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# Experimental comparison of two quantum computing architectures

Contributed by Christopher Monroe, February 1, 2017 (sent for review November 1, 2016; reviewed by Eric Hudson and Igor L. Markov)

### See related content:

- QnAs with Christopher Monroe- Apr 18, 2017

## Significance

Quantum computers are an emerging technology promising to be vastly more powerful at solving certain problems than any conventional computer. These devices are now moving out of the laboratory and becoming generally programmable. This allows identical quantum tasks or algorithms to be implemented on radically different technologies to inform further development and scaling. We run a series of algorithms on the two leading platforms: trapped atomic ions and superconducting circuits. Whereas the superconducting system offers faster gate clock speeds and a solid-state platform, the ion-trap system features superior qubits and reconfigurable connections. The performance of these systems is seen to reflect the topology of connections in the base hardware, supporting the idea that quantum computer applications and hardware should be codesigned.

## Abstract

We run a selection of algorithms on two state-of-the-art 5-qubit quantum computers that are based on different technology platforms. One is a publicly accessible superconducting transmon device (www.research.ibm.com/ibm-q) with limited connectivity, and the other is a fully connected trapped-ion system. Even though the two systems have different native quantum interactions, both can be programed in a way that is blind to the underlying hardware, thus allowing a comparison of identical quantum algorithms between different physical systems. We show that quantum algorithms and circuits that use more connectivity clearly benefit from a better-connected system of qubits. Although the quantum systems here are not yet large enough to eclipse classical computers, this experiment exposes critical factors of scaling quantum computers, such as qubit connectivity and gate expressivity. In addition, the results suggest that codesigning particular quantum applications with the hardware itself will be paramount in successfully using quantum computers in the future.

- quantum computing
- quantum information
- quantum information science
- quantum physics
- quantum computing architecture

Inspired by the vast computing power a universal quantum computer could offer, several candidate systems are being explored. They have allowed experimental demonstrations of quantum gates, operations, and algorithms of ever-increasing sophistication. Recently, two architectures, superconducting transmon qubits (1⇓⇓⇓–5) and trapped ions (6, 7), have reached a new level of maturity. They have become fully programmable multiqubit machines that provide the user with the flexibility to implement arbitrary quantum circuits from a high-level interface. This makes it possible for the first time to test quantum computers irrespective of their particular physical implementation.

Whereas the quantum computers considered here are still small scale and their capabilities do not currently reach beyond small demonstration algorithms, this line of inquiry can still provide useful insights into the performance of existing systems and the role of architecture in quantum computer design. These will be crucial for the realization of more advanced future incarnations of the present technologies.

The standard abstract model of quantum computation assumes that interactions between arbitrary pairs of qubits are available. However, physical architectures will in general have certain constraints on qubit connectivity, such as nearest-neighbor couplings only. These restrictions do not in principle limit the ability to perform arbitrary computations, because swap operations may be used to effect gates between arbitrary qubits using the connections available. For a general circuit, reducing a fully connected system to the more sparse star-shaped or linear nearest-neighbor connectivity requires an increase in the number of gates of

In this article, we make use of the public access recently granted by IBM to a *A*) via their “Quantum Experience” cloud service (www.research.ibm.com/ibm-q). This allows us to repeat algorithms that we perform in our own ion-trap experiment on an independent quantum computer of identical size and comparable capability but with a different physical implementation at its core.

## Physical Systems

The ion-trap system consists of five ^{171}Yb^{+} ions that are confined in a linear Paul trap and laser cooled close to their motional ground state (Fig. 1*B*) (6). The qubits are magnetic-field–insensitive pairs of states in the hyperfine-split *B*, *Inset*). The addressing during operations and the distinction between qubits during readout are both achieved by spatially resolving the ions. The fidelities for single- and 2-qubit gates are typically

In analogy to atoms given by nature, the man-made superconducting circuits in the IBM quantum computer can be thought of as “artificial atoms” (16). They are transmon qubits (17) or superconducting islands connected by Josephson junctions and shunt capacitors that provide superpositions of charge states that are insensitive to charge fluctuations. The device used here has a range of qubit frequencies between *A*, *Inset*), which are controlled-NOT (CNOT) gates targeting the central qubit. Single-qubit readout fidelities are typically

On these two machines, we compare a selection of composite gates and algorithms that represent a variety of circuit connectivities. In each case, we map the algorithms to the device by breaking them down into circuits made up of gates native to the specific hardware. We rely on an optimization protocol (21) to accomplish this task for the trapped ions and CNOT ^{a} algebra (22) with further manual optimization to compose the experiments for the IBM machine (23). The available gate set for the ion-trap system consists of the 2-qubit XX gate, as well as arbitrary single-qubit

In addition to the two systems considered here, Table 1 also gives the numbers for an LNN connectivity architecture as used, e.g., in superconducting qubits (1) as well as semiconductor gated quantum dots (24). The numbers in Table 1 show that the 2-qubit gate count strongly depends on the matching between the circuit and the qubit connectivity graph. The LNN architecture is as efficient as the fully connected system for the hidden shift algorithm, whereas the star-shaped system incurs overheads; the reverse is true for the Bernstein–Vazirani algorithm (Fig. 2).

## Algorithms

### Margolus and Toffoli Gates.

The Toffoli gate is a *A* and *B*. Note that for the Margolus gate, all entangling operations connect to the same qubit, which means that this circuit can be realized efficiently with star-shaped qubit connectivity. The systems perform this circuit at success probability *A1* and *B1*).

The full Toffoli circuit uses the same 3 qubits as the Margolus implementation so that preparation and measurement errors remain the same. The optimized circuit for the fully connected ion-trap system contains five 2-qubit gates and the additional operations lower the fidelity to *B2*). For the star-shaped system, an additional seven 2-qubit gates are needed to effect the swap operations necessary to go from the Margolus to the full Toffoli gate. This leads to a reduced success rate of *A2*). Note that the transformation

### Bernstein–Vazirani and Hidden Shift Algorithms.

In the Bernstein–Vazirani algorithm, an oracle implements the function **c** in a single shot. In the oracle, **c** is encoded in a pattern of CNOT gates, all of which target the ancilla qubit (29). As can be seen from the circuit in Fig. 2*C*, the entire algorithm maps well onto a star-shaped architecture. This algorithm is very similar to a parity check circuit used in error correction applications, and indeed the IBM system was laid out with this application in mind (5). The single-shot success probabilities are *A1* and *B1*).

To compare this to a similar algorithm with different connectivity requirements, we implement the hidden shift algorithm (30) for a black box bent function (31, 32). An oracle implements the shifted version **s** that constitutes the “hidden shift.” For a subset of Boolean functions, there exists a quantum algorithm that can solve this problem in a single oracle query, whereas classical algorithms require **s** using the circuit shown in Fig. 2*D*. The algorithm output state directly corresponds to the hidden shift **s**. The circuit involves gates between two disconnected pairs of qubits, which creates an overhead of six two-qubit gates for a star-shaped architecture. The results are shown in Fig. 4 *A2* and *B2*. The fidelity of the fully connected ion-trap implementation is

The errors in both devices appear concentrated in certain sets of states, leading to patterns in the off-diagonal elements of the result plots (Fig. 4). These highly structured signatures suggest that systematic errors dominate, especially readout errors. The grouped patterns such as in Fig. 4*A1* indicate flips of the least-significant bits, whereas parallel lines correspond to the most significant bits changing their state. In the trapped-ion results, these lines can be modulated in height due to readout crosstalk and are more pronounced on the lower-numbered state side due to

## Outlook

Comparing quantum computing architectures involves many interrelated factors. Quantum gate operation fidelities, qubit numbers, primitive gate speeds, and coherence times are obviously important low-level metrics in a large-scale quantum computer. The results presented here show higher absolute fidelities and coherence times in the trapped-ion system, with higher clock speeds for the superconducting system. However, these metrics are moving targets: Whereas these systems are the most advanced and versatile quantum computing platforms built to date, both technologies are currently advancing rapidly.

In any case, such metrics should not be considered in isolation. Our comparison points to important higher-level considerations in scaling a quantum computer. The overall performance of a quantum circuit and the “time to solution” will depend critically on architectural restrictions, qubit connectivity, gate reconfigurability, and gate expressivity, and these attributes will become ever more important as the system is scaled up. Even with 5-qubit systems, we find that the qubit connectivity graph is best codesigned to mirror the structure of the particular quantum circuit and that the choice of a more expressive gate library affects the efficiency of the computations.

The physical scaling of each of these leading technologies has many challenges, and how they will be connected and reconfigured at large scales is an open question. One of the biggest challenges is the management of the control complexity in larger systems and potential crosstalk from overlapping qubit interactions or control buses. In most superconducting designs, there are many current-carrying wires necessary for control and biasing the individual qubits, and this may be difficult to route through a large superconducting chip (1⇓⇓⇓–5). It will likely become a great challenge to manage the dilution refrigerator heat budget with such circuitry. Alternative modular superconducting architectures improve connectivity by integrating qubits with microwave cavity modes, at the expense of significant added volume per qubit (33). Ion-trap designs will hinge upon the stable and accurate delivery of laser beams (or near-field microwave sources) to address each qubit individually in a vacuum chamber. The fully connected nature of the ion-trap architecture may not scale to arbitrarily large numbers of qubits, owing to the spectral overlap of collective normal modes of motion. However, full connectivity between 20 and 100 trapped-ion qubits appears possible (6) and a modular approach for scaling to much larger systems with high connectivity and distance-independent operations seems promising (34, 35). In any hardware, an automated calibration procedure and powerful user interface will likely provide a higher level of integration. Such system-level attributes will become even more important as quantum circuits grow in complexity, regardless of physical platform.

Numeric probability values of the results presented in Figs. 3 and 4 in the main text are shown in Figs. S1–S3. The target populations, with nominal unit probabilities, are highlighted in yellow. The others, representing errors, show a bar graph scaled from 0 to 0.1 to emphasize the systematic error patterns.

Fig. S1 is a compilation of the Margolus/Toffoli gate data shown in Fig. 3 of the main text. Fig. S2 is a compilation of the Bernstein–Vazirani data shown in Fig. 4 *A1* and *B1* of the main text, and Fig. S3 is a compilation of the hidden shift data shown in Fig. 4 *A2* and *B2* of the main text.

## Acknowledgments

We thank D. L. Moehring, J. Kim, and K. R. Brown for key discussions; J. Gambetta and J. Chow at IBM for their assistance in interfacing with the IBM Quantum Experience project; and E. Edwards for the ion trap image. This work was supported by the Army Research Office with funds from the Intelligence Advanced Research Projects Activity (IARPA) LogiQ program, the Air Force Office of Scientific Research (AFOSR) Multidisciplinary University Research Initiative (MURI) program on Optimal Quantum Circuits, and the National Science Foundation (NSF) Physics Frontier Center at JQI. D.M. acknowledges support by the NSF. Any opinion, finding, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the NSF, IBM, or any of their employees.

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: monroe{at}umd.edu or linke{at}umd.edu.

This contribution is part of the special series of Inaugural Articles by members of the National Academy of Sciences elected in 2016.

Author contributions: N.M.L., D.M., M.R., S.D., C.F., and C.M. designed research; N.M.L., D.M., M.R., S.D., C.F., K.A.L., K.W., and C.M. performed research; N.M.L., D.M., M.R., S.D., C.F., K.A.L., K.W., and C.M. analyzed data; and N.M.L., D.M., M.R., S.D., C.F., K.A.L., K.W., and C.M. wrote the paper.

Reviewers: E.H., University of California, Los Angeles; and I.L.M., The University of Michigan.

Conflict of interest statement: C.M. is the cofounder and Chief Scientist of IonQ, Inc.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1618020114/-/DCSupplemental.

Freely available online through the PNAS open access option.

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