We introduce and experimentally characterize a general purpose device for signal processing in circuit quantum electrodynamics systems. The device is a broadband two-port microwave circuit element with three modes of operation: it can transmit, reflect, or invert incident signals between 4 and 8 GHz. This property makes it a versatile tool for lossless signal processing at cryogenic temperatures. In particular, rapid switching (≤$15\u2009ns$) between these operation modes enables several multiplexing readout protocols for superconducting qubits. We report the device's performance in a two-channel code domain multiplexing demonstration. The multiplexed data are recovered with fast readout times (up to $400\u2009ns$) and infidelities ≤$10\u22122$ for probe powers ≥$7\u2009fW$, in agreement with the expectation for binary signaling with Gaussian noise.

Superconducting qubits have recently emerged as a leading candidate for quantum information processing.^{1} These devices are straight-forward to fabricate^{2} and exhibit coherence times on the order of $100\u2009\mu s$ when embedded in 3D microwave cavities.^{3} Furthermore, with quantum-limited amplifiers,^{4–6} quantum non-demolition readout can be performed with fidelities exceeding 97%.^{7} Despite these promising characteristics of individual qubits, one major obstacle to scaling up quantum information processing is the requisite classical hardware.

Within the circuit quantum electrodynamics architecture, the state of a qubit (or several qubits) can be encoded in the phase of a microwave tone transmitted through a cavity that contains the qubit(s).^{34} Qubit states may then be measured by detecting the cavity's transmission. This readout method requires a microwave receiver consisting of multiple circulators, a directional coupler, a cryogenic amplifier, warm amplifiers, and mixers.^{8} Consequently, each cavity readout requires a copy of this bulky, power hungry, and expensive measurement chain.

Multiplexing is a conventional solution to such a scaling challenge. In addition to its widespread implementation in communication networks, multiplexing techniques are applied broadly in low-temperature physics. For example, in the last two decades, multiplexing has rapidly accelerated the readout of detector arrays used in astrophysics, high energy, and materials science.^{9–15,37}

The readout of many superconducting-qubits with frequency domain multiplexing has recently been demonstrated as a viable scheme for reducing hardware overhead.^{16–19} However, some applications (e.g., quantum simulators) require nearly identical qubit/cavity systems. In these cases, other multiplexing techniques such as time and code domain schemes^{20} are favorable.

In code domain multiplexing, each channel is modulated by distinct elements from an orthogonal set of functions. The orthogonality of the set is then exploited to recover the channel's original signal. This combines attractive features of frequency and time domain multiplexing, allowing for simultaneous measurement of all channels while maintaining the ability to dynamically allocate bandwidth. These considerations reflect the fact that in any practical application, hardware constraints typically require a combination of frequency, time, and code domain techniques to maximize bandwidth.

To enable a hybrid multiplexing approach for qubit readout, here we introduce a multipurpose device that we call a Tunable Inductor Bridge (TIB). Each TIB can be tuned to transmit, reflect, or invert a microwave tone (see Fig. 1(a)). Rapidly alternating between the transmit/reflect and transmit/invert operation modes allows the TIB to act as a fast switch or phase-chopper. This dual functionality makes the device ideal for implementing both time and code domain schemes, as shown in Fig. 1(b). For example, sequential qubit manipulation and readout can be implemented by positioning fast-switching TIBs upstream from each qubit. Alternatively, qubits can be simultaneously readout in a code domain scheme by positioning TIBs downstream from each qubit/cavity system and operating them as phase-choppers to spread the cavity output spectrum (beyond the qubit/cavity bandwidth). In this letter, we describe the design, layout, and theory of operation for the TIB. We characterize it experimentally and present a two-channel code domain multiplexing demonstration.

The TIB is a two-port microwave device composed of four tunable inductors arranged as a Wheatstone bridge (see Fig. 2(a)). The four inductors are split into two pairs that tune in tandem. Current in an on-chip bias line changes the inductance of the pairs in opposite directions, imbalancing the bridge. Examination of the transmission coefficient *T* reveals how changing this imbalance adjusts the TIB between its operation modes. When coupled to transmission lines of characteristic impedance *Z*_{0}, the forward scattering parameter at angular frequency *ω* is

Transmission is clearly nulled when the inductors *l*_{1} and *l*_{2} are equal, realizing the device's reflect operation mode. Switching to the transmit mode is accomplished by maximally imbalancing the bridge. Lastly, reversing the sense of this imbalance inverts the transmitted signal, as *T* is odd under exchange of *l*_{1} and *l*_{2}.

To realize tunable inductors for cryogenic microwave applications, we use series arrays of superconducting quantum interference devices (SQUIDs). When the geometric inductance of the SQUIDs is small with respect to their Josephson inductance, the critical current *I _{s}* of these SQUIDs tunes with the magnetic flux $\Phi $ that threads through them as

Here, $\varphi 0=\u210f/2e$ is the reduced flux quantum and *I*_{0} is the critical current of the Josephson junctions. When the current flowing through the arrays is small compared to *I _{s}*, the array inductance is

where $Nsq$ is the number of SQUIDs in the series array, and all junctions are assumed to be identical.

To layout a bridge of tunable inductors using SQUID arrays, we use a previously proposed figure-eight geometry.^{23} The simultaneous tuning of the inductor pairs is accomplished in two steps (depicted in Fig. 2(b)). First, an off-chip coil creates a background magnetic field of uniform strength across the chip. This threads a magnetic flux $\Phi \Sigma $ through all the SQUIDs, while the gradiometric layout of the figure-eight ensures that no net flux pierces the bridge. Second, an on-chip bias line carries a current that simultaneously threads a flux $\Phi \Delta $ through the two arrays on one side of the line, and a flux $\u2212\Phi \Delta $ through the other two arrays. Each SQUID in one pair of arrays is pierced by a sum of magnetic fluxes from the background coil and the bias line $\Phi \Sigma +\Phi \Delta $, while SQUIDs in the other pair of arrays are pierced by a total flux $\Phi \Sigma \u2212\Phi \Delta $. These differing fluxes result in different critical currents, as given in Eq. (2), and hence different inductances, as given in Eq. (3). The figure-eight ensures the connectivity of the inductors corresponds to Fig. 2(a), with each array opposite its equal in the bridge. A false-color image of the fabricated device is shown in Fig. 2(c).

To assess the TIB's performance as a phase-chopper, we performed a homodyne measurement on a microwave signal transmitted through the bridge. A schematic of the measurement is shown in Fig. 3(a). During the measurement, a small current in the TIB's bias line ($<200\u2009\mu A$) is modulated to tune the device between its transmit and invert modes. The resulting mixed-down voltage is then digitized and shown in Fig. 3(b). Each trace shows transmission with a different bias line modulation, in which the operation mode was switched 1, 2, 3, 4, or 64 times during the measurement period of $10\u2009\mu s$.

Switching between the transmit and invert modes occurs rapidly. Fig. 3(c) shows one such falling-edge in finer time-resolution. The observed switching time of $15\u2009ns$ (sampling interval time is $12.5\u2009ns$) is limited by an $80\u2009MHz$ low-pass filter after the mixer in the receiver chain.^{21} In principle, switching times on the order of 1 ns could be expected, constrained by the bandwidth of the Marchand baluns and the bandwidth over which the capacitors match the network to *Z*_{0}.

In addition to the switching time, other relevant specifications of the TIB are its linearity, on-off ratio, insertion loss, and phase balance.^{21} The $Nsq=20$ SQUIDs in each array form inductors with high-power handling.^{23} This is reflected in power-sweeps of the scattering parameters, which are linear up to powers of about $1\u2009pW$ in the three tested devices. For reference, dispersive-readout typically uses microwave tones with powers less than $1\u2009fW$.^{24} The on-off ratio (the ratio of the transmission coefficients in the transmit and reflect operation modes) can be tuned above $20\u2009dB$ over the entire $4\u20138\u2009GHz$ range, and $40\u2009dB$ at the designed center frequency of $6\u2009GHz$. The devices are also low-loss: the two TIBs used for the multiplexing demonstration in this letter have insertion losses below $0.5\u2009dB$. These specifications compare favorably with other recent realizations of fast, Josephson-junction based switches.^{25} Finally, the average magnitude of the phase imbalance between the transmit and invert modes is 5° over the $4\u20138\u2009GHz$ band.

A multiplexed readout of an *N*-qubit/cavity system is beyond the scope of this paper. To illustrate a proof of concept, here we present a multiplexed readout of an analogous two-channel system arranged to simulate the microwave signals that would be generated by periodically measuring two qubit/cavity systems simultaneously.

To modulate the channels in this code domain demonstration, two TIBs are programmed to switch between their transmit and invert modes according to distinct elements in the orthogonal and periodic (period *t _{w}*) set of functions ${wn}$ known as Walsh codes.

^{26}As the TIBs have finite bandwidth, modulation of the channels is not instantaneous. We denote our finite-bandwidth experimental realization of the Walsh codes as ${Wn}$, to distinguish them from the mathematical set ${wn}$.

^{27}The traces shown in Fig. 3(b) are five examples of elements in ${Wn}$ with $tw=10\u2009\mu s$.

In Fig. 4(a), a schematic of our homodyne measurement shows how two Walsh-modulated channels may be code domain multiplexed. First, we generate a string of *p* pseudorandom digital bits $D1$ with an arbitrary waveform generator and use these to modulate the phase of a 5.88 GHz microwave tone. The resulting waveform consists of a tone whose phase jumps pseudorandomly in time between 0 and *π*. This arrangement is duplicated with a second sequence of random digital voltages $D2$ to simulate the output of a second qubit/cavity system.^{28}

These microwave channels are then fed into a cryostat and modulated by separate TIBs. During one digital period, the two TIBs are programmed to transmit and invert according to distinct Walsh codes *w _{n}* and

*w*. The outputs of the TIBs are summed and directed out of the cryostat through a single microwave receiver. The final mixed down voltage is the sum of two Walsh-modulated analog square waves, where the upper and lower values of the two square waves correspond to the logical 1 and 0 of the original digital bits. We call the (mixed-down) analog versions of the two bit streams $A1$ and $A2$.

_{m}The time trace of the $k$th bit takes the form

Here, $Tch1$ ($Tch2$) is the transmission coefficient of the TIB in channel 1 (2). Reconstruction $Aq,kr$ of the $k$th bit in the $q$th channel follows from the orthogonality of the Walsh codes.^{29,35,36} For example,

This analog reconstruction may then be digitized with a thresholding procedure and compared to the transmitted digital bit to determine if a fault occurred.

To characterize our two-channel multiplexing demonstration, we ran the above protocol with $p=105$ digital bits passing through each channel. We then repeated that process for the $P42=12$ permutations for assigning Walsh codes from the set ${W1,W2,W3,W4}$ to a pair of channels. For each realization of the protocol, we recorded the number of faults in the reconstruction of each channel and divided this by the total number of transmitted bits to obtain a channel infidelity. Fig. 4(b) shows the infidelity in both channels as a function of the microwave power delivered to the TIBs.^{30} Three different bit durations of $tw=400\u2009ns,\u20092\u2009\mu s$, and $10\u2009\mu s$ are shown, with error bars indicating 95% confidence intervals, calculated for a binomial distribution. Solid lines are the predictions for a binary signaling process.^{31} The predictions have no adjustable parameters and are made from the measured spectral densities of the signal and noise in each channel.^{21}

We observe that the readout infidelities are independent of the permutation of Walsh codes for $tw>1\u2009\mu s$. For $tw\u22641\u2009\mu s$, timing considerations such as the relative delay between the drive and control lines and the imperfect orthogonality of $Wn$ become relevant. These factors cause cross-talk and intersymbol interference which lead to performance differences between the permutations. They also cause a discrepancy between the model's predictions and the measured infidelities, especially when the predicted infidelity is less than $10\u22123$. Fidelity may be improved in this regime with standard methods like pulse-shaping, equalization, and frame synchronization.^{20}

In summary, we present here an experimental demonstration of a broadband tri-state device that can rapidly ($\u226415\u2009ns$) switch between its transmit, reflect, and invert operation modes. The device is realized with a bridge of inductors built from SQUID arrays, making it a multipurpose tool for implementing time and code domain multiplexing schemes in a cryogenic microwave environment. To illustrate its capabilities, we have demonstrated a two-channel code domain multiplexed readout of a pair of random bit streams. With a $400\u2009ns$ readout time (compatible with demonstrated qubit readout times^{7,19,24,32,33}), infidelities $\u226410\u22122$ are achieved when the power entering the TIBs exceeds $7\u2009fW$. Although this power is larger than the $\u223c0.3$ fW used in typical qubit readout,^{24} our HEMT amplifier is about 20 times noisier than a quantum limited amplifier. To the degree that the measured infidelity is accurately predicted by a binary signaling model, the code domain scheme will not decrease qubit readout fidelity. This demonstration represents a step toward a flexible and scalable read-out architecture for superconducting qubits.

This work is supported by the ARO under Contract No. W911NF-14-1-0079 and the National Science Foundation under Grant No. 1125844.

## References

Unlike ${wn}$, the set ${Wn}$ is only approximately orthogonal, with $|1Tw\u222b0TwWn(t)Wm(t)dt|=\u03f5\u226a1$ when $n\u2260m$. We find $\u03f5\u22643\xd710\u22122$ when $Tw\u2265400\u2009ns$.

With dispersive-readout,^{34} in some cases the state-dependent phase of the transmitted tone varies by an angle $\theta <\pi $ as the qubit changes between its ground and excited states. Code-domain multiplexing still allows for the reconstruction of transmitted signals in this case. The signal power, however, scales with this phase difference as $\u2009sin2(\theta /2)$, as it depends on the square of the geometric distance between the signals in phase space.

The displayed infidelities are averaged over the 12 Walsh code permutations.