Freely Scalable Quantum Technologies Using Cells of 5-to-50 Qubits with Very Lossy and Noisy Photonic Links

Exquisite quantum control has now been achieved in small ion traps, in nitrogen-vacancy centers and in superconducting qubit clusters. We can regard such a system as a universal cell with diverse technological uses from communication to large-scale computing, provided that the cell is able to network with others and overcome any noise in the interlinks. Here, we show that loss-tolerant entanglement purification makes quantum computing feasible with the noisy and lossy links that are realistic today: With a modestly complex cell design, and using a surface code protocol with a network noise threshold of 13.3%, we find that interlinks that attempt entanglement at a rate of 2 MHz but suffer 98% photon loss can result in kilohertz computer clock speeds (i.e., rate of high-fidelity stabilizer measurements). Improved links would dramatically increase the clock speed. Our simulations employ local gates of a fidelity already achieved in ion trap devices.

Popular Summary

The last year has seen dramatic progress toward real-world quantum computers; prototype components are now behaving very well and very precisely, in one case with 99.9999% fidelity. Therefore, are these lab systems now good enough to work in the real world? We use powerful conventional number crunchers to simulate a model of a quantum computer, to discover (i) if it would be stable against imperfections if built from today’s components and (ii) how fast it would be. We encouragingly find that we are almost there—today’s lab systems are good enough if we can combine them in the right way.

The key to realizing a quantum computer is to use only small quantum systems or “cells,” the same systems that scientists can already control with exquisite accuracy (e.g., ion traps, superconducting qubits, or crystal impurity systems). Those small components need to be linked together into a network using, for instance, photons emitted from the cells (either photons of visible light, if the cells are ion traps or nitrogen centers in diamond, or microwave photons from superconducting cells). Such photonic links have already been demonstrated in labs, but they are noisy and lossy. Surprisingly, we find that these attributes are acceptable. A process called entanglement purification can handle these messy links, enabling the network of cells to collectively behave correctly. Here, we consider cells consisting of 5–50 qubits; the cells have spacings on the order of centimeters. Using numerical modeling and Monte Carlo simulations, we estimate that a “clock speed” of kilohertz is possible with today’s systems, even assuming a photon loss of 98%; this speed could be extended to megahertz in the not too distant future as links improve. These speeds seem slow compared to conventional machines, but because of the very different way that a quantum computer tackles problems, even modest speeds are enough to profoundly outperform conventional machines for code breaking, machine learning, optimization, and enhanced materials discovery.

We anticipate that full-scale, universal quantum computers may be achievable within a decade—sooner than previously thought.

Figure 1

A small, well-controlled quantum system interfaced to a noisy entanglement-sharing channel constitutes a universal cell if it can purify the entanglement to a high fidelity. Such cells enable secure communication; monogamy of entanglement [12] means the links need not be secure and may be either direct (i) or via repeater hubs (ii) or switches (iii). (b) Moreover, a dense array of cells bridged by short links constitutes a freely scalable computer, as we analyze here.

Figure 2

Monolithic versus network quantum computing. A piece of surface code is represented in the monolithic picture on the left, and in a network architecture on the right. (Here, for clarity we depict a 2D planar code; our network simulations will actually employ the toric code, i.e., periodic boundaries. These two variants are very similar; see Appendix pp2.) In a monolithic structure [24, 25], all qubits are contained within one physical system and two-qubit gates are performed directly between qubits via some physical interaction. In the network picture, the system is instead divided up into small cells, each of which contains a modest number of qubits that interact directly, while between cells only noisy and lossy interactions are possible. For the monolithic system, stabilizer measurements are performed by using a dedicated set of ancilla qubits (pink) interlaced with the data qubits (blue). In contrast, an efficient route to making stabilizer measurements in the network model is to first purify a shared GHZ state between the cells involved and then use this resource to evaluate the stabilizer: The parity of the four qubits measured out from the GHZ tells us the stabilizer outcome. In subsequent cycles, cells are grouped into different sets of four in order to evaluate a complementary set of stabilizers; see Appendix B 3.

Figure 3

Distilling a GHZ state with EPL-generated entanglement. (a) The symbols with gray-shaded stars represent the highly mixed Bell states ${\rho }_{\text{raw}}$ or ${\rho }_{\text{raw}}^{\prime }$ obtained when a single detector “click’ is seen. Following the the extreme photon loss protocol [26], we can combine two such pairs and measure one out, thus producing a single greatly improved pair. This process is represented by a symbol with an open star. (b) Circuit diagram showing the adapted GHZ distillation process for entanglement generated using the EPL protocol. Two GHZ states are produced and one is used to make a four-qubit parity projection onto the other. The three different protocols, basic, medium, and refined are shown.

Figure 4

Results of threshold calculations for the three protocols considered: (a) basic, (b) medium, and (c) refined. The logical error rate in the toric code is calculated for varying values of the network error rate $p_n$. The cells’ internal error rates are taken to be the best currently demonstrated in an ion trap system: a two-qubit gate error rate of 0.1% and a measurement error rate of 0.05% [1, 2]. We select $p_1=\frac{1}{4}\Rightarrow r\approx \frac{1}{4}$, and we take the phase shift in the EPL entanglement generation to be $p_{\text{drift}}$, of 1%. Details of the error model can be found in Appendix pp3. The three curves on each plot denote the results for increasing lattice sizes, where $L=8$, 12, and 16 (therefore containing $2L^2$ data qubits). The threshold is defined as the intersection of these curves from which we find the basic protocol has a threshold of $p_n=7.7\%$, medium has a threshold of $p_n=13.3\%$, and refined has a threshold of $p_n=19.4\%$. In Appendix pp5, we further explore the dependence of the network error threshold on the error rates for intracell two-qubit gates and measurements.

Figure 5

Example architectures relevant to the cellular network paradigm. In (a) and (b) there are 5 qubits available, the minimum required by the protocol: (a) An NV center with several ${}^{13}\mathrm{C}$ atoms within range of the core constitutes a five-qubit cell with one qubit (the electron spin) coupling to an optical channel. A simple ion trap (b) need only have five ions, but a more complex architecture (c) offers the advantage of temporarily storing, or “buffering,” the incomplete GHZ states. In this illustration, the eight independent entanglement sites further enhance the GHZ generation rate and thus increase the “clock speed” of the computer; this also obviates the need for optical switching, as illustrated in Fig. 6. The small square symbols indicate the generation of Bell pairs between cells and the subsequent synthesis of GHZ states out of those Bell pairs. A filled circle indicates an ion in this cell, whereas open circles are ions in neighboring traps; this may be more apparent from the multicell schematic in Fig. 11.

Figure 6

Illustration of the cellular architecture in 3D. See also the detailed schematics in Figs. 10 and 11.

Figure 7

Scheduling stabilizer measurements. Stabilizer measurements are split into four rounds, two of each type, plaquettes (orange) and stars (blue). The underlying lattice and qubit structure is shown here in black for a planar code of lattice dimension $L=4$. Decomposition of the stabilizer measurement procedure allows each round to be described as a round of perfect measurements either followed or preceded by errors. The stabilizer implementation including these errors is shown on the right.

Figure 8

The seven major factors contributing to the overall potential running speed of the device.

Figure 9

Thresholds with 1% of stabilizer outcomes missing: (a) Basic, (b) medium, (c) refined.

Figure 10

Four cells of the design shown in Fig. 5 with the connections achieved by optical switching. Note that the ion trap elements in this figure could equivalently be any other few-qubit, optically active system such as a NV center in diamond. The system shown here is equivalent to that shown in Fig. 11, which has more complex cells and dedicated (nonswitched) cell-cell links; those features increase the speed, but the error thresholds are the same.

Figure 11

Four cells of the design shown in Fig. 5 with the connections relevant to building their mutual GHZ states highlighted. The inset shows the abstract concept of of the networked computer from Fig. 1; this ion trap design and the design in Fig. 10 are specific realizations.

Figure 12

Threshold dependence on local error rates for the medium protocol. The threshold for fault tolerance is dependent on all the physical error rates in the system. In Fig. 4, we show the threshold in the network error rate $p_n$ when the local measurement and gate error rates $p_m$ and $p_g$ are fixed to 0.05% and 0.1%, respectively. Here, we demonstrate the robustness of the protocol to changes in these intracell error rates. In (a), we show the threshold’s dependence on $p_g$, while (b) reveals the dependence on $p_m$. Each plotted point represents an entire threshold calculation as shown in the inset of (a). The red data points correspond to the values discussed in the main text. We stress that as the errors deviate further from this point, it is likely that our protocol becomes suboptimal; thus, the threshold line should be regarded as a lower bound on the possible threshold.

Figure 13

Total error probability. Here, we take $r=\frac{1}{4}$ and $p_{\text{drift}}=0.01$. The five lines represent the total error probability corresponding, from bottom to top, to preparation error probabilities $p_n=\{0,0.025,0.05,0.075,0.1\}$. The error probability is close to the error rate without dark counts while $d$ is below approximately 0.01 but increases rapidly thereafter, passing through the distillation threshold even for the case $p_n=0$.

Figure 14

Influence of dark counts on effective preparation and drift error probabilities for $r=\frac{1}{4}$ and $p_{\text{drift}}=0.01$. The five upper lines (green) represent the effective preparation error weight ($p_n^{\prime }$) corresponding, from bottom to top, to preparation error probabilities $p_n=\{0,0.025,0.05,0.075,0.1\}$. The lower lines (blue), indistinguishable at this scale, correspond to the effective drift error weight ($p_{\text{drift}}$) for the same values of $p_n$. In the regions $d\le 0.007$ and $d\ge 0.469$, both $p_n^{\prime }$ and $p_{\text{drift}}$ correspond to valid probabilities, and hence, in these regimes dark counts are indistinguishable from other noise sources.