## Abstract

A critical ingredient for realising large-scale 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 same-frequency transmon qubits in a circuit QED chip with surface-code-compatible 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 surface-code error threshold for single-qubit 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.

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## Introduction

Building a fault-tolerant quantum computer requires the ability to efficiently address and control individual qubits in a large-scale 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 state-of-the-art in microwave-frequency quantum processors, such as those recently used for small-scale quantum error correction in superconducting circuits,^{4–6} it is beneficial to have a hardware-efficient 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 frequency-matched qubits and high-speed 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 fault-tolerant 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 frequency-crowding problem that arises when trying to fit many distinct-frequency qubits within the finite useful bandwidth of the circuit-based 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 field-programmable 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 same-frequency, solid-state qubits using shared microwave control hardware.

In this article, we demonstrate frequency reuse in an extensible solid-state 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 same-frequency qubits with relevant connectivity and sufficient isolation between same-frequency qubits. To this end, we show independent simultaneous control of two same-frequency qubits with a room-temperature 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 surface-code 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 second-excited 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 single-qubit control. This ensures that same-frequency qubits are never coupled through a single resonator. Independent microwave control of same-frequency 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 two-qubit gates only between different-frequency qubits coupled to the same resonator and never between same-frequency qubits. Our two-qubit gates will be implemented using fast flux-bias lines,^{27} but in this article we focus on VSM-based single-qubit control. The higher-order coupling between same-frequency 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 same-frequency 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 same-frequency qubits, we have designed a prototype room-temperature 4×2 (four input, two output) VSM that allows independent control of amplitude and phase for each of its input–output combinations. Fast marker-controlled 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} high-fidelity fast single-qubit control is achieved using the method of derivative-removal-via-adiabatic-gate (DRAG) pulsing, where the in-phase Gaussian pulse is combined with a quadrature derivative-of-Gaussian 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 same-frequency 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 high-precision control of one qubit while leaving the other qubit idle. To do this, we use the standard technique of single-qubit 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 same-frequency 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 so-called ‘atomic pulses’,^{36} where the 24 single-qubit 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 ground-state 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 two-level system, the measured ground- and excited-state 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 second-excited 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 first-excited-state 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 single-qubit 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 cross-excitation in the system. Dominant on-chip sources include residual exchange interaction *J* between Q_{T} and Q_{B} (mediated by the bus resonators and Q_{M}), and cross-driving, resulting from uncontrolled parasitic couplings between each drive line and the untargeted qubit. As detailed in the 'Materials and Methods' section, measurements of these on-chip sources found *J*/2*π*≤36±1 kHz, and an asymmetric cross-driving 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 off-chip source of cross-excitation, 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 on-chip cross-driving. The symmetric swapping of excitation under exchange is unlikely to explain the strong asymmetry in the amount of cross-excitation measured for the different qubits. Moreover, numerical simulations show that the observed effects are consistent with cross-driving alone (see Supplementary Material at [http://www.nature.com/npjqi] for additional data). This effect of cross-driving 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 single-qubit 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 cross-excitation effects do not dominate the error per Clifford.

The defining test of extensibility in our control architecture is to demonstrate the simultaneous, independent, single-qubit control over same-frequency 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 null-out the effect of cross-driving 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 cross-driving.

To demonstrate the full functionality of our control architecture, we implement all three selective broadcasting schemes and measure their performance using parallel single-qubit RB with independent Clifford sequences for each qubit. Figure 4 shows that the compiled scheme performs best, followed by the sequential and then 5-primitives 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 single-qubit control.

## Discussion

Our VSM allows efficient use of control equipment on same-frequency qubits. It enables high-precision single-qubit control of multiple qubits with a performance that surpasses the best-known surface-code fault-tolerance threshold for single-qubit 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 cross-excitation was measured in one qubit during the RB measurements (Figure 2h), which resulted primarily from on-chip cross-driving. In future experiments, this effect could be reduced by careful design of both the surface-code and physical chip layouts. In the first case, increasing the number of qubit frequencies used will result in larger lattice separations between same-frequency 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 on-chip 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 cross-driving effects can be reduced even at the control level. Specifically, as demonstrated here with the five-primitives 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 cross-driving in quantum algorithms. Furthermore, it is fully compatible with pulse-broadcasting 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 cross-driving, the measured cross-coupling (direct qubit–qubit quantum coupling) has no observable effect in the single-qubit RB measurements. Most likely, this is because rapid application of randomising Clifford pulses effectively decouples the cross-coupling 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 single-qubit RB on both qubits, in each case demonstrating performance that surpasses the fault-tolerance 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 five-primitives scheme for even a handful of qubits. By contrast, the five-primitives 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, marker-based 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, non-Clifford gate to the five pulse primitives, this can be extended to achieve universal single-qubit 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 single-qubit 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 fan-out, which we estimate should enter at ~100 same-frequency 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 VSM-based control and the fixed pulse overhead of the five-primitives broadcasting strategy, our experiment realises the simplest element of an extensible qubit control architecture. This design can be straightforwardly expanded to more same-frequency qubits without requiring any further microwave sources or arbitrary waveform generators. This experiment suggests that surface-code tiling with frequency reuse is a viable path towards large-scale 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 cross-coupling and cross-driving.

The use of control pulses for multiple qubits requires the qubits to be matched in frequency to avoid gate errors from off-resonant driving. Although Q_{T} and Q_{B} were designed to be identical, fabrication uncertainties resulted in a sweet-spot (maximum) frequency of 57 MHz higher than that of Q_{B}. With Q_{B} and Q_{M} kept at their respective sweet-spots (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 cross-coupling between Q_{T} and Q_{B}, we measure the evolution of excited-state 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 single-excitation swap experiments (Figure 5a,b) gives a residual exchange interaction^{12} between Q_{T} and Q_{B} with strength *J*/2*π*≤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 same-frequency qubits.

To characterise the residual on-chip cross-driving, 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 same-frequency 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 cross-driving 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 same-frequency superconducting qubits by selective broadcasting.
*npj Quantum Inf* **2**, 16029 (2016). https://doi.org/10.1038/npjqi.2016.29

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DOI: https://doi.org/10.1038/npjqi.2016.29