Scalable and High-Fidelity Quantum Random Access Memory in Spin-Photon Networks

A quantum random access memory (qRAM) is considered an essential computing unit to enable polynomial speedups in quantum information processing. Proposed implementations include the use of neutral atoms and superconducting circuits to construct a binary tree but these systems still require demonstrations of the elementary components. Here, we propose a photonic-integrated-circuit (PIC) architecture integrated with solid-state memories as a viable platform for constructing a qRAM. We also present an alternative scheme based on quantum teleportation and extend it to the context of quantum networks. Both implementations realize the two key qRAM operations, (1) quantum state transfer and (2) quantum routing, with already demonstrated components: electro-optic modulators, a Mach-Zehnder interferometer (MZI) network, and nanocavities coupled to artificial atoms for spin-based memory writing and retrieval. Our approaches furthermore benefit from built-in error detection based on photon heralding. Detailed theoretical analysis of the qRAM efficiency and query fidelity shows that our proposal presents viable near-term designs for a general qRAM.

Popular Summary

A random access memory (RAM) is an essential computing unit that allows on-demand data querying in modern computers. While a classical computer allows access to one data block per operation, a quantum random access memory (qRAM) allows the querying of data in superposition and is indispensable for many quantum algorithms. While various implementations have been proposed, they still require demonstration of elementary components that preclude any experimental realization. In this study, we propose a practical and efficient implementation of qRAM in a photonic integrated circuit based on existing technologies: electro-optic modulators, a Mach-Zehnder interferometer (MZI) network, and nanocavities strongly coupled to artificial atoms.

A qRAM is a binary tree the nodes of which need to perform two principal operations with high efficiency and fidelity: (1) transference of the quantum state from register qubits to stationary memories and (2) the performance of routing based on their transferred states. In our proposal, the register qubits are single photons that interact with long-lived spin memories coupled to optical cavities. In particular, the architecture uses photon detection to perform quantum state transfer, a technique known as heralding that removes qubit loss as a potential error.

Furthermore, we show that a qRAM can be mapped to quantum networks, which enable quantum teleportation that greatly improves the query rate by orders of magnitude. Detailed analyses of the query fidelity and efficiency show that our architecture is robust against device imperfections. Our scalable scheme presents a viable blueprint for a near-term qRAM necessary for quantum information processing tasks such as quantum searching and quantum machine learning.

Figure 1

An illustrative bucket-brigade model with a cavity-coupled $\mathrm{\Lambda }$-level atom at each tree node. (a) The address $|j⟩$ consisting of the register qubits $|k_0⟩|k_1⟩|k_2⟩$ arrives at the three-level binary tree containing $N=2^3$ memory cells. (b) Each register is a frequency-encoded photonic qubit in the $\{{\omega }_0,{\omega }_1\}$ basis. (c) For our implementation, each tree node is a $\mathrm{\Lambda }$-atom coupled to a single-sided nanocavity, the resonant frequency ${\omega }_c$ of which is tuned to the average of the two atomic transition frequencies, ${\omega }_0$ and ${\omega }_1$, which are separated by a Zeeman splitting $\mathrm{\Delta }$. For layer 1, the register $|k_1⟩$ sets the internal state of the node to $|{\psi }_A⟩={\alpha }_1|\downarrow ⟩+{\beta }_1|\uparrow ⟩$ that routes the successive register $|k_2⟩$. Two essential operations are (d) the setting mode via cavity reflection and (e) the routing mode.

Figure 2

The PIC implementation of qRAM. (a) The circuit representation of a quantum state transfer operation that maps the register qubit $|{\psi }_P⟩$ onto the atomic qubit $|{\psi }_A⟩$. (b) In the setting mode, the photon undergoes a $CZ$ operation to complete quantum state transfer. After passing through the MZI, the $|{\omega }_0⟩$ component resonantly couples to the add-drop filter that imparts a $\pi$ phase shift upon reflection off the mirror, while the $|{\omega }_1⟩$ component interacts with the atom-cavity system and acquires a spin-dependent phase shift. (c) In the routing mode, the MZI is set to a 50:50 beam splitter and the top waveguide of the add-drop filter is decoupled such that the ring resonator imparts a $\pi /2$ phase shift to the $|{\omega }_0⟩$ component upon a single pass. After cavity reflection, the returning photon reinterferes with itself and is routed to either the $|\downarrow ⟩$ path with probability $|\alpha {|}^2$ or the $|\uparrow ⟩$ path with probability $|\beta {|}^2$.

Figure 3

The quantum state transfer fidelities. The transferred state fidelity for a single setting operation is plotted against the atom-cavity cooperativity $C$ and the waveguide-cavity coupling strength ${\kappa }_{\mathrm{wg}}/\kappa$ for magnetic field deviations (a) $\delta B=-20\mathrm{\%}$, (b) $-10\mathrm{\%}$, (c) $0\mathrm{\%}$, and (d) $10\mathrm{\%}$. The contour lines denote the fidelity thresholds at $\mathcal{F}=0.985$, 0.99, 0.995, and 0.999.

Figure 4

The efficiency of the PIC qRAM. (a) The success rate (in hertz) is plotted against $N_{\mathrm{memories}}=2^n$ for a $n$-level qRAM for ${\kappa }_{\mathrm{wg}}/\kappa =0.95$, 0.965, 0.98, and 0.995 for schemes with (solid) and without (dashed) qubit loss detection (LD) with perfect routing operation ($ϵ=0$), as well as one with loss detection but with routing error probability $ϵ=5\times {10}^{-4}\ne 0$ (dashed dotted). On a log-log scale, the success rate rolls off polynomially with increasing $N_{\mathrm{memories}}=2^n$ due to an exponentially decreasing success probability of setting each layer $i$ (see Appendix pp6). (b) An enlargement of the black box in (a), highlighting the slight gain in efficiency for the cavity-assisted scheme with LD. (c) Both the success rate and transfer fidelity vary as a function of ${\kappa }_{\mathrm{wg}}/\kappa$ for $ϵ=0$. For a six-level qRAM with $C=100$, there exists a trade-off between $\overline{\mathrm{\Gamma }}$ and $\mathcal{F}$ after ${\kappa }_{\mathrm{wg}}/\kappa \approx 0.97$ where $\mathcal{F}$ is maximized by perfectly balancing losses.

Figure 5

The step-by-step procedure of the teleportation scheme. A quantum computer (QC) holds the query addresses that would be mapped onto a qRAM. (a) The QC and qRAM are remotely entangled (as represented by connecting gray lines) and the nodes of each qRAM layer are entangled in a GHZ state. (b) Local Bell-state measurements (BSMs) and subsequent Pauli transformations teleport the query addresses onto the binary tree (c) Then, in each node, the memory (red circle) and the broker (gray circle) qubits undergo a SWAP operation, leaving (d) the qRAM ready for the data retrieval process. (e) After the bus qubit has completed querying, the registers are swapped back onto the memory qubits to maintain coherence. (f) The QC and the qRAM are then remotely entangled again via their broker qubits. A subsequent local swap operation in the QC then result in entanglement between the memory qubits of the QC and the broker qubits of the qRAM. (g) Local BSMs in the qRAM then teleport the query addresses back onto the QC, returning (h) the binary tree to its original state.

Figure 6

An efficiency comparison between the conventional GLM scheme (dashed dot) and the teleportation scheme. For the teleportation scheme, the solid lines are analytical fits to the simulation data represented by the dashed lines (see Appendix 12). Each scheme is evaluated at different cavity-waveguide coupling strengths ${\kappa }_{\mathrm{wg}}/\kappa =0.95$, 0.965, 0.98, and 0.995.

Figure 7

The cavity reflection as a function of the probe frequency. The normalized probe frequency $\omega /\kappa$ is centered at the cavity resonance (black dashed line) ${\omega }_c$. The magnetic field is appropriately chosen such that the two atomic transition frequencies ${\omega }_0$ and ${\omega }_1$ coincide with the cavity-reflection maximum $r=+1$ and minimum $r=-1$. The reflection when (a) the spin is in the $|\downarrow ⟩$ state is the mirror of when (b) the spin is in the $|\uparrow ⟩$ state.

Figure 8

A schematic of the add-drop filter. (a) Each of the propagating fields in the add-drop filter is labeled for deriving the transfer matrices. The ring resonator (the resonance of which can be tuned by $\mathrm{\Delta }{\varphi }_R$) is coupled to the waveguides via balanced MZI, or interferometric, couplers, each containing a phase shifter $\mathrm{\Delta }{\varphi }_{i,m}$. When the top waveguide is coupled to the resonator, the ${\omega }_0$ component is routed to reflect off a Sagnac-loop reflector (mirror). (b) The output intensity toward the mirror $|s_m{|}^2$ as a function of $\mathrm{\Delta }{\varphi }_i$ and $\mathrm{\Delta }{\varphi }_m$. (c) The output intensity of the through component $|s_{\mathrm{out}}{|}^2$.

Figure 9

The decay rate of the ring resonator. (a) The total decay rate (line width) of the resonator is plotted as a function of $\mathrm{\Delta }{\varphi }_i$ and $\mathrm{\Delta }{\varphi }_m$ on a log scale. $\kappa$ reaches its minimum near $\mathrm{\Delta }{\varphi }_i=\pm \pi$ and $\mathrm{\Delta }{\varphi }_m=0$, at which the resonator is decoupled from the waveguides. $\kappa$ (in gigahertz) is plotted against $\mathrm{\Delta }{\varphi }_i$ for (b) the setting mode and (c) the routing mode.

Figure 10

A detailed PIC implementation of a tree node. Paths that are inactive in each mode are faded out.

Figure 11

The retrieval probability as a function of the number of memories on a log-log scale. $p_{\mathrm{succ}}$ displays a polynomial roll-off as $N_{\mathrm{memories}}$ increases for waveguide-cavity coupling ${\kappa }_{\mathrm{wg}}/\kappa =0.95$, 0.965, 0.98, and 0.995.

Figure 12

Operations to: (a) swap a Bell state between a pair of entangled electron spins and a pair of nuclear spins; (b) entangle two pairs of Bell states to form a four-qubit GHZ state in the nuclear spins.

Figure 13

A two-level qRAM is first entangled with a remote QC. Local BSMs in the QC complete quantum teleportation of the query addresses onto the binary tree. The memory layer is not shown in the schematic for simplicity.

Figure 14

The generation rates for the GHZ state and the remote entanglement link are evaluated at different cavity-waveguide coupling strengths ${\kappa }_{\mathrm{wg}}/\kappa =0.95$, 0.965, 0.98, and 0.995.

Figure 15

The proposed PIC architecture for the teleportation scheme. (a) The qRAM binary tree contains interspersed interconnect layers that enable intralayer connectivity. (b) Within each interconnect layer, a network of MZIs is classically controlled to direct the single photons to either the subsequent cavity or the detection system for heralding during GHZ-state creation. It is then switched to a transparent state during the data retrieval step.

Figure 16

The query fidelity as a function of the qRAM size. The nuclear and electron spin coherence times are, respectively, assumed to be: (a) $T_{2,n}={10}^0$ s and $T_{2,e}={10}^{-2}$ s, (b) $T_{2,n}={10}^1\mathrm{s}$ and $T_{2,e}={10}^{-1}$ s.