Benchmarking an 11-qubit quantum computer
K. Wright, K. M. Beck, S. Debnath, J. M. Amini, Y. Nam, N. Grzesiak,, J. -S. Chen, N. C. Pisenti, M. Chmielewski, C. Collins, K. M. Hudek, J., Mizrahi, J. D. Wong-Campos, S. Allen, J. Apisdorf, P. Solomon, M. Williams,, A. M. Ducore, A. Blinov, S. M. Kreikemeier, V. Chaplin
TL;DR
This paper presents an 11-qubit trapped ion quantum computer with high fidelity gates, demonstrating the execution of benchmark algorithms that outperform other existing hardware, showcasing its potential for quantum computing applications.
Contribution
The paper introduces a fully-connected 11-qubit trapped ion quantum computer with high gate fidelities and benchmarks its performance using standard algorithms.
Findings
Achieved 99.5% single-qubit gate fidelity
Achieved 97.5% two-qubit gate fidelity
Successfully executed benchmark algorithms with high success rates
Abstract
The field of quantum computing has grown from concept to demonstration devices over the past 20 years. Universal quantum computing offers efficiency in approaching problems of scientific and commercial interest, such as factoring large numbers, searching databases, simulating intractable models from quantum physics, and optimizing complex cost functions. Here, we present an 11-qubit fully-connected, programmable quantum computer in a trapped ion system composed of 13 Yb ions. We demonstrate average single-qubit gate fidelities of 99.5, average two-qubit-gate fidelities of 97.5, and state preparation and measurement errors of 0.7. To illustrate the capabilities of this universal platform and provide a basis for comparison with similarly-sized devices, we compile the Bernstein-Vazirani (BV) and Hidden Shift (HS) algorithms into our native gates and execute them…
| Ion | Gate Fidelity | SPAM from RB | SPAM from Microwave |
| 0 | 99.57(5) | 99.31(9) | 99.82(4) |
| 1 | 99.62(6) | 99.1(1) | 99.77(5) |
| 2 | 99.18(7) | 99.3(1) | 99.78(5) |
| 3 | 99.25(9) | 99.6(2) | 99.78(5) |
| 4 | 99.40(9) | 99.3(2) | 99.84(4) |
| 5 | 99.46(3) | 99.32(7) | 99.77(5) |
| 6 | 99.48(3) | 99.27(6) | 99.82(4) |
| 7 | 99.55(4) | 99.40(8) | 99.83(4) |
| 8 | 99.59(3) | 98.94(6) | 99.80(4) |
| 9 | 99.64(2) | 99.35(4) | 99.79(5) |
| 10 | 99.32(6) | 99.3(1) | 99.79(5) |
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
| 98.5 | 97.7 | 98.5 | 97.2 | 98.5 | 96.9 | 97.2 | 98.7 | 95.5 | 97.1 | 0 |
| 97.7 | 98.9 | 98.2 | 97.4 | 97.8 | 98.1 | 98.4 | 97.7 | 97.9 | 1 | |
| 98.0 | 97.5 | 96.5 | 98.4 | 98.0 | 97.2 | 97.3 | 96.0 | 2 | ||
| 96.4 | 97.4 | 97.1 | 98.9 | 96.0 | 98.0 | 97.7 | 3 | |||
| 98.6 | 97.3 | 97.3 | 98.3 | 97.8 | 96.5 | 4 | ||||
| 96.5 | 97.1 | 98.4 | 95.1 | 96.7 | 5 | |||||
| 96.2 | 97.2 | 98.1 | 98.2 | 6 | ||||||
| 97.3 | 98.5 | 97.3 | 7 | |||||||
| 96.7 | 97.0 | 8 | ||||||||
| 97.5 | 9 |
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Benchmarking an 11-qubit quantum computer
K. Wright
IonQ, Inc., College Park, MD 20740, USA
K. M. Beck
IonQ, Inc., College Park, MD 20740, USA
S. Debnath
IonQ, Inc., College Park, MD 20740, USA
J. M. Amini
IonQ, Inc., College Park, MD 20740, USA
Y. Nam
IonQ, Inc., College Park, MD 20740, USA
N. Grzesiak
IonQ, Inc., College Park, MD 20740, USA
J.-S. Chen
IonQ, Inc., College Park, MD 20740, USA
N. C. Pisenti
IonQ, Inc., College Park, MD 20740, USA
M. Chmielewski
IonQ, Inc., College Park, MD 20740, USA
Joint Quantum Institute and Department of Physics, University of Maryland, College Park, MD 20742, USA
C. Collins
IonQ, Inc., College Park, MD 20740, USA
K. M. Hudek
IonQ, Inc., College Park, MD 20740, USA
J. Mizrahi
IonQ, Inc., College Park, MD 20740, USA
J. D. Wong-Campos
IonQ, Inc., College Park, MD 20740, USA
S. Allen
IonQ, Inc., College Park, MD 20740, USA
J. Apisdorf
IonQ, Inc., College Park, MD 20740, USA
P. Solomon
IonQ, Inc., College Park, MD 20740, USA
M. Williams
IonQ, Inc., College Park, MD 20740, USA
A. M. Ducore
IonQ, Inc., College Park, MD 20740, USA
A. Blinov
IonQ, Inc., College Park, MD 20740, USA
S. M. Kreikemeier
IonQ, Inc., College Park, MD 20740, USA
V. Chaplin
IonQ, Inc., College Park, MD 20740, USA
M. Keesan
IonQ, Inc., College Park, MD 20740, USA
C. Monroe
IonQ, Inc., College Park, MD 20740, USA
Joint Quantum Institute and Department of Physics, University of Maryland, College Park, MD 20742, USA
J. Kim
IonQ, Inc., College Park, MD 20740, USA
Department of Electrical and Computer Engineering, Duke University, Durham, NC 27708, USA
Abstract
The field of quantum computing has grown from concept to demonstration devices over the past 20 years. Universal quantum computing offers efficiency in approaching problems of scientific and commercial interest, such as factoring large numbers,Shor1997 searching databases,Grover1997 simulating intractable models from quantum physics,Trabesinger2012 and optimizing complex cost functions.Farhi2014 Here, we present an 11-qubit fully-connected, programmable quantum computer in a trapped ion system composed of 13 171Yb*+* ions. We demonstrate average single-qubit gate fidelities of 99.5, average two-qubit-gate fidelities of 97.5, and state preparation and measurement errors of 0.7. To illustrate the capabilities of this universal platform and provide a basis for comparison with similarly-sized devices, we compile the Bernstein-Vazirani (BV)Bernstein1997 and Hidden Shift (HS)vanDam2003 ; Rotteler2010 algorithms into our native gates and execute them on the hardware with average success rates of 78 and 35, respectively. These algorithms serve as excellent benchmarks for any type of quantum hardware, and show that our system outperforms all other currently available hardware.
Small universal quantum computers that can execute textbook quantum circuits exist in both academicBarends2014 ; Monz2016 ; Debnath2016 ; Roy2018 ; Erhard2019 and industrialChow2012 ; Devitt2016 ; Pokharel2018 ; Hong2019 ; Nam2019 settings. With a range of two to seventy-two qubits and sufficient fidelity for only tens of entangling gates, these devices and the underlying qubit implementations can be difficult to compare. Even within the trapped ion platform, there is large diversity in atomic species, system architectures, and gate implementations. Trapped ion systems with one to two qubits have shown single-qubit gate fidelities of 99.9999Harty2014 with microwave-based operations and better than 99.99 fidelity with laser based operations,Gaebler2016 ; Ballance2016 state preparation and measurement (SPAM) error below ,Harty2014 ; Crain2019 and two-qubit gates with fidelities exceeding 99.9.Gaebler2016 ; Ballance2016 Algorithms have been executed on up to seven trapped-ion qubitsLandsman2018 and, while not optimized for universal quantum computing, quantum simulators with more than 50 ions have modeled fundamental quantum systems including Ising chainsZhang2017 and quantum magnetism.Bohnet2016
Benchmarking across implementations needs to be both universal across platforms and agnostic to the differences in the underlying hardware. In traditional computing, the performance of computers is measured by executing a set of benchmark problems representing various use-case scenarios, to provide users with an estimate of how the computers would perform in their specific applications. Canonical quantum algorithms demonstrate unambiguous advantage of quantum computers over classical computation, and provide verifiable outcomes to assess successful execution of the algorithm. Therefore, they can serve as ideal candidate problems for benchmarking the performance of any quantum computers. These benchmark algorithms exercise the full hardware/software stack. A hardware-specific compiler breaks down algorithms into the target hardware’s native gate set, optimizing for qubit connectivity, gate times, and coherenceLinke2017 to enhance the system’s performance. After execution on the hardware, the measurements can be directly compared with the expected output state to determine the accuracy of the device. This accuracy can then be compared with other devices that have compiled and run the same algorithm.Murali2019
We benchmark two algorithms on an IonQ trapped ion quantum computer, shown schematically in Figure 1. Our qubit register is comprised of a chain of trapped 171Yb*+* ions, spatially confined near a microfabricated surface electrode trap.HOA For this work, we load 13 ions, the middle 11 of which are used as qubits. The two end ions allow for a more uniform spacing of the central 11 ions. The ions are laser-cooled close to their motional ground state using a combination of Doppler and resolved sideband cooling. We encode quantum information into the hyperfine sublevels, and of the ground state. At the beginning of each computation, each qubit is initialized to via optical pumping with high accuracy. After qubit operations (described below), we read out the state of all of the qubits simultaneously by directing laser light resonant with the to transition, imaging each ion onto a detector and thresholding the photon counts to determine if each qubit was in the (spin up) or (spin down) state. Thresholding is done by taking a histogram of the collected photons and discriminating between collecting on average zero photons for the state and ten photons on average for the state.
A two-photon Raman transition drives single- and two-qubit coherent operations by applying a pair of counter-propagating beams from a mode-locked pulsed 355 nm laser.Hayes2010 One of these beams globally addresses all of the ions simultaneously, while the other beam addresses any of the ions individually (Figure 1). The individually-addressing beams pass through a multi-channel acousto-optic modulator (AOM), which allows for the simultaneous modulation of the phase, frequency and amplitude of each beam. To perform a single-qubit gate, we tune the frequency difference between Raman beams to resonantly drive a spin flip transition (). In order to perform a two-qubit gate, we off-resonantly drive motional sideband transitions to generate an XX-interaction.Sorensen2016 Both the global and individual beams are directed over the trap surface perpendicular to the axis of the ion chain to excite one principal axis of motion transverse to the chain axis. Individual addressing allows us to perform single- and two-qubit gates on any targeted qubits.
Native two-qubit entangling XX-gates are achieved by driving a spin dependent force.Sorensen2016 Using an amplitude modulated (AM) pulse on any selected pair of qubits, we address multiple transverse motional modes of the ion chain to mediate a spin-spin Ising interaction between qubits.Choi2014 To achieve high fidelity, the amplitude modulation is calculated to simultaneously decouple all motional modes from the spin at the end of the gate operation. Additionally, these pulse shapes are designed to provide robustness against frequency drift of motional modes and suppress residual off-resonant carrier excitation during the XX-gate.Choi2014 ; Wu2018 This gate, in conjunction with single-qubit rotations, forms a universal gate set for performing circuit model quantum computation. Since the XX-gates are mediated by the collective motion of the ion chain, we have all-to-all connectivity between qubits, allowing two-qubit gates to be executed between any qubit pair (Figure 2a).
We perform randomized benchmarkingKnill2008 to characterize the single-qubit operations on each ion of the 11-qubit chain. We apply a randomly chosen sequence of gates with length about the and axes. In between each of these gates, we either add a rotation about the , , or axis, or an identity operation (leaving the qubit idle for the duration of a gate). A final gate is chosen such that the final state is in the computational basis (i.e. or ). We measure the overlap between the measured and expected output states across 500 iterations for at least 24 sequences for each . The fidelity of our single-qubit gate is then determined by fitting the resulting overlap as a function of sequence length to a power law, . Here, the base is the gate fidelity and the intercept is the SPAM fidelity, equivalent to measuring the ion after a single rotation when it is in either state or . For a chain of 11 qubits, we measure an average single-qubit fidelity of 99.5 (Figure 2b) and an average SPAM fidelity of 99.3.
To quantify the performance of our two-qubit gates and estimate their fidelity, we measure the state fidelity of the Bell state prepared using a single XX-gate by performing partial tomography of the state.Ballance2016 ; Gaebler2016 The diagonal terms of the two-qubit density matrix are extracted by measuring the populations in the even parity states. The populations are measured when the overall AM pulse height for the XX-gate is tuned to achieve maximal entanglement such that the even-parity two-qubit states, and , are equal (). The off-diagonal elements are obtained from the amplitude of a parity oscillation, where the parity is given by ( and are the populations of the odd parity two-qubit states). The fidelity can then be calculated as ().Ballance2016 We use maximum likelihood estimation on experimentally observed data to extract the parameters of the fidelity expression.Ballance2016 We have performed this analysis for all 55 pairs of qubits in the 11-qubit chain (Figure 2c) and measure an average fidelity of 97.5 with a minimum and maximum fidelity of 95.1 and 98.9, respectively. The uncertainty here is determined by a statistical confidence interval on the maximum likelihood estimation. The reported fidelity represents a lower bound of the Bell state creation as we do not correct for SPAM errors on the two-qubit states or errors in single-qubit rotations used to observe the parity oscillations of the Bell state, which on average are 0.7 and 0.5 respectively.
To benchmark our system, we implement two well-known algorithms: the Bernstein-Vazirani (BV) and Hidden Shift (HS). Both of these algorithms have previously been run on trapped-ionDebnath2016 ; Fallek2016 ; Linke2017 and superconductingLinke2017 ; Roy2018 ; Murali2019 systems of up to 5 qubits. By comparing the results of this algorithm to the ideal result, we obtain a direct measure of the system performance, which accounts for our native gates, connectivity, coherence times, gate duration, and all other isolated metrics of system performance. These results can be used as part of a suite of algorithms to compare our hardware with other systems. The qubit number in these results is higher than any comparable published BV or HS results using a programmable quantum computer.Debnath2016 ; Fallek2016 ; Linke2017 ; Roy2018 ; Murali2019
The BV algorithm is an oracle problem in which the user tries to determine an unknown bit string of size , implemented by a specific oracle. The algorithm takes a binary input string and performs a controlled inversion of an ancillary bit or qubit based on the bit-wise product of the input and the unknown bit string modulo two, .Bernstein1997 For a quantum BV implementation (example shown in Figure 3a), a single quantum query is sufficient to determine the bit string .Grover1997 This is a linear improvement over the best classical algorithm, which requires queries. The BV algorithm was developed to help separate a class of problems that can be solved in polynomial time on a quantum computer with bounded errors (BQP) from its classical counterpart. For a problem to belong to BQP, a single quantum query success probability needs to be greater than 2/3.Bernstein1997
We compile the BV algorithm into our native gate set, comprised of single-qubit rotations and two-qubit XX-gates. Optimization during compilation reduces the number of needed gates compared to naively translating the textbook circuit from CNOT gates into rotations and XX-gates. The compilation exploits the full connectivity of our qubits, since we do not need SWAP operations. The implementation of BV requires a single-qubit ancilla and a register of qubits. There are possible bit strings, therefore for our 10-qubit register there are 1024 possible oracle implementations. We measured each implementation 500 times, conditioned upon on the measured ancilla state, and plot the output distribution in Figure 3b. Each oracle implementation has, depending on the unknown bit string , between zero and ten two-qubit gates between the ancilla and the qubit register, corresponding to the number of ones in the binary representation of the unknown bit string. The process matrix that maps the encoded oracle to the measured output state is nearly diagonal, resulting in a highly-peaked distribution at the encoded oracle. For our system, the average overlap between output state and unknown bit string is 78 (Figure 3c), where 87.8 of oracle implementations achieve the 2/3 success criteria defined by BQP. Conditioning the output on the ancilla state results in a 5.1 percentage point increase in the raw success probability of 73 to 78 and an 14.5 percentage point increase in the fraction of oracle implementations above the BQP threshold from 73.3 to 87.8. The average overlap in Figure 3c decreases with the number of two-qubit gates needed in the oracle. The off-diagonal components of the process matrix show errors since these should all have zero population. In Figure 3b, the dominant error is single-qubit bit-flips from to during the measurement process, which appear as faint diagonals in the lower left quadrant of the figure (see Methods). However, even for the oracle implementation where we have the lowest success probability, the next-most-probable state is still 4 times less likely than the correct string.
The HS algorithm consists of two N-bit to N-bit function oracles f and g, which are the same up to a shift by a hidden bit string , such that . The goal is to determine the hidden shift by querying the oracles. In our implementation Rotteler2010 of the HS algorithm, the oracles are inner product or “bent” functions and , where is the input and is the -th bit of (an example is shown in (Figure 4a). Classically it can be shown that determining the shift requires queries where is the length of the bit string . On a quantum computer, in principle, the shift can be read out in a single query.vanDam2003 ; Rotteler2010 In contrast to the BV algorithm, the quantum implementation of the HS algorithm shows an exponential reduction in the number of queries to the oracle compared to a classical computer.Rotteler2010
As with the BV algorithm, we compile the HS algorithm into our native gates. There are available oracle implementations corresponding to the possible hidden bit strings . We execute all 1024 possible implementations on our 10-qubit register (Figure 4b). The correct output state is the state corresponding to the hidden shift. The average overlap between the output state and was 35 (Figure 4c), and of the 1024 oracles, 1017 had most likely output states corresponding to the shift. The success probability for HS is lower and more uniform than that of BV because all of the oracles have the same number of two-qubit gates (10) and many more single-qubit gates (25-40). Every oracle implementation in HS has at least as many gates as the most challenging BV oracle implementation and therefore is more difficult. Given our average single- and two-qubit fidelities, we would not expect to surpass the BQP threshold for the HS oracles. However, the successful determination of the shift was achieved much more frequently than if we sampled a classical distribution where the success probability would have been 0.1.
In summary, we have constructed the largest programmable quantum computer to date that is capable of performing algorithms. We have used a trapped ion quantum computer to perform the largest quantum implementations of the Bernstein-Vazirani and Hidden Shift algorithms. Using a 10-qubit register, we implement all 1024 possible oracles for each algorithm. We exceed the BQP threshold for 87.8 of the oracle implementations in the Bernstein-Vazirani algorithm, an application designed to define this complexity class. We also demonstrate 35 overlap between the measured and expected output states in the implementation of the Hidden Shift algorithm, which is a more demanding application due to its higher gate count and exponential speed up over its classical analog. The success of both algorithms is a result of high-fidelity native gates and efficient gate compilation and compression in the fully-connected ion trap system. The demonstration of these two canonical algorithms is a starting point for benchmarking any quantum computer. Computing real problems on larger systems with more qubits will require even more gates in the future with even higher quality, and similar standard algorithms to those demonstrated here will likely play a crucial role in benchmarking quantum computers in the future.
Methods
The dominant error in our data is single-qubit bit flips driven by addressing crosstalk. In the BV algorithm, these errors show up predominantly as qubits that are in state when they should be in state in the output state. We model this as 3 probability of incorrectly preparing one qubit in the state, which we apply twice: first to model errors that occur in the implementation of the oracle, and then to model errors in the single-qubit gates that occur before readout. We estimate the 3 probability of incorrectly preparing one qubit by fitting by eye the relative amplitude of error states. Finally, detection mis-identification (measuring the state of one ion as other than it should be) is modeled at 0.2 probability on the resulting state, the measured SPAM infidelity measured with microwaves (Table 1). of our measurement results for the BV algorithm are shown in Figure 5a for comparison with the result of this error model (Figure 5b).
In the HS algorithm, the crosstalk errors show up as qubits that are measured to be in the wrong state (either or ). We model this as a 1 error of incorrectly preparing one qubit in either state, which we apply 5 times, followed 0.2 detection mis-identification. of our measurement results for the the HS algorithm are shown in Figure 5c for comparison with the result of this error model (Figure 5d).
Acknowledgements
The authors would like to thank David Moehring for his guidance in the construction of this apparatus, and the EURIQA team at the University of Maryland and Duke University for sharing their designs and for useful conversations.
Author Contributions
Experimental data collected and analyzed by K.W., K.M.B and S.D.; circuit compilation and gate design by Y.N., S.D. and N.G.; the apparatus was designed and built by K.W., K.M.B, J.M.A., S.D., JS.C., N.C.P., M.C., C.C., K.M.H, J.M., J.D.WC., S.A., J.A., P.S., M.W., A.D., A.B., S.M.K., V.C., M.K., C.M. and J.K.; K.M.B. and K.W. prepared the manuscript, with input from all authors.
Competing Interests
The authors declare no competing interests.
Correspondence
Correspondence and requests for materials should be addressed to Kenneth Wright (email: [email protected]).
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