Abstract
A critical ingredient for realising largescale quantum information processors will be the ability to make economical use of qubit control hardware. We demonstrate an extensible strategy for reusing control hardware on samefrequency transmon qubits in a circuit QED chip with surfacecodecompatible connectivity. A vector switch matrix enables selective broadcasting of input pulses to multiple transmons with individual tailoring of pulse quadratures for each, as required to minimise the effects of leakage on weakly anharmonic qubits. Using randomised benchmarking, we compare multiple broadcasting strategies that each pass the surfacecode error threshold for singlequbit gates. In particular, we introduce a selective broadcasting control strategy using five pulse primitives, which allows independent, simultaneous Clifford gates on arbitrary numbers of qubits.
Introduction
Building a faulttolerant quantum computer requires the ability to efficiently address and control individual qubits in a largescale system. Many leading experimental quantum information platforms, including trapped ions,^{1} electronic spins in impurities and quantum dots,^{2} and superconducting circuits,^{3} use qubits with level transitions in the microwave frequency domain. Addressing these transitions often involves expensive microwave electronics scaling linearly with the number of qubits. To move beyond the stateoftheart in microwavefrequency quantum processors, such as those recently used for smallscale quantum error correction in superconducting circuits,^{4–6} it is beneficial to have a hardwareefficient control strategy that harnesses economies of scale. One approach, as proposed by Hornibrook et al.,^{7} is to use microwave pulses from a single control source for multiple qubits, requiring frequencymatched qubits and highspeed routing of pulses to separate control lines. The linear scaling of control equipment could then be reduced to a constant overhead for the most expensive resources.
Using control equipment for multiple qubits has previously been demonstrated for optical addressing in atomic systems, where qubits naturally have the same frequency.^{8–11} Such frequency reuse also becomes possible in circuit quantum electrodynamics^{12} in the context of faulttolerant computation strategies that rely only on local interactions between qubits mediated by bus resonators.^{13–15} The natural isolation between different lattice sites allows the use of repeating patterns of qubit frequencies with selectivity provided by spatial separation. A tileable unit cell with a handful of qubit frequencies^{16} could, therefore, provide a promising route towards scalability. Crucially, this also solves the frequencycrowding problem that arises when trying to fit many distinctfrequency qubits within the finite useful bandwidth of the circuitbased devices, particularly for designs based on weakly anharmonic qubits where higher levels must also be avoided.^{17,18} Hornibrook et al.^{7} have recently demonstrated cryogenic switching matrix hardware for pulse distribution operating at 20 mK, triggered by a fieldprogrammable gate array at 4 K. Cryogenic control equipment may shorten feedback latency and reduce wiring complexity across temperature stages, but the isolation and operational frequency range reported to date are insufficient for typical circuit quantum electrodynamics experiments. Simultaneous driving of multiple qubits with distinct frequencies using individually dedicated control hardware has already been demonstrated in a range of previous circuit quantum electrodynamics experiments.^{19,20} To date, however, simultaneous driving has not been demonstrated on samefrequency, solidstate qubits using shared microwave control hardware.
In this article, we demonstrate frequency reuse in an extensible solidstate multiqubit architecture. An extensible architecture harnessing frequency reuse has two key requirements: a method for distributing control pulses to multiple qubits with economical means to adapt the pulses for each qubit, and a multiqubit device containing samefrequency qubits with relevant connectivity and sufficient isolation between samefrequency qubits. To this end, we show independent simultaneous control of two samefrequency qubits with a roomtemperature vector switch matrix (VSM) that we have developed. The VSM allows tailoring of control pulses to individual qubit properties, and routing of the pulses to either one or both of the qubits using fast digital markers. We develop several different approaches to selective pulse broadcasting, including a simple scheme for implementing independent Clifford control on an arbitrary number of qubits with a constant overhead in time. The device for this experiment is designed to allow testing in a circuit with the correct connectivity of a relevant surfacecode lattice.^{21,22} Using randomised benchmarking (RB), we show that all control schemes exceed the fidelity threshold for surface code and are dominated by qubit relaxation. We also develop a method for measuring leakage to the secondexcited state directly within the context of RB.^{23,24} We characterise the limitations of our system and find no major obstacles to scaling up to larger implementations.
Results
To demonstrate frequency reuse, we focus on a particular implementation of the surface code based on repeated tiling of a unit cell consisting of four qubits and two resonators^{13} (Figure 1). Each qubit couples to two resonators and each resonator couples to four qubits, requiring a minimum of four unique qubit frequencies at the operating point for singlequbit control. This ensures that samefrequency qubits are never coupled through a single resonator. Independent microwave control of samefrequency qubits requires a separate drive line for each. Such a line can be dedicated to each qubit,^{25} as used in this demonstration, or to each resonator^{26} to allow the driving of all four qubits coupled to it. This implementation of surface code requires twoqubit gates only between differentfrequency qubits coupled to the same resonator and never between samefrequency qubits. Our twoqubit gates will be implemented using fast fluxbias lines,^{27} but in this article we focus on VSMbased singlequbit control. The higherorder coupling between samefrequency qubits (mediated by at least two resonators and one qubit at a different frequency) is only ever a disturbance and error source. Our device contains a small block of this surface code implementation, consisting of two samefrequency transmon qubits (Q_{T} and Q_{B}), which are connected to a third qubit (Q_{M}) via separate bus resonators (Figure 1). Together, the buses and Q_{M}, although not actively used throughout the experiment, provide a realistic operating environment for Q_{T} and Q_{B}.
To efficiently control multiple samefrequency qubits, we have designed a prototype roomtemperature 4×2 (four input, two output) VSM that allows independent control of amplitude and phase for each of its input–output combinations. Fast markercontrolled digital switches enable routing of pulses to the qubits at nanosecond timescale, with approximately 50 dB isolation in the frequency range from 4 to 8 GHz (see Supplementary Material for additional data on VSM specifications). Because of the transmon’s weak anharmonicity,^{28} highfidelity fast singlequbit control is achieved using the method of derivativeremovalviaadiabaticgate (DRAG) pulsing, where the inphase Gaussian pulse is combined with a quadrature derivativeofGaussian pulse.^{29,30} By directing the two constituent pulses of DRAG control to two of the four inputs of the VSM, this allows independent, in situ DRAG tuning for both samefrequency qubits using four AWG channels (two channels for IQ modulation of each constituent pulse; see Supplementary Material at [http://www.nature.com/npjqi] for additional data).
The first critical test of our control architecture is to assess the VSM’s ability to implement highprecision control of one qubit while leaving the other qubit idle. To do this, we use the standard technique of singlequbit RB based on Clifford gates,^{31–33} which allows the characterisation of control performance independently of state preparation and measurement errors. After initialising all qubits in the ground state by relaxation, we use the VSM to selectively apply random sequences of Cliffords gates to only one of the samefrequency qubits (Figure 2a,b), in each case measuring the effect on both qubits simultaneously via multiplexed readout.^{34,35} We decompose each gate into the standard minimal sequence of π and ±π/2 pulses around the x and y axes, requiring on average 〈N_{p}〉=1.875 pulses per Clifford.^{24} This is in contrast to socalled ‘atomic pulses’,^{36} where the 24 singlequbit Cliffords can each be implemented with a single pulse. Each π and ±π/2 pulse is 16 ns long and separated by a 4 ns buffer from the next, for a total pulse time t_{p}=20 ns. After applying a final Clifford that inverts the cumulative effect of all m previous Cliffords, the driven qubit is ideally returned to the ground state, but as a result of imperfections such as gate errors and decoherence the final groundstate population decays as a function of m. The decay rate can be related to the average fidelity per Clifford F_{C}.^{31,32} From the RB data (Figure 2c,d), we extract the average Clifford fidelities for the two individually driven qubits to be 0.9982(2) (Q_{T}) and 0.9986(2) (Q_{B}). We compare these values with the expected average Clifford fidelities assuming only T_{1} decay:^{37}
The similarity (Figure 2c,d) shows that the results are predominately limited by relaxation effects. The difference in performance between the two qubits is consistent with their different T_{1} times. It is consistent with previous RB demonstrations^{19} that we do not see a strong signature of pure dephasing in the fidelity. Further measurements show that driving both qubits simultaneously by the same pulse sequence (both markers on) does not significantly impact the performance (see Supplementary Material at [http://www.nature.com/npjqi] for additional data).
In a strictly twolevel system, the measured ground and excitedstate populations averaged over many sequences (〈P_{0}〉 and 〈P_{1}〉) both converge to 0.5 for large m. For weakly anharmonic transmon qubits, leakage to the secondexcited state can be an important additional source of gate error, which can lead to a shift of the asymptotic values away from 0.5. We address this issue by performing the RB protocol both with and without an additional final π pulse,^{38} which allows us to explicitly estimate the populations of the first three transmon states (see Supplementary Material at [http://www.nature.com/npjqi] for additional data). From the measured leakage populations 〈P_{2}〉 (Figure 2e,f), we extract per Clifford leakage rates κ of 4.1(2)×10^{−6} (Q_{T}) and 1.3(4)×10^{−6} (Q_{B}) by fitting the following simple model to the data (see Supplementary Material at [http://www.nature.com/npjqi] for additional data):
where T_{2→1} is the second to firstexcitedstate relaxation time. As these leakage rates are much smaller than the gate errors (1−F_{C}), it is reasonable to neglect them when estimating the Clifford fidelity.
We next explore the effect of the singlequbit control pulses on the undriven qubit (Figure 2g,h). Although Q_{T} remains largely unaffected when driving Q_{B}, a substantial deviation from the ground state is measured in Q_{B} when driving Q_{T}. There are several possible mechanisms for crossexcitation in the system. Dominant onchip sources include residual exchange interaction J between Q_{T} and Q_{B} (mediated by the bus resonators and Q_{M}), and crossdriving, resulting from uncontrolled parasitic couplings between each drive line and the untargeted qubit. As detailed in the 'Materials and Methods' section, measurements of these onchip sources found J/2π≤36±1 kHz, and an asymmetric crossdriving that is stronger when Q_{B} is driven using the drive line of Q_{T} (−45 dB) than vice versa (−53 dB). This is larger than the dominant offchip source of crossexcitation, the finite VSM isolation, which is approximately −57 and −54 dB on Q_{T} and Q_{B}, respectively (see Supplementary Material at [http://www.nature.com/npjqi] for additional data), lower than the direct onchip crossdriving. The symmetric swapping of excitation under exchange is unlikely to explain the strong asymmetry in the amount of crossexcitation measured for the different qubits. Moreover, numerical simulations show that the observed effects are consistent with crossdriving alone (see Supplementary Material at [http://www.nature.com/npjqi] for additional data). This effect of crossdriving in the context of RB can be quantified using interleaved RB.^{39} In interleaved RB, the fidelity of an individual gate is estimated by interleaving repeated applications of that gate with a sequence of random Cliffords and comparing the performance with conventional RB. When using the VSM to implement individual qubit control, gates applied to one qubit should behave as an effective idling operation for the other. From this perspective, the sequential RB scheme can be viewed as interleaved RB for an idling operation. The idling fidelity can therefore be extracted by comparing the fidelities obtained for sequential RB with the fidelities for singlequbit RB. This yields an average idling fidelity for Q_{B} of 0.9986(5) (see Supplementary Material at [http://www.nature.com/npjqi] for additional data), consistent with the error due to additional T_{1} decay, confirming that crossexcitation effects do not dominate the error per Clifford.
The defining test of extensibility in our control architecture is to demonstrate the simultaneous, independent, singlequbit control over samefrequency qubits that is enabled by selective broadcasting using the VSM. We explore three paradigmatic schemes for implementing selective broadcasting of Cliffords on an arbitrary number of qubits n (Figure 3). In the most straightforward selective broadcasting scheme, the individual qubits are driven sequentially, with each pulse being directed to one qubit at a time. This results in a linear scaling of the average number of pulses per Clifford round (〈N_{p}〉=1.875×n). By contrast, the second paradigm takes best advantage of the VSM’s capability to broadcast simultaneously to multiple qubits by compiling the constituent Clifford pulses to minimise N_{p} for each Clifford combination in the sequence. However, compiling pulses comes at the cost of an exponentially increasing compilation time with the number of qubits before running the sequence.
This motivates our final broadcasting paradigm, where all Clifford gates can be implemented using the same fixed, ordered sequence of five pulse primitives (Figure 3). Independent Cliffords can be applied to all qubits, irrespective of n, by selectively directing the appropriate subset of pulses to each qubit, achieving a constant overhead in time for control of an arbitrary number of qubits. Although the number of pulse primitives must be at least five to produce the 24 unique Cliffords, the choice of the five pulses is not unique. We have chosen a combination of positive and negative rotations to partially nullout the effect of crossdriving on the undriven qubit (see Supplementary Material at [http://www.nature.com/npjqi] for additional data). We also alternate between two versions of the five pulse primitives, where in the second version they are in reverse order and opposite rotation to even further reduce crossdriving.
To demonstrate the full functionality of our control architecture, we implement all three selective broadcasting schemes and measure their performance using parallel singlequbit RB with independent Clifford sequences for each qubit. Figure 4 shows that the compiled scheme performs best, followed by the sequential and then 5primitives schemes, consistent with the average number of pulses required for each (Figure 3). In all cases, the average fidelity per Clifford is still dominated by relaxation (Table 1). The results are completely consistent with the values obtained in the test for isolated singlequbit control.
Discussion
Our VSM allows efficient use of control equipment on samefrequency qubits. It enables highprecision singlequbit control of multiple qubits with a performance that surpasses the bestknown surfacecode faulttolerance threshold for singlequbit gates of ~0.99^{40—42}, and is mainly limited by relaxation. Although the measurements show a nonnegligible amount of leakage from the computational subspace after many pulses (Figure 2e,f), the corresponding leakage errors are outweighed by other gate errors. A significant amount of crossexcitation was measured in one qubit during the RB measurements (Figure 2h), which resulted primarily from onchip crossdriving. In future experiments, this effect could be reduced by careful design of both the surfacecode and physical chip layouts. In the first case, increasing the number of qubit frequencies used will result in larger lattice separations between samefrequency qubits. This will provide better effective isolation at the cost of increased design and hardware complexity. In the second case, circuit QED provides naturally good circuit isolation due to the superconducting ground plane, but optimising the onchip coupling network to minimise spurious driving will still be one of the key problems in scaling up to larger systems. In addition, here we show that crossdriving effects can be reduced even at the control level. Specifically, as demonstrated here with the fiveprimitives pulse sequence (see Supplementary Material at [http://www.nature.com/npjqi] for additional data), we choose a sequence of control pulses in such a way that leakage pulses partially or completely cancel out. This technique is not limited to RB, but could also be used to minimise crossdriving in quantum algorithms. Furthermore, it is fully compatible with pulsebroadcasting methods, and allows straightforward scaling. This is not the case for compensation pulses, where the strength of each compensation pulse depends on the pulses applied to other qubits. By contrast with crossdriving, the measured crosscoupling (direct qubit–qubit quantum coupling) has no observable effect in the singlequbit RB measurements. Most likely, this is because rapid application of randomising Clifford pulses effectively decouples the crosscoupling dynamics. However, such a coupling may have a stronger effect in other contexts and may have to be addressed in future experiments.
We have introduced three selective broadcasting schemes for performing simultaneous, independent singlequbit RB on both qubits, in each case demonstrating performance that surpasses the faulttolerance threshold for the surface code for both driven qubits. Selective broadcasting is not limited to transmon qubits; it can be applied to any other qubit system in which qubits can be sufficiently matched in frequency, and where a VSM can be realised. The naive sequential broadcasting approach still performs acceptably with two qubits, but scales poorly with increasing numbers of qubits. On the other hand, the compiled scheme by definition always minimises the length of the pulse sequence, but at the cost of exponentially increasing difficulty of the compilation step with the number of qubits. Moreover, our estimates show that the number of pulses per Clifford round rapidly asymptotes to a total of five pulses, thus only providing negligible gain in time and fidelity over the fiveprimitives scheme for even a handful of qubits. By contrast, the fiveprimitives scheme combines both scalability and simplicity of implementation (Figure 3), selecting the target Clifford by routing a subset of five primitive pulses using digital markers. It is the minimal set of pulses required for independent Clifford control for an arbitrary number of qubits with no additional overhead in the number of sequence pulses. On a technical level, markerbased selection of Clifford gates can be useful when fast feedback has to be applied on multiple qubits, as is often the case in quantum error correction protocols. Furthermore, by adding a sixth, nonClifford gate to the five pulse primitives, this can be extended to achieve universal singlequbit control. We show that the fidelities are mainly limited by qubit relaxation for all broadcasting schemes, and are consistent with each other and with the singlequbit RB results once the average duration per Clifford round is taken into account. This confirms that selective pulse broadcasting does not decrease gate performance relative to that expected from the intrinsic pulse fidelities.
Owing to its small scale, our VSM prototype offers limited hardware savings. Although one microwave source is sufficient to control both Q_{T} and Q_{B}, full DRAG control still requires four AWG channels. However, no further channels will be required for controlling additional qubits, subject to the limitations of signal amplification and fanout, which we estimate should enter at ~100 samefrequency qubits. Although our VSM is designed to be compatible with the full DRAG control required for weakly anharmonic transmon qubits, it is by no means limited to this case. In fact, our VSM already enables precision control of strongly anharmonic qubits such as flux and fluxonium qubits^{43,44} without any hardware modification and using only one input per frequency. In this case, there would already be savings in AWG channels at the scale of our prototype.
Combining the connectivity of our device, the VSMbased control and the fixed pulse overhead of the fiveprimitives broadcasting strategy, our experiment realises the simplest element of an extensible qubit control architecture. This design can be straightforwardly expanded to more samefrequency qubits without requiring any further microwave sources or arbitrary waveform generators. This experiment suggests that surfacecode tiling with frequency reuse is a viable path towards largescale quantum processors.
Materials and Methods
This section provides further details on the circuit quantum electrodynamics device and VSM, and describes the methods used to measure crosscoupling and crossdriving.
The use of control pulses for multiple qubits requires the qubits to be matched in frequency to avoid gate errors from offresonant driving. Although Q_{T} and Q_{B} were designed to be identical, fabrication uncertainties resulted in a sweetspot (maximum) frequency of 57 MHz higher than that of Q_{B}. With Q_{B} and Q_{M} kept at their respective sweetspots (6.220 and 6.550 GHz, respectively), Q_{T} was then flux tuned to match Q_{B} with an accuracy of 50 kHz, determined using Ramsey measurements (see below). The coherence times at the operating point are provided in the Supplementary Material (see Supplementary Material at [http://www.nature.com/npjqi] for additional data).
The VSM was designed to accept multiple input pulses and selectively fan them out to multiple qubits with individual pulse tuning for each qubit (Figure 1). Each input signal is first split and then amplified back, before continuing on to a switch, phase shifter, tuneable attenuator and final amplifier. By ensuring that the line impedance does not depend on the state of the other switches, each VSM line can be controlled independently. Furthermore, the phase shifters are not implemented via delay lines to ensure the pulse timing is independent of the added phase.
To characterise the crosscoupling between Q_{T} and Q_{B}, we measure the evolution of excitedstate populations after a single excitation is injected at one of the qubits with a π pulse. To place a tight upper bound on the interaction strength J, the qubit frequencies must be matched as closely as possible. We achieve an accuracy of around 50 kHz using Ramsey experiments, limited by a combination of factors: the resolution of the flux tuning, the fitting resolution limit imposed by qubit T_{2} dephasing times, and also the frequency shifting induced by the qubit–qubit exchange interaction itself. The oscillation frequency in the singleexcitation swap experiments (Figure 5a,b) gives a residual exchange interaction^{12} between Q_{T} and Q_{B} with strength J/2π≤36±1 kHz. The pulse length of 16 ns used throughout these experiments results in an average drive pulse Rabi frequency almost three orders of magnitude larger than this residual coupling, ensuring that our drive pulses act locally. In addition, the first data points in Figure 5a,b confirm that, immediately following a local π pulse on one qubit, there is no excitation measured in the other qubit. This shows that the measurements are diagonal in the computational basis. These are two important sanity checks for any architecture utilising samefrequency qubits.
To characterise the residual onchip crossdriving, we disconnect the VSM and send driving pulses through the drive line for one of either Q_{T} or Q_{B}, and compare the amplitude required to implement a π pulse on each samefrequency qubit (Figure 5c,d). For this test, pulses are first amplified and then attenuated using a step attenuator to allow the large amplitude range required. The crossdriving on our chip is asymmetric, being stronger when Q_{B} is driven using the drive line of Q_{T} (−45 dB) than vice versa (−53 dB).
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Acknowledgements
We thank RN Schouten, W Vlothuizen and P Koobs de Hartog for experimental contributions and B Criger, T Chasseur and DJ Reilly for discussions. We acknowledge funding by the Dutch Organization for Fundamental Research on Matter (FOM), the Netherlands Organisation for Scientific Research (NWO/OCW and Vidi scheme), the EU FP7 project ScaleQIT, an ERC Synergy Grant and a Marie Curie Career Integration Grant (LDC).
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Asaad, S., Dickel, C., Langford, N. et al. Independent, extensible control of samefrequency superconducting qubits by selective broadcasting. npj Quantum Inf 2, 16029 (2016). https://doi.org/10.1038/npjqi.2016.29
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