algorithms necessitate data inputs, which are predominantly classical in today’s computing sector. Consequently, the creation of interfaces that bridge classical and quantum processors may be a crucial factor in achieving broad and applicable quantum advantages. To illustrate, for an algorithm that classically handles N data in polynomial time, we would need to transfer the classical data at an exponentially fast rate in N to ensure that the overall quantum speedup is exponential. The development of such a fast quantum memory is a critical component for the future of the quantum industry, akin to the quantum version of data centers that merge quantum networks and quantum memories4.

Quantum random access memory (QRAM) is a promising option to establish this efficient interface. For classical data of size N, QRAM circuits can achieve the upload of classical data within \({{{\mathcal{O}}}}(\log N)\) circuit depth and \({{{\mathcal{O}}}}(N)\) qubits. We primarily focus on the Bucket Brigade QRAM design due to its favorable circuit depth scaling. This design allows QRAM to serve as a fast quantum memory, as the coherence time of qubits does not scale rapidly with the qubit number N and does not heavily rely on quantum information propagation within the lattice. Hence, our main concern lies in the bounds of information propagation speed. Numerous quantum algorithms, such as the Harrow–Hassidim–Lloyd algorithm used for linear systems of equations3, may depend on QRAM to ensure significant accelerations when dealing with dense classical data. Indeed, QRAM is also a fundamental component in many quantum machine learning applications5,6,7. However, the implementation of large-scale, fault-tolerant QRAM presents a substantial challenge in quantum hardware8,9,10,11,12. While encouraging endeavors have been made in the development of QRAM, such as spin photon-based13 and hybrid quantum acoustic systems10, there still exists ambiguity surrounding the timeline and methodology for large-scale QRAM production. We highlight that one potential hindrance for QRAM at an asymptotic scale is intricately tied to core principles in quantum physics: relativity and causality.

Quantum hardware is realized by quantum many-body systems, whose low-energy descriptions are Lorentz-invariant, local, relativistic quantum field theories. There is no up-to-date evidence about superluminal information transfer and significant Lorentz violation14,15. Assuming that interactions are local, it seems that fundamentally one cannot ensure an \({{{\mathcal{O}}}}(\log N)\) circuit depth for \({{{\mathcal{O}}}}(N)\) qubits asymptotically. Here we make a simple estimation with the following two assumptions. Suppose we place qubits as particles split by the lattice spacing a = 10−6 m in one dimension. The total length of the qubit chain is thus given by L = aN, where N is the number of qubits. We further assume that running the total QRAM circuit takes time \(T=\Delta T\log N\), where ΔT = 10−3 s per clock cycle. From causality, for 1 dimension chain QRAM system, we simply have

$$\frac{L}{T}=\frac{Na}{\Delta T\log N}\le c\,,$$
(1)

where c is the speed of light. Taking c = 3 × 108 m/s, we get an upper bound on the number of qubits N ≤ 8.9 × 1012. That is, in the above setup, we can maximally have around 1000 billion qubits to deal with 1000 billion data in total. The number itself is comparable to the numbers of training parameters and training data size in a commercial large language model in machine learning. For instance, GPT-3 from OpenAI16 uses 175 billion training parameters to train 45,000 billion bytes of text data. If a quantum algorithm is somehow serving as part of the service with QRAM, it is approaching the bounds where causality is allowed to have exponential speedup5. The causality bound we describe is, as far as we know, original for generic quantum hardware.

The objective of this paper is to substantiate and provide a rigorous framework for the statement above. Causality constraints in quantum many-body physics can be derived from the Lieb–Robinson bound for a local Hamiltonian17,18,19. In the low-energy description, basic gates in QRAM circuits can be represented as local interactions within relativistic quantum field theory. In our study, we utilize both approaches as examples to depict maximum information transmission using commutators in hybrid quantum acoustic systems that actualize QRAM10. Not only will our work establish robust foundations for causal restrictions in QRAM, but it will also offer practical advice on quantum hardware design to enhance performance against causality bounds (refer to Fig. 1 and Fig. 2). Furthermore, our exploration of quantum gates as Feynman diagrams in relativistic quantum field theories (see Fig. 3) will contribute to our understanding of quantum hardware under extreme conditions.

Fig. 1: Bounds of QRAM size N for dimensions 1–3.
figure 1

Bounds of QRAM size N for dimensions 1–3. 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 \(\sqrt{d}\sqrt{\frac{{\lambda }^{(d)}}{\rho }}\). The definitions of d, ν, λj, and λ(1d) are defined previously under Eq. (2). This velocity is taken to be at most on the order of typical sound speed in solids: about 6000 m/s.

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Fig. 2: Heat plot for bounds of 1 dimension QRAM size N.
figure 2

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 \(\mathop{\sum }_{j = 1}^{\nu }\left(\frac{{\lambda }_{j}}{m}\right)\) or \(\frac{{\lambda }^{(1d)}}{\rho }\).

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Fig. 3: During the kth step of initialization, the kth address qubit \(\left\vert k\right\rangle\) follows the yellow branches in the figure, connecting the blue sites from left to right for each quantum router.
figure 3

\(\left\vert k\right\rangle\) is then routed with a controlled-SWAP gate, which exchanges the qubit with the right channel, if the control qubit is in state \(\left\vert 1\right\rangle\), and exchanges with the left channel, if the control qubit is in state \(\left\vert 0\right\rangle\). The \(\left\vert k\right\rangle\) qubit reaches the green site, and a SWAP gate between \(\left\vert k\right\rangle\), and the routing qubit will complete the initialization process.

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We will make two extra comments before we start detailed discussions. First, in the derivation of the bound Eq. (1), the fundamental time scale ΔT is considered to be the clock cycle time for implementing QRAM circuits. In the actual hardware designs, this clock cycle time might be related or bounded by the gate operation time or the decoherence time of quantum hardware, representing the operational time costs of each clock cycle. One can note that a smaller ΔT might lead to stronger bounds in N since a smaller time separation might lead to smaller space separation for the same velocity. However, this does not indicate that we should not improve the gate time. For instance, a larger choice of ΔT will lead to higher infidelities for QRAM12. Thus, a comprehensive design of QRAM hardware will lead to hybrid considerations including both causality bounds and fidelity. Second, we will primarily use the Hamiltonian of transmon and phonon in Eq. (3), where the speed limit is from the speed of sound determined by the hardware. However, one could also use classical communication, or alternative Hamiltonian constructions (and their corresponding quantum field theories), and the speed limit could be given by the speed of light. To introduce classical communication, corresponding classical fields will be considered to enable the QRAM coupling and could be treated as classical field backgrounds in the quantum systems. (Thus, classical communication is a way to improve the causality bound from sound speed to light speed, consistent with ref. 20.) Thus, our results will be generic for known quantum materials following causality, considering the speed of light as the upper bound. Both of the issues will be discussed in more detail in Supplementary Notes 1 and 2.

This section is structured as follows. In the section “QRAM hardware” we will discuss our approach to QRAM hardware models within the context of hybrid quantum acoustic systems. The section “Causality bounds from Lieb–Robinson bound” will provide a comprehensive discussion of causality bounds utilizing quantum field theory and Lieb–Robinson bounds, along with their implications for quantum hardware design. In the section “Discussion” we will conclude and offer future perspectives. Some detailed proofs and technical discussions are presented in Supplementary Notes 1 and 2.

Results

QRAM size limitations induced by locality

The speed limits imposed by Eq. (8) or Eq. (15) place constraints on the size of the QRAM, which can be encapsulated in Eq. (2). In this paper, we propose two methodologies to examine these constraints. Firstly, we explore a discrete localized model capable of implementing the quantum gates required for QRAM systems. We calculate its Lieb-Robinson bound to investigate the limitations imposed by locality. Secondly, we investigate a coarse-grained relativistic quantum field theory model and examine its light cone-like structure to derive similar constraints arising from locality.

In particular, we focus on a hardware-efficient QRAM model introduced in ref. 10. This model’s Hamiltonian describes a circuit quantum acoustodynamic (cQAD) system, where a transmon qubit is piezoelectrically coupled to a collection of acoustic modes of a PC resonator21. The coupling between the transmon qubit and the resonator, combined with external drives on the transmon, enables the initialization, manipulation, and measurement of phononic qubits. As the transmon can be simultaneously coupled to many high-Q modes on a single chip, this QRAM system offers hardware efficiency.

For the first method, we begin by introducing an upper limit denoted by ν, which serves as a threshold for potential couplings between sites that are widely separated. This limit is integrated into the definition of a localized Hamiltonian, expressed by Eq. (5), based on the proposed Hamiltonian outlined in ref. 10. Subsequently, we directly compute the Lieb–Robinson bound for the system governed by Eq. (10) to determine this limit.

In contrast, the second method involves the coarse-graining of Eq. (5) to yield Eq. (10). Following this, we compute the canonical commutation relations for the system. If the commutator of the field operator at x and its canonical momentum at y is 0, it indicates that x and y are “spacelike" separated. In the context of dealing with phonons, “spacelike" separation is defined relative to the speed of sound, not the speed of light. These two methodologies ultimately lead to the two results summarized in Eq. (2). These two methods are applicable for determining causality bounds in various QRAM designs, including unconventional ones like the 2D fractal QRAM discussed in ref. 20, which employs quantum teleportation for information transmission. The causality bounds of this system can also be evaluated using the methods outlined in Section ??. Additionally, if we examine QRAM designs based on photons, as discussed in ref. 22, the speed limit is evidently the speed of light. The calculations would resemble those of canonical commutation relations in standard QED systems, which are available in typical QFT textbooks such as23.

In a QRAM system comprising N qubits with a minimum separation of a between adjacent sites, the information stored in the database can propagate a maximum distance of N × a when subjected to a sequence of quantum gates generated by Eq. (10). Therefore, the total time required for the information to travel is given by the operation time calculated in Eq. (4). It is essential to ensure that the speed at which information travels is slower than the bound described by Eq. (2):

$$\frac{N}{{\log }^{2}N}\le \frac{4\sqrt{d}\left(\frac{\pi }{{g}_{1}}+\frac{\pi }{{g}_{2}}\right)}{a}{\left(\mathop{\sum }_{j = 1}^{\nu }\frac{{\lambda }_{j}}{m}\right)}^{1/2}\,\,{{\mbox{or}}}\,\,\sqrt{d}{\left(\frac{{\lambda }^{(1d)}}{\rho }\right)}^{1/2}\,,$$
(2)

where d represents the dimension of our QRAM model. Moreover, λj and λ(1d) are coupling constants for the discrete and continuous model defined in Eqs. (5) and (10). We illustrate the relationship between the material constants used in constructing QRAM and the size bound in Fig. 1.

We assume that the clock cycle time used in Fig. 1 is 10−3 s. The identification of this is not important, since what we care about is the total operation time. In terms of dissecting the total operation time into either individual gates or clock cycles will not make a big difference. The rough \(\log N\) scaling for time will work for both identifications. We also consider the speed limit to be approximately the sound speed in the resonator, which is around 6 × 103 m/s. However, it is possible to exceed this bound with alternative QRAM designs, in which case the ultimate speed limit would be the speed of light, c = 3 × 108 m/s.

The upper bounds for QRAM systems in higher dimensions are larger mainly because of the increased number of qubits they can accommodate (speed limits are also higher, as discussed before). In 2D systems, the total number of qubits is proportional to N2 and N3 for 3D. This difference in scaling allows for a significantly higher bound in higher-dimensional hardware designs, as illustrated in Fig. 1. The quadratic dependence of the total operation time on the circuit depth in this model does not significantly impact the overall results (at most up to one order of magnitude). It is specific to the model considered in this paper. For other models, one can calculate the corresponding dependence and extrapolate the results presented here.

Regarding differences between dimensions, we specifically mention an architecture design described in ref. 20, where quantum teleportation is used for qubit routing instead of SWAP actions. While most quantum gates in this design are confined by the sound speed of the 2D solid, the routing steps employing quantum teleportation are constrained by the speed of light. As the system size increases, a larger proportion of qubit distances are covered by quantum teleportation, resulting in only a few steps being confined by sound speed. Based on this observation, we estimate the maximum bounds for the QRAM size N in this 2D design to be on the order of ~1020 to ~1022 qubits, whose performance is somewhere between 2D and 3D in Fig. 1. For explicit calculations, please see the Supplementary Discussion.

In Fig. 2, the horizontal axis represents the coupling constants, while the vertical axis represents the square of the speed limit. We consider g1 and g2 to have approximately the same value, both <1. This assumption is fundamental in perturbation theory, as used in ref. 10, and in quantum field theory in our analysis.

With more realistic clock cycle times, the restriction on the number of qubits, N, becomes even larger, extending to up to 1014 qubits. However, it is crucial not to excessively lean toward the left in Fig. 2, as doing so would result in exceptionally prolonged operation times for QRAM. To elevate the upper bound on N, one would need to identify a solid where atoms are strongly coupled or, alternatively, decrease the coupling constants between the transmon qubit and the resonator to extend the clock cycle times. Nevertheless, the upper bounds depicted in Figs. 1 and 2 already suffice for the requirements of most quantum algorithms involving large quantities of qubits.

Methods

QRAM hardware

Generally speaking, in both classical and quantum random access memories, each memory location is identified by a distinct binary address. When reading from the memory, an address is supplied as input, and the memory element situated at that address is retrieved at the output. In classical RAM, transistors serve as the physical components of the addressing mechanism, functioning as classical routers that guide electrical signals to the designated memory location specified by the address bits. Similarly, in the quantum realm, quantum routers serve as the foundational building blocks of the addressing mechanism, directing quantum information to the intended memory location in the QRAM24. A detailed illustration of how to implement quantum routers in QRAM is shown in Fig. 4. Our study employs QRAM hardware, the technical details of which have been meticulously outlined in refs. 9,10. For a broader understanding of designs of QRAM from the cQAD system, please see refs. 21,25,26,27,28. We direct interested readers to these sources for a comprehensive analysis. In this section, we offer a succinct overview of the procedure used for implementing quantum gates to extract data from a QRAM system comprising N qubits, and we compute the total time of operation. The Hamiltonian that controls this QRAM system is as follows:

$$\begin{array}{l}H={\omega }_{q}{q}^{{\dagger} }q-\frac{\alpha }{2}{q}^{{\dagger} }{q}^{{\dagger} }qq\\\quad +\,\mathop{\sum}\limits_{k}\left({\omega }_{k}{m}_{k}^{{\dagger} }{m}_{k}+{g}_{k}{q}^{{\dagger} }{m}_{k}+{g}_{k}^{* }q{m}_{k}^{{\dagger} }\right)\\\quad +\,\mathop{\sum}\limits_{j}\left({\Omega }_{j}{q}^{{\dagger} }{e}^{i{\omega }_{j}t}+{{{\rm{H.}}}}{{{\rm{C.}}}}\right)\,.\end{array}$$
(3)

Here q represents the annihilation operator for the transmon qubit, while mk represents the annihilation operator for the kth phonon mode. The transmon is described as an anharmonic oscillator with a Kerr nonlinearity parameter α, and it is coupled to the kth phonon mode with a coupling strength of gk. The external drives applied to the transmon with frequencies ωj, which are represented as \({H}_{d}={\sum }_{j}{\Omega }_{j}{q}^{{\dagger} }{{\rm {e}}}^{i{\omega }_{j}t}+{{{\rm{H.}}}}{{{\rm{C.}}}}\). In this content, Ωj denotes the amplitude of the drive for the jth frequency, and H.C. stands for the Hermitian conjugate.

Fig. 4
figure 4

Correspondence of Feynman diagrams of Eq. (16) as quantum gates for realization of QRAM.

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The operation of the QRAM system is underpinned by two pivotal stages: initialization and routing. The initialization phase engages a SWAP gate, whereas the routing phase necessitates the integration of a controlled-SWAP gate and a SWAP gate. Throughout the initialization phase, the address qubit is systematically integrated into the system for individual initialization. For a QRAM comprising N qubits, \(\log N\) steps are required to accomplish comprehensive initialization. Each step, denoted as k in Fig. 4, encompasses routing the kth address qubit for (k−1) times, followed by a SWAP operation to facilitate initialization.

Upon comprehensive initialization of the QRAM system, information extraction from the database becomes feasible. The precise extraction protocol is contingent on the type of information being accessed, as elaborated in ref. 10. At the leading order, the overall operation times for extracting both classical and quantum information are essentially equivalent, with only insignificant discrepancies in numerical factors, where we use the time calculated in ref. 10:

$${T}_{c\,{{\mbox{or}}}\,q} \sim {\tau }_{0}\times {\log }^{2}N\,.$$
(4)

Here we define \(\frac{\pi }{{g}_{1}}+\frac{\pi }{{g}_{2}}\equiv {\tau }_{0}\). Unlike the linear dependence on \(\log N\) in the total operation time of the simple example (Eq. (1)), the specific example we discuss in this paper exhibits a quadratic dependence on the circuit depth \(\log N\).

Causality bounds from Lieb–Robinson bound

Considering the locality constraint inherent in QRAM, employing a local Hamiltonian is essential. However, the one referenced in ref. 10 may not be ideally suited in scenarios around the speed of light (or sound) because of the unfettered coupling between the transmon qubit and phonon modes. As a result, we commence by formulating a local Hamiltonian that mirrors the system’s properties:

$$\begin{array}{l}H={H}_{{{{\mathcal{U}}}}}+{H}_{\phi }+{H}_{I}\\\quad =\mathop{\sum}\limits_{\overrightarrow{r}}\left(\frac{{{{{\mathcal{P}}}}}_{\overrightarrow{r}}^{2}}{2m}+\mathop{\sum }\limits_{j}^{\nu }\mathop{\sum}\limits_{\alpha ,\beta }\frac{{\lambda }_{j}}{2}{({{{{\mathcal{U}}}}}_{\alpha }(\overrightarrow{r})-{{{{\mathcal{U}}}}}_{\alpha }(\overrightarrow{r}+j\overrightarrow{{e}_{\beta }}))}^{2}\right)\\\quad\;\;\; +\int\,{d}^{d}{{{\bf{x}}}}\frac{1}{2}\left(\dot{\phi }{(t,{{{\bf{x}}}})}^{2}+| \nabla \phi (t,{{{\bf{x}}}}){| }^{2}\right)\\\quad\;\;\; +\int\,{d}^{d}{{{\bf{x}}}}\left[\mathop{\sum}\limits_{\overrightarrow{r}}\mathop{\sum}\limits_{\alpha }\left({{{\mathcal{C}}}}\phi (t,{{{\bf{x}}}}){{{{\mathcal{U}}}}}_{\alpha }(\overrightarrow{r})\right)+\frac{h}{4!}{\phi }^{4}(t,{{{\bf{x}}}})\right]\,.\\ \end{array}$$
(5)

We consider the general case for d = 1, 2, 3 dimensions, where the indices α and β refer to the directions in these dimensions. The lattice sites in the solid are denoted by the vector \(\overrightarrow{r}\), and distortions in the α direction at site \(\overrightarrow{r}\) are represented as \({{{{\mathcal{U}}}}}_{\alpha }(\overrightarrow{r})\). Additionally, \({\overrightarrow{e}}_{\beta }\) denotes the unit vector in the β direction, and λj represents the coupling constant between sites \(\overrightarrow{r}\) and \(\overrightarrow{r}+j\overrightarrow{{e}_{\beta }}\). The construction of \({H}_{{{{\mathcal{U}}}}}\) is detailed in Supplementary Note 1. ϕ is a massless scalar field, and Hϕ corresponds to the standard Hamiltonian in quantum field theory textbooks23. The interaction Hamiltonian is analogous to the interactions in Eq. (3), as explained also in Supplementary Note 1.

For mathematical simplicity, throughout most of the paper, we assume the isotropy of the solid. Otherwise, we would just have different frequencies for modes traveling in different directions. As demonstrated in Eq. (5), the coupling constants λj remain the same for different values of α. The construction of this Hamiltonian is fundamentally anchored in reintroducing locality into the system. We represent the phonon modes utilizing the position distortion operator, denoted as \({{{{\mathcal{U}}}}}_{\alpha }\). To confine couplings between distantly located sites, we impose a ceiling limit symbolized by ν. This approach intuitively aligns with the concept that atoms distanced further apart are likely to be weakly coupled. Unlike the approach of modeling the transmon qubit as a single harmonic oscillator with a drive, as depicted in Eq. (3), we now portray it as a massless scalar field. Through transforming the drive frequency into the frequency of the ϕ operator, we further endorse locality, guided by the principles of special relativity and the architectural groundwork of quantum field theory. It is also important to note that we are working with a two-level system, hence qubits.

Thus, we present the Lieb–Robinson bound for the Hamiltonian given by Eq. (5), which provides us with a bound on the velocity of information propagation. Define Weyl operator:

$$W(f)\equiv \exp \left\{i\mathop{\sum}\limits_{n\in {\Lambda }_{L}}{{\mathrm{Re}}}\,[f(n)]{q}_{n}+{{\mathrm{Im}}}\,[f(n)]{p}_{n}\right\}.$$

Here f(n) is a complex function and n denote sites in the whole system represented as ΛL here.

Proposition 1

For Hamiltonian Eq. (5), we have the following Lieb–Robinson bound for the operator commutator norm:

$$\begin{array}{l}| | [{\tau }_{t}^{\Lambda }(W(f)),W(g)]| | \\ \le C\mathop{\sum}\limits_{x,y\in {\Lambda }_{L}}| f(x)| | g(y)| \\ {{\rm {e}}}^{-\mu m\left[d(X,Y)-{c}_{\omega ,\lambda }\max \left(\frac{2}{\mu },{{\rm {e}}}^{(\mu /2)+1}\right)| t| \right]}\,,\end{array}$$
(6)

where

$${c}_{\omega ,\lambda }={\left(d\mathop{\sum }\limits_{j}^{\nu }\frac{{\lambda }_{j}}{m}\right)}^{1/2}\,.$$
(7)

A meticulous and rigorous elaboration of the aforementioned causality statement is provided in Supplementary Note 1. Upon performing optimization with respect to μ, we derive the subsequent Lieb–Robinson velocity:

$${v}_{h}\le 4{\left(d\mathop{\sum }\limits_{j = 1}^{\nu }\frac{{\lambda }_{j}}{m}\right)}^{1/2}\,.$$
(8)

This bound acts as a maximum speed for transmitting information in a system described by Eq. (5). Interestingly, the couplings to the transmon qubit do not affect this bound because they commute with the chosen Weyl operator. Additionally, Eq. (8) is directly influenced by the parameter ν, indicating that the presence of couplings between distant sites increases the speed limit. This is intuitive as stronger interactions between distant sites pose a challenge to locality. We note that the stronger the coupling between each site of the solid, the faster information can travel. Also noteworthy is the relationship between the speed limits in different dimensions (for d = 1, 2, 3):

$${v}_{h}^{(d)}=\sqrt{d}{v}_{h}^{(1d)}=4\sqrt{d}{\left(\mathop{\sum }\limits_{j = 1}^{\nu }\frac{{\lambda }_{j}}{m}\right)}^{1/2}\,.$$
(9)

Hence, a direct relationship exists between the material’s dimensionality and the speed limit. Materials with higher dimensionality possess increased speed limits due to enhanced coupling opportunities and degrees of freedom. Furthermore, more pronounced dependencies are observable when considering the ratio between the number of qubits and the clock cycle time, a topic we will delve into in the subsequent sections.

Causality bounds in relativistic quantum field theory

Alternatively, we propose a fresh perspective to comprehend the system by examining it in a low-temperature context. In this regime, only the long-wavelength modes with small k are consequential. We implement a coarse-graining of the system by setting k → 0 and subsequently perform a Legendre transformation to derive the subsequent Lagrangian. A comprehensive discussion on the coarse-graining procedure, as well as the reason for the selection of this specific Lagrangian density, can be found in Supplementary Note 2.

$$\begin{array}{l}{{{\mathscr{L}}}}\,=\,\frac{\rho }{2}| \dot{{{{\mathcal{U}}}}}{| }^{2}-\mathop{\sum }\limits_{\alpha ,\beta }^{d}\frac{{\lambda }^{(d)}}{2}{\left(\frac{\partial {{{{\mathcal{U}}}}}_{\alpha }}{\partial {x}^{\beta }}\right)}^{2}+\frac{1}{2}{\partial }_{\mu }\phi {\partial }^{\mu }\phi \\\qquad\;-\mathop{\sum}\limits_{\alpha }{{{\mathcal{C}}}}\phi {{{{\mathcal{U}}}}}_{\alpha }-\frac{h}{4!}{\phi }^{4}\,.\end{array}$$
(10)

As a result of the coarse-graining procedure, the site distortions \({{{{\mathcal{U}}}}}_{\alpha }(\overrightarrow{r})\) metamorphose into a continuous function \({{{{\mathcal{U}}}}}_{\alpha }(t,\overrightarrow{x})\). The coupling constant λ(d) in d dimensions bears a direct correlation to the original couplings λj in d dimensions. By performing an expansion up to the first order in k, a relationship is established between the two coupling constants, as follows:

$${\lambda }^{(d)}=d\mathop{\sum }\limits_{j=1}^{\nu }{\lambda }_{j}a{j}^{2}\,.$$
(11)

Our attention is devoted primarily to the causal bound of the system, a state attained via the development of the Lagrangian. By hypothesizing that the Lagrangian can be expressed as an integral of a density-dependent on the derivatives of \({{{{\mathcal{U}}}}}_{\alpha }\), we can curtail interactions between distantly located sites. This is accomplished by analyzing the commutator \(\left[{\pi }_{\alpha }(x),{{{{\mathcal{U}}}}}_{\beta }(y)\right]\), an approach akin to the computation involved in the Lieb–Robinson bound. We initially underscore the fact that this quantity invariably amounts to zero when x and y are spacelike separated, a phenomenon attributed to Lorentz symmetry:

$$[{\pi }_{\alpha }(x),{{{{\mathcal{U}}}}}_{\beta }(y)]=0\,\,{{\mbox{for}}}\,\,{(x-y)}^{2}\, <\, 0\,.$$
(12)

This ensures that special relativity is respected and information cannot travel faster than the speed of light. To get more information, we focus on the analytical structure of \(\left[{\pi }_{\alpha }(x),{{{{\mathcal{U}}}}}_{\beta }(y)\right]\) when x and y are timelike separated, specifically for the case d = 1 as given in Eq. (13) up to leading order (for explicit calculation, see in the Supplementary Note 2):

$$[{{{{\mathcal{U}}}}}_{\beta }(y),{\pi }_{\alpha }(x)] < \frac{1}{\pi }\frac{{{\rm {e}}}^{-(r-\sqrt{{\lambda }^{(1d)}/\rho }t)\epsilon }}{\left(r-\sqrt{{\lambda }^{(1d)}/\rho }t\right)}\,.$$
(13)

In 1D, the commutator is exponentially suppressed if \((r-\sqrt{{\lambda }^{(1d)}/\rho }t) \,>\, 0\). This sets the velocity limit for the system in 1D:

$${v}^{(1d)}=\sqrt{{\lambda }^{(1d)}/\rho }\,.$$
(14)

The velocity limits in different dimensions (d = 1, 2, 3) follow a relation similar to that in the Lieb–Robinson bound:

$${v}^{(d)}={\left(\frac{{\lambda }^{(d)}}{\rho }\right)}^{1/2}=\sqrt{d}{v}^{(1d)}=\sqrt{d}{\left(\frac{{\lambda }^{(1d)}}{\rho }\right)}^{1/2}\,.{v}^{(d)}={\left(\frac{{\lambda }^{(d)}}{\rho }\right)}^{1/2}=\sqrt{d}{v}^{(1d)}=\sqrt{d}{\left(\frac{{\lambda }^{(1d)}}{\rho }\right)}^{1/2}\,.$$
(15)

It is not surprising that the bound is independent of ϕ and interaction terms, as the interaction parts commute with \({{{{\mathcal{U}}}}}_{\alpha }\). By relating the coupling constants in the discrete and continuous cases through coarse-graining, we observe that the two-speed limits are effectively the same. The numerical factor j differs because of the imprecision of coarse-graining when considering farther site couplings and ignoring higher derivative terms in Eq. (10). However, if we consider only couplings between nearest neighbors, the two-speed limits are equal. This can be understood intuitively, as \(\partial {{{{\mathcal{U}}}}}_{\alpha }/\partial {x}^{\beta }\) is the continuous counterpart of the coupling between nearest neighbors. Couplings between more distant sites can be accurately described only by including higher derivatives of \({{{{\mathcal{U}}}}}_{\alpha }\) in the Lagrangian, which challenges our assumptions of locality.

Scattering process as quantum gates

The Feynman diagrams corresponding to Eq. (10) can reproduce the wave mixing process described in ref. 10. By utilizing the Kerr nonlinearity in Eq. (3), the authors achieved wave mixing between external drives and phonon modes, resulting in the realization of quantum gates for QRAM.

We observe that the interaction term \(\phi {{{{\mathcal{U}}}}}_{\alpha }\) allows us to convert any incoming ϕ field into a phonon mode in the Feynman diagrams generated by Eq. (10). By attaching \({{{{\mathcal{U}}}}}_{\alpha }\) to the tree-level diagram via ϕ4 interactions, we can mix the transmon waves and phonon modes, reproducing the wave mixing process described in ref. 10.

Since ϕ is a scalar field, its propagator at the tree-level process, for dimensions d = 1, 2, 3, is a classical number. We can incorporate the propagators into the vertices and introduce effective coupling constants g1 and g2. Consequently, we construct the following effective Lagrangian density. We put l0 in just to keep g1 and g2 dimensions fixed for different d. (Details of this wave-mixing processes are presented in Supplementary Note 2.)

$$\begin{array}{l}{{{\mathscr{L}}}}\,=\,\frac{\rho }{2}| \dot{{{{\mathcal{U}}}}}{| }^{2}-\mathop{\sum }\limits_{\alpha ,\beta }^{d}\frac{{\lambda }^{(d)}}{2}{\left(\frac{\partial {{{{\mathcal{U}}}}}_{\alpha }}{\partial {x}^{\beta }}\right)}^{2}\\\qquad\,\,+\frac{1}{2}{\partial }_{\mu }\phi {\partial }^{\mu }\phi -\frac{{g}_{1}}{4{l}_{0}^{2-d}}\mathop{\sum}\limits_{\alpha }{\phi }^{2}{{{{\mathcal{U}}}}}_{\alpha }^{2}\\\qquad\,\,-\frac{{g}_{1}}{2{l}_{0}^{2-d}}\mathop{\sum}\limits_{\alpha \ne \beta }{\phi }^{2}{{{{\mathcal{U}}}}}_{\alpha }{{{{\mathcal{U}}}}}_{\beta }-\frac{{g}_{2}}{3!{l}_{0}^{2-d}}\mathop{\sum}\limits_{\alpha }\phi {{{{\mathcal{U}}}}}_{\alpha }^{3}\\\qquad\,\,-\frac{{g}_{2}}{{l}_{0}^{2-d}}\mathop{\sum}\limits_{\alpha \ne \beta \ne \gamma }\phi {{{{\mathcal{U}}}}}_{\alpha }{{{{\mathcal{U}}}}}_{\beta }{{{{\mathcal{U}}}}}_{\gamma }-\frac{{g}_{2}}{2{l}_{0}^{2-d}}\mathop{\sum}\limits_{\alpha \ne \beta }\phi {{{{\mathcal{U}}}}}_{\alpha }^{2}{{{{\mathcal{U}}}}}_{\beta }\,.\end{array}$$
(16)

Hence, there exists a correspondence between the Feynman diagrams generated by Eq. (16) and the quantum gates employed in QRAM.

Through direct calculations, as presented in Supplementary Note 2, we determine the time scale for the three gates that will be utilized. The time scale for the SWAP gates, beam-splitter, and CZ gates are determined, respectively as follows: \({t}_{{{{\rm{sw}}}}}=\frac{\pi }{2{g}_{1}}\), \({t}_{{{{\rm{bs}}}}}=\frac{\pi }{4{g}_{1}}\), and \({t}_{{{{\rm{cz}}}}}=\frac{\pi }{{g}_{2}}\). Importantly, these time scales are in the identical form as the times calculated in ref. 10.

To construct QRAM, an essential building block is the controlled-SWAP gate, as depicted in Fig. 4. In our specific example, the controlled-SWAP gate is realized using a combination of basic gates: beam-splitter + CZ + beam-splitter, illustrated in Fig. 5.

Fig. 5
figure 5

Realization of the controlled-SWAP gate from basic gates: beam-splitter and CZ.

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The detailed demonstration of this construction can be found in Supplementary Note 2 and ref. 10. It is important to note that the decomposition of controlled-SWAP gates shown in Fig. 5 is a specific case, assuming channel 1 is a qubit and channels 2 and 3 can be bosonic modes. Under the assumption that all modes are two-level systems, there might be even simpler decompositions of the controlled-SWAP gates, such as using 3 Toffoli gates.

Reference 29 provides experimental demonstrations for the controlled-SWAP gates in the specific design shown in Fig. 5. However, that paper did not demonstrate the coherence of the controlled-SWAP due to significant dephasing in channel 1. On the other hand, ref. 30 uses a slightly different method and improved device coherence properties to demonstrate deterministic controlled-SWAP operation.

For classical communication systems, electromagnetic fields can be treated classically with interactions based on specific designs. The key is to identify the medium through which the information is carried. In the case of classical light-speed communication, Maxwell’s equations can be manipulated to derive the wave equation for electromagnetic waves, which reveals the speed limit of c = 3 × 108 m/s for these waves. This result is well-documented in classical electromagnetic textbooks31,32,33. The upper bound of the memory system can be estimated by analyzing the specific design of the communication system, similar to the previous sections.

Discussion

In this paper we establish a universal bound on the maximum number of qubits permissible in quantum random access memory (QRAM), derived from causality and relativity principles. We substantiate the causality bounds through rigorous applications of relativistic quantum field theories and Lieb–Robinson bounds in quantum apparatus, using hybrid quantum acoustic systems as illustrative examples. While the bounds are elucidated by using specific examples, we propose that similar bounds would likely be present in general quantum devices constructed by local quantum many-body physics systems that implement QRAM. Our research not only offers essential insights into the constraints of quantum devices under extreme circumstances but also proposes strategies to optimize hardware designs, ensuring rapid quantum memory in the asymptotic context.

Our work initiates an avenue of research that delves into the fundamental aspects of quantum memories and their computational constraints. Future research trajectories may encompass the development of strategies to optimize and circumvent causality bounds, comprehension of the ultimate capabilities of quantum memories for specific quantum algorithm applications, the conception of innovative alternative quantum hardware models with augmented connectivity and nonlocal interactions, and the exploration of implications on the causality bound from hybrid quantum-classical co-designs that take into account data contents. We present these potential research themes as fertile ground for future investigation.