Quantum Random Access Memory Architectures Using 3D Superconducting Cavities

Quantum random access memory (QRAM) is a common architecture resource for algorithms with many proposed applications, including quantum chemistry, windowed quantum arithmetic, unstructured search, machine learning, and quantum cryptography. Here, we propose two bucket-brigade QRAM architectures based on high-coherence superconducting resonators, which differ in their realizations of the conditional-routing operations. In the first, we directly construct cavity-controlled controlled-$\mathrm{SWAP}$ ($\mathrm{CSWAP}$) operations, while in the second, we utilize the properties of giant-unidirectional emitters (GUEs). For both architectures, we analyze single- and dual-rail implementations of a bosonic qubit. In the single-rail encoding, we can detect first-order ancilla errors, while the dual-rail encoding additionally allows for the detection of photon losses. For parameter regimes of interest, the postselected infidelity of a QRAM query in a dual-rail architecture is nearly an order of magnitude below that of a corresponding query in a single-rail architecture. These findings suggest that dual-rail encodings are particularly attractive as architectures for QRAM devices in the era before fault tolerance.

Popular Summary

A device capable of implementing quantum random-access memory (QRAM) is essential for many contemporary quantum algorithms. This device functions much like a classical RAM; however, queries may now be performed in superposition. Due in part to the noise inherent in today’s quantum devices, no QRAMs have yet been experimentally realized.

Here, we propose architectures for a QRAM based on high-coherence (long-lived) superconducting cavities. Our proposal is based on a new gate that allows for the detection of the vast majority of errors that occur during the operation of the QRAM. If we detect an error, we simply start over and try again. Thus, for runs where no errors were detected, we can be reasonably confident the QRAM functioned as intended. In this way, we suppress the effects of detrimental noise and provide a guideline for experimentalists to build the first small-scale QRAM.

Looking ahead, we envision devising schemes for not just detecting errors in a QRAM but also correcting them. Some errors are more detrimental than others and focusing on correcting the worst ones is a possible path toward eventually scaling up to a larger-scale QRAM.

Figure 1

A comparison of fan-out and bucket-brigade QRAM with two-level routers. (a) By constructing a tree of quantum routers, one can access data in superposition. We consider the state of the tree in a query to $D_4$ and $D_5$ (highlighted) where the indicated router experiences an amplitude-damping error (the general argument also holds for other error types; see Ref. [18]). (b) In a fan-out approach [2], all routers are set at each level according to the state of the address at that level, leading to a GHZ-like state. The resulting state of the QRAM tree is thus highly entangled and susceptible to decoherence. An amplitude-damping event on the indicated router causes the state of the tree to collapse and the query to fail. (c) In the case of bucket-brigade QRAM [15, 18], only routers participating in a query to a given branch are activated. This leads to a less entangled state of the QRAM tree as compared to fan-out QRAM. Amplitude damping now cannot occur on the router indicated in (a) and the superposition state is not disrupted.

Figure 2

Single- and dual-rail $\mathrm{CSWAP}$ QRAM. (a) We include a legend explaining our coloring scheme, the notation for hardware elements, and gate schematics for this figure as well as later figures. For the single-rail (b)–(e) and dual-rail (f)–(i) implementations, we detail the gates required for (b),(f) setting the state of the router, (c),(g) performing the ${\mathrm{c}}_0\mathrm{SWAP}$ and ${\mathrm{c}}_1\mathrm{SWAP}$ operations, and (d),(h) copying classical data into the state of the bus. The cavities outlined in dashed orange or black indicate that setting the state of the router occurs both at the top of the tree (involving the address cavity) as well as at intermediate nodes (involving output cavities). We show example hardware layouts for 3-bit (e) single-rail and (i) dual-rail QRAM. The full quantum circuit associated with a QRAM query on the hardware of (e) is shown in Appendix pp1. The naming scheme for routers and input-output elements is, e.g., router $ij$, with $i$ corresponding to the layer and $j$ to the position going from left to right. We emphasize that in our proposed bucket-brigade QRAM schemes, the classical data are not stored in any quantum elements before they are copied into the state of the bus. The data instead determine which quantum gates to perform. Thus instead of “memory cells” at the bottom of the tree (which would require additional expensive quantum hardware), we have routers that allow access to data stored at two distinct locations; see (d) and (h).

Figure 3

The infidelity, the postselected (p.s.) infidelity, and the failure probability of the logical $\mathrm{cz}$ operation as a function of the parameter set for both single rail (SR) and dual rail (DR). Utilizing postselection and parameter sets 2 and 3, infidelities on the order of ${10}^{-5}$ and ${10}^{-6}$ are possible for single and dual rail, respectively.

Figure 4

The overall infidelity and flag probability of a QRAM query as a function of $n$. (a) For the three parameter sets shown in Table 1, we calculate the postselected infidelity for both the single- and dual-rail architectures (showing only values for which the fidelity is nonzero). In the inset, we plot the fidelity without utilizing postselection. The dual-rail implementation enjoys a decrease in the postselected infidelity with respect to the single-rail case. (b) This comes at the price of increasing the flag probability.

Figure 5

The 3-bit QRAM implemented using GUEs. We show explicitly the single-rail architecture, with the right inset detailing the structure of an internal node in the dual-rail case. The left inset indicates the coupling mechanism of the data cavities to the waveguide via transfer resonators.

Figure 6

QRAM operations using GUEs. We have omitted the waveguide and transfer resonators for clarity. The three required primitives for bucket-brigade QRAM are (a),(d) setting the state of the router, (b),(e) controlled routing (the following state transfer into the waveguide is not shown) and (c),(f) copying classical data onto the bus, in the cases of single rail and dual rail, respectively. The gates are labeled $t_1,t_2,\dots$ to indicate their temporal ordering. The gate schematics and notation utilized here are explained in Fig. 2.

Figure 7

The postselected infidelity and failure probability of a QRAM query using GUEs. (a) In the single-rail case, a high-fidelity query is only possible for relatively small $n$ limited by the infidelity of state transfer. In the dual-rail case, we may postselect for photon loss enabling higher-fidelity queries for larger values of $n$. (b) The flag probability in the single-rail case is due to the ability to postselect away ancilla errors during the $\mathrm{cz}$ operation preceding the state-transfer protocol.

Figure 8

The quantum circuit for a query on the 3-bit QRAM of Fig. 2. The gates required to route in address 0, address 1,address 2, and the bus are colored as purple, green, blue, and orange, respectively. The gates in black copy classical data into the bus. Referring to the routing gates (purple, green, blue, and orange gates) as $U_{\mathrm{route}}$ and the data-copy gates (black) as $U_{\mathrm{data}}$, a full QRAM query is given by the unitary $U_{\mathrm{route}}^{†}{U}_{\mathrm{data}}{U}_{\mathrm{route}}$. The gates in $U_{\mathrm{route}}^{†}$, which disentangle the bus from the routers, are not shown due to space limitations. This quantum circuit has been prepared using the QCircuit package [53].

Figure 9

The postselected gate infidelity and failure probability of the $\mathrm{cz}$ operation. We sweep over transmon (a) relaxation and (b) dephasing times as well as cavity (c) relaxation and (d) dephasing. To observe the scaling of the infidelity and failure probability with each parameter, we set to zero all decay rates that are not being swept. In the single-rail case, postselection allows for the detection of first-order ancilla decay and dephasing errors [(a) and (b)]. Thus the postselected infidelities scale approximately quadratically with the coherence times. In the dual-rail case, we may now additionally postselect on first-order cavity-decay errors (c).

Figure 10

A schematic of multiple GUEs coupled to a common waveguide. The physics of the state-transfer protocol can be understood by considering only two pairs of GUEs coupled to the waveguide, e.g., GUEs $b$ and $c$.

Figure 11

The state-transfer protocol between two GUEs. (a) Population swaps from GUE $b$ to GUE $c$ due to the control pulses (b) applied between the GUEs and the transfer resonators. (c) The transfer resonators are minimally occupied during the course of the operation [the area plotted in (c) is highlighted by a dashed box in (a)].

Figure 12

The state-transfer infidelity and failure probability. We sweep over data cavity (a) $T_1^{\mathrm{cav}}$ and (b) $T_{\varphi }^{\mathrm{cav}}$ as well as (c) $T_{1,\mathrm{nr}}^{\mathrm{tran}}$ and (d) $T_{\varphi }^{\mathrm{tran}}$. The plateau of the single-rail infidelity (and failure probability) is due to the inherent infidelity associated with our state-transfer protocol, at the level of ${10}^{-4}$ for our parameters. In the dual-rail case, decay to the vacuum state is detectable; thus the postselected infidelity scales as $(T_1^{\mathrm{cav}}{)}^{-2},(T_{1,\mathrm{nr}}^{\mathrm{tran}}{)}^{-2}$.

Figure 13

The reflection and transmission probabilities during pass through and absorption. (a) The pass-through protocol is relatively insensitive to decay-rate asymmetries. The reflection probability and deviation of the transmission probability from unity are both less than $0.005\mathrm{\%}$ for $\delta \gamma /\gamma \le 20\mathrm{\%}$. (b) For the absorption process, the transmission probability is nearly constant as a function of $\delta \gamma$. This should ideally be zero and is not due to slight violation of the dark-state condition. The reflection probability varies quadratically with $\delta \gamma$ [see Eqs. (E19)–(E20)] and remains below the transmission probability for $\delta \gamma /\gamma \lesssim 10\mathrm{\%}$.